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

2003 China Team Selection Test, 1
Tags: trigonometry , inequalities
Let $g(x)= \sum_{k=1}^{n} a_k \cos{kx}$, $a_1,a_2, \cdots, a_n, x \in R$. If $g(x) \geq -1$ holds for every $x \in R$, prove that $\sum_{k=1}^{n}a_k \leq n$.

2022 Germany Team Selection Test, 3
Tags: number theory
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]

2008 AMC 10, 7
Tags: geometry , ratio
An equilateral triangle of side length $ 10$ is completely filled in by non-overlapping equilateral triangles of side length $ 1$. How many small triangles are required? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 100 \qquad \textbf{(D)}\ 250 \qquad \textbf{(E)}\ 1000$

2017 ISI Entrance Examination, 3
Tags: calculus computations , analytic geometry , combinatorics , function , algebra
Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function given by $$f(x) =\begin{cases} 1 & \mbox{if} \ x=1 \\ e^{(x^{10}-1)}+(x-1)^2\sin\frac1{x-1} & \mbox{if} \ x\neq 1\end{cases}$$ (a) Find $f'(1)$ (b) Evaluate $\displaystyle \lim_{u\to\infty} \left[100u-u\sum_{k=1}^{100} f\left(1+\frac{k}{u}\right)\right]$.

2012 IMAR Test, 1
Tags: vector , combinatorial geometry , set , vector geometry , geometry
Let $K$ be a convex planar set, symmetric about a point $O$, and let $X, Y , Z$ be three points in $K$. Show that $K$ contains the head of one of the vectors $\overrightarrow{OX} \pm \overrightarrow{OY} , \overrightarrow{OX} \pm \overrightarrow{OZ}, \overrightarrow{OY} \pm \overrightarrow{OZ}$.

2021 Science ON grade VII, 2
Tags: angle , geometry
In triangle $ABC$, we have $\angle ABC=\angle ACB=44^o$. Point $M$ is in its interior such that $\angle MBC=16^o$ and $\angle MCB=30^o$. Prove that $\angle MAC=\angle MBC$. [i] (Andra Elena Mircea)[/i]

2016 Iran MO (2nd Round), 1
Tags: inequalities , algebra
If $0<a\leq b\leq c$ prove that $$\frac{(c-a)^2}{6c}\leq \frac{a+b+c}{3}-\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$$

2021 Kyiv Mathematical Festival, 5
Tags: combinatorics
Bilbo composes a number triangle of zeroes and ones in such a way: he fills the topmost row with any $n$ digits, and in other rows he always writes $0$ under consecutive equal digits and writes $1$ under consecutive distinct digits. (An example of a triangle for $n=5$ is shown below.) In how many ways can Bilbo fill the topmost row for $n=100$ so that each of $n$ rows of the triangle contains even number of ones?\[\begin{smallmatrix}1\,0\,1\,1\,0\\1\,1\,0\,1\\0\,1\,1\\1\,0\\1\end{smallmatrix}\] (O. Rudenko and V. Brayman)

2014 ASDAN Math Tournament, 2
Tags:
Let $RICE$ be a quadrilateral with an inscribed circle $O$ such that every side of $RICE$ is tangent to $O$. Given taht $RI=3$, $CE=8$, and $ER=7$, compute $IC$.

2004 Austria Beginners' Competition, 1
Tags: digit , number theory
Find the smallest four-digit number that when divided by $3$ gives a four-digit number with the same digits. (Note: Four digits means that the thousand Unit digit must not be $0$.)

2015 Iran Team Selection Test, 4
Tags: geometry
Ali puts $5$ points on the plane such that no three of them are collinear. Ramtin adds a sixth point that is not collinear with any two of the former points.Ali wants to eventually construct two triangles from the six points such that one can be placed inside another. Can Ali put the 5 points in such a manner so that he would always be able to construct the desired triangles? (We say that triangle $T_1$ can be placed inside triangle $T_2$ if $T_1$ is congruent to a triangle that is located completely inside $T_2$.)

1989 Tournament Of Towns, (231) 5
Tags: tiling , combinatorial geometry , combinatorics
A rectangular $M \times N$ board is divided into $1 \times $ cells. There are also many domino pieces of size $1 \times 2$. These pieces are placed on a board so that each piece occupies two cells. The board is not entirely covered, but it is impossible to move the domino pieces (the board has a frame, so that the pieces cannot stick out of it). Prove that the number of uncovered cells is (a) less than $\frac14 MN$, (b) less than $\frac15 MN$.

2024 May Olympiad, 1
Tags: algebra
Determine all the two-digit numbers that satisfy the following condition: if we multiply their two digits, the result is equal to half the number. For example, $24$ does not satisfy the condition, because $2 \times 4 = 8$ and $8$ is not half of $24$.

2015 Indonesia MO Shortlist, N8
Tags: diophantine equation , infinitely many solutions , number theory
The natural number $n$ is said to be good if there are natural numbers $a$ and $b$ that satisfy $a + b = n$ and $ab | n^2 + n + 1$. (a) Show that there are infinitely many good numbers. (b) Show that if $n$ is a good number, then $7 \nmid n$.

