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

2009 Thailand Mathematical Olympiad, 6
Tags: polynomial , algebra
Find all polynomials of the form $P(x) = (-1)^nx^n + a_1x^{n-1} + a_2x^{n-2} + ...+ a_{n-1}x + a_n$ with the following two properties: (i) $\{a_1, a_2, . . . , a_n-1, a_n\} =\{0, 1\}$, and (ii) all roots of $P(x)$ are distinct real numbers

2025 Harvard-MIT Mathematics Tournament, 9
Tags: combinatorics
Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon.

2000 All-Russian Olympiad Regional Round, 11.8
Tags: combinatorics
There are $2000$ cities in the country, some pairs of cities are connected by roads. It is known that no more than $N$ different non-self-intersecting cyclic routes of odd length. Prove that the country can be divided into $N + 2$ republics so that no two cities from the same republic are connected by a road.

2006 Silk Road, 1
Tags: algebra , function
Found all functions $f: \mathbb{R} \to \mathbb{R}$, such that for any $x,y \in \mathbb{R}$, \[f(x^2+xy+f(y))=f^2(x)+xf(y)+y.\]

2025 CMIMC Combo/CS, 6
Tags: computer science , combinatorics
Consider a $4 \times 4$ grid of squares. We place coins in some of the grid squares so that no two coins are orthogonally adjacent, and each $2 \times 2$ square in the grid has at least one coin. How many ways are there to place the coins?

2016 ASDAN Math Tournament, 13
Tags:
Suppose $\{a_n\}_{n=1}^\infty$ is a sequence. The partial sums $\{s_n\}_{n=1}^\infty$ are defined by $$s_n=\sum_{i=1}^na_i.$$ The Cesàro sums are then defined as $\{A_n\}_{n=1}^\infty$, where $$A_n=\frac{1}{n}\cdot\sum_{i=1}^ns_i.$$ Let $a_n=(-1)^{n+1}$. What is the limit of the Cesàro sums of $\{a_n\}_{n=1}^\infty$ as $n$ goes to infinity?

2012 Tournament of Towns, 4
Tags: equilateral , incenter , geometry
Given a triangle $ABC$. Suppose I is its incentre, and $X, Y, Z$ are the incentres of triangles $AIB, BIC$ and $AIC$ respectively. The incentre of triangle $XYZ$ coincides with $I$. Is it necessarily true that triangle $ABC$ is regular?

2012 Olympic Revenge, 4
Tags: number theory
Say that two sets of positive integers $S, T$ are $\emph{k-equivalent}$ if the sum of the $i$th powers of elements of $S$ equals the sum of the $i$th powers of elements of $T$, for each $i= 1, 2, \ldots, k$. Given $k$, prove that there are infinitely many numbers $N$ such that $\{1,2,\ldots,N^{k+1}\}$ can be divided into $N$ subsets, all of which are $k$-equivalent to each other.

2002 AMC 10, 4
Tags:
What is the value of \[ (3x \minus{} 2)(4x \plus{} 1) \minus{} (3x \minus{} 2)4x \plus{} 1\]when $ x \equal{} 4$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 12$

PEN Q Problems, 3
Tags: polynomial
Let $n \ge 2$ be an integer. Prove that if $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq \sqrt{\frac{n}{3}}$, then $k^2 +k + n$ is prime for all integers $k$ such that $0 \leq k \leq n - 2$.

1994 Baltic Way, 5
Tags: algebra , polynomial
Let $p(x)$ be a polynomial with integer coefficients such that both equations $p(x)=1$ and $p(x)=3$ have integer solutions. Can the equation $p(x)=2$ have two different integer solutions?

1983 National High School Mathematics League, 5
Tags: function
Function $F(x)=|\cos^2x+2\sin x\cos x-\sin^2x+Ax+B|$, where $A,B$ are two real numbers, $x\in[0,\frac{3}{2}\pi]$. $M$ is the maximun value of $F(x)$. Find the minumum value of $M$.

