Found problems: 131
1987 Czech and Slovak Olympiad III A, 3
Let $f:(0,\infty)\to(0,\infty)$ be a function satisfying $f\bigl(xf(y)\bigr)+f\bigl(yf(x)\bigr)=2xy$ for all $x,y>0$. Show that $f(x) = x$ for all positive $x$.
2016 Spain Mathematical Olympiad, 5
From all possible permutations from $(a_1,a_2,...,a_n)$ from the set $\{1,2,..,n\}$, $n\geq 1$, consider the sets that satisfies the $2(a_1+a_2+...+a_m)$ is divisible by $m$, for every $m=1,2,...,n$. Find the total number of permutations.
1987 Czech and Slovak Olympiad III A, 6
Let $AA',BB',CC'$ be parallel lines not lying in the same plane. Denote $U$ the intersection of the planes $A'BC,AB'C,ABC'$ and $V$ the intersection of the planes $AB'C',A'BC',A'B'C$. Show that the line $UV$ is parallel with $AA'$.
2007 Flanders Math Olympiad, 3
Let $ABCD$ be a square with side $10$. Let $M$ and $N$ be the midpoints of $[AB]$ and $[BC]$ respectively. Three circles are drawn: one with midpoint $D$ and radius $|AD|$, one with midpoint $M$ and radius $|AM|$, and one with midpoint $N$ and radius $|BN|$. The three circles intersect in the points $R, S$ and $T$ inside the square. Determine the area of $\triangle RST$.
Brazil L2 Finals (OBM) - geometry, 2013.5
Let ABC be a scalene triangle and AM is the median relative to side BC. The diameter circumference AM intersects for the second time the side AB and AC at points P and Q, respectively, both different from A. Assuming that PQ is parallel to BC, determine the angle measurement <BAC.
Any solution without trigonometry?
2020 Kosovo National Mathematical Olympiad, 2
Let $a_1,a_2,...,a_n$ be integers such that $a_1^{20}+a_2^{20}+...+a_n^{20}$ is divisible by $2020$. Show that $a_1^{2020}+a_2^{2020}+...+a_n^{2020}$ is divisible by $2020$.
2025 Kosovo National Mathematical Olympiad`, P3
Let $m$ and $n$ be natural numbers such that $m^3-n^3$ is a prime number. What is the remainder of the number $m^3-n^3$ when divided by $6$?
2022 Ecuador NMO (OMEC), 3
A polygon is [b]gridded[/b] if the internal angles of the polygon are either $90$ or $270$, it has integer side lengths and its sides don't intersect with each other.
Prove that for all $n \ge 8$, it exist a gridded polygon with area $2n$ and perimeter $2n$.
2011 Bangladesh Mathematical Olympiad, HS
[size=130][b]Higher Secondary: 2011[/b]
[/size]
Time: 4 Hours
[b]Problem 1:[/b]
Prove that for any non-negative integer $n$ the numbers $1, 2, 3, ..., 4n$ can be divided in tow mutually exclusive classes with equal number of members so that the sum of numbers of each class is equal.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=709
[b]Problem 2:[/b]
In the first round of a chess tournament, each player plays against every other player exactly once. A player gets $3, 1$ or $-1$ points respectively for winning, drawing or losing a match. After the end of the first round, it is found that the sum of the scores of all the players is $90$. How many players were there in the tournament?
