Found problems: 85335
2001 India IMO Training Camp, 2
Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.
2023 CCA Math Bonanza, L2.2
For a positive integer $n$ let $f(n)$ denote the number of ways to put $n$ objects into pairs if the only thing that matters is which object each object gets paired with. Find the sum of all $f(f(2k))$, where $k$ ranges from 1 to 2023.
[i]Lightning 2.2[/i]
2007 China Team Selection Test, 1
When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.
1993 APMO, 4
Determine all positive integers $n$ for which the equation
\[ x^n + (2+x)^n + (2-x)^n = 0 \]
has an integer as a solution.
1990 IMO, 1
Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that
\[ f(xf(y)) \equal{} \frac {f(x)}{y}
\]
for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.
2009 QEDMO 6th, 4
Let $a$ and $b$ be two real numbers and let $n$ be a nonnegative integer. Then prove that
$$\sum_{k=0}^{n} {n \choose k} (a + k)^k (b - k)^{n-k} = \sum_{k=0}^{n} \frac{n!}{t!} (a + b)^t $$
2007 National Olympiad First Round, 22
Let $n$ and $m$ be integers such that $n\leq 2007 \leq m$ and $n^n \equiv -1 \equiv m^m \pmod 5$. What is the least possible value of $m-n$?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ 8
$
2014 CentroAmerican, 3
A positive integer $n$ is [i]funny[/i] if for all positive divisors $d$ of $n$, $d+2$ is a prime number. Find all funny numbers with the largest possible number of divisors.
2012 India Regional Mathematical Olympiad, 1
Let $ABC$ be a triangle and $D$ be a point on the segment $BC$ such that $DC = 2BD$. Let $E$ be the mid-point of $AC$. Let $AD$ and $BE$ intersect in $P$. Determine the ratios $BP:PE$ and $AP:PD$.
2013 Canadian Mathematical Olympiad Qualification Repechage, 5
For each positive integer $k$, let $S(k)$ be the sum of its digits. For example, $S(21) = 3$ and $S(105) = 6$. Let $n$ be the smallest integer for which $S(n) - S(5n) = 2013$. Determine the number of digits in $n$.
2022 Yasinsky Geometry Olympiad, 4
Let $X$ be an arbitrary point on side $BC$ of triangle $ABC$. Triangle $T$ is formed by the angle bisectors of the angles $\angle ABC$, $\angle ACB$ and $\angle AXC$. Prove that the circle circumscribed around the triangle $T$, passes through the vertex $A$.
(Dmytro Prokopenko)
2007 Gheorghe Vranceanu, 3
$ \lim_{n\to\infty } \frac{1}{2^n}\left( \left( \frac{a}{a+b}+\frac{b}{b+c} \right)^n +\left( \frac{b}{b+c}+\frac{c}{c+a} \right)^n +\left( \frac{c}{c+a}+\frac{a}{a+b} \right)^n \right) ,\quad a,b,c>0 $
1999 Junior Balkan Team Selection Tests - Moldova, 1
Solve in $R$ the system:
$$\begin{cases} \dfrac{xyz}{x + y + 1}= 1998000\\ \\
\dfrac{xyz}{y + z - 1}= 1998000 \\ \\
\dfrac{xyz}{z+x}= 1998000 \end{cases}$$
2008 Saint Petersburg Mathematical Olympiad, 2
In a kingdom, there are roads open between some cities with lanes both ways, in such a way, that you can come from one city to another using those roads. The roads are toll, and the price for taking each road is distinct. A minister made a list of all routes that go through each city exactly once. The king marked the most expensive road in each of the routes and said to close all the roads that he marked at least once. After that, it became impossible to go from city $A$ to city $B$, from city $B$ to city $C$, and from city $C$ to city $A$. Prove that the kings order was followed incorrectly.
