Found problems: 33
2023 Israel Olympic Revenge, P4
Let $c$ be a positive real and $a_1, a_2, \dots$ be a sequence of nonnegative integers satisfying the following conditions for every positive integer $n$:
[b](i)[/b]$\frac{2^{a_1}+2^{a_2}+\cdots+2^{a_n}}{n}$ is an integer;
[b](ii)[/b]$\textbullet 2^{a_n}\leq cn$.
Prove that the sequence $a_1, a_2, \dots$ is eventually constant.
2020 Olympic Revenge, 5
Let $n$ be a positive integer. Given $n$ points in the plane, prove that it is possible to draw an angle with measure $\frac{2\pi}{n}$ with vertex as each one of the given points, such that any point in the plane is covered by at least one of the angles.
2019 Israel Olympic Revenge, P2
A $5779$-dimensional polytope is call a [b]$k$-tope[/b] if it has exactly $k$ $5778$-dimensional faces.
Find all sequences $b_{5780}, b_{5781}, \dots, b_{11558}$ of nonnegative integers, not all $0$, such that the following condition holds:
It is possible to tesselate every $5779$-dimensional polytope with [u]convex[/u] $5779$-dimensional polytopes, such that the number of $k$-topes in the tessellation is proportional to $b_k$, while there are no $k$-topes in the tessellation if $k\notin \{5780, 5781, \dots, 11558\}$.
2020 Israel Olympic Revenge, P1
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in \mathbb{R}$ one has
\[f(f(x)+y)=f(x+f(y))\]
and in addition the set $f^{-1}(a)=\{b\in \mathbb{R}\mid f(b)=a\}$ is a finite set for all $a\in \mathbb{R}$.
2023 Turkey Olympic Revenge, 2
Let $ABC$ be a triangle. A point $D$ lies on line $BC$ and points $E,F$ are taken on $AC,AB$ such that $DE \parallel AB$ and $DF\parallel AC$. Let $G = (AEF) \cap (ABC) \neq A$ and $I = (DEF) \cap BC\neq D$. Let $H$ and $O$ denote the orthocenter and the circumcenter of triangle $DEF$. Prove that $A,O,I$ are collinear if and only if $G,H,I$ are collinear.
[i]Proposed by Kaan Bilge[/i]
2015 Olympic Revenge, 2
Given $v = (a,b,c,d) \in \mathbb{N}^4$, let $\Delta^{1} (v) = (|a-b|,|b-c|,|c-d|,|d-a|)$ and $\Delta^{k} (v) = \Delta(\Delta^{k-1} (v))$ for $k > 1$. Define $f(v) = \min\{k \in \mathbb{N} : \Delta^k (v) = (0,0,0,0)\}$ and $\max(v) = \max\{a,b,c,d\}.$ Show that $f(v) < 1000\log \max(v)$ for all sufficiently large $v$ and $f(v) > 0.001 \log \max (v)$ for infinitely many $v$.
2015 Olympic Revenge, 4
Consider a game in the integer points of the real line, where an Angel tries to escape from a Devil. A positive integer $k$ is chosen, and the Angel and the Devil take turns playing. Initially, no point is blocked. The Angel, in point $A$, can move to any point $P$ such that $|AP| \le k$, as long as $P$ is not blocked. The Devil may block an arbitrary point. The Angel loses if it cannot move and wins if it does not lose in finitely many turns. Let $f(k)$ denote the least number of rounds the Devil takes to win. Prove that $$0.5 k \log_2 (k) (1 + o(1)) \le f(k) \le k \log_2(k) (1 +o(1)).$$
Note: $a(x) = b(x) (1+o(1))$ if $\lim_{x \to \infty} \frac{b(x)}{a(x)} = 1$.
2020 Olympic Revenge, 1
Let $n$ be a positive integer and $a_1, a_2, \dots, a_n$ non-zero real numbers. What is the least number of non-zero coefficients that the polynomial $P(x) = (x - a_1)(x - a_2)\cdots(x - a_n)$ can have?
2023 Turkey Olympic Revenge, 5
There are $10$ cups, each having $10$ pebbles in them. Two players $A$ and $B$ play a game, repeating the following in order each move:
$\bullet$ $B$ takes one pebble from each cup and redistributes them as $A$ wishes.
