B RITISH M ATHEMATICAL O LYMPIAD
Round1:Wednesday,14January1998
Time allowed Three and a half hours.
Instructions?Full written solutions-not just answers-are
required,with complete proofs of any assertions
you may make.Marks awarded will depend on the
clarity of your mathematical presentation.Work
in rough?rst,and then draft your?nal version
carefully before writing up your best attempt.
Do not hand in rough work.
?One complete solution will gain far more credit
than several un?nished attempts.It is more
important to complete a small number of questions
than to try all?ve problems.
?Each question carries10marks.
?The use of rulers and compasses is allowed,but
calculators and protractors are forbidden.
?Start each question on a fresh sheet of paper.Write
on one side of the paper only.On each sheet of
working write the number of the question in the
top left hand corner and your name,initials and
school in the top right hand corner.
?Complete the cover sheet provided and attach it to
the front of your script,followed by the questions
1,2,3,4,5in order.
?Staple all the pages neatly together in the top left
hand corner.
Do not turn over until told to do so.
B RITISH M ATHEMATICAL O LYMPIAD
1.A5×5square is divided into25unit squares.One of the
numbers1,2,3,4,5is inserted into each of the unit squares in such a way that each row,each column and each of the two diagonals contains each of the?ve numbers once and only once.The sum of the numbers in the four squares immediately below the diagonal from top left to bottom right is called the score.
Show that it is impossible for the score to be20.
What is the highest possible score?
2.Let a1=19,a2=98.For n≥1,de?ne a n+2to be the
remainder of a n+a n+1when it is divided by100.What is the remainder when
a21+a22+···+a21998
is divided by8?
3.ABP is an isosceles triangle with AB=AP and P AB acute.
P C is the line through P perpendicular to BP,and C is a point on this line on the same side of BP as A.(You may assume that C is not on the line AB.)D completes the parallelogram ABCD.P C meets DA at M.
Prove that M is the midpoint of DA.
4.Show that there is a unique sequence of positive integers(a n)
satisfying the following conditions:
a1=1,a2=2,a4=12,
a n+1a n?1=a2n±1for n=2,3,4,....
5.In triangle ABC,D is the midpoint of AB and E is the point
of trisection of BC nearer to C.Given that ADC=BAE ?nd BAC.
B RITISH M ATHEMATICAL O LYMPIAD
Round2:Thursday,26February1998
Time allowed Three and a half hours.
Each question is worth10marks. Instructions?Full written solutions-not just answers-are
required,with complete proofs of any assertions
you may make.Marks awarded will depend on the
clarity of your mathematical presentation.Work
in rough?rst,and then draft your?nal version
carefully before writing up your best attempt.
Rough work should be handed in,but should be
clearly marked.
?One or two complete solutions will gain far more
credit than partial attempts at all four problems.
?The use of rulers and compasses is allowed,but
calculators and protractors are forbidden.
?Staple all the pages neatly together in the top left
hand corner,with questions1,2,3,4in order,and
the cover sheet at the front.
In early March,twenty students will be invited
to attend the training session to be held at
Trinity College,Cambridge(2-5April).On the
?nal morning of the training session,students sit
a paper with just3Olympiad-style problems.The
UK Team-six members plus one reserve-for this
summer’s International Mathematical Olympiad
(to be held in Taiwan,13-21July)will be chosen
immediately thereafter.Those selected will be
expected to participate in further correspondence
work between April and July,and to attend a
short residential session in early July before leaving
for Taiwan.
Do not turn over until told to do so.
B RITISH M ATHEMATICAL O LYMPIAD
1.A booking o?ce at a railway station sells tickets to200
destinations.One day,tickets were issued to3800passengers.
Show that
(i)there are(at least)6destinations at which the passenger
arrival numbers are the same;
(ii)the statement in(i)becomes false if‘6’is replaced by‘7’.
2.A triangle ABC has BAC>BCA.A line AP is drawn
so that P AC=BCA where P is inside the triangle.
A point Q outside the triangle is constructed so that P Q
is parallel to AB,and BQ is parallel to AC.R is the point on BC(separated from Q by the line AP)such that P RQ=BCA.
Prove that the circumcircle of ABC touches the circumcircle of P QR.
3.Suppose x,y,z are positive integers satisfying the equation
1
x?
1
y
=
1
z
,
and let h be the highest common factor of x,y,z.
Prove that hxyz is a perfect square.
Prove also that h(y?x)is a perfect square.
4.Find a solution of the simultaneous equations
xy+yz+zx=12
xyz=2+x+y+z
in which all of x,y,z are positive,and prove that it is the only such solution.
Show that a solution exists in which x,y,z are real and distinct.
B RITISH M ATHEMATICAL O LYMPIAD
Round1:Wednesday,13January1999
Time allowed Three and a half hours.
Instructions?Full written solutions-not just answers-are
required,with complete proofs of any assertions
you may make.Marks awarded will depend on the
clarity of your mathematical presentation.Work
in rough?rst,and then draft your?nal version
carefully before writing up your best attempt.
Do not hand in rough work.
?One complete solution will gain far more credit
than several un?nished attempts.It is more
important to complete a small number of questions
than to try all?ve problems.
?Each question carries10marks.
?The use of rulers and compasses is allowed,but
calculators and protractors are forbidden.
?Start each question on a fresh sheet of paper.Write
on one side of the paper only.On each sheet of
working write the number of the question in the
top left hand corner and your name,initials and
school in the top right hand corner.
?Complete the cover sheet provided and attach it to
the front of your script,followed by the questions
1,2,3,4,5in order.
?Staple all the pages neatly together in the top left
hand corner.
Do not turn over until told to do so.
B RITISH M ATHEMATICAL O LYMPIAD
1.I have four children.The age in years of each child is a
positive integer between2and16inclusive and all four ages are distinct.A year ago the square of the age of the oldest child was equal to the sum of the squares of the ages of the other three.In one year’s time the sum of the squares of the ages of the oldest and the youngest will be equal to the sum of the squares of the other two children.
Decide whether this information is su?cient to determine their ages uniquely,and?nd all possibilities for their ages.
2.A circle has diameter AB and X is a?xed point of AB lying
between A and B.A point P,distinct from A and B,lies on the circumference of the circle.Prove that,for all possible positions of P,
tan AP X
tan P AX
remains constant.
3.Determine a positive constant c such that the equation
xy2?y2?x+y=c
has precisely three solutions(x,y)in positive integers.
4.Any positive integer m can be written uniquely in base3form
as a string of0’s,1’s and2’s(not beginning with a zero).For example,
98=(1×81)+(0×27)+(1×9)+(2×3)+(2×1)=(10122)3.
Let c(m)denote the sum of the cubes of the digits of the base 3form of m;thus,for instance
c(98)=13+03+13+23+23=18.
Let n be any?xed positive integer.De?ne the sequence(u r) by
u1=n and u r=c(u r?1)for r≥2.
Show that there is a positive integer r for which u r=1,2 or17.
5.Consider all functions f from the positive integers to the
positive integers such that
(i)for each positive integer m,there is a unique positive
integer n such that f(n)=m;
(ii)for each positive integer n,we have
f(n+1)is either4f(n)?1or f(n)?1.
Find the set of positive integers p such that f(1999)=p for some function f with properties(i)and(ii).
B RITISH M ATHEMATICAL O LYMPIAD
Round2:Thursday,25February1999
Time allowed Three and a half hours.
Each question is worth10marks. Instructions?Full written solutions-not just answers-are
required,with complete proofs of any assertions
you may make.Marks awarded will depend on the
clarity of your mathematical presentation.Work
in rough?rst,and then draft your?nal version
carefully before writing up your best attempt.
Rough work should be handed in,but should be
clearly marked.
?One or two complete solutions will gain far more
credit than partial attempts at all four problems.
?The use of rulers and compasses is allowed,but
calculators and protractors are forbidden.
?Staple all the pages neatly together in the top left
hand corner,with questions1,2,3,4in order,and
the cover sheet at the front.
