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

1985 Iran MO (2nd round), 1
Tags: trigonometry , induction , strong induction , algebra proposed , algebra
Let $\alpha $ be an angle such that $\cos \alpha = \frac pq$, where $p$ and $q$ are two integers. Prove that the number $q^n \cos n \alpha$ is an integer.

2002 USAMTS Problems, 4
Tags: USAMTS
Two overlapping triangles could divide a plane into up to eight regions, and three overlapping triangles could divide the plane into up to twenty regions. Find, with proof, the maximum number of regions into which six overlapping triangles could divide the plane. Describe or draw an arrangement of six triangles that divides the plane into that many regions.

2022 VN Math Olympiad For High School Students, Problem 5
Tags: geometry
Given a convex quadrilateral $MNPQ$. Assume that there exists 2 points $U, V$ inside $MNPQ$ satifying:$$\angle MUN = \angle MUV = \angle NUV = \angle QVU = \angle PVU = \angle PVQ$$Consider another 2 points $X, Y$ in the plane. Prove that the sum$$XM + XN + XY + YP + YQ$$get its minimum value iff $X\equiv U, Y\equiv V$.

1966 All Russian Mathematical Olympiad, 076
Tags: algebra
A rectangle $ABCD$ is drawn on the cross-lined paper with its sides laying on the lines, and $|AD|$ is $k$ times more than $|AB|$ ($k$ is an integer). All the shortest paths from $A$ to $C$ coming along the lines are considered. Prove that the number of those with the first link on $[AD]$ is $k$ times more then of those with the first link on $[AB]$.

1992 Flanders Math Olympiad, 3
Tags: geometry , 3D geometry , sphere , conics
a conic with apotheme 1 slides (varying height and radius, with $r < \frac12$) so that the conic's area is $9$ times that of its inscribed sphere. What's the height of that conic?

1995 AMC 8, 14
Tags:
A team won $40$ of its first $50$ games. How many of the remaining $40$ games must this team win so it will have won exactly $70 \%$ of its games for the season? $\text{(A)}\ 20 \qquad \text{(B)}\ 23 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35$

2006 Purple Comet Problems, 7
Tags: number theory , relatively prime , 2006
Heather and Kyle need to mow a lawn and paint a room. If Heather does both jobs by herself, it will take her a total of nine hours. If Heather mows the lawn and, after she finishes, Kyle paints the room, it will take them a total of eight hours. If Kyle mows the lawn and, after he finishes, Heather paints the room, it will take them a total of seven hours. If Kyle does both jobs by himself, it will take him a total of six hours. It takes Kyle twice as long to paint the room as it does for him to mow the lawn. The number of hours it would take the two of them to complete the two tasks if they worked together to mow the lawn and then worked together to paint the room is a fraction $\tfrac{m}{n}$where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2018 Costa Rica - Final Round, N3
Tags: number theory , diophantine
Let $a$ and $b$ be positive integers such that $2a^2 + a = 3b^2 + b$. Prove that $a-b$ is a perfect square.

1997 Belarusian National Olympiad, 4
Tags: geometry , construction , projective geometry , orthocenter , incenter
A triangle $A_1B_1C_1$ is a parallel projection of a triangle $ABC$ in space. The parallel projections $A_1H_1$ and $C_1L_1$ of the altitude $AH$ and the bisector $CL$ of $\vartriangle ABC$ respectively are drawn. Using a ruler and compass, construct a parallel projection of : (a) the orthocenter, (b) the incenter of $\vartriangle ABC$.

2004 Iran Team Selection Test, 5
Tags: combinatorics proposed , combinatorics
This problem is generalization of [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=5918]this one[/url]. Suppose $G$ is a graph and $S\subset V(G)$. Suppose we have arbitrarily assign real numbers to each element of $S$. Prove that we can assign numbers to each vertex in $G\backslash S$ that for each $v\in G\backslash S$ number assigned to $v$ is average of its neighbors.

2024 Irish Math Olympiad, P2
Tags: Integers , irmo
A non-negative integer $p$ is a [i]3-choice[/i] if $\dfrac{k(k-1)(k-2)}{6}$ for some positive integer $k$. Let $p$ and $q$ be 3-choices with $p<q$. Show there is an integer $n$ such that $p \leq n^2 < q$.

2019 Latvia Baltic Way TST, 16
Tags: algebra , system of equations , factorial
Determine all tuples of positive integers $(x, y, z, t)$ such that: $$ xyz = t!$$ $$ (x+1)(y+1)(z+1) = (t+1)!$$ holds simultaneously.