2019 Iran Team Selection Test, 6
Tags: inequalities
$x,y$ and $z$ are real numbers such that $x+y+z=xy+yz+zx$. Prove that $$\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.$$ [i]Proposed by Navid Safaei[/i]

2013 National Olympiad First Round, 15
Tags: algebra
No matter how $n$ real numbers on the interval $[1,2013]$ are selected, if it is possible to find a scalene polygon such that its sides are equal to some of the numbers selected, what is the least possible value of $n$? $ \textbf{(A)}\ 14 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 10 $

2002 District Olympiad, 2
Tags: trigonometry , geometry
In the $xOy$ system, consider the points $A_n(n,n^3)$ with $n\in \mathbb{N}^*$ and the point $B(0,1)$. Prove that a) for any positive integers $k>j>i\ge 1$, the points $A_i,A_j,A_k$ cannot be collinear. b) for any positive integers $i_k>i_{k-1}>\ldots>i_1\ge 1$, we have \[\mu(\widehat{A_{i_1}OB})+\mu(\widehat{A_{i_2}OB})+\cdots+\mu(\widehat{A_{i_k}OB})<\frac{\pi}{2}\] [i]***[/i]

2012 Silk Road, 1
Tags: circumcircle , trapezoid , geometry
Trapezium $ABCD$, where $BC||AD$, is inscribed in a circle, $E$ is midpoint of the arc $AD$ of this circle not containing point $C$ . Let $F$ be the foot of the perpendicular drawn from $E$ on the line tangent to the circle at the point $C$ . Prove that $BC=2CF$.

2015 AMC 12/AHSME, 13
Tags:
A league with $12$ teams holds a round-robin tournament, with each team playing every other team once. Games either end with one team victorious or else end in a draw. A team scores $2$ points for every game it wins and $1$ point for every game it draws. Which of the following is $\textbf{not}$ a true statement about the list of $12$ scores? $\textbf{(A) }\text{There must be an even number of odd scores.}$ $\textbf{(B) }\text{There must be an even number of even scores.}$ $\textbf{(C) }\text{There cannot be two scores of 0.}$ $\textbf{(D) }\text{The sum of the scores must be at least 100.}$ $\textbf{(E) }\text{The highest score must be at least 12.}$

2024 TASIMO, 2
Tags: sequence , algebra
Find all positive integers $(r,s)$ such that there is a non-constant sequence $a_n$ os positive integers such that for all $n=1,2,\dots$ \[ a_{n+2}= \left(1+\frac{{a_2}^r}{{a_1}^s} \right ) \left(1+\frac{{a_3}^r}{{a_2}^s} \right ) \dots \left(1+\frac{{a_{n+1}}^r}{{a_n}^s} \right ).\] Proposed by Navid Safaei, Iran

2005 Today's Calculation Of Integral, 54
Tags: partial fractions , algebra , integration , calculus computations , trigonometry , calculus
evaluate \[\int_{-1}^0 \sqrt{\frac{1+x}{1-x}}dx\]

2020 Israel Olympic Revenge, G
Tags: nine point center , angle bisector , apollonius circle , geometry
Let $ABC$ be an acute triangle with $AB\neq AC$. The angle bisector of $\angle BAC$ intersects with $BC$ at a point $D$. $BE,CF$ are the altitudes of the triangle and $Ap_1,Ap_2$ are the isodynamic points of triangle $ABC$.Let the $A$-median of $ABC$ intersect $EF$ at $T$. Show that the line connecting $T$ with the nine-point center of $ABC$ is perpendicular to $BC$ if and only if $\angle Ap_1DAp_2=90^\circ$.

1976 Chisinau City MO, 121
Tags: odd , integer , algebra , polynomial
Prove that the polynomial $P (x)$ with integer coefficients, taking odd values for $x = 0$ and $x= 1$, has no integer roots.

2020 Paraguay Mathematical Olympiad, 1
Tags: number theory
José has the following list of numbers: $100, 101, 102, ..., 118, 119, 120$. He calculates the sum of each of the pairs of different numbers that you can put together. How many different prime numbers can you get calculating those sums?

2014 ASDAN Math Tournament, 6
Tags: team test
Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$.

2020 Putnam, A2
Tags: binomial identites
Let $k$ be a nonnegative integer. Evaluate \[ \sum_{j=0}^k 2^{k-j} \binom{k+j}{j}. \]

2018 Romania Team Selection Tests, 2
Tags: series summation , floor function , number theory
Given a square-free integer $n>2$, evaluate the sum $\sum_{k=1}^{(n-2)(n-1)} \lfloor ({kn})^{1/3} \rfloor$.