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

1983 IMO Longlists, 46
Tags: function , limit , algebra , logarithm
Let $f$ be a real-valued function defined on $I = (0,+\infty)$ and having no zeros on $I$. Suppose that \[\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty.\] For the sequence $u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|$, prove that $u_n \to +\infty$ as $n \to +\infty.$

Bangladesh Mathematical Olympiad 2020 Final, #12
Tags: number theory
$2^{2921}$ has $581$ digits and starts with a $4$. How many $2^n$'s starts with a $4$, where $0$ is the last digit?

2021/2022 Tournament of Towns, P3
Tags: geometry , incircle
The intersection of two triangles is a hexagon. If this hexagon is removed, six small triangles remain. These six triangles have the same in-radii. Prove the in-radii of the original two triangles are also equal. Spoiler: This is one of the highlights of TT. Also SA3

2021 AMC 12/AHSME Spring, 14
Tags:
What is the value of $$\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?$$ $\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2,200\qquad \textbf{(E) }21,000$

1962 All Russian Mathematical Olympiad, 020
Tags: pentagon , minimum , maximum , geometry
Given regular pentagon $ABCDE$. $M$ is an arbitrary point inside $ABCDE$ or on its side. Let the distances $|MA|, |MB|, ... , |ME|$ be renumerated and denoted with $$r_1\le r_2\le r_3\le r_4\le r_5.$$ Find all the positions of the $M$, giving $r_3$ the minimal possible value. Find all the positions of the $M$, giving $r_3$ the maximal possible value.

2022 South East Mathematical Olympiad, 6
Tags: geometry
$H$ is the orthocenter of $\triangle ABC$,the circle with center $H$ passes through $A$,and it intersects with $AC,AB$ at two other points $D,E$.The orthocenter of $\triangle ADE$ is $H'$,line $AH'$ intersects with $DE$ at point $F$.Point $P$ is inside the quadrilateral $BCDE$,so that $\triangle PDE\sim\triangle PBC$.Let point $K$ be the intersection of line $HH'$ and line $PF$.Prove that $A,H,P,K$ lie on one circle. [img]https://i.ibb.co/mcyhxRM/graph.jpg[/img]

1999 Junior Balkan Team Selection Tests - Romania, 1
Tags: algebra , polynomial
Let be a natural number $ n. $ Prove that there is a polynomial $ P\in\mathbb{Z} [X,Y] $ such that $ a+b+c=0 $ implies $$ a^{2n+1}+b^{2n+1}+c^{2n+1}=abc\left( P(a,b)+P(b,c)+P(c,a)\right) $$ [i]Dan Brânzei[/i]

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.