In early March,twenty students will be invited
to attend the training session to be held at
Trinity College,Cambridge(8-11April).On the
?nal morning of the training session,students sit
a paper with just3Olympiad-style problems.The
UK Team-six members plus one reserve-for this
summer’s International Mathematical Olympiad
(to be held in Bucharest,Romania,13-22July)
will be chosen immediately thereafter.Those
selected will be expected to participate in further
correspondence work between April and July,and
to attend a short residential session(3-7July)in
Birmingham before leaving for Bucharest.
Do not turn over until told to do so.
B RITISH M ATHEMATICAL O LYMPIAD
1.For each positive integer n,let S n denote the set consisting of
the?rst n natural numbers,that is
S n={1,2,3,4,...,n?1,n}.
(i)For which values of n is it possible to express S n as
the union of two non-empty disjoint subsets so that the
elements in the two subsets have equal sums?
(ii)For which values of n is it possible to express S n as the union of three non-empty disjoint subsets so that the
elements in the three subsets have equal sums?
2.Let ABCDEF be a hexagon(which may not be regular),
which circumscribes a circle S.(That is,S is tangent to each of the six sides of the hexagon.)The circle S touches AB,CD,EF at their midpoints P,Q,R respectively.Let X,Y,Z be the points of contact of S with BC,DE,F A respectively.Prove that P Y,QZ,RX are concurrent.
3.Non-negative real numbers p,q and r satisfy p+q+r=1.
Prove that
7(pq+qr+rp)≤2+9pqr.
4.Consider all numbers of the form3n2+n+1,where n is a
positive integer.
(i)How small can the sum of the digits(in base10)of such
a number be?
(ii)Can such a number have the sum of its digits(in base10) equal to1999?
B RITISH M ATHEMATICAL O LYMPIAD
Round1:Wednesday,12January2000
Time allowed Three and a half hours.
Instructions?Full written solutions-not just answers-are
required,with complete proofs of any assertions
you may make.Marks awarded will depend on the
clarity of your mathematical presentation.Work
in rough?rst,and then draft your?nal version
carefully before writing up your best attempt.
Do not hand in rough work.
?One complete solution will gain far more credit
than several un?nished attempts.It is more
important to complete a small number of questions
than to try all?ve problems.
?Each question carries10marks.
?The use of rulers and compasses is allowed,but
calculators and protractors are forbidden.
?Start each question on a fresh sheet of paper.Write
on one side of the paper only.On each sheet of
working write the number of the question in the
top left hand corner and your name,initials and
school in the top right hand corner.
?Complete the cover sheet provided and attach it to
the front of your script,followed by the questions
1,2,3,4,5in order.
?Staple all the pages neatly together in the top left
hand corner.
Do not turn over until told to do so.
B RITISH M ATHEMATICAL O LYMPIAD
1.Two intersecting circles C1and C2have a common tangent
which touches C1at P and C2at Q.The two circles intersect at M and N,where N is nearer to P Q than M is.The line P N meets the circle C2again at R.Prove that MQ bisects angle P MR.
2.Show that,for every positive integer n,
121n?25n+1900n?(?4)n
is divisible by2000.
3.Triangle ABC has a right angle at A.Among all points P on
the perimeter of the triangle,?nd the position of P such that
AP+BP+CP
is minimized.
4.For each positive integer k>1,de?ne the sequence{a n}by
a0=1and a n=kn+(?1)n a n?1for each n≥1.
Determine all values of k for which2000is a term of the sequence.
5.The seven dwarfs decide to form four teams to compete in the
Millennium Quiz.Of course,the sizes of the teams will not all be equal.For instance,one team might consist of Doc alone, one of Dopey alone,one of Sleepy,Happy&Grumpy,and one of Bashful&Sneezy.In how many ways can the four teams be made up?(The order of the teams or of the dwarfs within the teams does not matter,but each dwarf must be in exactly one of the teams.)
Suppose Snow-White agreed to take part as well.In how many ways could the four teams then be formed?
B RITISH M ATHEMATICAL O LYMPIAD
Round2:Wednesday,23February2000
Time allowed Three and a half hours.
Each question is worth10marks. Instructions?Full written solutions-not just answers-are
required,with complete proofs of any assertions
you may make.Marks awarded will depend on the
clarity of your mathematical presentation.Work
in rough?rst,and then draft your?nal version
carefully before writing up your best attempt.
Rough work should be handed in,but should be
clearly marked.
?One or two complete solutions will gain far more
credit than partial attempts at all four problems.
?The use of rulers and compasses is allowed,but
calculators and protractors are forbidden.
?Staple all the pages neatly together in the top left
hand corner,with questions1,2,3,4in order,and
the cover sheet at the front.
In early March,twenty students will be invited
to attend the training session to be held at
Trinity College,Cambridge(6-9April).On the
?nal morning of the training session,students
sit a paper with just3Olympiad-style problems.
The UK Team-six members plus one reserve
-for this summer’s International Mathematical
Olympiad(to be held in South Korea,13-24July)
will be chosen immediately thereafter.Those
selected will be expected to participate in further
correspondence work between April and July,and
to attend a short residential session before leaving
for South Korea.
Do not turn over until told to do so.
B RITISH M ATHEMATICAL O LYMPIAD
1.Two intersecting circles C1and C2have a common tangent
which touches C1at P and C2at Q.The two circles intersect at M and N,where N is nearer to P Q than M is.Prove that the triangles MNP and MNQ have equal areas.
2.Given that x,y,z are positive real numbers satisfying
xyz=32,?nd the minimum value of
x2+4xy+4y2+2z2.
3.Find positive integers a and b such that
(3
√a+3√b?1)2=49+203√6.
4.(a)Find a set A of ten positive integers such that no six
distinct elements of A have a sum which is divisible by6.
(b)Is it possible to?nd such a set if“ten”is replaced by
“eleven”?
B RITISH M ATHEMATICAL O LYMPIAD
Round1:Wednesday,17January2001
Time allowed Three and a half hours.
Instructions?Full written solutions-not just answers-are
required,with complete proofs of any assertions
you may make.Marks awarded will depend on the
clarity of your mathematical presentation.Work
in rough?rst,and then draft your?nal version
carefully before writing up your best attempt.
Do not hand in rough work.
?One complete solution will gain far more credit
than several un?nished attempts.It is more
important to complete a small number of questions
than to try all?ve problems.
?Each question carries10marks.
?The use of rulers and compasses is allowed,but
calculators and protractors are forbidden.
?Start each question on a fresh sheet of paper.Write
on one side of the paper only.On each sheet of
working write the number of the question in the
top left hand corner and your name,initials and
school in the top right hand corner.
?Complete the cover sheet provided and attach it to
the front of your script,followed by the questions
1,2,3,4,5in order.
?Staple all the pages neatly together in the top left
hand corner.
Do not turn over until told to do so.
2001British Mathematical Olympiad
Round1
1.Find all two-digit integers N for which the sum of the digits of10N?N is divisible by170.
2.Circle S lies inside circle T and touches it at A.From a
point P(distinct from A)on T,chords P Q and P R of T are drawn touching S at X and Y respectively.Show that QAR=2XAY.
3.A tetromino is a?gure made up of four unit squares connected
by common edges.
(i)If we do not distinguish between the possible rotations of
a tetromino within its plane,prove that there are seven
distinct tetrominoes.
(ii)Prove or disprove the statement:It is possible to pack all seven distinct tetrominoes into a4×7rectangle without
overlapping.
4.De?ne the sequence(a n)by
a n=n+{
√n},
where n is a positive integer and{x}denotes the nearest integer to x,where halves are rounded up if necessary.
Determine the smallest integer k for which the terms
a k,a k+1,...,a k+2000form a sequence of2001consecutive
integers.
5.A triangle has sides of length a,b,c and its circumcircle has
radius R.Prove that the triangle is right-angled if and only if a2+b2+c2=8R2.
B RITISH M ATHEMATICAL O LYMPIAD
Round2:Tuesday,27February2001
Time allowed Three and a half hours.
Each question is worth10marks. Instructions?Full written solutions-not just answers-are
required,with complete proofs of any assertions
you may make.Marks awarded will depend on the
clarity of your mathematical presentation.Work
in rough?rst,and then draft your?nal version
carefully before writing up your best attempt.