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=708
[b]Problem 3:[/b]
$E$ is the midpoint of side $BC$ of rectangle $ABCD$. $A$ point $X$ is chosen on $BE$. $DX$ meets extended $AB$ at $P$. Find the position of $X$ so that the sum of the areas of $\triangle BPX$ and $\triangle DXC$ is maximum with proof.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=683
[b]Problem 4:[/b]
Which one is larger 2011! or, $(1006)^{2011}$? Justify your answer.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=707
[b]Problem 5:[/b]
In a scalene triangle $ABC$ with $\angle A = 90^{\circ}$, the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R$. The lines $RS$ and $BC$ intersect at $N$ while the lines $AM$ and $SR$ intersect at $U$. Prove that the triangle $UMN$ is isosceles.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=706
[b]Problem 6:[/b]
$p$ is a prime and sum of the numbers from $1$ to $p$ is divisible by all primes less or equal to $p$. Find the value of $p$ with proof.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=693
[b]Problem 7:[/b]
Consider a group of $n > 1$ people. Any two people of this group are related by mutual friendship or mutual enmity. Any friend of a friend and any enemy of an enemy is a friend. If $A$ and $B$ are friends/enemies then we count it as $1$ [b]friendship/enmity[/b]. It is observed that the number of friendships and number of enmities are equal in the group. Find all possible values of $n$.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=694
[b]Problem 8:[/b]
$ABC$ is a right angled triangle with $\angle A = 90^{\circ}$ and $D$ be the midpoint of $BC$. A point $F$ is chosen on $AB$. $CA$ and $DF$ meet at $G$ and $GB \parallel AD$. $CF$ and $AD$ meet at $O$ and $AF = FO$. $GO$ meets $BC$ at $R$. Find the sides of $ABC$ if the area of $GDR$ is $\dfrac{2}{\sqrt{15}}$
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=704
[b]Problem 9:[/b]
The repeat of a natural number is obtained by writing it twice in a row (for example, the repeat of $123$ is $123123$). Find a positive integer (if any) whose repeat is a perfect square.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=703
[b]Problem 10:[/b]
Consider a square grid with $n$ rows and $n$ columns, where $n$ is odd (similar to a chessboard). Among the $n^2$ squares of the grid, $p$ are black and the others are white. The number of black squares is maximized while their arrangement is such that horizontally, vertically or diagonally neighboring black squares are separated by at least one white square between them. Show that there are infinitely many triplets of integers $(p, q, n)$ so that the number of white squares is $q^2$.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=702
The problems of the Junior categories are available in [url=http://matholympiad.org.bd/forum/]BdMO Online forum[/url]:
http://matholympiad.org.bd/forum/viewtopic.php?f=25&t=678
2016 Spain Mathematical Olympiad, 1
Two real number sequences are guiven, one arithmetic $\left(a_n\right)_{n\in \mathbb {N}}$ and another geometric sequence $\left(g_n\right)_{n\in \mathbb {N}}$ none of them constant. Those sequences verifies $a_1=g_1\neq 0$, $a_2=g_2$ and $a_{10}=g_3$. Find with proof that, for every positive integer $p$, there is a positive integer $m$, such that $g_p=a_m$.
2024 Israel National Olympiad (Gillis), P6
Quadrilateral $ABCD$ is inscribed in a circle. Let $\omega_A$, $\omega_B$, $\omega_C$, $\omega_D$ be the incircles of triangles $DAB$, $ABC$, $BCD$, $CDA$ respectively. The common external common tangent of $\omega_A$, $\omega_B$, different from line $AB$, meets the external common tangent of $\omega_A$, $\omega_D$, different from $AD$, at point $A'$. Similarly, the external common tangent of $\omega_B$, $\omega_C$ different from $BC$ meets the external common tangent of $\omega_C$, $\omega_D$ different from $CD$ at $C'$.
Prove that $AA'\parallel CC'$.
2020 Kosovo National Mathematical Olympiad, 3
Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ?
2018 Czech and Slovak Olympiad III A, 1
In a group of people, there are some mutually friendly pairs. For positive integer $k\ge3$ we say the group is $k$-great, if every (unordered) $k$-tuple of people from the group can be seated around a round table it the way that all pairs of neighbors are mutually friendly. [i](Since this was the 67th year of CZE/SVK MO,)[/i] show that if the group is 6-great, then it is 7-great as well.
[b]Bonus[/b] (not included in the competition): Determine all positive integers $k\ge3$ for which, if the group is $k$-great, then it is $(k+1)$-great as well.
2015 Spain Mathematical Olympiad, 3
Let $ABC$ be a triangle. $M$, and $N$ points on $BC$, such that $BM=CN$, with $M$ in the interior of $BN$. Let $P$ and $Q$ be points in $AN$ and $AM$ respectively such that $\angle PMC= \angle MAB$, and $\angle QNB= \angle NAC$. Prove that $ \angle QBC= \angle PCB$.
