MathDB

1999 IMO Shortlist

Part of IMO Shortlist

Subcontests

(8)
8
1
7
2

IMO ShortList 1999, geometry problem 7

The point MM is inside the convex quadrilateral ABCDABCD, such that MA = MC, \hspace{0,2cm} \widehat{AMB} = \widehat{MAD} + \widehat{MCD}   \textnormal{and}   \widehat{CMD} = \widehat{MCB} + \widehat{MAB}. Prove that AB⋅CM=BC⋅MDAB \cdot CM = BC \cdot MD and BM⋅AD=MA⋅CD.BM \cdot AD = MA \cdot CD.

IMO ShortList 1999, combinatorics problem 7

Let p>3p >3 be a prime number. For each nonempty subset TT of {0,1,2,3,…,p−1}\{0,1,2,3, \ldots , p-1\}, let E(T)E(T) be the set of all (p−1)(p-1)-tuples (x1,…,xp−1)(x_1, \ldots ,x_{p-1} ), where each xi∈Tx_i \in T and x1+2x2+…+(p−1)xp−1x_1+2x_2+ \ldots + (p-1)x_{p-1} is divisible by pp and let ∣E(T)∣|E(T)| denote the number of elements in E(T)E(T). Prove that ∣E({0,1,3})∣≥∣E({0,1,2})∣|E(\{0,1,3\})| \geq |E(\{0,1,2\})| with equality if and only if p=5p = 5.
1
2

IMO ShortList 1999, geometry problem 1

Let ABC be a triangle and MM be an interior point. Prove that
min⁡{MA,MB,MC}+MA+MB+MC<AB+AC+BC. \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.

IMO ShortList 1999, combinatorics problem 1

Let n≥1n \geq 1 be an integer. A path from (0,0)(0,0) to (n,n)(n,n) in the xyxy plane is a chain of consecutive unit moves either to the right (move denoted by EE) or upwards (move denoted by NN), all the moves being made inside the half-plane x≥yx \geq y. A step in a path is the occurence of two consecutive moves of the form ENEN. Show that the number of paths from (0,0)(0,0) to (n,n)(n,n) that contain exactly ss steps (n≥s≥1)(n \geq s \geq 1) is 1s(n−1s−1)(ns−1).\frac{1}{s} \binom{n-1}{s-1} \binom{n}{s-1}.
6
3

IMO ShortList 1999, number theory problem 6

Prove that for every real number MM there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds MM.

IMO ShortList 1999, algebra problem 6

For n≥3n \geq 3 and a1≤a2≤…≤ana_{1} \leq a_{2} \leq \ldots \leq a_{n} given real numbers we have the following instructions: - place out the numbers in some order in a ring; - delete one of the numbers from the ring; - if just two numbers are remaining in the ring: let SS be the sum of these two numbers. Otherwise, if there are more the two numbers in the ring, replace Afterwards start again with the step (2). Show that the largest sum SS which can result in this way is given by the formula Smax=∑k=2n(n−2[k2]−1)ak.S_{max}= \sum^n_{k=2} \begin{pmatrix} n -2 \\ [\frac{k}{2}] - 1\end{pmatrix}a_{k}.

IMO ShortList 1999, combinatorics problem 6

Suppose that every integer has been given one of the colours red, blue, green or yellow. Let xx and yy be odd integers so that ∣x∣≠∣y∣|x| \neq |y|. Show that there are two integers of the same colour whose difference has one of the following values: x,y,x+yx,y,x+y or x−yx-y.
4
4
5
2

IMO ShortList 1999, geometry problem 5

Let ABCABC be a triangle, Ω\Omega its incircle and Ωa,Ωb,Ωc\Omega_{a}, \Omega_{b}, \Omega_{c} three circles orthogonal to Ω\Omega passing through (B,C),(A,C)(B,C),(A,C) and (A,B)(A,B) respectively. The circles Ωa\Omega_{a} and Ωb\Omega_{b} meet again in C′C'; in the same way we obtain the points B′B' and A′A'. Prove that the radius of the circumcircle of A′B′C′A'B'C' is half the radius of Ω\Omega.

IMO ShortList 1999, number theory problem 5

Let n,kn,k be positive integers such that n is not divisible by 3 and k≥nk \geq n. Prove that there exists a positive integer mm which is divisible by nn and the sum of its digits in decimal representation is kk.
3
3

IMO ShortList 1999, number theory problem 3

Prove that there exists two strictly increasing sequences (an)(a_{n}) and (bn)(b_{n}) such that an(an+1)a_{n}(a_{n}+1) divides bn2+1b^{2}_{n}+1 for every natural n.

IMO ShortList 1999, algebra problem 3

A game is played by nn girls (n≥2n \geq 2), everybody having a ball. Each of the (n2)\binom{n}{2} pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice nice if at the end nobody has her own ball and it is called tiresome if at the end everybody has her initial ball. Determine the values of nn for which there exists a nice game and those for which there exists a tiresome game.

Recurrence giving periods of rest before catching flies

A biologist watches a chameleon. The chameleon catches flies and rests after each catch. The biologist notices that: [*]the first fly is caught after a resting period of one minute; [*]the resting period before catching the 2mth2m^\text{th} fly is the same as the resting period before catching the mthm^\text{th} fly and one minute shorter than the resting period before catching the (2m+1)th(2m+1)^\text{th} fly; [*]when the chameleon stops resting, he catches a fly instantly.
[*]How many flies were caught by the chameleon before his first resting period of 99 minutes in a row? [*]After how many minutes will the chameleon catch his 98th98^\text{th} fly? [*]How many flies were caught by the chameleon after 1999 minutes have passed?
2
4