Rough work should be handed in,but should be
clearly marked.
?One or two complete solutions will gain far more
credit than partial attempts at all four problems.
?The use of rulers and compasses is allowed,but
calculators and protractors are forbidden.
?Staple all the pages neatly together in the top left
hand corner,with questions1,2,3,4in order,and
the cover sheet at the front.
In early March,twenty students will be invited
to attend the training session to be held at
Trinity College,Cambridge(8-11April).On the
?nal morning of the training session,students sit a
paper with just3Olympiad-style problems,and
8students will be selected for further training.
Those selected will be expected to participate
in correspondence work and to attend another
meeting in Cambridge(probably26-29May).The
UK Team of6for this summer’s International
Mathematical Olympiad(to be held in Washington
DC,USA,3-14July)will then be chosen.
Do not turn over until told to do so.
2001British Mathematical Olympiad
Round2
1.Ahmed and Beth have respectively p and q marbles,with
p>q.
Starting with Ahmed,each in turn gives to the other as many marbles as the other already possesses.It is found that after 2n such transfers,Ahmed has q marbles and Beth has p marbles.
Find
p
q
in terms of n.
2.Find all pairs of integers(x,y)satisfying
1+x2y=x2+2xy+2x+y.
3.A triangle ABC has ACB>ABC.
The internal bisector of BAC meets BC at D.
The point E on AB is such that EDB=90?.
The point F on AC is such that BED=DEF.
Show that BAD=F DC.
4.N dwarfs of heights1,2,3,...,N are arranged in a circle.
For each pair of neighbouring dwarfs the positive di?erence between the heights is calculated;the sum of these N di?erences is called the“total variance”V of the arrangement.
Find(with proof)the maximum and minimum possible values of V.
British Mathematical Olympiad
Round1:Wednesday,5December2001
Time allowed Three and a half hours.
Instructions?Full written solutions-not just answers-are
required,with complete proofs of any assertions
you may make.Marks awarded will depend on the
clarity of your mathematical presentation.Work
in rough?rst,and then draft your?nal version
carefully before writing up your best attempt.
Do not hand in rough work.
?One complete solution will gain far more credit
than several un?nished attempts.It is more
important to complete a small number of questions
than to try all?ve problems.
?Each question carries10marks.
?The use of rulers and compasses is allowed,but
calculators and protractors are forbidden.
?Start each question on a fresh sheet of paper.Write
on one side of the paper only.On each sheet of
working write the number of the question in the
top left hand corner and your name,initials and
school in the top right hand corner.
?Complete the cover sheet provided and attach it to
the front of your script,followed by the questions
1,2,3,4,5in order.
?Staple all the pages neatly together in the top left
hand corner.
Do not turn over until told to do
so.
2001British Mathematical Olympiad
Round1
1.Find all positive integers m,n,where n is odd,that satisfy
1
m
+
4
n
=
1
12
.
2.The quadrilateral ABCD is inscribed in a circle.The diagonals
AC,BD meet at Q.The sides DA,extended beyond A,and CB, extended beyond B,meet at P.
Given that CD=CP=DQ,prove that CAD=60?.
3.Find all positive real solutions to the equation
x+ x6 = x2 + 2x3 ,
where t denotes the largest integer less than or equal to the real number t.
4.Twelve people are seated around a circular table.In how many ways
can six pairs of people engage in handshakes so that no arms cross?
(Nobody is allowed to shake hands with more than one person at once.)
5.f is a function from Z+to Z+,where Z+is the set of non-negative
integers,which has the following properties:-
a)f(n+1)>f(n)for each n∈Z+,
b)f(n+f(m))=f(n)+m+1for all m,n∈Z+.
Find all possible values of f(2001).
British Mathematical Olympiad
Round2:Tuesday,26February2002
Time allowed Three and a half hours.
Each question is worth10marks. Instructions?Full written solutions-not just answers-are
required,with complete proofs of any assertions
you may make.Marks awarded will depend on the
clarity of your mathematical presentation.Work
in rough?rst,and then draft your?nal version
carefully before writing up your best attempt.
Rough work should be handed in,but should be
clearly marked.
?One or two complete solutions will gain far more
credit than partial attempts at all four problems.
?The use of rulers and compasses is allowed,but
calculators and protractors are forbidden.
?Staple all the pages neatly together in the top left
hand corner,with questions1,2,3,4in order,and
the cover sheet at the front.
In early March,twenty students will be invited
to attend the training session to be held at
Trinity College,Cambridge(4–7April).On the
?nal morning of the training session,students sit a
paper with just3Olympiad-style problems,and
8students will be selected for further training.
Those selected will be expected to participate
in correspondence work and to attend another
meeting in Cambridge.The UK Team of6for this
summer’s International Mathematical Olympiad
(to be held in Glasgow,22–31July)will then be
chosen.
Do not turn over until told to do
so.
2002British Mathematical Olympiad
Round2
1.The altitude from one of the vertices of an acute-angled
triangle ABC meets the opposite side at D.From D perpendiculars DE and DF are drawn to the other two sides.
Prove that the length of EF is the same whichever vertex is chosen.
2.A conference hall has a round table wth n chairs.There are
n delegates to the conference.The?rst delegate chooses his or her seat arbitrarily.Thereafter the(k+1)th delegate sits k places to the right of the k th delegate,for1≤k≤n?1.
(In particular,the second delegate sits next to the?rst.)No chair can be occupied by more than one delegate.
Find the set of values n for which this is possible.
3.Prove that the sequence de?ned by
y0=1,y n+1=
1
2 3y n+
5y2n?4 ,(n≥0) consists only of integers.
4.Suppose that B1,...,B N are N spheres of unit radius
arranged in space so that each sphere touches exactly two others externally.Let P be a point outside all these spheres, and let the N points of contact be C1,...,C N.The length of the tangent from P to the sphere B i(1≤i≤N)is denoted by t i.Prove the product of the quantities t i is not more than the product of the distances P C i.
Supported
by
British Mathematical Olympiad
Round1:Wednesday,11December2002
Time allowed Three and a half hours.
Instructions?Full written solutions-not just answers-are
required,with complete proofs of any assertions
you may make.Marks awarded will depend on the
clarity of your mathematical presentation.Work
in rough?rst,and then draft your?nal version
carefully before writing up your best attempt.
Do not hand in rough work.
?One complete solution will gain far more credit
than several un?nished attempts.It is more
important to complete a small number of questions
than to try all?ve problems.
?Each question carries10marks.
?The use of rulers and compasses is allowed,but
calculators and protractors are forbidden.
?Start each question on a fresh sheet of paper.Write
on one side of the paper only.On each sheet of
working write the number of the question in the
top left hand corner and your name,initials and
school in the top right hand corner.
?Complete the cover sheet provided and attach it to
the front of your script,followed by the questions
1,2,3,4,5in order.
?Staple all the pages neatly together in the top left
hand corner.
Do not turn over until told to do
so.
Supported
by
2002/3British Mathematical Olympiad
Round1
1.Given that
34!=295232799cd96041408476186096435ab000000, determine the digits a,b,c,d.