2022 Israel National Olympiad, P2
Real nonzero numbers $a,b,c,d,e,f,k,m$ satisfy the equations
\[\frac{a}{b}+\frac{c}{d}+\frac{e}{f}=k\]
\[\frac{b}{c}+\frac{d}{e}+\frac{f}{a}=m\]
\[ad=be=cf\]
Express $\frac{a}{c}+\frac{c}{e}+\frac{e}{a}+\frac{b}{d}+\frac{d}{f}+\frac{f}{b}$ using $m$ and $k$.
2015 Bangladesh Mathematical Olympiad, 4
There are $36$ participants at a BdMO event. Some of the participants shook hands with each other. But no two participants shook hands with each other more than once. Each participant recorded the number of handshakes they made. It was found that no two participants with the same number of handshakes made, had shaken hands with each other. Find the maximum possible number of handshakes at the party with proof. (When two participants shake hands with each other, this will be counted as one handshake.)
2022 Bosnia and Herzegovina Junior BMO TST, 3
Let $ABC$ be an acute triangle. Tangents on the circumscribed circle of triangle $ABC$ at points $B$ and $C$ intersect at point $T$. Let $D$ and $E$ be a foot of the altitudes from $T$ onto $AB$ and $AC$ and let $M$ be the midpoint of $BC$. Prove:
A) Prove that $M$ is the orthocenter of the triangle $ADE$.
B) Prove that $TM$ cuts $DE$ in half.
1957 Czech and Slovak Olympiad III A, 3
Find all real numbers $\alpha$ such that both values $\cot(\alpha)$ and $\cot(2\alpha)$ are integers.
2018 Czech and Slovak Olympiad III A, 2
Let $x,y,z$ be real numbers such that the numbers $$\frac{1}{|x^2+2yz|},\quad\frac{1}{|y^2+2zx|},\quad\frac{1}{|z^2+2xy|}$$ are lengths of sides of a (non-degenerate) triangle. Determine all possible values of $xy+yz+zx$.
2024 Israel National Olympiad (Gillis), P2
A positive integer $x$ satisfies the following:
\[\{\frac{x}{3}\}+\{\frac{x}{5}\}+\{\frac{x}{7}\}+\{\frac{x}{11}\}=\frac{248}{165}\]
Find all possible values of
\[\{\frac{2x}{3}\}+\{\frac{2x}{5}\}+\{\frac{2x}{7}\}+\{\frac{2x}{11}\}\]
where $\{y\}$ denotes the fractional part of $y$.
1987 Czech and Slovak Olympiad III A, 5
Consider a table with three rows and eleven columns. There are zeroes prefilled in the cell of the first row and the first column and in the cell of the second row and the last column. Determine the least real number $\alpha$ such that the table can be filled with non-negative numbers and the following conditions hold simultaneously:
(1) the sum of numbers in every column is one,
(2) the sum of every two neighboring numbers in the first row is at most one,
(3) the sum of every two neighboring numbers in the second row is at most one,
(4) the sum of every two neighboring numbers in the third row is at most $\alpha$.
1956 Czech and Slovak Olympiad III A, 4
Let a semicircle $AB$ be given and let $X$ be an inner point of the arc. Consider a point $Y$ on ray $XA$ such that $XY=XB$. Find the locus of all points $Y$ when $X$ moves on the arc $AB$ (excluding the endpoints).
2020 Kosovo National Mathematical Olympiad, 3
Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of triangle $\triangle ABC$ and $P$ a variable point in line segment $BC$. The circle with center $P$ and radius $PA$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $R$ and $RP$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $Q$. Show that points $A$, $O$, $P$ and $Q$ are concyclic.
2022 Ecuador NMO (OMEC), 1
Prove that it is impossible to divide a square with side length $7$ into exactly $36$ squares with integer side lengths.
1956 Czech and Slovak Olympiad III A, 1
Find all $x,y\in\left(0,\frac{\pi}{2}\right)$ such that
\begin{align*}
\frac{\cos x}{\cos y}&=2\cos^2 y, \\
\frac{\sin x}{\sin y}&=2\sin^2 y.
\end{align*}