Found problems: 15925
2009 Saint Petersburg Mathematical Olympiad, 6
$(x_n)$ is sequence, such that $x_{n+2}=|x_{n+1}|-x_n$. Prove, that it is periodic.
2021 Science ON grade IX, 3
Real numbers $a,b,c$ with $0\le a,b,c\le 1$ satisfy the condition
$$a+b+c=1+\sqrt{2(1-a)(1-b)(1-c)}.$$
Prove that
$$\sqrt{1-a^2}+\sqrt{1-b^2}+\sqrt{1-c^2}\le \frac{3\sqrt 3}{2}.$$
[i] (Nora Gavrea)[/i]
2015 Dutch IMO TST, 3
Let $n$ be a positive integer.
Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$.
Consider for $1 \le i \le k$ the number $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\
b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$
Show that $c_1 + c_2 + ... + c_k = n^2 - 1$.
2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 1
Let $A=\left(
\begin{array}{ccc}
1 & 1& 0 \\
0 & 1& 0 \\
0 &0 & 2
\end{array}
\right),\ B=\left(
\begin{array}{ccc}
a & 1& 0 \\
b & 2& c \\
0 &0 & a+1
\end{array}
\right)\ (a,\ b,\ c\in{\mathbb{C}}).$
(1) Find the condition for $a,\ b,\ c$ such that ${\text{rank} (AB-BA})\leq 1.$
(2) Under the condition of (1), find the condition for $a,\ b,\ c$ such that $B$ is diagonalizable.
2016 Baltic Way, 6
The set $\{1, 2, . . . , 10\}$ is partitioned to three subsets $A, B$ and $C.$ For each subset the sum of its elements, the product of its elements and the sum of the digits of all its elements are calculated.
Is it possible that $A$ alone has the largest sum of elements, $B$ alone has the largest product of elements, and $C$ alone has the largest sum of digits?
2019 Korea Junior Math Olympiad., 6
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfies the followings. (Note that $\mathbb{R}$ stands for the set of all real numbers)
(1) For each real numbers $x$, $y$, the equality $f(x+f(x)+xy) = 2f(x)+xf(y)$ holds.
(2) For every real number $z$, there exists $x$ such that $f(x) = z$.
1990 IMO Longlists, 95
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.
2006 Bundeswettbewerb Mathematik, 2
Find all functions $f: Q^{+}\rightarrow R$ such that
$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$ for all $x,y\in Q^{+}$
2024 Indonesia MO, 7
Suppose $P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0$ where $a_0, a_1, \ldots, a_{n-1}$ are reals for $n\geq 1$ (monic $n$th-degree polynomial with real coefficients). If the inequality
\[ 3(P(x)+P(y)) \geq P(x+y) \] holds for all reals $x,y$, determine the minimum possible value of $P(2024)$.
2005 Greece National Olympiad, 1
Find the polynomial $P(x)$ with real coefficients such that $P(2)=12$ and $P(x^2)=x^2(x^2+1)P(x)$ for each $x\in\mathbb{R}$.
2017 Latvia Baltic Way TST, 3
Find all functions $f (x) : Z \to Z$ defined on integers, take integer values, and for all $x,y \in Z$ satisfy $$f(x+y)+f(xy)=f(x)f(y)+1$$
1996 May Olympiad, 5
In an electronic game of questions and answers, for each correct answer the player adds $5$ points on the screen, for each incorrect answer $2$ points are subtracted and when the player does not answer, no score is added or subtracted. Each game has $30$ questions. Francisco played $5$ games and in all of them he obtained the same number of points, greater than zero, but the number of correct answers, errors and unanswered questions in each game was different. Give all the possible scores that Francisco could obtain.
2008 China Girls Math Olympiad, 2
Let $ \varphi(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$ be a polynomial with real coefficients. Given that $ \varphi(x)$ has three positive real roots and that $ \varphi(0) < 0$, prove that
\[ 2b^3 \plus{} 9a^2d \minus{} 7abc \leq 0.