2025 Vietnam National Olympiad, 2
Tags: modular arithmetic , number theory
For each non-negative integer $n$, let $u_n = \left( 2+\sqrt{5} \right)^n + \left( 2-\sqrt{5} \right)^n$. a) Prove that $u_n$ is a positive integer for all $n \geq 0$. When $n$ changes, what is the largest possible remainder when $u_n$ is divided by $24$? b) Find all pairs of positive integers $(a, b)$ such that $a, b < 500$ and for all odd positive integers $n$, $u_n \equiv a^n - b^n \pmod {1111}$.

2013 Vietnam National Olympiad, 2
Tags: circumcircle , geometry
Let $ABC$ be a cute triangle.$(O)$ is circumcircle of $\triangle ABC$.$D$ is on arc $BC$ not containing $A$.Line $\triangle$ moved through $H$($H$ is orthocenter of $\triangle ABC$ cuts circumcircle of $\triangle ABH$,circumcircle $\triangle ACH$ again at $M,N$ respectively. a.Find $\triangle$ satisfy $S_{AMN}$ max b.$d_{1},d_{2}$ are the line through $M$ perpendicular to $DB$,the line through $N$ perpendicular to $DC$ respectively. $d_{1}$ cuts $d_{2}$ at $P$.Prove that $P$ move on a fixed circle.

2021 Ukraine National Mathematical Olympiad, 6
Tags: tangent circle , geometry
Circles $w_1$ and $w_2$ intersect at points $P$ and $Q$ and touch a circle $w$ with center at point $O$ internally at points $A$ and $B$, respectively. It is known that the points $A,B$ and $Q$ lie on one line. Prove that the point $O$ lies on the external bisector $\angle APB$. (Nazar Serdyuk)

2019 Purple Comet Problems, 17
Tags: geometry
The following diagram shows equilateral triangle $\vartriangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\vartriangle ABC$ a distance $\frac18 AB$ along a side of $\vartriangle ABC$ in the directions from $A$ to $B$, from $B$ to $C$, and from $C$ to $A$. The shaded region inside all four of the triangles has area $300$. Find the area of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/3/a/8d724563c7411547d3161076015b247e882122.png[/img]

2012 India PRMO, 16
Tags: sum , function , algebra
Let $N$ be the set of natural numbers. Suppose $f: N \to N$ is a function satisfying the following conditions: (a) $f(mn) =f(m)f(n)$ (b) $f(m) < f(n)$ if $m < n$ (c) $f(2) = 2$ What is the sum of $\Sigma_{k=1}^{20}f(k)$?

2018 CCA Math Bonanza, I6
Tags:
A lumberjack is building a non-degenerate triangle out of logs. Two sides of the triangle have lengths $\log 101$ and $\log 2018$. The last side of his triangle has side length $\log n$, where $n$ is an integer. How many possible values are there for $n$? [i]2018 CCA Math Bonanza Individual Round #6[/i]

2020-IMOC, A6
Tags: polynomial , algebra
$\definecolor{A}{RGB}{255,0,0}\color{A}\fbox{A6.}$ Let $ P (x)$ be a polynomial with real coefficients such that $\deg P \ge 3$ is an odd integer. Let $f : \mathbb{R}\rightarrow\mathbb{Z}$ be a function such that $$\definecolor{A}{RGB}{0,0,200}\color{A}\forall_{x\in\mathbb{R}}\ f(P(x)) = P(f(x)).$$ $\definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(a)}$ Prove that the range of $f$ is finite. $\definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(b)}$ Show that for any positive integer $n$, there exist $P$, $f$ that satisfies the above condition and also that the range of $f$ has cardinality $n$. [i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b]. [color=#3D9186]#1735[/color]

2016 Saint Petersburg Mathematical Olympiad, 5
Tags: coloring , winning positions , game , game strategy , combinatorial geometry , combinatorics
Kostya and Sergey play a game on a white strip of length 2016 cells. Kostya (he plays first) in one move should paint black over two neighboring white cells. Sergey should paint either one white cell either three neighboring white cells. It is forbidden to make a move, after which a white cell is formed the doesn't having any white neighbors. Loses the one that can make no other move. However, if all cells are painted, then Kostya wins. Who will win if he plays the right game (has a winning strategy)?

2025 Nordic, 1
Tags: function , fe , algebra , functional equation
Let $n$ be a positive integer greater than $2$. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying: $(f(x+y))^{n} = f(x^{n})+f(y^{n}),$ for all integers $x,y$

2021 Belarusian National Olympiad, 9.7
Tags: number theory
It is known that $(x-y)^3 \vdots 6x^2-2y^2$, where $x,y$ are some integers. Prove that then also $(x+y)^3 \vdots 6x^2-2y^2$.

1999 National High School Mathematics League, 11
Tags:
Line $l:ax+by+c=0$, where $a,b,c\in\{-3,-2,-1,0,1,2,3\}$, and $a,b,c$ are different. If the bank angle of $l$ is an acute angle, then the number of such lines is________.