imo2000p.pdf

Problems Shortlisted to the 2000 IMO Jury

41 st International Mathematical Olympiad, Washington, DC, USA

Combinatorics

C1. A magician has one hundred cards numbered 1 to 100. He puts them into three

boxes, a red one, a white one and a blue one, so that each box contains at least one

card. A member of the audience draws two cards from two different boxes and

announces the sum of numbers on those cards. Given this information, the magician

locates the box from which no card has been drawn. How many ways are there to put

the cards in the three boxes so that the trick works?

C2. A staircase box with 3 steps of width 2 is made of 12 unit cubes. Determine all

integers n for which it is possible to build a cube of side n using such bricks.

4 a positive integer. Given a set S = {P 1 , » , P n } of n points in the plane

such that no three are collinear and no four concyclic, let a t with 1

n be the

number of circles P i P j P k that contain P t in their interior, and let m (S) = a 1 + » + a n .

Prove that there exists a positive integer f(n) , depending only on n such that the points

of S are the vertices of a convex polygon if and only if m (S) = f(n) .

2n/3 . Find the least m for which

it is possible to place m pawns on a square of an n x n chessboard so that no column

or row contains a block of k adjacent unoccupied squares.

C5. On a plane we have n rectangles with parallel sides. The sides of distinct

rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into

connected regions. A region is said to be nice if it has at least one of the vertices of

the original n squares on its boundary. Prove that the sum of the numbers of the

vertices of all nice regions is less than 40 n . (nonconvex regions are allowed)

C6. Let p and q be pairwise prime positive integers. A subset S of the set of

nonnegative integers is called an ideal if 0

S and if n

S then n+p, n+q

S. Find

the number of ideals.

Algebra

A1. Let a, b, c be positive real numbers so that abc =1. Prove that

C3. Let n

C4. Let n and k be positive integers so that n/2 < k

A2. Let a, b, c be positive integers satisfying the conditions b>2a and c>2b. Show

that there exists a real number

with the property that {

a}, {

b}, {

(1/3,2/3].

A3. Find all pairs of real valued functions f, g defined on the set of real numbers such

that

(

))

(

)

(

)

(

)

for all real x and y .

A4. The function F is defined in the set of nonnegative integers and takes nonnegative

integer values satisfying the conditions: for every n

i. F(4 n ) = F(2 n ) + F( n )

ii. F(4 n+ 2) = F(4 n ) +1

iii. F(2 n +1) = F(2 n ) + 1

Prove that for each positive integer m , the number of integers n with 0

F(4 n ) = F(3 n ) is F(2 m+1 ).

n < 2 m and

a positive real number. Initially there are n

fleas on a horizontal line, not all at the same point. We define a move as choosing two

fleas at some points A and B, with A to the left of B, and letting the flea from A jump

over the flea from B to the point C so that BC/AB =

2 be a positive integer and

such that, for any point M on the line and for any initial

position of the n fleas, there exists a sequence of moves that will take them all to the

position right of M.

A6. A nonempty set A of real numbers is called a B 3 -set if the conditions a 1 , », a 6

and a 1 + a 2 + a 3 = a 4 + a 5 + a 6 imply that the sequences ( a 1 , a 2 , a 3 ) and ( a 4 , a 5 , a 6 ) are

identical to a permutation. Let A = { a (0) = 0 < a (1) < a (2) <»} and B = { b (0) = 0 <

b (1) < b (2) <»} be infinite sequences with D(A) = D(B), where, for a set X of real

numbers, D(X) denotes the difference set

. Prove that if A is a B 3 -set

then A = B.

A7. For a polynomial of degree 2000 with distinct real coefficients let M(P) be the set

of all polynomials that can be produced from P by a permutation of its coefficients. A

polynomial will be called n independent if P( n )=0 and we can get from any Q in M(P)

a polynomial Q 1 such that Q 1 ( n ) = 0 by interchanging at most one pair of the

coefficients of Q. Find all integers n for which n independent polynomials exist.

Number Theory

2 that satisfy the following condition: for all a

and b relatively prime to n we have a = b ( mod n ) if an only if ab = 1 ( mod n) .

N2. For a positive integer n let d( n ) be the number of positive divisors of n . Find all

positive integers n so that

(

)

N3. Does there exist a positive integer n such that n has exactly 2000 prime divisors

and n divides 2 n + 1.

N4. Determine all triples of positive integers ( a, m, n ) so that a m + 1 divides ( a+ 1) n .

A5. Let n

Determine all values of

N1. Determine all positive integers n

N5. Prove that there exist infinitely many positive integers n so that p = nr , where p

and r are respectively the semiperimeter and the inradius of a triangle with integer

side lengths.

N6. Show that the set of positive integers which can not be represented as a sum of

distinct squares is finite.

Geometry

G1. In the plane we are given two circles intersecting at X and Y. Prove that there

exist four points with the following property:

For every circle touching the two given circles at A and B, and meeting the line XY at

C and D, each of the lines AC, AD, BC, BD passes through one of these points.

G2. Two circles G 1 and G 2 intersect at M and N. Let AB be the line tangent to these

circles at A and B, respectively, so that M lies closer to AB than N. Let CD be the line

parallel to AB and passing through M, with C on G 1 and D on G 2 . Lines AC and BD

meet at E; lines AN and CD meet at P; lines BN and CD meet at Q. Show that EP =

EQ.

G3. Let O be the circumcenter and H the orthocenter of an acute triangle ABC. Show

that there exist points D, E and F on sides BC, CA and AB respectively such that OD

+ DH = OE + EH = OF + FH, and the lines AD, BE and CF be concurrent.

4. Prove that A is cyclic if and only

if one can assign each vertex A i a pair ( b i , c i ) of real numbers so that

G5. The tangents at B and A to the circumcircle of an acute angled triangle ABC meet

the tangent at C at T and U respectively. AT meets BC at P and Q is the midpoint of

AP; BU meets CA at R and S is the midpoint of BR. Prove that

ABQ =

CBX < 90. If Y is

the point of intersection of the perpendicular bisectors of AB and CD prove that

ADX =

BCX < 90 and

DAX =

AYB = 2

ADX.

G7. Ten gangsters are standing on a flat surface, and the distances among them are all

distinct. At twelve oÆclock, when the church bells start chiming, each of them shoots

at the one among the other nine gangsters who is closest to him. What is the smallest

number of gangsters to be killed?

G8. Let AH 1 , BH 2 , CH 3 be the altitudes of an acute angled triangle ABC. Its incircle

touches the sides BC, AC and AB at T 1 , T 2 and T 3 respectively. Consider the

symmetric images of the lines H 1 H 2 , H 2 H 3 and H 3 H 1 with respect to the lines T 1 T 2 ,

T 2 T 3 and T 3 T 1 . Prove that these images forma triangle whose vertices lie on the

incircle of ABC.

G4. Let A = A 1 A 2 »A n be a convex polygon, n

BAS.

Determine, in terms of ratios of side-lengths, the triangles for which this angle is a

maximum.

G6. Let ABCD be a convex quadrilateral with AB not parallel to CD, and let X a

point inside ABCD such that

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