2.The triangle ABC,where AB perpendicular from A to BC meets S again at P.The point X lies on the line segment AC,and BX meets S again at Q. Show that BX=CX if and only if P Q is a diameter of S. 3.Let x,y,z be positive real numbers such that x2+y2+z2=1. Prove that x2yz+xy2z+xyz2≤13. 4.Let m and n be integers greater than1.Consider an m×n rectangular grid of points in the plane.Some k of these points are coloured red in such a way that no three red points are the vertices of a right-angled triangle two of whose sides are parallel to the sides of the grid. Determine the greatest possible value of k. 5.Find all solutions in positive integers a,b,c to the equation a!b!=a!+b!+c! Supported by British Mathematical Olympiad Round2:Tuesday,25February2003 Time allowed Three and a half hours. Each question is worth10marks. Instructions?Full written solutions-not just answers-are required,with complete proofs of any assertions you may make.Marks awarded will depend on the clarity of your mathematical presentation.Work in rough?rst,and then draft your?nal version carefully before writing up your best attempt. Rough work should be handed in,but should be clearly marked. ?One or two complete solutions will gain far more credit than partial attempts at all four problems. ?The use of rulers and compasses is allowed,but calculators and protractors are forbidden. ?Staple all the pages neatly together in the top left hand corner,with questions1,2,3,4in order,and the cover sheet at the front. In early March,twenty students will be invited to attend the training session to be held at Trinity College,Cambridge(3-6April).On the ?nal morning of the training session,students sit a paper with just3Olympiad-style problems,and 8students will be selected for further training. Those selected will be expected to participate in correspondence work and to attend further training.The UK Team of6for this summer’s International Mathematical Olympiad(to be held in Japan,7-19July)will then be chosen. Do not turn over until told to do so. Supported by 2003British Mathematical Olympiad Round2 1.For each integer n>1,let p(n)denote the largest prime factor of n. Determine all triples x,y,z of distinct positive integers satisfying (i)x,y,z are in arithmetic progression,and (ii)p(xyz)≤3. 2.Let ABC be a triangle and let D be a point on AB such that 4AD=AB.The half-line is drawn on the same side of AB as C, starting from D and making an angle ofθwith DA whereθ=ACB. If the circumcircle of ABC meets the half-line at P,show that P B=2P D. 3.Let f:N→N be a permutation of the set N of all positive integers. (i)Show that there is an arithmetic progression of positive integers a,a+d,a+2d,where d>0,such that f(a) (ii)Must there be an arithmetic progression a,a+d,..., a+2003d,where d>0,such that f(a) [A permutation of N is a one-to-one function whose image is the whole of N;that is,a function from N to N such that for all m∈N there exists a unique n∈N such that f(n)=m.] 4.Let f be a function from the set of non-negative integers into itself such that for all n≥0 (i) f(2n+1) 2? f(2n) 2=6f(n)+1,and (ii)f(2n)≥f(n). How many numbers less than2003are there in the image of f? 最大与最小 知识要点 在日常生活和工作中，经常会遇到这样一类问题：怎样安排时间最省、怎样行走路线最短、怎样管理费用最低、怎样设计面积最大、怎样合作效率最高、怎样加工利用率最大等等，它们都可以归结为在一定条件下的最大值或最小值方面的数学问题。 最大和最小都是在某一固定范围內比较的结果。固定的范围就是一个定值，抓住这个“定值”就抓住了解题的关键。 解决极值问题的策略，常常因题而异，归纳起来主要有以下四个“突破口”： ①从极端情况入手；②用枚举比较入手；③由分析推理入手；④凭构造方程入手。 最小 1.（2008年4月13日第六届小学“希望杯”全国数学邀请赛五年级第2试第4题）有一排椅子有27个 座位，为了使后去的人随意坐在哪个位置都有人与他相邻，则至少要先坐_______人。 2.圆桌周围恰好有12把椅子，现在已经有一些人在桌边就坐。当再有一人入座时，就必须和已就坐的 某人相邻。问：已就坐的最少有多少人？ 3.阶梯教室座位有10排，每排有16个座位，当有150个人就座时，某些排坐着的人数就一样多。我们希 望人数一样的排数尽可能少，这样的排数至少有多少排？ 4.（2007年台湾第十一届小学数学世界邀请赛个人赛第6题）商店里销售的铅笔有两种包装，五支包装 的每包售价6元，七支包装的每包售价7元。某校至少要购买铅笔111支，请问至少要花费_______元。 5.若干名家长（爸爸或妈妈，他们都不是老师）和老师陪同一些小学生参加某次数学竞赛，已知家长和 老师共有22人，家长比老师多，妈妈比爸爸多，女老师比妈妈多2人，至少有1名男老师，那么在这22人中，爸爸有多少人？ 6.（2007年“我爱数学夏令营”综合测试题第7题）一个小公司有5个职工，月平均工资为2700元。 已知最高工资是最低工资的2倍，那么最高月工资最少为_______元。 7.（1999年第八届日本小学数学奥林匹克大赛决赛第7题）有一批货物，它们的总重量是19500千克， 不知道每一件货物的重量，但没有一件货物的重量超过350千克。现在有若干辆大卡车，每辆最多可运1500千克货物，想一次把这批货物全部运走。不管每一件货物的重量是多少，为了必须一次运完所有的货物，至少需要多少辆大卡车？（不考虑货物的体积） 2020年中国数学奥林匹克试题和详细解答word 版 一、给定锐角三角形PBC ，PC PB ≠．设A ，D 分不是边PB ，PC 上的点，连接AC ，BD ，相交于点O. 过点O 分不作OE ⊥AB ，OF ⊥CD ，垂足分不为E ，F ，线段BC ，AD 的中点分不为M ，N ． 