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.

AND:
OR:
NO:

Found problems: 85335

1967 IMO Longlists, 28
Tags: parameterization , calculus , trigonometry , algebra , trigonometric identities
Find values of the parameter $u$ for which the expression \[y = \frac{ \tan(x-u) + \tan(x) + \tan(x+u)}{ \tan(x-u)\tan(x)\tan(x+u)}\] does not depend on $x.$

2001 District Olympiad, 2
Tags: algebra
Two numbers $(z_1,z_2)\in \mathbb{C}^*\times \mathbb{C}^*$ have the property $(P)$ if there is a real number $a\in [-2,2]$ such that $z_1^2-az_1z_2+z_2^2=0$. Prove that if $(z_1,z_2)$ have the property $(P)$, then $(z_1^n,z_2^n)$ satisfy this property, for any positive integer $n$. [i]Dorin Andrica[/i]

2003 China Team Selection Test, 2
Tags: modular arithmetic , quadratic , number theory
Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.

2014 ELMO Shortlist, 1
Tags: modular arithmetic , mod 13 , number theory
Does there exist a strictly increasing infinite sequence of perfect squares $a_1, a_2, a_3, ...$ such that for all $k\in \mathbb{Z}^+$ we have that $13^k | a_k+1$? [i]Proposed by Jesse Zhang[/i]

2023 Iranian Geometry Olympiad, 3
Tags: combinatorics , combinatorial geometry , square , geometry
Let $ABCD$ be a square with side length $1$. How many points $P$ inside the square (not on its sides) have the property that the square can be cut into $10$ triangles of equal area such that all of them have $P$ as a vertex? [i]Proposed by Josef Tkadlec - Czech Republic[/i]

2016 Fall CHMMC, 9
Tags: geometry
In quadrilateral $ABCD$, $AB = DB$ and $AD = BC$. If $\angle ABD = 36^{\circ}$ and $\angle BCD = 54^{\circ}$, find $\angle ADC$ in degrees.

2013 NIMO Problems, 2
Tags: geometry , perimeter , probability , analytic geometry , graphing lines , slope , geometric transformation
Square $\mathcal S$ has vertices $(1,0)$, $(0,1)$, $(-1,0)$ and $(0,-1)$. Points $P$ and $Q$ are independently selected, uniformly at random, from the perimeter of $\mathcal S$. Determine, with proof, the probability that the slope of line $PQ$ is positive. [i]Proposed by Isabella Grabski[/i]

2015 ASDAN Math Tournament, 15
Tags:
Let $ABCD$ be a regular tetrahedron of side length $12$. Select points $E,F,G,H$ on $AC,BC,BD,AD$, respectively, such that $AE=BF=BG=AH=3$. Compute the area of quadrilateral $EFGH$.

2024 Spain Mathematical Olympiad, 3
Tags: geometry
Let $ABC$ be a scalene triangle and $P$ be an interior point such that $\angle PBA=\angle PCA$. The lines $PB$ and $PC$ intersect the internal and external bisectors of $\angle BAC$ at $Q$ and $R$, respectively. Let $S$ be the point such that $CS$ is parallel to $AQ$ and $BS$ is parallel to $AR$. Prove that $Q$, $R$ and $S$ are colinear.

2014 European Mathematical Cup, 4
Tags: function , induction , number theory
Find all functions $f$ from positive integers to themselves such that: 1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$ 2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.

2021 Ukraine National Mathematical Olympiad, 4
Tags: geometry , circumcenter
Let $O, I, H$ be the circumcenter, the incenter, and the orthocenter of $\triangle ABC$. The lines $AI$ and $AH$ intersect the circumcircle of $\triangle ABC$ for the second time at $D$ and $E$, respectively. Prove that if $OI \parallel BC$, then the circumcenter of $\triangle OIH$ lies on $DE$. (Fedir Yudin)

1999 Romania National Olympiad, 1
Tags: linear algebra
Let $A \in \mathcal{M}_2(\mathbb{C})$ and $C(A)=\{B \in \mathcal{M}_2(\mathbb{C}) : AB=BA \}.$ Prove that $$|\det(A+B)| \ge |\det B|,$$ for any $B \in C(A),$ if and only if $A^2=O_2.$

PEN S Problems, 10
Tags: geometry , perimeter
Let $p$ be an odd prime. Show that there is at most one non-degenerate integer triangle with perimeter $4p$ and integer area. Characterize those primes for which such triangle exist.

