Found problems: 85335
2018 ELMO Shortlist, 1
Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$?
[i]Proposed by Daniel Liu[/i]
1958 Miklós Schweitzer, 2
[b]2.[/b] Let $A(x)$ denote the number of positive integers $n$ not greater than $x$ and having at least one prime divisor greater than $\sqrt[3]{n}$. Prove that $\lim_{x\to \infty} \frac {A(x)}{x}$ exists. [b](N. 15)[/b]
2020 CCA Math Bonanza, L3.3
Compute the largest prime factor of $111^2+11^3+1^1$.
[i]2020 CCA Math Bonanza Lightning Round #3.3[/i]
2022 Germany Team Selection Test, 3
Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$
[i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]
2017 Saudi Arabia BMO TST, 2
Let $R^+$ be the set of positive real numbers. Find all function $f : R^+ \to R$ such that, for all positive real number $x$ and $y$, the following conditions are satisfied:
i) $2f (x) + 2f (y) \le f (x + y)$
ii) $(x + y)[y f (x) + x f (y)] \ge x y f (x + y)$
2007 Croatia Team Selection Test, 1
Find integral solutions to the equation \[(m^{2}-n^{2})^{2}=16n+1.\]
1950 AMC 12/AHSME, 45
The number of diagonals that can be drawn in a polygon of 100 sides is:
$\textbf{(A)}\ 4850 \qquad
\textbf{(B)}\ 4950\qquad
\textbf{(C)}\ 9900 \qquad
\textbf{(D)}\ 98 \qquad
\textbf{(E)}\ 8800$
2017 Dutch Mathematical Olympiad, 5
The eight points below are the vertices and the midpoints of the sides of a square. We would like to draw a number of circles through the points, in such a way that each pair of points lie on (at least) one of the circles.
Determine the smallest number of circles needed to do this.
[asy]
unitsize(1 cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((0,1));
dot((2,1));
dot((0,2));
dot((1,2));
dot((2,2));
[/asy]
2021 Final Mathematical Cup, 4
Let $P$ is a regular $(2n+1)$-gon in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ , if the line segment $\overline{ES}$ contains no other points that lie on the sides of $P$ except $S$ . We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$ , at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$ , we consider them colorless). Find the largest positive integer $n$ for which such a coloring is possible.
Kyiv City MO Seniors 2003+ geometry, 2019.11.2
In an acute-angled triangle $ABC$, in which $AB<AC$, the point $M$ is the midpoint of the side $BC, K$ is the midpoint of the broken line segment $BAC$ . Prove that $\sqrt2 KM > AB$.
(George Naumenko)
2011 Costa Rica - Final Round, 5
Given positive integers $a,b,c$ which are pairwise relatively prime, show that \[2abc-ab-bc-ac \] is the biggest number that can't be expressed in the form $xbc+yca+zab$ with $x,y,z$ being natural numbers.
1950 AMC 12/AHSME, 50
A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two hour chase, the top sail of the privateer is carried away; she can now make only $17$ miles while the merchantman makes $15$. The privateer will overtake the merchantman at:
$\textbf{(A)}\ 3\text{:}45\text{ p.m.} \qquad
\textbf{(B)}\ 3\text{:}30\text{ p.m.} \qquad
\textbf{(C)}\ 5\text{:}00\text{ p.m.} \qquad
\textbf{(D)}\ 2\text{:}45\text{ p.m.} \qquad
\textbf{(E)}\ 5\text{:}30\text{ p.m.}$
2010 Sharygin Geometry Olympiad, 5
A point $E$ lies on the altitude $BD$ of triangle $ABC$, and $\angle AEC=90^\circ.$ Points $O_1$ and $O_2$ are the circumcenters of triangles $AEB$ and $CEB$; points $F, L$ are the midpoints of the segments $AC$ and $O_1O_2.$ Prove that the points $L,E,F$ are collinear.
2023 Indonesia TST, 3
Prove that $5^n-3^n$ is not divisible by $2^n+65$ for any positive integer $n$.
1999 Federal Competition For Advanced Students, Part 2, 3
Find all pairs $(x, y)$ of real numbers such that
\[y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999\]
where $f(x)=[x]$ is the floor function.
2023 Olympic Revenge, 1
Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous functions such that $\lim_{x\rightarrow \infty} f(x) =\infty$ and $\forall x,y\in \mathbb{R}, |x-y|>\varphi, \exists n<\varphi^{2023}, n\in \mathbb{N}$ such that
$$f^n(x)+f^n(y)=x+y$$
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.
2024 Vietnam Team Selection Test, 3
Let $ABC$ be an acute scalene triangle. Incircle of $ABC$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $X,Y,Z$ be feet the altitudes of from $A,B,C$ to the sides $BC,CA,AB$ respectively. Let $A',B',C'$ be the reflections of $X,Y,Z$ in $EF,FD,DE$ respectively. Prove that triangles $ABC$ and $A'B'C'$ are similar.
2015 Denmark MO - Mohr Contest, 5
For which numbers $n$ is it possible to put marks on a stick such that all distances $1$ cm, $2$ cm, . . . , $n$ cm each appear exactly once as the distance between two of the marks, and no other distance appears as such a distance?
2022 Yasinsky Geometry Olympiad, 4
Let $BM$ be the median of triangle $ABC$. On the extension of $MB$ beyond $B$, the point $K$ is chosen so that $BK =\frac12 AC$. Prove that if $\angle AMB=60^o$, then $AK=BC$.
(Mykhailo Standenko)
2020-21 IOQM India, 4
Let $ABCD$ be a rectangle in which $AB + BC + CD = 20$ and $AE = 9$ where $E$ is the midpoint of the side $BC$. Find the area of the rectangle.
2013 Today's Calculation Of Integral, 871
Define sequences $\{a_n\},\ \{b_n\}$ by
\[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\]
(1) Find $b_n$.
(2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$
(3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$
1992 Brazil National Olympiad, 6
Given a set of n elements, find the largest number of subsets such that no subset is contained in any other
2005 Germany Team Selection Test, 2
If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that
\[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]
2017 Singapore Senior Math Olympiad, 1
Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$