\]
2021 Kyiv City MO Round 1, 11.4
For positive real numbers $a, b, c$ with sum $\frac{3}{2}$, find the smallest possible value of the following expression:
$$\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} + \frac{1}{abc}$$
[i]Proposed by Serhii Torba[/i]
1990 IMO Longlists, 92
Let $n$ be a positive integer and $m = \frac{(n+1)(n+2)}{2}$. In coordinate plane, there are $n$ distinct lines $L_1, L_2, \ldots, L_n$ and $m$ distinct points $A_1, A_2, \ldots, A_m$, satisfying the following conditions:
[b][i]i)[/i][/b] Any two lines are non-parallel.
[b][i]ii)[/i][/b] Any three lines are non-concurrent.
[b][i]iii)[/i][/b] Only $A_1$ does not lies on any line $L_k$, and there are exactly $k + 1$ points $A_j$'s that lie on line $L_k$ $(k = 1, 2, \ldots, n).$
Prove that there exist a unique polynomial $p(x, y)$ with degree $n$ satisfying $p(A_1) = 1$ and $p(A_j) = 0$ for $j = 2, 3, \ldots, m.$
2020 Brazil Team Selection Test, 5
Let $n \geq 3$ be a fixed integer. The number $1$ is written $n$ times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers $a$ and $b$, replacing them with the numbers $1$ and $a+b$, then adding one stone to the first bucket and $\gcd(a, b)$ stones to the second bucket. After some finite number of moves, there are $s$ stones in the first bucket and $t$ stones in the second bucket, where $s$ and $t$ are positive integers. Find all possible values of the ratio $\frac{t}{s}$.
2007 Romania Team Selection Test, 1
Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find
\[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]
2024 ELMO Shortlist, A6
Let $\mathbb R^+$ denote the set of positive real numbers. Find all functions $f:\mathbb R^+\to\mathbb R$ and $g:\mathbb R^+\to\mathbb R$ such that for all $x,y\in\mathbb R^+$, $g(x)-g(y)=(x-y)f(xy)$.
[i]Linus Tang[/i]
2004 Finnish National High School Mathematics Competition, 2
$a, b$ and $c$ are positive integers and \[\frac{a\sqrt{3} + b}{b\sqrt{3} + c}\]
is a rational number.
Show that \[\frac{a^2 + b^2 + c^2}{a + b + c}\] is an integer.
2016 Hanoi Open Mathematics Competitions, 2
Given an array of numbers $A = (672, 673, 674, ..., 2016)$ on table.
Three arbitrary numbers $a,b,c \in A$ are step by step replaced by number $\frac13 min(a,b,c)$.
After $672$ times, on the table there is only one number $m$, such that
(A): $0 < m < 1$ (B): $m = 1$ (C): $1 < m < 2$ (D): $m = 2$ (E): None of the above.
TNO 2023 Senior, 6
The points inside a circle \( \Gamma \) are painted with \( n \geq 1 \) colors. A color is said to be dense in a circle \( \Omega \) if every circle contained within \( \Omega \) has points of that color in its interior. Prove that there exists at least one color that is dense in some circle contained within \( \Gamma \).
2001 Saint Petersburg Mathematical Olympiad, 11.6
Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that for any $x,y$ the following is true:
$$f(x+y+f(y))=f(x)+2y$$
[I]proposed by F. Petrov[/i]
2013 Tournament of Towns, 6
There are fi ve distinct real positive numbers. It is known that the total sum of their squares and the total sum of their pairwise products are equal.
(a) Prove that we can choose three numbers such that it would not be possible to make a triangle with sides' lengths equal to these numbers.
(b) Prove that the number of such triples is at least six (triples which consist of the same numbers in different order are considered the same).
Kvant 2024, M2808
Some participants of the tournament are friends with each other, and everyone has at least one friend. Each participant of the tournament was given a T-shirt with the number of his friends at the tournament written on it. Prove that at least one participant has the arithmetic mean of the numbers written on his friends' T-shirts, not less than the arithmetic mean of the numbers on all T-shirts.
[i] From Czech-Slovak Olympiad 1991 [/i]
2018 Turkey Team Selection Test, 6
$a_0, a_1, \ldots, a_{100}$ and $b_1, b_2,\ldots, b_{100}$ are sequences of real numbers, for which the property holds: for all $n=0, 1, \ldots, 99$, either
$$a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n,$$
or
$$a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n.$$
Given $a_{100}\leq a_0$, what is the maximal value of $b_1+b_2+\cdots+b_{100}$?