Found problems: 4776
1968 IMO Shortlist, 1
Two ships sail on the sea with constant speeds and fixed directions. It is known that at $9:00$ the distance between them was $20$ miles; at $9:35$, $15$ miles; and at $9:55$, $13$ miles. At what moment were the ships the smallest distance from each other, and what was that distance ?
2008 AMC 12/AHSME, 12
A function $ f$ has domain $ [0,2]$ and range $ [0,1]$. (The notation $ [a,b]$ denotes $ \{x: a\le x\le b\}$.) What are the domain and range, respectively, of the function $ g$ defined by $ g(x)\equal{}1\minus{}f(x\plus{}1)$?
$ \textbf{(A)}\ [\minus{}1,1],[\minus{}1,0] \qquad
\textbf{(B)}\ [\minus{}1,1],[0,1] \qquad
\textbf{(C)}\ [0,2],[\minus{}1,0] \qquad
\textbf{(D)}\ [1,3],[\minus{}1,0] \qquad
\textbf{(E)}\ [1,3],[0,1]$
2005 MOP Homework, 1
Let $n$ be a natural number and $f_1$, $f_2$, ..., $f_n$ be polynomials with integers coeffcients. Show that there exists a polynomial $g(x)$ which can be factored (with at least two terms of degree at least $1$) over the integers such that $f_i(x)+g(x)$ cannot be factored (with at least two terms of degree at least $1$) over the integers for every $i$.
1998 IberoAmerican Olympiad For University Students, 1
The definite integrals between $0$ and $1$ of the squares of the continuous real functions $f(x)$ and $g(x)$ are both equal to $1$.
Prove that there is a real number $c$ such that
\[f(c)+g(c)\leq 2\]
2016 Balkan MO Shortlist, A3
Find all injective functions $f: \mathbb R \rightarrow \mathbb R$ such that for every real number $x$ and every positive integer $n$,$$ \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016$$
[i](Macedonia)[/i]
1987 IMO Longlists, 77
Find the least positive integer $k$ such that for any $a \in [0, 1]$ and any positive integer $n,$
\[a^k(1 - a)^n < \frac{1}{(n+1)^3}.\]
2016 District Olympiad, 4
Let $ I $ be an open real interval, and let be two functions $ f,g:I\longrightarrow\mathbb{R} $ satisfying the identity:
$$ x,y\in I\wedge x\neq y\implies\frac{f(x)-g(y)}{x-y} +|x-y|\ge 0. $$
[b]a)[/b] Prove that $ f,g $ are nondecreasing.
[b]b)[/b] Give a concrete example for $ f\neq g. $
2010 Switzerland - Final Round, 4
Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that
\[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]
2007 Kazakhstan National Olympiad, 1
Convex quadrilateral $ABCD$ with $AB$ not equal to $DC$ is inscribed in a circle. Let $AKDL$ and $CMBN$ be rhombs with same side of $a$. Prove that the points $K, L, M, N$ lie on a circle.
2023 Rioplatense Mathematical Olympiad, 5
Let $\mathbb{R}^{+}$ be the set of positive real numbers. Determine all non-negative real number $\alpha$ such that there exist a function $f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$f(x^{\alpha}+y)=(f(x+y))^{\alpha}+f(y)$$
for any $x,y$ positive real numbers.
2000 China Team Selection Test, 1
Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.
Today's calculation of integrals, 855
Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$.
For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$
2025 Romanian Master of Mathematics, 4
Let $\mathbb{Z}$ denote the set of integers and $S \subset \mathbb{Z} $ be the set of integers that are at least $10^{100}$. Fix a positive integer $c$. Determine all functions $f: S \rightarrow \mathbb{Z} $ satisfying
$f(xy+c)=f(x)+f(y)$, for all $x,y \in S$
2007 Turkey Team Selection Test, 3
Let $a, b, c$ be positive reals such that their sum is $1$. Prove that \[\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.\]
2009 South africa National Olympiad, 6
Let $A$ denote the set of real numbers $x$ such that $0\le x<1$. A function $f:A\to \mathbb{R}$ has the properties:
(i) $f(x)=2f(\frac{x}{2})$ for all $x\in A$;
(ii) $f(x)=1-f(x-\frac{1}{2})$ if $\frac{1}{2}\le x<1$.
Prove that
(a) $f(x)+f(1-x)\ge \frac{2}{3}$ if $x$ is rational and $0<x<1$.
(b) There are infinitely many odd positive integers $q$ such that equality holds in (a) when $x=\frac{1}{q}$.
1995 North Macedonia National Olympiad, 5
Let $ a, b, c, d \in \mathbb {R}, $ $ b \neq0. $
Find the functions of the $ f: \mathbb{R} \to \mathbb{R} $ such that $ f (x + d \cdot f (y)) = ax + by + c, $ for all $ x, y \in \mathbb{R}. $
2008 Iran MO (3rd Round), 2
Let $ g,f: \mathbb C\longrightarrow\mathbb C$ be two continuous functions such that for each $ z\neq 0$, $ g(z)\equal{}f(\frac1z)$. Prove that there is a $ z\in\mathbb C$ such that $ f(\frac1z)\equal{}f(\minus{}\bar z)$
2020 Korea National Olympiad, 1
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$
for all $x,y\in\mathbb{R}$.
1959 AMC 12/AHSME, 29
On a examination of $n$ questions a student answers correctly $15$ of the first $20$. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is $50\%$, how many different values of $n$ can there be?
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ \text{the problem cannot be solved} $
2013 Hong kong National Olympiad, 1
Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1$. Prove that
\[\sqrt[4]{\frac{\sqrt{3}}{a}+6\sqrt{3}b}+\sqrt[4]{\frac{\sqrt{3}}{b}+6\sqrt{3}c}+\sqrt[4]{\frac{\sqrt{3}}{c}+6\sqrt{3}a}\le\frac{1}{abc}\]
When does inequality hold?
2015 Azerbaijan IMO TST, 2
Find all functions $f:[0,1] \to \mathbb{R}$ such that the inequality \[(x-y)^2\leq|f(x) -f(y)|\leq|x-y|\] is satisfied for all $x,y\in [0,1]$
2006 IMC, 5
Let $a, b, c, d$ three strictly positive real numbers such that \[a^{2}+b^{2}+c^{2}=d^{2}+e^{2},\] \[a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.\] Compare \[a^{3}+b^{3}+c^{3}\] with \[d^{3}+e^{3},\]
1992 IMO, 2
Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]
2014 Chile TST Ibero, 1
Consider a function $f: \mathbb{R} \to \mathbb{R}$ satisfying for all $x \in \mathbb{R}$:
\[
f(x+1) = \frac{1}{2} + \sqrt{f(x) - f(x)^2}.
\]
Prove that there exists a $b > 0$ such that $f(x + b) = f(x)$ for all $x \in \mathbb{R}$.
2011 China Team Selection Test, 2
Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.