$\bullet$ After $B$ distributes the pebbles, he tells how many pebbles are in each cup to $A$. Then $B$ destroys all the cups having no pebbles.
$\bullet$ $B$ switches the places of two cups without telling $A$.
After finitely many moves, $A$ can guarantee that $n$ cups are destroyed. Find the maximum possible value of $n$.
(Note that $A$ doesn't see the cups while playing.)
[i]Proposed by Emre Osman[/i]
2002 Olympic Revenge, 3
Show that if $x,y,z,w$ are positive reals, then
\[
\frac{3}{2}\sqrt{(x^2+y^2)(w^2+z^2)} + \sqrt{(x^2+w^2)(y^2+z^2) - 3xyzw} \geq (x+z)(y+w)
\]
2023 Turkey Olympic Revenge, 3
Find all polynomials $P$ with integer coefficients such that $$s(x)=s(y) \implies s(|P(x)|)=s(|P(y)|).$$
for all $x,y\in \mathbb{N}$.
Note: $s(x)$ denotes the sum of digits of $x$.
[i]Proposed by Şevket Onur YILMAZ[/i]
2021 Olympic Revenge, 1
Let $a$, $b$, $c$, $k$ be positive reals such that $ab+bc+ca \leq 1$ and $0 < k \leq \frac{9}{2}$. Prove that:
\[\sqrt[3]{ \frac{k}{a} + (9-3k)b} + \sqrt[3]{\frac{k}{b} + (9-3k)c} + \sqrt[3]{\frac{k}{c} + (9-3k)a } \leq \frac{1}{abc}.\]
[i]Proposed by Zhang Yanzong and Song Qing[/i]
2015 Olympic Revenge, 3
For every $n \in \mathbb{N}$, there exist integers $k$ such that $n | k$ and $k$ contains only zeroes and ones in its decimal representation. Let $f(n)$ denote the least possible number of ones in any such $k$. Determine whether there exists a constant $C$ such that $f(n) < C$ for all $n \in \mathbb{N}$.
2002 Olympic Revenge, 1
Show that there is no function \(f:\mathbb{N}^* \rightarrow \mathbb{N}^*\) such that \(f^n(n)=n+1\) for all \(n\) (when \(f^n\) is the \(n\)th iteration of \(f\))
2020 Israel Olympic Revenge, P3
For each positive integer $n$, define $f(n)$ to be the least positive integer for which the following holds:
For any partition of $\{1,2,\dots, n\}$ into $k>1$ disjoint subsets $A_1, \dots, A_k$, [u]all of the same size[/u], let $P_i(x)=\prod_{a\in A_i}(x-a)$. Then there exist $i\neq j$ for which
\[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\]
a) Prove that there is a constant $c$ so that $f(n)\le c\cdot \sqrt{n}$ for all $n$.
b) Prove that for infinitely many $n$, one has $f(n)\ge \ln(n)$.
2020 Israel Olympic Revenge, P2
Let $A, B\subset \mathbb{Z}$ be two sets of integers. We say that $A,B$ are [u]mutually repulsive[/u] if there exist positive integers $m,n$ and two sequences of integers $\alpha_1, \alpha_2, \dots, \alpha_n$ and $\beta_1, \beta_2, \dots, \beta_m$, for which there is a [b]unique[/b] integer $x$ such that the number of its appearances in the sequence of sets $A+\alpha_1, A+\alpha_2, \dots, A+\alpha_n$ is [u]different[/u] than the number of its appearances in the sequence of sets $B+\beta_1, \dots, B+\beta_m$.
For a given quadruple of positive integers $(n_1,d_1, n_2, d_2)$, determine whether the sets
\[A=\{d_1, 2d_1, \dots, n_1d_1\}\]
\[B=\{d_2, 2d_2, \dots, n_2d_2\}\]
are mutually repulsive.
For a set $X\subset \mathbb{Z}$ and $c\in \mathbb{Z}$, we define $X+c=\{x+c\mid x\in X\}$.
2020 Olympic Revenge, 3
Let $ABC$ be a triangle and $\omega$ its circumcircle. Let $D$ and $E$ be the feet of the angle bisectors relative to $B$ and $C$, respectively. The line $DE$ meets $\omega$ at $F$ and $G$. Prove that the tangents to $\omega$ through $F$ and $G$ are tangents to the excircle of $\triangle ABC$ opposite to $A$.