PEN A Problems, 43
Tags: modular arithmetic , Divisibility Theory , pen
Suppose that $p$ is a prime number and is greater than $3$. Prove that $7^{p}-6^{p}-1$ is divisible by $43$.

2013 Purple Comet Problems, 3
Tags:
Let $N$ be the sum of the first four three-digit prime numbers. Find the sum of the prime factors of $\tfrac{N}2$.

2020 Ukrainian Geometry Olympiad - December, 5
Tags: geometry , parallel , Centroid , Circumcenter , circumcircle
Let $ABC$ be an acute triangle with $\angle ACB = 45^o$, $G$ is the point of intersection of the medians, and $O$ is the center of the circumscribed circle. If $OG =1$ and $OG \parallel BC$, find the length of $BC$.

2021 Balkan MO Shortlist, A5
Tags: Balkan , shortlist , 2021 , algebra , functional equation
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(xf(x + y)) = yf(x) + 1$$ holds for all $x, y \in \mathbb{R}^{+}$. [i]Proposed by Nikola Velov, North Macedonia[/i]

2021 LMT Spring, A8
Tags:
Isosceles $\triangle{ABC}$ has interior point $O$ such that $AO = \sqrt{52}$, $BO = 3$, and $CO = 5$. Given that $\angle{ABC}=120^{\circ}$, find the length $AB$. [i]Proposed by Powell Zhang[/i]

2019 Thailand TST, 2
Tags: algebra , Sequences , IMO Shortlist , maximum value , IMO shortlist 2018
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

2011 Ukraine Team Selection Test, 8
Tags: Sequence , algebra , Natural Numbers
Is there an increasing sequence of integers $ 0 = {{a} _{0}} <{{a} _{1}} <{{a} _{2}} <\ldots $ for which the following two conditions are satisfied simultaneously: 1) any natural number can be given as $ {{a} _{i}} + {{a} _{j}} $ for some (possibly equal) $ i \ge 0 $, $ j \ge 0$ , 2) $ {{a} _ {n}}> \tfrac {{{n} ^ {2}}} {16} $ for all natural $ n $?

1995 Abels Math Contest (Norwegian MO), 1b
Tags: algebra
Prove that if  $(x+\sqrt{x^2 +1})(y+\sqrt{y^2 +1})= 1$ for real numbers $x,y$, then $x+y = 0$.

1996 Taiwan National Olympiad, 3
Tags: geometry , perpendicular bisector , angle bisector , geometry unsolved
Let be given points $A,B$ on a circle and let $P$ be a variable point on that circle. Let point $M$ be determined by $P$ as the point that is either on segment $PA$ with $AM=MP+PB$ or on segment $PB$ with $AP+MP=PB$. Find the locus of points $M$.

2024 Baltic Way, 8
Tags: combinatorics , combinatorics proposed , grid , game , winning strategy
Let $a$, $b$, $n$ be positive integers such that $a + b \leq n^2$. Alice and Bob play a game on an (initially uncoloured) $n\times n$ grid as follows: - First, Alice paints $a$ cells green. - Then, Bob paints $b$ other (i.e.uncoloured) cells blue. Alice wins if she can find a path of non-blue cells starting with the bottom left cell and ending with the top right cell (where a path is a sequence of cells such that any two consecutive ones have a common side), otherwise Bob wins. Determine, in terms of $a$, $b$ and $n$, who has a winning strategy.

1997 Canada National Olympiad, 2
Tags: combinatorics unsolved , combinatorics
The closed interval $A = [0, 50]$ is the union of a finite number of closed intervals, each of length $1$. Prove that some of the intervals can be removed so that those remaining are mutually disjoint and have total length greater than $25$. Note: For reals $a\le b$, the closed interval $[a, b] := \{x\in \mathbb{R}:a\le x\le b\}$ has length $b-a$; disjoint intervals have [i]empty [/i]intersection.

2008 Saint Petersburg Mathematical Olympiad, 6
Tags: number theory , greatest common divisor
$a+b+c \leq 3000000$ and $a\neq b \neq c \neq a$ and $a,b,c$ are naturals. Find maximum $GCD(ab+1,ac+1,bc+1)$

1964 Bulgaria National Olympiad, Problem 2
Tags:
Find all $n$-tuples of reals $x_1,x_2,\ldots,x_n$ satisfying the system: $$\begin{cases}x_1x_2\cdots x_n=1\\x_1-x_2x_3\cdots x_n=1\\x_1x_2-x_3x_4\cdots x_n=1\\\vdots\\x_1x_2\cdots x_{n-1}-x_n=1\end{cases}$$