Found problems: 15925
1995 China Team Selection Test, 3
Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.)
Does the same result hold with a degree 3 or degree 5 polynomial?
2008 Macedonia National Olympiad, 1
Find all injective functions $ f : \mathbb{N} \to \mathbb{N}$ which satisfy
\[ f(f(n)) \le\frac{n \plus{} f(n)}{2}\]
for each $ n \in \mathbb{N}$.
2020 New Zealand MO, 5
Find all functions $f:\mathbb R \to \mathbb R$ such that for all $x,y\in \mathbb R$
$f(x+f(y))=2x+2f(y+1)$
2023 Hong Kong Team Selection Test, Problem 3
Let $n\ge 4$ be a positive integer. Consider any set $A$ formed by $n$ distinct real numbers such that the following condition holds: for every $a\in A$, there exist distinct elements $x, y, z \in A$ such that $\left| x-a \right|, \left| y-a \right|, \left| z-a \right| \ge 1$. For each $n$, find the greatest real number $M$ such that $\sum_{a\in A}^{}\left| a \right|\ge M$.
1963 Dutch Mathematical Olympiad, 3
Twenty numbers $a_1,a_2,..,a_{20}$ satisfy:
$$a_k \ge 7k \,\,\,\,\, for \,\,\,\,\, k = 1,2,..., 20$$
$$a_1+a_2+...+a_{20}=1518$$
Prove that among the numbers $k = 1,2,... ,20$ there are no more than seventeen, for which $a_k \ge 20k -2k^2$.
V Soros Olympiad 1998 - 99 (Russia), 11.3
For each value of parameter $a$, solve the the equation $$ x - \sqrt{x^2-a^2} = \frac{(x-a)^2}{2(x+a)}$$
2010 Hanoi Open Mathematics Competitions, 6
Find the greatest integer less than $(2 +\sqrt3)^5$ .
(A): $721$ (B): $722$ (C): $723$ (D): $724$ (E) None of the above.
2002 IMO Shortlist, 2
Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.
1985 Balkan MO, 2
Let $a,b,c,d \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ be real numbers such that
$\sin{a}+\sin{b}+\sin{c}+\sin{d}=1$ and $\cos{2a}+\cos{2b}+\cos{2c}+\cos{2d}\geq \frac{10}{3}$.
Prove that $a,b,c,d \in [0, \frac{\pi}{6}]$
2022 Turkey Junior National Olympiad, 1
$x, y, z$ are positive reals such that $x \leq 1$. Prove that
$$xy+y+2z \geq 4 \sqrt{xyz}$$
1957 Moscow Mathematical Olympiad, 347
a) Let $ax^3 + bx^2 + cx + d$ be divisible by $5$ for given positive integers $a, b, c, d$ and any integer $x$. Prove that $a, b, c$ and $d$ are all divisible by $5$.
b) Let $ax^4 + bx^3 + cx^2 + dx + e$ be divisible by $7$ for given positive integers $a, b, c, d, e$ and all integers $x$. Prove that $a, b, c, d$ and $e$ are all divisible by $7$.
1999 Junior Balkan Team Selection Tests - Moldova, 1
Solve in $R$ the system:
$$\begin{cases} \dfrac{xyz}{x + y + 1}= 1998000\\ \\
\dfrac{xyz}{y + z - 1}= 1998000 \\ \\
\dfrac{xyz}{z+x}= 1998000 \end{cases}$$
2019 India IMO Training Camp, P1
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
2017 Macedonia National Olympiad, Problem 1
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for each natural integer $n>1$ and for all $x,y \in \mathbb{N}$ the following holds:
$$f(x+y) = f(x) + f(y) + \sum_{k=1}^{n-1} \binom{n}{k}x^{n-k}y^k$$
2024 Rioplatense Mathematical Olympiad, 5
Let $S = \{2, 3, 4, \dots\}$ be the set of positive integers greater than 1. Find all functions $f : S \to S$ that satisfy
\[
\text{gcd}(a, f(b)) \cdot \text{lcm}(f(a), b) = f(ab)
\]
for all pairs of integers $a, b \in S$.
Clarification: $\text{gcd}(a,b)$ is the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ is the least common multiple of $a$ and $b$.
2015 Irish Math Olympiad, 5
Suppose a doubly infinite sequence of real numbers $. . . , a_{-2}, a_{-1}, a_0, a_1, a_2, . . .$ has the property that $$a_{n+3} =\frac{a_n + a_{n+1} + a_{n+2}}{3},$$ for all integers $n .$ Show that if this sequence is bounded (i.e., if there exists a number $R$ such that $|a_n| \leq R$ for all $n$), then $a_n$ has the same value for all $n.$
2024 Argentina Iberoamerican TST, 5
Let \( \mathbb R \) be the set of real numbers. Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that, for all real numbers \( x \) and \( y \), the following equation holds:$$\big (x^2-y^2\big )f\big (xy\big )=xf\big (x^2y\big )-yf\big (xy^2\big ).$$
2003 IMO Shortlist, 5
Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ that satisfy the following conditions:
- $f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx})$ for all $x,y,z\in\mathbb{R}^+$;
- $f(x)<f(y)$ for all $1\le x<y$.
[i]Proposed by Hojoo Lee, Korea[/i]
2001 China Team Selection Test, 1
Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer
2012 Iran MO (3rd Round), 3
Suppose $p$ is a prime number and $a,b,c \in \mathbb Q^+$ are rational numbers;
[b]a)[/b] Prove that $\mathbb Q(\sqrt[p]{a}+\sqrt[p]{b})=\mathbb Q(\sqrt[p]{a},\sqrt[p]{b})$.
[b]b)[/b] If $\sqrt[p]{b} \in \mathbb Q(\sqrt[p]{a})$, prove that for a nonnegative integer $k$ we have $\sqrt[p]{\frac{b}{a^k}}\in \mathbb Q$.
[b]c)[/b] If $\sqrt[p]{a}+\sqrt[p]{b}+\sqrt[p]{c} \in \mathbb Q$, then prove that numbers $\sqrt[p]{a},\sqrt[p]{b}$ and $\sqrt[p]{c}$ are rational.
2017 Purple Comet Problems, 3
When Phil and Shelley stand on a scale together, the scale reads $151$ pounds. When Shelley and Ryan stand on the same scale together, the scale reads $132$ pounds. When Phil and Ryan stand on the same scale together, the scale reads $115$ pounds. Find the number of pounds Shelley weighs.
2020 AMC 12/AHSME, 8
How many ordered pairs of integers $(x, y)$ satisfy the equation$$x^{2020}+y^2=2y?$$
$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{infinitely many}$
2019 JBMO Shortlist, A4
Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that
$a^4 - 2019a = b^4 - 2019b = c$.
Prove that $- \sqrt{c} < ab < 0$.
2012 Vietnam National Olympiad, 2
Consider two odd natural numbers $a$ and $b$ where $a$ is a divisor of $b^2+2$ and $b$ is a divisor of $a^2+2.$ Prove that $a$ and $b$ are the terms of the series of natural numbers $\langle v_n\rangle$ defined by
\[v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.\]
1978 IMO Shortlist, 6
Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$