2022 Israel Olympic Revenge, 1
For each positive integer $n$, decide whether it is possible to tile a square with exactly $n+1$ similar rectangles, each with a positive area and aspect ratio $1:n$.
2023 Israel Olympic Revenge, P1
Armadillo and Badger are playing a game. Armadillo chooses a nonempty tree (a simple acyclic graph) and places apples at some of its vertices (there may be several apples on a single vertex). First, Badger picks a vertex $v_0$ and eats all its apples. Next, Armadillo and Badger take turns alternatingly, with Armadillo starting. On the $i$-th turn the animal whose turn it is picks a vertex $v_i$ adjacent to $v_{i-1}$ that hasn't been picked before and eats all its apples. The game ends when there is no such vertex $v_i$.
Armadillo's goal is to have eaten more apples than Badger once the game ends. Can she fulfill her wish?
2022 Israel Olympic Revenge, 3
Determine if there exist positive real numbers $x, \alpha$, so that for any non-empty finite set of positive integers $S$, the inequality
\[\left|x-\sum_{s\in S}\frac{1}{s}\right|>\frac{1}{\max(S)^\alpha}\]
holds, where $\max(S)$ is defined as the maximum element of the finite set $S$.
2020 Olympic Revenge, 4
Let $n$ be a positive integer and $A$ a set of integers such that the set $\{x = a + b\ |\ a, b \in A\}$ contains $\{1^2, 2^2, \dots, n^2\}$. Prove that there is a positive integer $N$ such that if $n \ge N$, then $|A| > n^{0.666}$.
2022 Israel Olympic Revenge, 4
A (not necessarily regular) tetrahedron $A_1A_2A_3A_4$ is given in space. For each pair of indices $1\leq i<j\leq 4$, an ellipsoid with foci $A_i,A_j$ and string length $\ell_{ij}$, for positive numbers $\ell_{ij}$, is given (in all 6 ellipsoids were built).
For each $i=1,2$, a pair of points $X_i\neq X'_i$ was chosen so that $X_i, X'_i$ both belong to all three ellipsoids with $A_i$ as one of their foci. Prove that the lines $X_1X'_1, X_2X'_2$ share a point in space if and only if
\[\ell_{13}+\ell_{24}=\ell_{14}+\ell_{23}\]
[i]Remark: An [u]ellipsoid[/u] with foci $P,Q$ and string length $\ell>|PQ|$ is defined here as the set of points $X$ in space for which $|XQ|+|XP|=\ell$.[/i]
2023 Turkey Olympic Revenge, 1
Find all $c\in \mathbb{R}$ such that there exists a function $f:\mathbb{R}\to \mathbb{R}$ satisfying $$(f(x)+1)(f(y)+1)=f(x+y)+f(xy+c)$$ for all $x,y\in \mathbb{R}$.
[i]Proposed by Kaan Bilge[/i]
2023 Israel Olympic Revenge, P2
Triangle $\Delta ABC$ is inscribed in circle $\Omega$. The tangency point of $\Omega$ and the $A$-mixtilinear circle of $\Delta ABC$ is $T$. Points $E$, $F$ were chosen on $AC$, $AB$ respectively so that $EF\parallel BC$ and $(TEF)$ is tangent to $\Omega$. Let $\omega$ denote the $A$-excircle of $\Delta AEF$, which is tangent to sides $EF$, $AE$, $AF$ at $K$, $Y$, $Z$ respectively. Line $AT$ intersects $\omega$ at two points $P$, $Q$ with $P$ between $A$ and $Q$. Let $QK$ and $YZ$ intersect at $V$, and let the tangent to $\omega$ at $P$ and the tangent to $\Omega$ at $T$ intersect at $U$. Prove that $UV\parallel BC$.
2015 Olympic Revenge, 1
For $n \in \mathbb{N}$, let $P(n)$ denote the product of distinct prime factors of $n$, with $P(1) = 1$. Show that for any $a_0 \in \mathbb{N}$, if we define a sequence $a_{k+1} = a_k + P(a_k)$ for $k \ge 0$, there exists some $k \in \mathbb{N}$ with $a_k/P(a_k) = 2015$.