〔1〕假设A ，B ，C ，D 四点共圆，求证：EM FN EN FM ?=?； 〔2〕假设 EM FN EN FM ?=?，是否一定有A ，B ，C ，D 四点共圆？证明你的结论． 解〔1〕设Q ，R 分不是OB ，OC 的中点，连接 EQ ，MQ ，FR ，MR ，那么 11 ,22EQ OB RM MQ OC RF ====， 又OQMR 是平行四边形，因此 OQM ORM ∠=∠， 由题设A ，B ，C ，D 四点共圆，因此 ABD ACD ∠=∠， 因此 图1 22EQO ABD ACD FRO ∠=∠=∠=∠， 因此 EQM EQO OQM FRO ORM FRM ∠=∠+∠=∠+∠=∠， 故 EQM MRF ???， 因此 EM ＝FM ， 同理可得 EN ＝FN ， 因此 EM FN EN FM ?=?． 〔2〕答案是否定的． 当AD ∥BC 时，由于B C ∠≠∠，因此A ，B ，C ，D 四点不共圆，但现在仍旧有 EM FN EN FM ?=?，证明如下： 如图2所示，设S ，Q 分不是OA ，OB 的中点，连接ES ，EQ ，MQ ，NS ，那么 11 ,22 NS OD EQ OB ==， C B 因此 NS OD EQ OB =．①又 11 , 22 ES OA MQ OC ==，因此 ES OA MQ OC =．② 而AD∥BC，因此 OA OD OC OB =，③ 由①，②，③得NS ES EQ MQ =． 因为2 NSE NSA ASE AOD AOE ∠=∠+∠=∠+∠， ()(1802) EQM MQO OQE AOE EOB EOB ∠=∠+∠=∠+∠+?-∠ (180)2 AOE EOB AOD AOE =∠+?-∠=∠+∠， 即NSE EQM ∠=∠， 因此NSE ?～EQM ?， 故 EN SE OA EM QM OC ==〔由②〕．同理可得， FN OA FM OC =， 因此EN FN EM FM =， 从而EM FN EN FM ?=?． C B 2020最新第二届华博士小学数学奥林匹克网上竞赛试题及答案 （五年级） (红色为正确答案) 选择正确的答案： (1)在下列算式中加一对括号后，算式的最大值是（）。 7 ×9 + 12 ÷ 3 - 2 A 75 B 147 C 89 D 90 (2)已知三角形的内角和是180度.一个五边形的内角和应是( )度. A 500 B 540 C 360 D 480 (3)甲乙两个数的和是15.95,甲数的小数点向右移动一位就等于乙数,那么 甲数是( ). A 1.75 B 1.47 C 1.45 D 1.95 (4)一个顾客买了6瓶酒,每瓶付1.3元,退空瓶时,售货员说,每只空瓶钱比酒钱 少1.1元,顾客应退回的瓶钱是( )元. A 0.8 B 0.4 C 0.6 D 1.2 (5)两数相除得3余10,被除数,除数,商与余数之和是143,这两个数分别是( ) 和( ). A 30和100 B 110和30 C 100和34 D 95和40 (6) 今年爸爸和女儿的年龄和是44岁,10年后,爸爸的年龄是女儿的3倍,今年女儿是多少岁? A16 B11 C9 D10 (7)一个两位数除250,余数是37,这样的两位数是( ). A 17 B38 C 71 D 91 (8)把一条细绳先对折,再把它所折成相等的三折,接着再对折,然后用剪刀在折过三次的绳中间剪一刀,那么这条绳被剪成( )段. A 13 B 12 C 14 D 15 (9) 把两个表面积都是6平方厘米的正方体拼成一个长方体,这个长方体的表面积( ). A 12 B 18 C10D11 (10)一昼夜钟面上的时针和分针重叠( )次. A 23 B 12 C 20 D13 (11)某车间四月份实际生产机器76台,其中原计划生产的台数比超产台数多60台, 求四月份比原计划超产多少台机器? A 16 B 8 C 10 D 12 (12)一块红砖长25厘米,宽15厘米,用这样的红砖拼成一个正方形最少需要多少块? A 15 B 12 C 75 D 8 (13)图中ABCD 是长方形,已知AB=4厘米,BC=6厘米,三角形EFD 的面积 比三角形ABF 的面积大6平方厘米,求ED=? A 9 B 7 C 8 D 6 (14)一天,甲乙丙三人去郊外钓鱼已知甲比乙多钓6条,丙钓的是甲的2 倍,比乙多钓22条,问他们三人一共钓了多少条? A 48 B 50 C 52 D 58 (15)张师傅以1元钱4个苹果的价格买进苹果若干个,又以2元钱5个苹果有价格把这些苹果卖出,如果他要赚得15元钱的利润,那么他必须卖出苹果多少个? A 10 B 100 C 20 D 160 E D C B 全国小学数学奥林匹克竞赛简介 奥数就是奥林匹克数学的简称，即国际数学竞赛，取名仿自于奥林匹克运动会。 1934年和1935年，前苏联开始在列宁格勒和莫斯科举办中学数学竞赛，并冠以数学奥林匹克的名称。1959年罗马尼亚数学物理学会邀请东欧国家中学生参加在布加勒斯特举办的第一届国际数学奥林匹克竞赛。从此每年一次，至今已举办了50届。 奥数的出题范围超出了所有国家的义务教育水平，有些题目的难度大大超过了大学入学考试，有些题目甚至数学家也感到棘手。通过这样高水平的比赛，可以及早发现数学人才，然后进行培养，使其脱颖而出。 近年，国内外很多名牌大学和重点中学比较注重奥数人才，通常通过奥数选拔优秀生源。北京大学、清华大学、复旦大学等高校对奥数优秀的学生偏爱有佳，每年有很多全国高中数学竞赛成绩优异的学生直接免试进入北大数学系。 由于，高校和重点中学对奥数人才的重视，近年来，又出现了小学奥数一词。小学奥数全称叫"小学奥林匹克数学"，或叫"小学数学奥林匹克"，称呼起源于"数学是思维的体操"它体现了数学与奥林匹克体育运动精神的共通性：更快、更高、更强。其实它更准确应称为"小学竞赛数学"。 从1986年起，中国中学生在国际数学奥林匹克连续几年取得优异成绩；1990年7月，在我国北京成功地举办了第31届国际数学奥林匹克，我国代表队再次取得总分第一。中国学生在学习数学上的潜力被发现了，大大激发了全国中、小学生学习数学的兴趣，数学课外活动蓬勃地开展，中、小学数学竞赛活动受到广大师生和家长的欢迎，也得到了社会各界人士的更多关心和支持。1990年11月，在湖南宁乡召开的中国数学会普及工作委员会第六次全国工作会议上，与会同仁一致认识到，为了顺应群众积极高涨的形势，更要坚持"在普及的基础上不断提高"的方针，要引导数学竞赛这一群众性的课外活动健康地发展，为了统筹安排高中、初中、小学的数学课外活动，处理好相互的衔接关系。会议决定，从1991年起，每年春季举行一次"小学数学奥林匹克"，会议还特别强调，中国数学会举办的高中联赛、初中联赛、小学数学奥林匹克都是普及型、大众化的数学竞赛。为了使"小学数学奥林匹克"的试题能适合多数学生的实际水平，在举办1991年"小学数学奥林匹克"时，主试委员会向全国发出一份试题样卷，广泛征求意见，另外，把初赛试卷，分成A，B，C三种不同水平的试卷，供合地选择采用，同时还宣布了两条命题原则："一、试题涉及的知识范围不超出现行的小学数学教学大纲；二、每一道题一定有一种简单的算术解法。"并且声明，抽屉原则、容斥原理、运筹学等离课堂教学内容较远的内容，一定不在试题中出现。我们就是希望，不要过多的课外辅导，尽可能减轻学生的学习负担。经过若干年的实践，全国反映较好，普遍认为试题有利于启迪思维和智力开发，也有利于课堂教学水平的提高。参加者十分踊跃，人数逐年增加。事实上，试题难度逐年在降低，一年比一年容易些，获得高分的人数大幅度增加。以1993年来说，参加决赛的16万学生中，全国有500多人获满分(十二道试题都做对)，有10%的人做对九道题以上，有40%以上学生能做对六道以上，可以说试题的难易程度是比较适当的。这项赛事分为初赛和决赛，分别在每年的三月份和四份，从1993年开始我们又举办了这项赛事的后继活动---"小学数学奥林匹克总决赛"，后来称为"我爱数学少年夏令营"。 "全国小学数学奥林匹克"(创办于1991年)每年3、4月中国数学会普及工作委员会为有关省份提供了一份"小学数学奥林匹克"初赛和决赛试卷，目的在于引导学有余力的小学生的数学课外活动的方向。目前包括"三段式"--小学数学奥林匹克初赛、决赛、我爱数学夏令营。初赛(每年3月份)、决赛(每年4月份)和夏令营(每年暑期)。组织这项活动的原则：一是要把它办成一个"大众化、普及型"的活动；二是要使所出的题目"不超前、不超纲"；三是要尽可能给每个题目一个小学生看得懂的算术解法；四是要充分认识到地区发展不平衡的特点。 “我爱数学少年夏令营”简介 权威性：★★★★★ 举办方：中国数学会普及工作委员会 三年级“奥林匹克”数学指导 时刻、时间与钟表 同学们，你一定知道钟表是用来记时的，爸爸妈妈当你很小时就会教你如何看钟表、报时间，可钟表里有许多有趣的数学问题。 什么叫“时间”它有两层意思： 1. 表示某一种特定时候。 如：北京时间八点整。每天早上六点起床等等，为了区别别一种含义，我们把表示某一种特定的时候，叫时刻。（也叫点） 2. 表示两个不同时刻的间隔。 如：从早上8时到10时，花了2个小时的时间写作业，从杭州到上海火车运行的时间是2小时30分。这叫做时间。 我们可以从单位名称上来区分时刻与时间的差异。 时刻，一般用“时”如：飞机上午8时起航，指飞机离开机场时刻。时间一般用“小时”共飞行了8小时，指飞机从上午8时起飞到下午4时降落，在空中飞行了8个小时。 同学们不仅要会读钟面上显示的时刻，还要学会观察钟面所表示的不同的时刻之间的时间关系。找出规律。 如：长短针位置的判断时刻，确定长，短针互换位置后的时刻，反射到镜面上的钟面的时刻等等。有利于培养自己观察能力。 例1 根据前3个钟面的规律，画出第4个钟面的长、短针。 3 分析：前面三个钟表所表示的时刻分别是1时，3时30分，6时，相邻两个钟的时间差都是2小时30分。