Found problems: 15925
1987 All Soviet Union Mathematical Olympiad, 456
Every evening uncle Chernomor (see Pushkin's tales) appoints either $9$ or $10$ of his 33 "knights" in the "night guard". When it can happen, for the first time, that every knight has been on duty the same number of times?
2010 Junior Balkan MO, 1
The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations
\[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.
1979 Austrian-Polish Competition, 6
A positive integer $n$ and a real number $a$ are given. Find all $n$-tuples $(x_1, ... ,x_n)$ of real numbers that satisfy the system of equations $$\sum_{i=1}^{n} x_i^k= a^k \,\,\,\, for \,\,\,\, k = 1,2, ... ,n$$
2007 Iran Team Selection Test, 2
Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication.
[i]By Mohsen Jamali[/i]
1969 German National Olympiad, 5
Prove that for all real numbers $x$ holds:
$$\sin 5x = 16 \sin x \cdot \sin \left(x -\frac{\pi}{5} \right) \cdot \sin\left(x -\frac{2\pi}{5} \right) \sin \left(x +\frac{2\pi}{5} \right) $$
1982 All Soviet Union Mathematical Olympiad, 341
Prove that the following inequality is valid for the positive $x$:
$$2^{x^{1/12}}+ 2^{x^{1/4}} \ge 2^{1 + x^{1/6} }$$
2020 Poland - Second Round, 1.
Assume that for pairwise distinct real numbers $a,b,c,d$ holds:
$$ (a^2+b^2-1)(a+b)=(b^2+c^2-1)(b+c)=(c^2+d^2-1)(c+d).$$
Prove that $ a+b+c+d=0.$
1993 Balkan MO, 1
Let $a,b,c,d,e,f$ be six real numbers with sum 10, such that \[ (a-1)^2+(b-1)^2+(c-1)^2+(d-1)^2+(e-1)^2+(f-1)^2 = 6. \] Find the maximum possible value of $f$.
[i]Cyprus[/i]
2003 China Team Selection Test, 2
Let $S$ be a finite set. $f$ is a function defined on the subset-group $2^S$ of set $S$. $f$ is called $\textsl{monotonic decreasing}$ if when $X \subseteq Y\subseteq S$, then $f(X) \geq f(Y)$ holds. Prove that: $f(X \cup Y)+f(X \cap Y ) \leq f(X)+ f(Y)$ for $X, Y \subseteq S$ if and only if $g(X)=f(X \cup \{ a \}) - f(X)$ is a $\textsl{monotonic decreasing}$ funnction on the subset-group $2^{S \setminus \{a\}}$ of set $S \setminus \{a\}$ for any $a \in S$.
2023 Iran MO (3rd Round), 2
Does there exist bijections $f,g$ from positive integers to themselves st:
$$g(n)=\frac{f(1)+f(2)+ \cdot \cdot \cdot +f(n)}{n}$$
holds for any $n$?
2021 Malaysia IMONST 1, 18
How many real numbers $x$ are solutions to the equation $|x - 2| - 4 =\frac{1}{|x - 3|}$ ?
1992 Irish Math Olympiad, 1
Describe in geometric terms the set of points $(x,y)$ in the plane such that $x$ and $y$ satisfy the condition $t^2+yt+x\ge 0$ for all $t$ with $-1\le t\le 1$.
1977 Spain Mathematical Olympiad, 7
The numbers $A_1 , A_2 ,... , A_n$ are given. Prove, without calculating derivatives, that the value of $X$ that minimizes the sum $(X - A_1)^2 + (X -A_2)^2 + ...+ (X - A_n)^2$ is precisely the arithmetic mean of the given numbers.
1995 Vietnam National Olympiad, 2
Find all poltnomials $ P(x)$ with real coefficients satisfying: For all $ a>1995$, the number of real roots of $ P(x)\equal{}a$ (incuding multiplicity of each root) is greater than 1995, and every roots are greater than 1995.
2012 Romania National Olympiad, 4
[color=darkred] Let $m$ and $n$ be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial $\prod_{k=1}^m\, (f+k)$ , when $f$ covers the set of $n^{\text{th}}$ - degree polynomials with complex coefficients.
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2018 Iran MO (2nd Round), 4
Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that:
$$f(x+y)f(x^2-xy+y^2)=x^3+y^3$$
for all reals $x, y $.
1986 China Team Selection Test, 3
Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that:
[b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$.
[b]II.[/b] Find such $A_k$ for $19^{86}.$
1988 China Team Selection Test, 2
Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying
(i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$.
(ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.
2019 LIMIT Category B, Problem 2
Let $\mathbb C$ denote the set of all complex numbers. Define
$$A=\{(z,w)|z,w\in\mathbb C\text{ and }|z|=|w|\}$$$$B=\{(z,w)|z,w\in\mathbb C\text{ and }z^2=w^2\}$$$\textbf{(A)}~A=B$
$\textbf{(B)}~A\subset B\text{ and }A\ne B$
$\textbf{(C)}~B\subset A\text{ and }B\ne A$
$\textbf{(D)}~\text{None of the above}$
PEN F Problems, 8
Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.
2004 Nicolae Coculescu, 3
Let be three nonzero complex numbers $ a,b,c $ satisfying
$$ |a|=|b|=|c|=\left| \frac{a+b+c-abc}{ab+bc+ca-1} \right| . $$
Prove that these three numbers have all modulus $ 1 $ or there are two distinct numbers among them whose sum is $ 0. $
[i]Costel Anghel[/i]
2015 Saudi Arabia Pre-TST, 2.2
Find all functions $f : R \to R$ that satisfy $f(x + y^2 - f(y)) = f(x)$ for all $x,y \in R$.
(Vo Quoc Ba Can)
2008 JBMO Shortlist, 9
Consider an integer $n \ge 4 $ and a sequence of real numbers $x_1, x_2, x_3,..., x_n$. An operation consists in eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3. x_7. x_{11}, ...$(for example, the sequence $4,5,9,3,6, 6,1, 8$ produces the sequence $9,1$). Upon the sequence $1, 2, 3, ..., 1024 $ the operation is performed successively for $5$ times. Show that at the end only one number remains and fi nd this number.
1988 China Team Selection Test, 4
There is a broken computer such that only three primitive data $c$, $1$ and $-1$ are reserved. Only allowed operation may take $u$ and $v$ and output $u \cdot v + v.$ At the beginning, $u,v \in \{c, 1, -1\}.$ After then, it can also take the value of the previous step (only one step back) besides $\{c, 1, -1\}$. Prove that for any polynomial $P_{n}(x) = a_0 \cdot x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ with integer coefficients, the value of $P_n(c)$ can be computed using this computer after only finite operation.
2011 USA Team Selection Test, 3
Let $p$ be a prime. We say that a sequence of integers $\{z_n\}_{n=0}^\infty$ is a [i]$p$-pod[/i] if for each $e \geq 0$, there is an $N \geq 0$ such that whenever $m \geq N$, $p^e$ divides the sum
\[\sum_{k=0}^m (-1)^k {m \choose k} z_k.\]
Prove that if both sequences $\{x_n\}_{n=0}^\infty$ and $\{y_n\}_{n=0}^\infty$ are $p$-pods, then the sequence $\{x_ny_n\}_{n=0}^\infty$ is a $p$-pod.