Found problems: 15925
1954 Moscow Mathematical Olympiad, 269
a) Given $100$ numbers $a_1, ..., a_{100}$ such that $\begin{cases}
a_1 - 3a_2 + 2a_3 \ge 0, \\
a_2 - 3a_3 + 2a_4 \ge 0, \\
a_3 - 3a_4 + 2a_5 \ge 0, \\
... \\
a_{99} - 3a_{100} + 2a_1 \ge 0, \\
a_{100} - 3a_1 + 2a_2 \ge 0 \end{cases}$
prove that the numbers are equal.
b) Given numbers $a_1=1, ..., a_{100}$ such that $\begin{cases}
a_1 - 4a_2 + 3a_3 \ge 0, \\
a_2 - 4a_3 + 3a_4 \ge 0, \\
a_3 - 4a_4 + 3a_5 \ge 0, \\
... \\
a_{99} - 4a_{100} + 3a_1 \ge 0, \\
a_{100} - 4a_1 + 3a_2 \ge 0 \end{cases}$
Find $a_2, a_3, ... , a_{100}.$
2014 China Western Mathematical Olympiad, 5
Given a positive integer $m$, Prove that there exists a positive integers $n_0$ such that all first digits after the decimal points of $\sqrt{n^2+817n+m}$ in decimal representation are equal, for all integers $n>n_0$.
2017 Abels Math Contest (Norwegian MO) Final, 1b
Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.
2021-IMOC qualification, A1
Prove that if positive reals $x,y$ satisfy $x+y= 3$, $x,y \ge 1$ then $$9(x- 1)(y- 1) + (y^2 + y+ 1)(x + 1) + (x^2-x+ 1)(y- 1) \ge 9$$
VII Soros Olympiad 2000 - 01, 9.1
Draw on the plane a set of points whose coordinates $(x,y)$ satisfy the equation $x^3 + y^3 = x^2y^2 + xy$.
2013 Brazil Team Selection Test, 4
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
\[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\]
and $f(-1) \neq 0$.
2006 QEDMO 3rd, 3
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $ x,y$:
$ x f(x)\minus{}yf(y)\equal{}(x\minus{}y)f(x\plus{}y)$.
2016 IOM, 2
Let $a_1, . . . , a_n$ be positive integers satisfying the inequality
$\sum_{i=1}^{n}\frac{1}{a_n}\le \frac{1}{2}$.
Every year, the government of Optimistica publishes its Annual Report with n economic indicators. For each $i = 1, . . . , n$,the possible values of the $i-th$ indicator are $1, 2, . . . , a_i$. The Annual Report is said to be optimistic if at least $n - 1$ indicators have higher values than in the previous report. Prove that the government can publish optimistic Annual Reports in an infinitely long sequence.
2015 Taiwan TST Round 3, 1
Let $\mathbb{Q}^+$ be the set of all positive rational numbers. Find all functions $f:\mathbb{Q}^+\rightarrow \mathbb{Q}^+$ satisfying $f(1)=1$ and
\[ f(x+n)=f(x)+nf(\frac{1}{x}) \forall n\in\mathbb{N},x\in\mathbb{Q}^+\]
2010 Contests, 4
Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that
\[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]
2008 AIME Problems, 1
Let $ N\equal{}100^2\plus{}99^2\minus{}98^2\minus{}97^2\plus{}96^2\plus{}\cdots\plus{}4^2\plus{}3^2\minus{}2^2\minus{}1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $ N$ is divided by $ 1000$.
2019 India PRMO, 16
Let $N$ denote the number of all natural numbers $n$ such that $n$ is divisible by a prime $p> \sqrt{n}$ and $p<20$. What is the value of $N$ ?
2002 Spain Mathematical Olympiad, Problem 1
Find all the polynomials $P(t)$ of one variable that fullfill the following for all real numbers $x$ and $y$:
$P(x^2-y^2) = P(x+y)P(x-y)$.
2018 Ramnicean Hope, 1
Show that $ 2/3+\sin 2018^{\circ } >0. $
[i]Costică Ambrinoc[/i]
2022 CMIMC, 1.7
Let $f(n)$ count the number of values $0\le k\le n^2$ such that $43\nmid\binom{n^2}{k}$. Find the least positive value of $n$ such that $$43^{43}\mid f\left(\frac{43^{n}-1}{42}\right)$$
[i]Proposed by Adam Bertelli[/i]
1978 IMO Longlists, 12
The equation $x^3 + ax^2 + bx + c = 0$ has three (not necessarily distinct) real roots $t, u, v$. For which $a, b, c$ do the numbers $t^3, u^3, v^3$ satisfy the equation $x^3 + a^3x^2 + b^3x + c^3 = 0$?
1998 Belarus Team Selection Test, 2
Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.
2017 China Team Selection Test, 4
An integer $n>1$ is given . Find the smallest positive number $m$ satisfying the following conditions： for any set $\{a,b\}$ $\subset \{1,2,\cdots,2n-1\}$ ,there are non-negative integers $ x, y$ ( not all zero) such that $2n|ax+by$ and $x+y\leq m.$
1994 Swedish Mathematical Competition, 1
$x\sqrt8 + \frac{1}{x\sqrt8} = \sqrt8$ has two real solutions $x_1, x_2$. The decimal expansion of $x_1$ has the digit $6$ in place $1994$. What digit does $x_2$ have in place $1994$?
1956 Moscow Mathematical Olympiad, 332
Prove that the system of equations $\begin{cases} x_1 - x_2 = a \\
x_3 - x_4 = b \\
x_1 + x_2 + x_3 + x_4 = 1\end{cases}$ has at least one solution in positive numbers ($x_1 ,x_2 ,x_3 ,x_4>0$) if and only if $|a| + |b| < 1$.
2012 AMC 8, 8
A shop advertises everything is "half price in today's sale." In addition, a coupon gives a $20\%$ discount on sale prices. Using the coupon, the price today represents what percentage off the original price?
$\textbf{(A)}\hspace{.05in}10 \qquad \textbf{(B)}\hspace{.05in}33 \qquad \textbf{(C)}\hspace{.05in}40 \qquad \textbf{(D)}\hspace{.05in}60 \qquad \textbf{(E)}\hspace{.05in}70 $
2012 Saint Petersburg Mathematical Olympiad, 1
$a,b,c$ are reals, such that every pair of equations of $x^3-ax^2+b=0,x^3-bx^2+c=0,x^3-cx^2+a=0$ has common root.
Prove $a=b=c$
2018 Korea - Final Round, 5
Determine whether or not two polynomials $P, Q$ with degree no less than 2018 and with integer coefficients exist such that $$P(Q(x))=3Q(P(x))+1$$ for all real numbers $x$.
1953 Moscow Mathematical Olympiad, 257
Let $x_0 = 10^9$, $x_n = \frac{x^2_{n-1}+2}{2x_{n-1}}$ for $n > 0$. Prove that $0 < x_{36} - \sqrt2 < 10^{-9}$.
PEN N Problems, 4
Show that if an infinite arithmetic progression of positive integers contains a square and a cube, it must contain a sixth power.