因此第4个钟也应是在第3个钟6点的基础上增加2小时30分，应显示出的时刻是8点30分 例2 按次序观察图中各钟面所表示的时刻，找出各种钟面所表示的时间规律，请在第5只钟面上标出符合规律的时刻 分析：把各钟面表示的时刻依次排列起来 11点30分→12点5分→12点40分→1点15分→（）→2点25分 发现它们相邻两钟的间隔时间都是35分钟，因此第5个钟面的时刻应是1点50分。 例3 见图：是反射在镜面上的两只钟面的长针和短针的位置，请说出各钟面的时刻？ 分析：同学们我们只要用镜子实践一下，就会发现任何物体经过镜面反射，它的位置发生了变化。左边的在镜子反射后成为右边，右边的在镜子反射后变为左边了，因此，要从镜面上反射出来的钟面时刻推出原钟面的时刻，只要将镜面上的钟面左右翻转半圈，这两只钟面表示的时刻分别为6点40分和8点15分 第36届国际数学奥林匹克试题 1．（保加利亚） 设A 、B 、C 、D 是一条直线上依次排列的四个不同的点，分别以AC 、BD 为直径的圆相交于X 和Y ，直线XY 交BC 于Z 。若P 为XY 上异于Z 的一点，直线CP 与以AC 为直径的圆相交于C 和M ，直线BP 与以BD 为直径的圆相交于B 和N 。试证：AM 、DN 和XY 三线共点。 证法一：*设AM 交直线XY 于点Q ，而DN 交直线XY 于点Q ′（如图95-1，注意：这里只画出了点P 在线段XY 上的情形，其他情况可类似证明）。须证：Q 与Q ′重合。 由于XY 为两圆的根轴，故XY ⊥AD ，而AC 为直径，所以 ∠QMC=∠PZC=90° 进而，Q ，M ，Z ，B 四点共圆。 同理Q ′，N ，Z ，B 四点共圆。 这样，利用圆幂定理，可知 QP ·PZ=MP ·PC=XP ·PY ， Q ′P ·PZ=NP ·PB=XP ·PY 。 所以，QP= Q ′P 。而Q 与Q ′都在直线XY 上且在直线AD 同侧，从而，Q 与Q ′重合。命题获证。 分析二* 如图95-2，以XY 为弦的任意圆O ， 只需证明当P 确定时，S 也确定。 证法二：设X （0，m ）,P （0，y 0）， ∠PCA=α， m 、y 0是定值。有2 0.yx x x ctg y x C A c =?-=但α， 则.0 2 αtg y m x A -= 因此，AM 的方程为 ).(0 2 ααtg y m x ctg y ?+= 令0 2,0y m y x s ==得，即点S 的位置取决于点P 的位置，与⊙O 无关，所以AM 、DN 和ZY 三条直线共点。 2．（俄罗斯）设a 、b 、c 为正实数且满足abc=1。试证： .2 3)(1)(1)(1333≥+++++b a c a c b c b a 证法一：**设γβα++=++=++=---------1111111112,2,2b a c a c b c b a ， 有.0=++γβα于是， ) (4)(4)(4333b a c a c b c b a +++++ )(4)(4)(4333b a c a b c a c b a b c c b a a b c +++++= 112 111121111211)()()(------------+++++++++++=b a b a c c b c b c b γαβα 21112 1112111111)()()()(2)(2γβαγβα------------+++++++++++=b a a c c b c b a .6132)111(23=?≥++≥abc c b a ∴原不等式成立。 背景资料：陕西省永寿县中学安振平老师在《证明不等式的若干代换技巧》一文中运用“增量代换”给出证法一，还用增量代换法给出第 6届IMO 试题的证明。什么是增量代换法？—— 由α≤+=≥0,,其中令a b a b a 称为增量。运用这种方法来论证问题，我们称为增量代换法。 题1 设c b a ,,是某一三角形三边长。求证： .3)()()(222abc c b a c b a c b a c b a ≤-++-++-+ （第6届IMO 试题） 证明 不失一般性，设.,0,0,0,,,y x z y x z y x c y x b x a >≥≥>++=+==且 abc c b a c b a c b a c b a 3)()()(222--++-++-+则 + ++++-+++++-++++=x z y x y x x z y x y x x z y x y x x [)()]()[()(])()[(222 匈牙利数学奥林匹克试题(1975-2002) 75th Kürschák Competition 1975 Problem1.Transform the equation ab2(1/(a + c)2 + 1/(a - c)2) = (a - b) into a simpler form, given that a > c ≥ 0, b > 0. Problem2.Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral Problem3.Let x0 = 5, x n+1 = x n + 1/x n. Prove that 45 < x1000 < 45.1. 76th Kürschák Competition 1976 Problem1 ABCD is a parallelogram. P is a point outside the parallelogram such that angles PAB and PCB have the same value but opposite orientation. Show that angle APB = angle DPC. Problem2 A lottery ticket is a choice of 5 distinct numbers from 1, 2, 3, ... , 90. Suppose that 55 distinct lottery tickets are such that any two of them have a common number. Prove that one can find four numbers such that every ticket contains at least one of the four Problem3 Prove that if the quadratic x2 + ax + b is always positive (for all real x) then it can be written as the quotient of two polynomials whose coefficients are all positive. 77th Kürschák Competition 1977 Problem1 Show that there are no integers n such that n4 + 4n is a prime greater than 5. Problem2ABC is a triangle with orthocenter H. The median from A meets the circumcircle again at A1, and A2 is the reflection of A1 in the midpoint of BC. The points B2 and C2 are defined similarly. Show that H, A2, B2 and C2 lie on a circle. Problem3 Three schools each have n students. Each student knows a total of n+1 students at the other two schools. Show that there must be three students, one from each school, who know each other. 78th Kürschák Competition 1978 Problem1 a and b are rationals. Show that if ax2 + by2 = 1 has a rational solution (in x and y), then it must have infinitely many. Problem2 The vertices of a convex n-gon are colored so that adjacent vertices have different colors. Prove that if n is odd, then the polygon can be divided into triangles with non-intersecting diagonals such that no diagonal has its endpoints the same color. Problem3 A triangle has inradius r and circumradius R. Its longest altitude has length H. Show that if the triangle does not have an obtuse angle, then H ≥ r + R. When does equality hold? 79th Kürschák Competition 1979 Problem1 The base of a convex pyramid has an odd number of edges. The lateral edges of the pyramid are all equal, and the angles between neighbouring faces are all equal. Show that the base must be a regular polygon. Problem 2f is a real-valued function defined on the reals such that f(x) ≤ x and f(x+y) ≤ f(x) + f(y) for all x, y. Prove that f(x) = x for all x. Solution Suppose f(0) < 0. Then f(y) ≤ f(0) + f(y) < f(y). Contradiction. So f(0) ≥ 0 and hence f(0) = 0. Now suppose f(x) < x. Then f(0) ≤ f(x) + f(-x) < x + f(-x) ≤ x - x = 0. Contradiction. Hence f(x) = x. 2009中国数学奥林匹克解答 一、给定锐角三角形PBC ，PC PB ≠．