2017-IMOC, A5
Find all functions $f:\mathbb Z\to\mathbb Z$ such that
$$f(mf(n+1))=f(m+1)f(n)+f(f(n))+1$$for all integer pairs $(m,n)$.
2022 VJIMC, 4
In a box there are $31$, $41$ and $59$ stones coloured, respectively, red, green and blue. Three players,
having t-shirts of these three colours, play the following game. They sequentially make one of two moves:
(I) either remove three stones of one colour from the box,
(II) or replace two stones of different colours by two stones of the third colour.
The game ends when all the stones in the box have the same colour and the winner is the player whose t-shirt
has this colour. Assuming that the players play optimally, is it possible to decide whether the game ends and
who will win, depending on who the starting player is?
2001 Romania National Olympiad, 3
Let $f:[-1,1]\rightarrow\mathbb{R}$ be a continuous function. Show that:
a) if $\int_0^1 f(\sin (x+\alpha ))\, dx=0$, for every $\alpha\in\mathbb{R}$, then $f(x)=0,\ \forall x\in [-1,1]$.
b) if $\int_0^1 f(\sin (nx))\, dx=0$, for every $n\in\mathbb{Z}$, then $f(x)=0,\ \forall x\in [-1,1]$.
2021 USAMTS Problems, 5
Let $a$, $b$, $c$, $d$ be positive real numbers. Prove that $d$ is an integer [b]if and only if[/b] there are positive real numbers $e$, $f$ satisfying $$\left \lfloor \dfrac{\left\lfloor\frac{x + a}{b}\right\rfloor + c} {d} \right\rfloor = \left \lfloor \dfrac{x + e}{f} \right \rfloor$$ for all real numbers $x$. (For a real $y$, $\lfloor y \rfloor$ is the greatest integer less than or equal to $y$.)
2017 CMIMC Individual Finals, 1
Let $ABCD$ be an isosceles trapezoid with $AD\parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP\perp CD$ and $BQ\perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX=3$ and $XQ=1$. Compute the largest possible area of $ABCD$.
2007 Indonesia MO, 8
Let $ m$ and $ n$ be two positive integers. If there are infinitely many integers $ k$ such that $ k^2\plus{}2kn\plus{}m^2$ is a perfect square, prove that $ m\equal{}n$.
2017 AIME Problems, 15
The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths $2\sqrt3$, $5$, and $\sqrt{37}$, as shown, is $\tfrac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$.
[asy]
size(5cm);
pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0);
real t = .385, s = 3.5*t-1;
pair R = A*t+B*(1-t), P=B*s;
pair Q = dir(-60) * (R-P) + P;
fill(P--Q--R--cycle,gray);
draw(A--B--C--A^^P--Q--R--P);
dot(A--B--C--P--Q--R);
[/asy]
2021 Purple Comet Problems, 17
For real numbers $x$ let $$f(x)=\frac{4^x}{25^{x+1}}+\frac{5^x}{2^{x+1}}.$$ Then $f\left(\frac{1}{1-\log_{10}4}\right)=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2013 Junior Balkan Team Selection Tests - Romania, 4
Consider acute triangles $ABC$ and $BCD$, with $\angle BAC = \angle BDC$, such that $A$ and $D$ are on opposite sides of line $BC$. Denote by $E$ the foot of the perpendicular line to $AC$ through $B$ and by $F$ the foot of the perpendicular line to $BD$ through $C$. Let $H_1$ be the orthocenter of triangle $ABC$ and $H_2$ be the orthocenter of $BCD$. Show that lines $AD, EF$ and $H_1H_2$ are concurrent.
2011 Tournament of Towns, 5
In the plane are $10$ lines in general position, which means that no $2$ are parallel and no $3$ are concurrent. Where $2$ lines intersect, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?
2024 Putnam, A2
For which real polynomials $p$ is there a real polynomial $q$ such that
\[
p(p(x))-x=(p(x)-x)^2q(x)
\]
for all real $x$?