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

1999 Putnam, 2
Tags: algebra , polynomial
Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),f_2(x),\ldots,f_k(x)$ such that \[p(x)=\sum_{j=1}^k(f_j(x))^2.\]

2020 May Olympiad, 2
Tags: number theory
a) Determine if there are positive integers $a, b$ and $c$, not necessarily distinct, such that $a+b+c=2020$ and $2^a+2^b+2^c$ it's a perfect square. b) Determine if there are positive integers $a, b$ and $c$, not necessarily distinct, such that $a+b+c=2020$ and $3^a+3^b+3^c$ it's a perfect square.

2001 Irish Math Olympiad, 1
Tags: modular arithmetic , number theory
Find all positive integer solutions $ (a,b,c,n)$ of the equation: $ 2^n\equal{}a!\plus{}b!\plus{}c!$.

2007 AMC 8, 20
Tags:
Before district play, the Unicorns had won $45\%$ of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all? $\textbf{(A)}\ 48 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 52 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 60$

2009 239 Open Mathematical Olympiad, 1
Tags:
Kostya drives a car from a village to a city, driving along three roads. Moreover, on each of these roads, he drives at a constant speed. Is it possible that a third of the distance traveled was completed earlier than a third of the time, half of the distance traveled later than half of the time, and two-thirds of the distance was earlier than two-thirds of the time?

2021 LMT Fall, 10
Tags: combinatorics
There are $15$ people attending math team: $12$ students and $3$ captains. One of the captains brings $33$ identical snacks. A nonnegative number of names (students and/or captains) are written on the NO SNACK LIST. At the end of math team, all students each get n snacks, and all captains get $n +1$ snacks, unless the person’s name is written on the board. After everyone’s snacks are distributed, there are none left. Find the number of possible integer values of $n$.

2020-2021 OMMC, 10
Tags:
Positive integers $a,b,c$ exist such that $a+b+c+1$, $a^2+b^2+c^2 +1$, $a^3+b^3+c^3+1,$ and $a^4+b^4+c^4+7459$ are all multiples of $p$ for some prime $p$. Find the sum of all possible values of $p$ less than $1000$.

2025 Turkey Team Selection Test, 1
Tags: graph theory , combinatorics , edge coloring
In a complete graph with $2025$ vertices, each edge has one of the colors $r_1$, $r_2$, or $r_3$. For each $i = 1,2,3$, if the $2025$ vertices can be divided into $a_i$ groups such that any two vertices connected by an edge of color $r_i$ are in different groups, find the minimum possible value of $a_1 + a_2 + a_3$.

2011 JHMT, 4
Tags: geometry
Compute the largest value of $r$ such that three non-overlapping circles of radius $r$ can be inscribed in a unit square.

Oliforum Contest III 2012, 2
Tags: polynomial , sum of digits , number theory
Show that for every polynomial $f(x)$ with integer coefficients, there exists a integer $C$ such that the set $\{n \in Z :$ the sum of digits of $f(n)$ is $C\}$ is not finite.

2014 Saint Petersburg Mathematical Olympiad, 7
Tags: combinatorics , graph theory
Some cities in country are connected with oneway road. It is known that every closed cyclic route, that don`t break traffic laws, consists of even roads. Prove that king of city can place military bases in some cities such that there are not roads between these cities, but for every city without base we can go from city with base by no more than $1$ road. [hide=PS]I think it should be one more condition, like there is cycle that connect all cities [/hide]

1978 Swedish Mathematical Competition, 2
Tags: sum , number theory , digit , algebra
Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find \[ s_1 + s_2 + \cdots + s_n \]

1991 Arnold's Trivium, 91
Tags: algebra , polynomial , linear algebra , matrix
Find the Jordan normal form of the operator $e^{d/dt}$ in the space of quasi-polynomials $\{e^{\lambda t}p(t)\}$ where the degree of the polynomial $p$ is less than $5$, and of the operator $\text{ad}_A$, $B\mapsto [A, B]$, in the space of $n\times n$ matrices $B$, where $A$ is a diagonal matrix.

2013 Czech And Slovak Olympiad IIIA, 6
Tags: inequalities , algebra
Find all positive real numbers $p$ such that $\sqrt{a^2 + pb^2} +\sqrt{b^2 + pa^2} \ge a + b + (p - 1) \sqrt{ab}$ holds for any pair of positive real numbers $a, b$.

