Found problems: 85335
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 $a_1,a_2,...,a_{2n}$
2013 Dutch IMO TST, 5
Let $ABCDEF$ be a cyclic hexagon satisfying $AB\perp BD$ and $BC=EF$.Let $P$ be the intersection of lines $BC$ and $AD$ and let $Q$ be the intersection of lines $EF$ and $AD$.Assume that $P$ and $Q$ are on the same side of $D$ and $A$ is on the opposite side.Let $S$ be the midpoint of $AD$.Let $K$ and $L$ be the incentres of $\triangle BPS$ and $\triangle EQS$ respectively.Prove that $\angle KDL=90^0$.
2005 AMC 12/AHSME, 4
A store normally sells windows at $ \$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?
$ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 200 \qquad \textbf{(C)}\ 300 \qquad \textbf{(D)}\ 400 \qquad \textbf{(E)}\ 500$
2000 Putnam, 2
Prove that the expression \[ \dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m} \] is an integer for all pairs of integers $ n \ge m \ge 1 $.