Olympiad problems

1985 AIME Problems, 14

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?

Estonia Open Junior - geometry, 2019.1.5

Tags: ratio , parallelogram , geometry

Point $M$ lies on the diagonal $BD$ of parallelogram $ABCD$ such that $MD = 3BM$. Lines $AM$ and $BC$ intersect in point $N$. What is the ratio of the area of triangle $MND$ to the area of parallelogram $ABCD$?

2010 Romania National Olympiad, 3

Tags: ramsey theory , combinatorics

In the plane are given $100$ points, such that no three of them are on the same line. The points are arranged in $10$ groups, any group containing at least $3$ points. Any two points in the same group are joined by a segment. a) Determine which of the possible arrangements in $10$ such groups is the one giving the minimal numbers of triangles. b) Prove that there exists an arrangement in such groups where each segment can be coloured with one of three given colours and no triangle has all edges of the same colour. [i]Vasile Pop[/i]

1999 Bosnia and Herzegovina Team Selection Test, 1

Tags: algebra , root , equation , real root

Let $a$, $b$ and $c$ be lengths of sides of triangle $ABC$. Prove that at least one of the equations $$x^2-2bx+2ac=0$$ $$x^2-2cx+2ab=0$$ $$x^2-2ax+2bc=0$$ does not have real solutions

2021 Saudi Arabia Training Tests, 19

Tags: tangent , circumcircle , geometry

Let $ABC$ be a triangle with $AB < AC$ inscribed in $(O)$. Tangent line at $A$ of $(O)$ cuts $BC$ at $D$. Take $H$ as the projection of $A$ on $OD$ and $E,F$ as projections of $H$ on $AB,AC$.Suppose that $EF$ cuts $(O)$ at $R,S$. Prove that $(HRS)$ is tangent to $OD$

1989 China National Olympiad, 3

Tags: function , complex number , algebra

Let $S$ be the unit circle in the complex plane (i.e. the set of all complex numbers with their moduli equal to $1$). We define function $f:S\rightarrow S$ as follow: $\forall z\in S$, $ f^{(1)}(z)=f(z), f^{(2)}(z)=f(f(z)), \dots,$ $f^{(k)}(z)=f(f^{(k-1)}(z)) (k>1,k\in \mathbb{N}), \dots$ We call $c$ an $n$-[i]period-point[/i] of $f$ if $c$ ($c\in S$) and $n$ ($n\in\mathbb{N}$) satisfy: $f^{(1)}(c) \not=c, f^{(2)}(c) \not=c, f^{(3)}(c) \not=c, \dots, f^{(n-1)}(c) \not=c, f^{(n)}(c)=c$. Suppose that $f(z)=z^m$ ($z\in S; m>1, m\in \mathbb{N}$), find the number of $1989$-[i]period-point[/i] of $f$.

2008 Irish Math Olympiad, 2

Tags: inequalities

For positive real numbers $ a$, $ b$, $ c$ and $ d$ such that $ a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2 \equal{} 1$ prove that $ a^2b^2cd \plus{} \plus{}ab^2c^2d \plus{} abc^2d^2 \plus{} a^2bcd^2 \plus{} a^2bc^2d \plus{} ab^2cd^2 \le 3/32,$ and determine the cases of equality.

2017-IMOC, N6

Tags: number theory

A mouse walks on a plane. At time $i$, it could do nothing or turn right, then it moves $p_i$ meters forward, where $p_i$ is the $i$-th prime. Is it possible that the mouse moves back to the starting point?

1980 IMO Shortlist, 9

Tags: pascal triangle , modular arithmetic , number theory , representation

Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

2018 Poland - Second Round, 5

Tags: combinatorics , set theory , pigeonhole principle , counting

Let $A_1, A_2, ..., A_k$ be $5$-element subsets of set $\{1, 2, ..., 23\}$ such that, for all $1 \le i < j \le k$ set $A_i \cap A_j$ has at most three elements. Show that $k \le 2018$.

2009 JBMO Shortlist, 5

Tags: geometry

Let ${A, B, C}$ and ${O}$ be four points in plane, such that $\angle ABC>{{90}^{{}^\circ }}$ and ${OA=OB=OC}$.Define the point ${D\in AB}$ and the line ${l}$ such that ${D\in l, AC\perp DC}$ and ${l\perp AO}$. Line ${l}$ cuts ${AC}$at ${E}$ and circumcircle of ${ABC}$ at ${F}$. Prove that the circumcircles of triangles ${BEF}$and ${CFD}$are tangent at ${F}$.

1991 IMTS, 5

Tags: induction , combinatorics

Two people, $A$ and $B$, play the following game with a deck of 32 cards. With $A$ starting, and thereafter the players alternating, each player takes either 1 card or a prime number of cards. Eventually all of the cards are chosen, and the person who has none to pick up is the loser. Who will win the game if they both follow optimal strategy?

1998 AMC 8, 5

Tags:

Which of the following numbers is largest? $ \text{(A)}\ 9.12344\qquad\text{(B)}\ 9.123\overline{4}\qquad\text{(C)}\ 9.12\overline{34}\qquad\text{(D)}\ 9.1\overline{234}\qquad\text{(E)}\ 9.\overline{1234} $

2017 EGMO, 5

Tags: number theory , equation

Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$ a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple. b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple. [i]There are exactly $n$ factors in the product on the left hand side.[/i]

1987 Traian Lălescu, 2.3

Tags: geometry , 3d geometry

Let be a cube $ ABCDA'BC'D' $ such that $ AB=1, $ and let $ M,N,P,Q $ be points on the segments $ A'B',C'D',A'D', $ respectively, $ BC, $ excluding their extremities. [b]a)[/b] If $ MN $ is perpendicular to $ PQ, $ then $ AM+A'P+CQ+CN=3. $ [b]b)[/b] If $ MN $ and $ PQ $ are concurrent, then $ AM\cdot CQ=A'P\cdot CN. $

1977 All Soviet Union Mathematical Olympiad, 248

Tags: combinatorics , algebra , sum

Given natural numbers $x_1,x_2,...,x_n,y_1,y_2,...,y_m$. The following condition is valid: $$(x_1+x_2+...+x_n)=(y_1+y_2+...+y_m)<mn \,\,\,\, (*)$$ Prove that it is possible to delete some terms from (*) (not all and at least one) and to obtain another valid condition.

1973 IMO, 2

Tags: function , algebra , fixed point , group theory

$G$ is a set of non-constant functions $f$. Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$. If $f$ and $g$ are in $G$, then so is $fg$, where $fg$ is defined by $fg(x)=f(g(x))$. If $f$ is in $G$, then so is the inverse $f^{-1}$. If $f(x)=ax+b$, then $f^{-1}(x)= \frac{x-b}{a}$. Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$. Prove that all the functions in $G$ have a common fixed point.

2018 Saudi Arabia BMO TST, 1

Tags: number theory , power of 2

Find the smallest positive integer $n$ which can not be expressed as $n =\frac{2^a - 2^b}{2^c - 2^d}$ for some positive integers $a, b, c, d$

1958 Poland - Second Round, 2

Tags: geometry , combinatorics , combinatorial geometry

Six equal disks are placed on a plane so that their centers lie at the vertices of a regular hexagon with sides equal to the diameter of the disks. How many revolutions will a seventh disk of the same size make when rolling in the same plane externally over the disks before returning to its initial position?