MathDB

1994 IMO Shortlist

Part of IMO Shortlist

Subcontests

(7)
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N is inside the quadrilateral abcd

ABCD ABCD is a quadrilateral with BC BC parallel to AD AD. M M is the midpoint of CD CD, P P is the midpoint of MA MA and Q Q is the midpoint of MB MB. The lines DP DP and CQ CQ meet at N N. Prove that N N is inside the quadrilateral ABCD ABCD.

Determine uniquely the ages of all the female citizens

In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens xx and x′x' either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens x=x0,x1,…,xn=x′x = x_0, x_1, \ldots, x_n = x' for some integer n≥2n \geq 2 such that xi−1 x_{i-1} and xix_i know each other. In a census, all male citizens declare their ages, and there is at least one male citizen. Each female citizen provides only the information that her age is the average of the ages of all the citizens she knows. Prove that this is enough to determine uniquely the ages of all the female citizens.
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Nice sequence

Define the sequence a1,a2,a3,... a_1, a_2, a_3, ... as follows. a1 a_1 and a2 a_2 are coprime positive integers and a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1. Show that for every m>1 m > 1 there is an n>m n > m such that amm a_m^m divides ann a_n^n. Is it true that a1 a_1 must divide ann a_n^n for some n>1 n > 1?

Preventing first player from getting 11 Xs in a row

Two players play alternatively on an infinite square grid. The first player puts an XX in an empty cell and the second player puts an OO in an empty cell. The first player wins if he gets 1111 adjacent XX's in a line - horizontally, vertically or diagonally. Show that the second player can always prevent the first player from winning.
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Functions f(x)f(y) = y^a*f(x/2) + x^b*f(y/2)

Let R \mathbb{R} denote the set of all real numbers and \mathbb{R}^\plus{} the subset of all positive ones. Let α \alpha and β \beta be given elements in R, \mathbb{R}, not necessarily distinct. Find all functions f: \mathbb{R}^\plus{} \mapsto \mathbb{R} such that f(x)f(y) \equal{} y^{\alpha} f \left( \frac{x}{2} \right) \plus{} x^{\beta} f \left( \frac{y}{2} \right) \forall x,y \in \mathbb{R}^\plus{}.

Find the number of positive integers

Define the sequences an,bn,cn a_n, b_n, c_n as follows. a_0 \equal{} k, b_0 \equal{} 4, c_0 \equal{} 1. If an a_n is even then a_{n \plus{} 1} \equal{} \frac {a_n}{2}, b_{n \plus{} 1} \equal{} 2b_n, c_{n \plus{} 1} \equal{} c_n. If an a_n is odd, then a_{n \plus{} 1} \equal{} a_n \minus{} \frac {b_n}{2} \minus{} c_n, b_{n \plus{} 1} \equal{} b_n, c_{n \plus{} 1} \equal{} b_n \plus{} c_n. Find the number of positive integers k<1995 k < 1995 such that some a_n \equal{} 0.

Cards and algorithm

There are n \plus{} 1 cells in a row labeled from 0 0 to n n and n \plus{} 1 cards labeled from 0 0 to n n. The cards are arbitrarily placed in the cells, one per cell. The objective is to get card i i into cell i i for each i i. The allowed move is to find the smallest h h such that cell h h has a card with a label k>h k > h, pick up that card, slide the cards in cells h \plus{} 1, h \plus{} 2, ... , k k one cell to the left and to place card k k in cell k k. Show that at most 2^n \minus{} 1 moves are required to get every card into the correct cell and that there is a unique starting position which requires 2^n \minus{} 1 moves. [For example, if n \equal{} 2 and the initial position is 210, then we get 102, then 012, a total of 2 moves.]
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Recursively defined function

Let f(x) \equal{} \frac{x^2\plus{}1}{2x} for x≠0. x \neq 0. Define f^{(0)}(x) \equal{} x and f^{(n)}(x) \equal{} f(f^{(n\minus{}1)}(x)) for all positive integers n n and x≠0. x \neq 0. Prove that for all non-negative integers n n and x \neq \{\minus{}1,0,1\} \frac{f^{(n)}(x)}{f^{(n\plus{}1)}(x)} \equal{} 1 \plus{} \frac{1}{f \left( \left( \frac{x\plus{}1}{x\minus{}1} \right)^{2n} \right)}.

circle with tangent lines

A circle C C with center O. O. and a line L L which does not touch circle C. C. OQ OQ is perpendicular to L, L, Q Q is on L. L. P P is on L, L, draw two tangents L1,L2 L_1, L_2 to circle C. C. QA,QB QA, QB are perpendicular to L1,L2 L_1, L_2 respectively. (A A on L1, L_1, B B on L2 L_2). Prove that, line AB AB intersect QO QO at a fixed point. Original formulation: A line l l does not meet a circle ω \omega with center O. O. E E is the point on l l such that OE OE is perpendicular to l. l. M M is any point on l l other than E. E. The tangents from M M to ω \omega touch it at A A and B. B. C C is the point on MA MA such that EC EC is perpendicular to MA. MA. D D is the point on MB MB such that ED ED is perpendicular to MB. MB. The line CD CD cuts OE OE at F. F. Prove that the location of F F is independent of that of M. M.

The game cannot terminate

1994 1994 girls are seated at a round table. Initially one girl holds n n tokens. Each turn a girl who is holding more than one token passes one token to each of her neighbours. a.) Show that if n<1994 n < 1994, the game must terminate. b.) Show that if n \equal{} 1994 it cannot terminate.