设A ，D 分别是边PB ，PC 上的点，连接AC ，BD ，相交于点O. 过点O 分别作OE ⊥AB ，OF ⊥CD ，垂足分别为E ，F ，线段BC ，AD 的中点分别为M ，N ． （1）若A ，B ，C ，D 四点共圆，求证：EM FN EN FM ?=?； （2）若 EM FN EN FM ?=?，是否一定有A ，B ，C ，D 四点共圆？证明你的结论． 解（1）设Q ，R 分别是OB ，OC 的中点，连接 EQ ，MQ ，FR ，MR ，则 11 ,22 EQ OB RM MQ OC RF ====， 又OQMR 是平行四边形，所以 OQM ORM ∠=∠， 由题设A ，B ，C ，D 四点共圆，所以 ABD ACD ∠=∠， 于是 图1 22EQO ABD ACD FRO ∠=∠=∠=∠， 所以 E Q M E Q O O Q M F R O O R M ∠=∠+∠=∠+∠=∠， 故 E Q M M R F ???， 所以 EM ＝FM ， 同理可得 EN ＝FN ， 所以 E M F N E N F M ?=?． （2）答案是否定的． 当AD ∥BC 时，由于B C ∠≠∠，所以A ，B ，C ，D 四点不共圆，但此时仍然有 EM FN EN FM ?=?，证明如下： 如图2所示，设S ，Q 分别是OA ，OB 的中点，连接ES ，EQ ，MQ ，NS ，则 11 ,22 NS OD EQ OB ==， 所以 N S O D E Q O B =． ① C B 又 11 , 22 ES OA MQ OC ==，所以 ES OA MQ OC =．② 而AD∥BC，所以 OA OD OC OB =，③ 由①，②，③得NS ES EQ MQ =． 因为2 NSE NSA ASE AOD AOE ∠=∠+∠=∠+∠， ()(1802) EQM MQO OQE AOE EOB EOB ∠=∠+∠=∠+∠+?-∠ (180)2 AOE EOB AOD AOE =∠+?-∠=∠+∠， 即NSE EQM ∠=∠， 所以NSE ?～EQM ?， 故 EN SE OA EM QM OC ==（由②）．同理可得， FN OA FM OC =， 所以EN FN EM FM =， 从而EM FN EN FM ?=?． C B 智巧趣题 知识要点 数学问题中有许多趣题，它们充分地体现了数学思维和方法的神奇魅力，学习这些趣题，并掌握其中的数学原理，有利于我们思维的拓展，同时激发对数学的兴趣。 本讲主要考察学生对于所学知识的活学活用能力，注意观察生活中的各类事实，学会用数学方法巧解各类问题。旨在锻炼学生的灵活思考、创新思考的能力，鼓励学生多多动手、动脑，从解决问题的过程中感受学习的乐趣。 翻硬币 【例 1】（2003年4月20日第一届小学“希望杯”全国数学邀请赛五年级2试第6题）桌面上4枚硬币向上的一面都是“数字”，另一面都是“国徽”，如果每次翻转3枚硬币，至少_______次可使向上的一面都是“国徽”。 【例 2】桌上放有345枚正面朝下的硬币，第1次翻动其中1枚，第2次翻动其中2枚，第3次翻动其中3枚，……，第345次翻动其中345枚。经过345次翻动后，能否使这345枚硬币都正面朝上？ 倒墨水 【例 3】（2005年第三届小学“希望杯”全国数学邀请赛四年级培训题）甲杯中有200毫升红墨水，乙杯中有100毫升蓝墨水，从甲杯倒出50毫升到乙杯里，搅匀后，又从乙杯倒出50毫升到甲杯里。 这时，甲杯中混入的蓝墨水与乙杯中混入的红墨水的多少关系是_______（填“前者少”、“前者多”、“相同”或“不确定的”）。 【例 4】（2005年3月13日第三届小学“希望杯”全国数学邀请赛四年级第1试第18题）小华和爸爸分享“红、黑甜品”（红豆沙加芝麻糊）。方法是：小华先将两勺红豆沙倒进盛载芝麻糊的碗中，搅匀后再取回两勺放入原先盛载红豆沙的碗中，混成后，爸爸问小华：“如果混合前红豆沙与芝麻糊的体积一样，那么混合后红豆沙含芝麻糊的分量与芝麻糊含红豆沙的分量比较，哪一个多？”。 小华的正确答案是_______。 【例 5】欣欣喝一杯牛奶，第一次喝了这杯牛奶的1 3 ，用水加满；第二次又喝了杯里的 1 3 ，又用水加满； 第三次又喝了杯里的1 3 ，又用水加满；之后她把这一杯全部喝完。想想欣欣喝的牛奶多还是水多？ 【例 6】有一个注入了1999升的容器A和一个与A大小相同的空着的容器B。第一回把A的1 2 移入B； 第二回把B的1 3 移入A；第三回把A的 1 4 移入B；然后把B的 1 5 移入A……就这样不断地移下 去。请问：当第1999回把A中的水移入B中时，B容器中有多少升水？ 波兰数学奥林匹克试题及解答 1 设为正整数，证明：所有与互质且不超过的自然数的立方和是的倍数。 2 在锐角三角形中，点是边上一点，使得 。证明：。 3 已知正实数的和等于1，证明：。 4 圆周上的点都被染上了某三种颜色中的一种，证明：在这个圆周上存在三个点，它们是某个等腰三角形的顶点，且它们同色。 5 求所有的正整数对（），使得与都是完全立方数。 6 点是内部或边界上一点，点分别是点在边 上的垂足，证明：的充要条件是点在边 上。 7 证明：对任意正整数，和每一个实数，存在实数，使得。 8 关于非负整数的函数定义如下：对任意 ；对。证明：对 均有。 9 设为给定的自然数，且，证明：是一个完全平方数。 10 设是三维空间中彼此垂直的三个单位向量，设是过点的一个平面，分别是在平面上的投影。对任意平面，求数 构成的集合。 11 设为正整数，是具有下述性质的个自然数构成的集合：中任意 个元素中，必有两个数，使得其中一个是另一个的倍数。证明：中存在 个数，使得对，均有。 12 点分别是锐角三角形的边上的点， 的外接圆交于一点，证明：若 ，则为三角形的三条高。 解答或提示 1 利用结论：若，则，将与配对即可证明此题。 2 记，则，利用正弦定理可知，， ，从而，要证的式子等价于，最后一式是显然的。 3 注意到，，所以， ， 故。 于是，我们有：。 即：。结合，可知命题成立。 4 可以证明：该圆周的内接正十三边形的13个顶点中，必有同色的三个点，它们是一个等腰三角形的顶点。 5 设是满足条件的正整数对，不失一般性，设， 则：，故，这表明 ，将之代入，可知是一个完全立方数，从而，是一个完全立方数。设 ，展开可知，于是。注意到： ， 故或，分别求解，可知只能是，进而 。所求数对。 6 利用勾股定理易证：等价于。 7 任给，及，令待定， 则： 第32届中国数学奥林匹克获奖名单 一等奖（116人，按省市自治区排列） 编号姓名地区学校 M16001 吴蔚琰安徽合肥一六八 M16002 考图南安徽安师大附中 M16003 徐名宇安徽合肥一中 M16004 吴作凡安徽安师大附中 M16005 周行健北京人大附中 M16006 王阳昇北京北京四中 M16007 陈远洲北京北师大附属实验中学M16008 杨向谦北京人大附中 M16009 夏晨曦北京北师大二附 M16010 谢卓凡北京清华附中 M16011 薛彦钊北京人大附中 M16012 胡宇征北京北京四中 M16013 徐天杨北京北京101中学 M16014 董昕妍北京人大附中 M16015 冯韫禛北京人大附中 M16016 林挺福建福建师范大学附属中学M16017 任秋宇广东华南师大附中 M16018 何天成广东华南师大附中 M16019 戴悦浩广东华南师大附中 M16020 谭健翔广东华南师大附中 M16021 王迩东广东华南师大附中 M16022 程佳文广东深圳中学 M16023 李振广东深圳外国语学校 M16024 张坤隆广东深圳中学 M16025 齐文轩广东深圳中学 M16026 卜辰璟贵州贵阳一中 M16027 顾树锴河北衡水第一中学 M16028 袁铭泽河北衡水第一中学 M16029 卢梓潼河北石家庄二中 M16030 赵振华河南郑州外国语学校 M16031 陈泰杰河南郑州外国语学校 M16032 迟舒乘黑龙江哈尔滨市第三中学 M16033 黄桢黑龙江哈尔滨市第三中学 M16034 姚睿湖北华中师范大学第一附属中学M16035 魏昕湖北武汉二中 M16036 黄楚昊湖北武钢三中 M16037 刘鹏飞湖北武汉二中 M16038 赵子源湖北华中师范大学第一附属中学M16039 徐行知湖北武钢三中 M16040 吴金泽湖北武汉二中 M16041 李弘梓湖北武汉二中 M16042 施奕成湖北华中师范大学第一附属中学M16043 袁睦苏湖北武汉二中 M16044 王子迎湖北武汉二中 M16045 袁昕湖北华中师范大学第一附属中学M16046 陈子瞻湖北湖北省黄冈中学 M16047 詹立宸湖北华中师范大学第一附属中学M16048 严子恒湖北武钢三中 M16049 陈贵显湖北华中师范大学第一附属中学M16050 张騄湖南长沙市长郡中学 M16051 刘哲成湖南长沙市雅礼中学 M16052 仝方舟湖南长沙市长郡中学 M16053 谢添乐湖南长沙市雅礼中学 M16054 尹龙晖湖南长沙市雅礼中学 M16055 黄磊湖南长沙市雅礼中学 M16056 肖煜湖南长沙市长郡中学 M16057 吴雨澄湖南湖南师范大学附属中学M16058 方浩湖南长沙市第一中学 