1992 Turkey Team Selection Test, 1
Tags: prime number , number theory
Is there $14$ consecutive positive integers such that each of these numbers is divisible by one of the prime numbers $p$ where $2\leq p \leq 11$.

2014 Contests, 3
Tags: number theory , divisibility
Let $1000 \leq n = \text{ABCD}_{10} \leq 9999$ be a positive integer whose digits $\text{ABCD}$ satisfy the divisibility condition: $$1111 | (\text{ABCD} + \text{AB} \times \text{CD}).$$ Determine the smallest possible value of $n$.

1970 Swedish Mathematical Competition, 5
Tags: geometry , square
A $3\times 1$ paper rectangle is folded twice to give a square side $1$. The square is folded along a diagonal to give a right-angled triangle. A needle is driven through an interior point of the triangle, making $6$ holes in the paper. The paper is then unfolded. Where should the point be in order to maximise the smallest distance between any two holes?

2015 May Olympiad, 1
Tags: number theory , digit
The teacher secretly thought of a three-digit $S$ number. Students $A, B, C$ and $D$ tried to guess, saying, respectively, $541$, $837$, $291$ and $846$. The teacher told them, “Each of you got it right exactly one digit of $S$ and in the correct position ”. What is the number $S$?

2021 CMIMC, 2.3 1.1
Tags: combinatorics
Adam has a box with $15$ pool balls in it, numbered from $1$ to $15$, and picks out $5$ of them. He then sorts them in increasing order, takes the four differences between each pair of adjacent balls, and finds exactly two of these differences are equal to $1$. How many selections of $5$ balls could he have drawn from the box? [i]Proposed by Adam Bertelli[/i]

2020 AIME Problems, 15
Tags: geometry
Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT=CT=16$, $BC=22$, and $TX^2+TY^2+XY^2=1143$. Find $XY^2$.

2019 Finnish National High School Mathematics Comp, 1
Tags: algebra , equation
Solve $x(8\sqrt{1-x}+\sqrt{1+x}) \le 11\sqrt{1+x}-16\sqrt{1-x}$ when $0<x\le 1$

2020-IMOC, N6
Tags: number theory , greatest common divisor
$\textbf{N6.}$ Let $a,b$ be positive integers. If $a,b$ satisfy that \begin{align*} \frac{a+1}{b} + \frac{b+1}{a} \end{align*} is also a positive integer, show that \begin{align*} \frac{a+b}{gcd(a,b)^2} \end{align*} is a Fibonacci number. [i]Proposed by usjl[/i]

2014 Czech and Slovak Olympiad III A, 6
Tags: inequalities
For arbitrary non-negative numbers $a$ and $b$ prove inequality $\frac{a}{\sqrt{b^2+1}}+\frac{b}{\sqrt{a^2+1}}\ge\frac{a+b}{\sqrt{ab+1}}$, and find, where equality occurs. (Day 2, 6th problem authors: Tomáš Jurík, Jaromír Šimša)

2012 France Team Selection Test, 2
Tags: circumcircle , trigonometry , parallelogram , geometric transformation , reflection , ratio , geometry
Let $ABC$ be an acute-angled triangle with $AB\not= AC$. Let $\Gamma$ be the circumcircle, $H$ the orthocentre and $O$ the centre of $\Gamma$. $M$ is the midpoint of $BC$. The line $AM$ meets $\Gamma$ again at $N$ and the circle with diameter $AM$ crosses $\Gamma$ again at $P$. Prove that the lines $AP,BC,OH$ are concurrent if and only if $AH=HN$.

2023 Turkey Junior National Olympiad, 1
Tags: combinatorics
Initially, there are $n$ red boxes numbered with the numbers $1,2,\dots ,n$ and $n$ white boxes numbered with the numbers $1,2,\dots ,n$ on the table. At every move, we choose $2$ different colored boxes and put a ball on each of them. After some moves, every pair of the same numbered boxes has the property of either the number of balls from the red one is $6$ more than the number of balls from the white one or the number of balls from the white one is $16$ more than the number of balls from the red one. With that given information find all possible values of $n$