This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2007 Portugal MO, 4
Tags: combinatorics , combinatorial geometry
Fernanda decided to decorate a square blanket with a ribbon and buttons, placing a button in the center of each square where the ribbon passes and forming the design indicated in the figure. If Fernanda sews the first button in the shaded square on line $0$, on which line does she sew the $2007$th button? [img]https://cdn.artofproblemsolving.com/attachments/2/9/0c9c85ec6448ee3f6f363c8f4bcdd5209f53f6.png[/img]

2018 Brazil EGMO TST, 3
Tags: geometry , equilateral , concurrency
An equilateral triangle $ABC$ is inscribed in a circle $\Omega$ and has incircle $\omega$. Points $P$ and $Q$ are in segments $AC$ and $AB$, respectively, such that $PQ$ is tangent to $\omega$. The circle $\Omega_B$ has center $P$ and radius $PB$ and the circle $\Omega_C$ is defined similarly. Prove that $\Omega$, $\Omega_B$ and $\Omega_C$ have a common point.

1954 Moscow Mathematical Olympiad, 274
Tags: system of equations , algebra
Solve the system $\begin{cases} 10x_1 + 3x_2 + 4x_3 + x_4 + x_5 = 0 \\ 11x_2 + 2x_3 + 2x_4 + 3x_5 + x_6 = 0 \\ 15x_3 + 4x_4 + 5x_5 + 4x_6 + x_7 = 0 \\ 2x_1 + x_2 - 3x_3 + 12x_4 - 3x_5 + x_6 + x_7 = 0 \\ 6x_1 - 5x_2 + 3x_3 - x_4 + 17x_5 + x_6 = 0 \\ 3x_1 + 2x_2 - 3x_3 + 4x_4 + x_5 - 16x_6 + 2x_7 = 0\\ 4x_1 - 8x_2 + x_3 + x_4 + 3x_5 + 19x_7 = 0 \end{cases}$

2004 Tournament Of Towns, 1
Tags: geometry
In triangle $ABC$ the bisector of angle $A$, the perpendicular to side $AB$ from its midpoint, and the altitude from vertex $B$, intersect in the same point. Prove that the bisector of angle $A$, the perpendicular to side $AC$ from its midpoint, and the altitude from vertex $C$ also intersect in the same point.

2014 JBMO TST - Turkey, 2
Tags: ceiling function , combinatorics
$3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$.

2022-2023 OMMC, 9
Tags:
An ant lies on each corner of a $20 \times 23$ rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After $10$ seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants.

2024 Bulgarian Autumn Math Competition, 11.2
Tags: geometry , incenter
Let $ABC$ be a triangle with $\angle ABC = 60^{\circ}$. Find the angles of the triangle if $\angle BHI = 60^{\circ}$, where $H$ and $I$ are the orthocenter and incenter of $ABC$

2021-IMOC, G3
Tags: geometry , incenter
Let $I$ be the incenter of the acute triangle $\triangle ABC$, and $BI$, $CI$ intersect the altitude of $\triangle ABC$ through $A$ at $U$, $V$, respectively. The circle with $AI$ as a diameter intersects $\odot(ABC)$ again at $T$, and $\odot(TUV)$ intersects the segment $BC$ and $\odot(ABC)$ at $P$, $Q$, respectively. Let $R$ be another intersection of $PQ$ and $\odot(ABC)$. Show that $AR\parallel BC$.

2010 Slovenia National Olympiad, 5
Tags: geometry , rectangle , combinatorics
Let $ABCD$ be a square with the side of $20$ units. Amir divides this square into $400$ unit squares. Reza then picks $4$ of the vertices of these unit squares. These vertices lie inside the square $ABCD$ and define a rectangle with the sides parallel to the sides of the square $ABCD.$ There are exactly $24$ unit squares which have at least one point in common with the sides of this rectangle. Find all possible values for the area of a rectangle with these properties. [hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]

1983 Miklós Schweitzer, 8
Tags: superior algebra , combinatorial geometry
Prove that any identity that holds for every finite $ n$-distributive lattice also holds for the lattice of all convex subsets of the $ (n\minus{}1)$-dimensional Euclidean space. (For convex subsets, the lattice operations are the set-theoretic intersection and the convex hull of the set-theoretic union. We call a lattice $ n$-$ \textit{distributive}$ if \[ x \wedge (\bigvee_{i\equal{}0}^n y_i)\equal{}\bigvee_{j\equal{}0}^n(x \wedge (\bigvee_{0\leq i \leq n, \;i \not\equal{} j\ }y_i))\] holds for all elements of the lattice.) [i]A. Huhn[/i]

2025 Belarusian National Olympiad, 9.8
Tags: combinatorics
In some galaxy there are $1000000$ planets and on each of them there are at least $101$ portals. Each portal allows to teleport between some two planets, no two planets are connected by more than one portal. It is known that starting from any planet using portals you can get to any other planet, while it is impossible to return to that planet using once at most 5 different portals. Prove that starting from any planet you can get to any other planet within a year, using at most one portal daily (a year consists of 365 days). [i]M. Zorka[/i]

2023 Baltic Way, 13
Tags: geometry
Let $ABC$ be an acute triangle with $AB<AC$ and incenter $I$. Let $D$ be the projection of $I$ onto $BC$. Let $H$ be the orthocenter of $ABC$ and suppose that $\angle IDH=\angle CBA-\angle ACB$. Prove that $AH=2ID$.

