Found problems: 15925
2009 Romania Team Selection Test, 1
Given an integer $n\geq 2$, determine the maximum value the sum $x_1+\cdots+x_n$ may achieve, as the $x_i$ run through the positive integers, subject to $x_1\leq x_2\leq \cdots \leq x_n$ and $x_1+\cdots+x_n=x_1 x_2\cdots x_n$.
2015 Czech-Polish-Slovak Junior Match, 5
Find the smallest real constant $p$ for which the inequality holds $\sqrt{ab}- \frac{2ab}{a + b} \le p \left( \frac{a + b}{2} -\sqrt{ab}\right)$ with any positive real numbers $a, b$.
2016 Dutch IMO TST, 4
Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.
2022 Junior Balkan Team Selection Tests - Moldova, 6
The non-negative numbers $x,y,z$ satisfy the relation $x + y+ z = 3$. Find the smallest possible numerical value and the largest possible numerical value for the expression
$$E(x,y, z) = \sqrt{x(y + 3)} + \sqrt{y(z + 3)} + \sqrt{z(x + 3)} .$$
1989 Putnam, A3
Prove that all roots of $ 11z^{10} \plus{} 10iz^9 \plus{} 10iz \minus{}11 \equal{} 0$ have unit modulus (or equivalent $ |z| \equal{} 1$).
1990 Tournament Of Towns, (267) 1
Given $$a=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99}}}}, \,\,and\,\,\,
b=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99+\dfrac{1}{100}}}}}$$
Prove that $$|a-b| <\frac{1}{99! 100!}$$
(G Galperin, Moscow)
1987 Iran MO (2nd round), 2
Let $f$ be a real function defined in the interval $[0, +\infty )$ and suppose that there exist two functions $f', f''$ in the interval $[0, +\infty )$ such that
\[f''(x)=\frac{1}{x^2+f'(x)^2 +1} \qquad \text{and} \qquad f(0)=f'(0)=0.\]
Let $g$ be a function for which
\[g(0)=0 \qquad \text{and} \qquad g(x)=\frac{f(x)}{x}.\]
Prove that $g$ is bounded.
2023 Middle European Mathematical Olympiad, 2
If $a, b, c, d>0$ and $abcd=1$, show that $$\frac{ab+1}{a+1}+\frac{bc+1}{b+1}+\frac{cd+1}{c+1}+\frac{da+1}{d+1} \geq 4. $$ When does equality hold?
2018 Germany Team Selection Test, 1
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
2013 Romania Team Selection Test, 3
Given an integer $n\geq 2$, determine all non-constant polynomials $f$ with complex coefficients satisfying the condition
\[1+f(X^n+1)=f(X)^n.\]
1988 IMO Longlists, 90
Does there exist a number $\alpha, 0 < \alpha < 1$ such that there is an infinite sequence $\{a_n\}$ of positive numbers satisfying \[ 1 + a_{n+1} \leq a_n + \frac{\alpha}{n} \cdot \alpha_n, n = 1,2, \ldots? \]
1963 Swedish Mathematical Competition., 4
Given the real number $k$, find all differentiable real-valued functions $f(x)$ defined on the reals such that $f(x+y) = f(x) + f(y) + f(kxy)$ for all $x, y$.
1989 Tournament Of Towns, (203) 1
The positive numbers $a, b$ and $c$ satisfy $a \ge b \ge c$ and $a + b + c \le 1$ . Prove that $a^2 + 3b^2 + 5c^2 \le 1$ .
(F . L . Nazarov)
2006 Thailand Mathematical Olympiad, 9
Compute the largest integer not exceeding $$\frac{2549^3}{2547\cdot 2548} -\frac{2547^3}{2548\cdot 2549}$$
2011 Kosovo National Mathematical Olympiad, 1
It is given the function $f:\mathbb{R} \to \mathbb{R}$ such that it holds $f(\sin x)=\sin (2011x)$. Find the value of $f(\cos x)$.
2017 CHMMC (Fall), 4
Jordan has an infinite geometric series of positive reals whose sum is equal to $2\sqrt2 + 2$. It turns out that if Jordan squares each term of his geometric series and adds up the resulting numbers, he get a sum equal to $4$. If Jordan decides to take the fourth power of each term of his original geometric series and add up the resulting numbers, what sum will he get?
2016 BMT Spring, 5
Find
$$\frac{\tan 1^o}{1 + \tan 1^o }+\frac{\tan 2^o}{1 + \tan 2^o } + ... + \frac{\tan 89^o}{1 + \tan 89^o}$$
2019 Thailand TST, 2
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
2017 Regional Olympiad of Mexico Northeast, 6
Find all triples of real numbers $(a, b, c)$ that satisfy the system of equations $$\begin{cases} b^2 = 4a(\sqrt{c} - 1) \\ c^2 = 4b (\sqrt{a} - 1) \\ a^2 = 4c(\sqrt{b} - 1) \end{cases}$$
2016 Balkan MO Shortlist, A2
For all $x,y,z>0$ satisfying $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}\le x+y+z$, prove that
$$\frac{1}{x^2+y+z}+\frac{1}{y^2+z+x}+\frac{1}{z^2+x+y} \le 1$$
2021 APMO, 2
For a polynomial $P$ and a positive integer $n$, define $P_n$ as the number of positive integer pairs $(a,b)$ such that $a<b \leq n$ and $|P(a)|-|P(b)|$ is divisible by $n$. Determine all polynomial $P$ with integer coefficients such that $P_n \leq 2021$ for all positive integers $n$.
1998 IMO Shortlist, 3
Let $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that
\[
\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)}
\geq \frac{3}{4}.
\]
2007 Baltic Way, 1
For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$. In each subset $P_i$, let $p_i$ be the product of the two numbers in $P_i$. Prove that
\[\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1 \]
2021 HMNT, 10
Real numbers $x, y, z$ satisfy $$x + xy + xyz = 1, y + yz + xyz = 2, z + xz + xyz = 4.$$
The largest possible value of $xyz$ is $\frac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, $d$ are integers, $d$ is positive, $c$ is square-free, and gcd$(a,b, d) = 1$. Find $1000a + 100b + 10c + d$.
2024 Abelkonkurransen Finale, 2b
Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying
\[xf(f(x)+y)=f(xy)+x^2\]
for all $x,y \in \mathbb{R}$.