M16059 郭鹏吉林东北师大附中 M16060 丁力煌江苏南京外国语学校 M16061 朱心一江苏南京外国语学校 M16062 高轶寒江苏南京外国语学校 M16063 彭展翔江西高安二中 M16064 刘鸿骏江西江西省吉安市第一中学M16065 孔繁淏辽宁大连二十四中 M16066 孔繁浩辽宁东北育才学校 M16067 孟响辽宁大连24中 M16068 毕梦达辽宁辽宁省实验中学 学 奥 数 这里总有一本适合你 奥数图书出版大事记 2000年 《奥数教程》（10种）第一版问世 2001年 《奥数教程》获优秀畅销书奖 2002年 《奥数教程》在香港出版繁体字版和网络版 2002年 《奥数测试》（第一版）出版 2003年 《奥数教程》（第二版）出版，并开展“有奖订正”、“巧解共享”活动 2003年 《奥数教程》（3~6年级）VCD出版 2003年~ 陆续出版由IMO中国国家集训队教练组编写的《走向IMO：数学奥林匹克试题集锦》 2005年 “奥数”图书累计销量近1000万册 2005年 出版《数学奥林匹克小丛书》（30种） 2006年 《奥数教程》（第三版）、《奥数测试》（第二版）出版 2006年 《数学奥林匹克小丛书》（12种）繁体字版在台湾出版 2007~2008年 《多功能题典》丛书中的小学、初中和高中数学竞赛相继出版 2008年 《日本小学数学奥林匹克（六年级）》出版 2009年~ 《高中数学联赛备考手册（预赛试题集锦）》陆续出版 2009年 《Mathematical Olympiad in China》、《Problems of Number Theory in Mathematical Competitions》和《Graph Theory》相继与新加坡世界科技出版公司联合出版 2010年 《全俄中学生数学奥林匹克（1993~2006）》出版 2010~2011年 《高思学校竞赛数学课本》和《高思学校竞赛数学导引》（3~6年级）相继出版 2011年 《从课本到奥数》（1~9年级A、B版）出版 2011年 《初中数学联赛考前辅导》和《高中数学联赛考前辅导》出版 小学数学奥林匹克竞赛试题及答案 （三年级） (红色为正确答案) 1、根据下列数中的规律在括号里填入合适的数： 17、2、14、2、11、2、（ ）、（ ）。 A 2、8 B 8、2 C 5、4 D 2、2 2、甲乙丙三个数平均数是150，甲数48，乙数与丙数相同，那么乙数是（ ）。 A 201 B 402 C 51 D 102 3、同学们做操,排成一个正方形的队伍,从前,后,左右数,小红都是第5 个,问一共有( )人. A 81 B25 C 32 D120 4、在“A ÷9=B …..C ”算式里,其中B 、C 都是一位数，那么A 最大是多少？ A 90 B 91 C 89 D 87 5、妈妈从蛋糕店买来一块方形蛋糕，（如图），让小红动手分成8块，最小要切（ ）刀。 A 2 B 4 C 3 D 5 6、在所有四位数中，各位数字之和等于35的数共有（ ）个。 A 4 B 5 C 3 D 6 7、如图，在小方格里最多放入一个?,要想使得同一行、同一列或对角连线上的三个小方格最多不出现三个?，那么在这九个小方格里最多能放入（ ）个?。（） A 4 B7 C 6 D 5 8、甲乙二人买同一种杂志，甲买一本差2角8分，乙买一本差2角6分，而他俩的钱合起来买一本还剩2角6分，那么这种杂志每本价钱是（ ）。 A 1元 B 7角 C 8角 D 9角 9、从1—9中选出6个数填在算式： ÷??（ + ）?（ - ），使结果最大。那么这个结果是（ ）。 A 190 B 702 C 630 D 890 10、夏令营基地小买部规定：每三个空汽水瓶可一瓶汽水。李明如果买6瓶汽水，那么他最多可以让（ ）位小伙伴喝到汽水。 A 11 B 8 C 10 D 9个 11、图中阴影部分是一个正方形，那么最大长方形的周长是（ A 26 B 28 C 24 D 25 41st IMO2000 Problem1.AB is tangent to the circles CAMN and NMBD.M lies between C and D on the line CD,and CD is parallel to AB.The chords NA and CM meet at P;the chords NB and MD meet at Q.The rays CA and DB meet at E.Prove that P E=QE. Problem2.A,B,C are positive reals with product1.Prove that(A?1+ 1 B )(B?1+1 C )(C?1+1 A )≤1. Problem3.k is a positive real.N is an integer greater than1.N points are placed on a line,not all coincident.A move is carried out as follows. Pick any two points A and B which are not coincident.Suppose that A lies to the right of B.Replace B by another point B to the right of A such that AB =kBA.For what values of k can we move the points arbitrarily far to the right by repeated moves? Problem4.100cards are numbered1to100(each card di?erent)and placed in3boxes(at least one card in each box).How many ways can this be done so that if two boxes are selected and a card is taken from each,then the knowledge of their sum alone is always su?cient to identify the third box? Problem5.Can we?nd N divisible by just2000di?erent primes,so that N divides2N+1?[N may be divisible by a prime power.] Problem6.A1A2A3is an acute-angled triangle.The foot of the altitude from A i is K i and the incircle touches the side opposite A i at L i.The line K1K2is re?ected in the line L1L2.Similarly,the line K2K3is re?ected in L2L3and K3K1is re?ected in L3L1.Show that the three new lines form a triangle with vertices on the incircle. 1 2012年IMO国际数学奥林匹克试题解答 第一题 设J是三角形ABC顶点A所对旁切圆的圆心. 该旁切圆与边BC相切于点M, 与直线AB和AC分别相切于点K和L. 直线LM和BJ相交于点F, 直线KM与CJ相交于点G. 设S是直线AF和BC的交点, T是直线AG和BC 的交点. 证明: M是线段ST的中点. 2012年IMO国际数学奥林匹克试题第一题 解答: 因为 ∠JFL=∠JBM?∠FMB=∠JBM?∠CML=12(∠A+∠C)?12∠C=12∠A= ∠JAL, 所以A、F、J、L四点共圆. 由此可得AF⊥FJ, 而BJ是∠ABS的角平分线, 于是三角形ABS的角平分线与高重合, 从而AB=BS; 同理可得AC=CT. 综上, 有 SM=SB+BM=AB+BK=AK=AL=AC+CL=CT+CM=MT, 即M是线段ST的中点. 第二题 设n?3, 正实数a2,a3,?,a n满足a2?a3???a n=1, 证明: (a2+1)2(a3+1)3?(a n+1)n>n n. 解答:由均值不等式, 我们有 (a k+1)k=?(a k+1k?1+?+1k?1)k(ka k?(1k?1)k?1?? ? ? ? ? ? ? ? ? ? ? ?√k)k=k k(k?1)k?1a k, 当a k=1k?1时等号成立, 其中k=2,3,?,n. 于是 (a2+1)2(a3+1)3?(a n+1)n?221a2?3322a3???n n(n?1)n?1a n=n n. 当对任意的k=2,3,?,n时, 若恒有a k=1k?1, 此时由n?3知 a2?a3???a n=1(n?1)!≠1, 因此上述不等式等号不成立, 从而不等式得证. 第三题 "欺诈猜数游戏" 在两个玩家甲和乙之间进行, 游戏依赖于两个甲和乙都知道的正整数k和n. 游戏开始时甲先选定两个整数x和N, 1?x?N. 甲如实告诉乙N的值, 但对x 守口如瓶. 乙现在试图通过如下方式的提问来获得关于x的信息: 每次提问, 乙任选一个由若干正整数组成的集合S(可以重复使用之前提问中使用过的集合), 问甲"x是否属于S?". 乙可以提任意数量的问题. 在乙每次提问之后, 家必须对乙的提问立刻回答"是" 或"否", 甲可以说谎话, 并且说谎的次数没有限制, 唯一的限制是甲在任意连续k+1次回答中至少又一次回答是真话. 在乙问完所有想问的问题之后, 乙必须指出一个至多包含n个正整数的集合X, 若x属于X, 则乙获胜; 否则甲获胜. 证明:五年级最大与最小学生版
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