2017 Latvia Baltic Way TST, 7
Tags: number theory , digit , combinatorics
All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of$ k$ for which the same $k$-digit number can be found in at least two different places in this string?

Durer Math Competition CD 1st Round - geometry, 2019.C3
Tags: geometry , area
The best parts of grandma’s $30$ cm $ \times 30$ cm square shaped pie are the edges. For this reason grandma’s three grandchildren would like to split the pie between each other so that everyone gets the same amount (of the area) of the pie, but also of the edges. Can they cut the pie into three connected pieces like that?

2016 Poland - Second Round, 3
Tags: algebra , function
Determine, whether exists function $f$, which assigns each integer $k$, nonnegative integer $f(k)$ and meets the conditions: $f(0) > 0$, for each integer $k$ minimal number of the form $f(k - l) + f(l)$, where $l \in \mathbb{Z}$, equals $f(k)$.

2024 Girls in Mathematics Tournament, 4
Tags: number theory , phi function
Find all integers $a$ such that there are infinitely many positive integers $n$ such that $n$ divides $\phi(n)!+a$.

1974 Poland - Second Round, 6
Tags: combinatorics , algebra
There is a sequence of integers $ a_1, a_2, \ldots, a_{2n+1} $ with the following property: after eliminating any term, the remaining ones can be divided into two groups of $ n $ terms such that the sum of the terms in the first group is equal to the sum words in the second. Prove that all terms of the sequence are equal.

2001 Estonia National Olympiad, 3
Tags: sum , geometry , angle
There are three squares in the picture. Find the sum of angles $ADC$ and $BDC$. [img]https://cdn.artofproblemsolving.com/attachments/c/9/885a6c6253fca17e24528f8ba8a5d31a18c845.png[/img]

2010 Today's Calculation Of Integral, 651
Tags: calculus , integration , limit , floor function , calculus computations
Find \[\lim_{n\to\infty}\int _0^{2n} e^{-2x}\left|x-2\lfloor\frac{x+1}{2}\rfloor\right|\ dx.\] [i]1985 Tohoku University entrance exam/Mathematics, Physics, Chemistry, Biology[/i]

2024 CCA Math Bonanza, I15
Tags:
Let $ABC$ be a triangle with side lengths $AB=13$, $BC=15$, $CA=14$. Let $\ell$ be the line passing through $A$ parallel to $BC$. Define $H$ as the orthocenter of $\triangle ABC$, and extend $BH$ to intersect $AC$ at $E$ and $\ell$ at $G$. Similarly, extend $CH$ to intersect $AB$ at $F$ and $\ell$ at $D$. Let $M$ be the midpoint of $BC$, and let $AM$ intersect the circumcircle of $AEF$ again at $P$. The ratio $\frac{PD}{PG}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Individual #15[/i]

2023 Indonesia MO, 5
Tags: number theory , greatest common divisor , least common multiple
Let $a$ and $b$ be positive integers such that $\text{gcd}(a, b) + \text{lcm}(a, b)$ is a multiple of $a+1$. If $b \le a$, show that $b$ is a perfect square.

2020 CHMMC Winter (2020-21), 1
Tags: geometry
A unit circle is centered at $(0, 0)$ on the $(x, y)$ plane. A regular hexagon passing through $(1, 0)$ is inscribed in the circle. Two points are randomly selected from the interior of the circle and horizontal lines are drawn through them, dividing the hexagon into at most three pieces. The probability that each piece contains exactly two of the hexagon's original vertices can be written as \[ \frac{2\left(\frac{m\pi}{n}+\frac{\sqrt{p}}{q}\right)^2}{\pi^2} \] for positive integers $m$, $n$, $p$, and $q$ such that $m$ and $n$ are relatively prime and $p$ is squarefree. Find $m+n+p+q$.

2022 Moldova EGMO TST, 6
Tags: geometry
Let $ABC$ be a triangle with $\angle ABC=130$. Point $D$ on side $AC$ is the foot of the perpendicular from $B$. Points $E$ and $F$ are on sides $(AB)$ and $(BC)$ such that $DE=DF$ and $AEFC$ is cyclic. Find $\angle EDF$.

2014 IFYM, Sozopol, 6
Tags: inequalities , algebra
The positive real numbers $a,b,c$ are such that $21ab+2bc+8ca\leq 12$. Find the smallest value of $\frac{1}{a}+\frac{2}{b}+\frac{3}{c}$.

1980 Bulgaria National Olympiad, Problem 5
Tags: combinatorics
Prove that the number of ways of choosing $6$ among the first $49$ positive integers, at least two of which are consecutive, is equal to $\binom{49}6-\binom{44}6$.