Found problems: 85335
2025 Poland - First Round, 7
Circles $o_1, o_2$ with equal radii intersect at points $A, B$. Points $C, D, E, F$ lie in this order on one line, with $C, E$ lying on $o_1$ and $D, F$ on $o_2$. Perpendicular bisectors of $CD$ and $EF$ intersect $AB$ at $X, Y$ respectively. Prove that $AX=BY$.
1985 Greece National Olympiad, 2
Conside the continuous $ f: \mathbb{R}\to\mathbb{R}$ . It is also know that equation $f(f(f(x)))=x$ has solution in $\mathbb{R}$. Prove that equation $f(x)=x$ has solution in $\mathbb{R}$.
2019 IFYM, Sozopol, 6
Find all odd numbers $n\in \mathbb{N}$, for which the number of all natural numbers, that are no bigger than $n$ and coprime with $n$, divides $n^2+3$.
2021 Turkey Team Selection Test, 9
For which positive integer couples $(k,n)$, the equality
$\Bigg|\Bigg\{{a \in \mathbb{Z}^+: 1\leq a\leq(nk)!, gcd \left(\binom{a}{k},n\right)=1}\Bigg\}\Bigg|=\frac{(nk)!}{6}$
holds?
1967 AMC 12/AHSME, 40
Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is:
$\textbf{(A)}\ 159\qquad
\textbf{(B)}\ 131\qquad
\textbf{(C)}\ 95\qquad
\textbf{(D)}\ 79\qquad
\textbf{(E)}\ 50$
2022 APMO, 1
Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.
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 $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$
2020 Iran RMM TST, 4
In a trapezoid $ABCD$ with $AD$ parallel to $BC$ points $E, F$ are on sides $AB, CD$ respectively. $A_1, C_1$ are on $AD,BC$ such that $A_1, E, F, A$ lie on a circle and so do $C_1, E, F, C$. Prove that lines $A_1C_1, BD, EF$ are concurrent.
2010 Kazakhstan National Olympiad, 2
On sides of convex quadrilateral $ABCD$ on external side constructed equilateral triangles $ABK, BCL, CDM, DAN$. Let $P,Q$- midpoints of $BL, AN$ respectively and $X$- circumcenter of $CMD$.
Prove, that $PQ$ perpendicular to $KX$
2020 GQMO, 7
Each integer in $\{1, 2, 3, . . . , 2020\}$ is coloured in such a way that, for all positive integers $a$ and $b$
such that $a + b \leq 2020$, the numbers $a$, $b$ and $a + b$ are not coloured with three different colours.
Determine the maximum number of colours that can be used.
[i]Massimiliano Foschi, Italy[/i]
2016 Indonesia TST, 1
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
2015 ASDAN Math Tournament, 2
Nick is taking a $10$ question test where each answer is either true or false with equal probability. Nick forgot to study, so he guesses randomly on each of the $10$ problems. What is the probability that Nick answers exactly half of the questions correctly?
2003 AMC 12-AHSME, 1
What is the difference between the sum of the first $ 2003$ even counting numbers and the sum of the first $ 2003$ odd counting numbers?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 2003 \qquad
\textbf{(E)}\ 4006$
2000 AIME Problems, 4
The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.
[asy]
defaultpen(linewidth(0.7));
draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36));
draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61));
draw((34,36)--(34,45)--(25,45));
draw((36,36)--(36,38)--(34,38));
draw((36,38)--(41,38));
draw((34,45)--(41,45));[/asy]
2006 Tournament of Towns, 3
Let $a$ be some positive number. Find the number of integer solutions $x$ of inequality $2 < xa < 3$ given that inequality $1 < xa < 2$ has exactly $3$ integer solutions. Consider all possible cases.
[i](4 points)[/i]
1984 IMO Longlists, 15
Consider all the sums of the form
\[\displaystyle\sum_{k=1}^{1985} e_kk^5=\pm 1^5\pm 2^5\pm\cdots\pm1985^5\]
where $e_k=\pm 1$. What is the smallest nonnegative value attained by a sum of this type?
1998 Canada National Olympiad, 1
Determine the number of real solutions $a$ to the equation:
\[ \left[\,\frac{1}{2}\;a\,\right]+\left[\,\frac{1}{3}\;a\,\right]+\left[\,\frac{1}{5}\;a\,\right] = a. \]
Here, if $x$ is a real number, then $[\,x\,]$ denotes the greatest integer that is less than or equal to $x$.
2017 NMTC Junior, 1
(a) Find all prime numbers $p$ such that $4p^2+1$ and $6p^2+1$ are also primes.
(b)Find real numbers $x,y,z,u$ such that \[xyz+xy+yz+zx+x+y+z=7\]\[yzu+yz+zu+uy+y+z+u=10\]\[zux+zu+ux+xz+z+u+x=10\]\[uxy+ux+xy+yu+u+x+y=10\]
2007 Princeton University Math Competition, 6
Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of sidelengths $9$ and $10$ or of sidelengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of sidelengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?
2006 Denmark MO - Mohr Contest, 2
Determine all sets of real numbers $(x,y,z)$ which fulfills
$$\begin{cases} x + y =2 \\ xy -z^2= 1\end{cases}$$
2010 Tournament Of Towns, 2
In a quadrilateral $ABCD$ with an incircle, $AB = CD; BC < AD$ and $BC$ is parallel to $AD$. Prove that the bisector of $\angle C$ bisects the area of $ABCD$.
2011 Korea National Olympiad, 1
Find the number of positive integer $ n < 3^8 $ satisfying the following condition.
"The number of positive integer $k (1 \leq k \leq \frac {n}{3})$ such that $ \frac{n!}{(n-3k)! \cdot k! \cdot 3^{k+1}} $ is not a integer" is $ 216 $.
2023 Sharygin Geometry Olympiad, 10.8
A triangle $ABC$ is given. Let $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ be circles centered at points $X$, $Y$, $Z$, $T$ respectively such that each of lines $BC$, $CA$, $AB$ cuts off on them four equal chords. Prove that the centroid of $ABC$ divides the segment joining $X$ and the radical center of $\omega_2$, $\omega_3$, $\omega_4$ in the ratio $2:1$ from $X$.
2017 Thailand Mathematical Olympiad, 1
Let $p$ be a prime. Show that $\sqrt[3]{p} +\sqrt[3]{p^5} $ is irrational.
2012 Tournament of Towns, 6
(a) A point $A$ is marked inside a circle. Two perpendicular lines drawn through $A$ intersect the circle at four points. Prove that the centre of mass of these four points does not depend on the choice of the lines.
(b) A regular $2n$-gon ($n \ge 2$) with centre $A$ is drawn inside a circle (A does not necessarily coincide with the centre of the circle). The rays going from $A$ to the vertices of the $2n$-gon mark $2n$ points on the circle. Then the $2n$-gon is rotated about $A$. The rays going from $A$ to the new locations of vertices mark new $2n$ points on the circle. Let $O$ and $N$ be the centres of gravity of old and new points respectively. Prove that $O = N$.