2012 Online Math Open Problems, 7
Tags: probability , trigonometry , function , inequalities , number theory , relatively prime
Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m+n$. [i]Ray Li.[/i]

1983 Putnam, B2
Tags: algebra , combinatorics , polynomial
For positive integers $n$, let $C(n)$ be the number of representation of $n$ as a sum of nonincreasing powers of $2$, where no power can be used more than three times. For example, $C(8)=5$ since the representations of $8$ are: $$8,4+4,4+2+2,4+2+1+1,\text{ and }2+2+2+1+1.$$Prove or disprove that there is a polynomial $P(x)$ such that $C(n)=\lfloor P(n)\rfloor$ for all positive integers $n$.

2010 Saudi Arabia IMO TST, 1
Tags: number theory , divide , divisible
Find all pairs $(m,n)$ of integers, $m ,n \ge 2$ such that $mn - 1$ divides $n^3 - 1$.

2022 BMT, 13
Tags: algebra , system of equations
Real numbers $x$ and $y$ satisfy the system of equations $$x^3 + 3x^2 = -3y - 1$$ $$y^3 + 3y^2 = -3x - 1.$$ What is the greatest possible value of $x$?

2013 Princeton University Math Competition, 5
Tags:
Find the number of pairs $(n,C)$ of positive integers such that $C\leq 100$ and $n^2+n+C$ is a perfect square.

2005 Belarusian National Olympiad, 2
Tags: geometry
A line parallel to the side $AC$ of a triangle $ABC$ with $\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$, so that $CN/BN = AC/BC = 2/1$. The segments $CM$ and $AN$ meet at $O$. Let $K$ be a point on the segment $ON$ such that $MO+OK = KN$. The bisector of $\angle ABC$ meets the line through $K$ perpendicular to $AN$ at point $T$. Determine $\angle MTB$.

2000 Harvard-MIT Mathematics Tournament, 20
Tags: geometry , perimeter , hmmt , trigonometry
What is the minimum possible perimeter of a triangle two of whose sides are along the x- and y-axes and such that the third contains the point $(1,2)$?

2005 MOP Homework, 2
Tags: combinatorics
Set $S=\{1,2,...,2004\}$. We denote by $d_1$ the number of subset of $S$ such that the sum of elements of the subset has remainder $7$ when divided by $32$. We denote by $d_2$ the number of subset of $S$ such that the sum of elements of the subset has remainder $14$ when divided by $16$. Compute $\frac{d_1}{d_2}$.

2020 AMC 10, 25
Tags: factorization
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product$$n = f_1\cdot f_2\cdots f_k,$$where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$? $\textbf{(A) } 112 \qquad\textbf{(B) } 128 \qquad\textbf{(C) } 144 \qquad\textbf{(D) } 172 \qquad\textbf{(E) } 184$

2018-2019 SDML (High School), 5
Tags:
The graph of the equation $y = ax^2 + bx + c$ is shown in the diagram. Which of the following must be positive? [DIAGRAM NEEDED] $ \mathrm{(A) \ } a \qquad \mathrm{(B) \ } ab^2 \qquad \mathrm {(C) \ } b - c \qquad \mathrm{(D) \ } bc \qquad \mathrm{(E) \ } c - a$

2025 Malaysian APMO Camp Selection Test, 4
Tags: number theory
Find all pairs of distinct primes $(p,q)$ such that $p$ and $q$ are both prime factors of $p^3+q^2+1$, and are the only such prime factors. [i]Proposed by Takeda Shigenori[/i]

2020 Saint Petersburg Mathematical Olympiad, 1.
Tags: combinatorics
Andryusha has $100$ stones of different weight and he can distinguish the stones by appearance, but does not know their weight. Every evening, Andryusha can put exactly $10$ stones on the table and at night the brownie will order them in increasing weight. But, if the drum also lives in the house then surely he will in the morning change the places of some $2$ stones.Andryusha knows all about this but does not know if there is a drum in his house. Can he find out?