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

2023 Chile TST Ibero., 3
Tags: number theory
Determine the smallest positive integer \( n \) with the following property: for every triple of positive integers \( x, y, z \), with \( x \) dividing \( y^3 \), \( y \) dividing \( z^3 \), and \( z \) dividing \( x^3 \), it also holds that \( (xyz) \) divides \( (x + y + z)^n \).

1995 AIME Problems, 7
Tags: trigonometry , quadratic , number theory , relatively prime , algebra , quadratic formula
Given that $(1+\sin t)(1+\cos t)=5/4$ and \[ (1-\sin t)(1-\cos t)=\frac mn-\sqrt{k}, \] where $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$

2010 National Olympiad First Round, 3
Tags:
How many real pairs $(x,y)$ are there such that \[ x^2+2y = 2xy \\ x^3+x^2y = y^2 \] $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None} $

2002 China Western Mathematical Olympiad, 2
Tags: parallelogram , circumcircle , geometry
Let $ O$ be the circumcenter of acute triangle $ ABC$. Point $ P$ is in the interior of triangle $ AOB$. Let $ D,E,F$ be the projections of $ P$ on the sides $ BC,CA,AB$, respectively. Prove that the parallelogram consisting of $ FE$ and $ FD$ as its adjacent sides lies inside triangle $ ABC$.

2013 China Team Selection Test, 2
Tags: arithmetic sequence , number theory , algebra
Find the greatest positive integer $m$ with the following property: For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.

2006 Croatia Team Selection Test, 2
Tags: inequalities
Assume that $a, b,$ and $c$ are positive real numbers for which $(a+b)(a+c)(b+c) = 1$. Prove that $ab+bc+ca \leq\frac{3 }{4}.$

2017 Bulgaria JBMO TST, 3
Tags:
Prove that for all positive real $m, n, p, q$ and $t=\frac{m+n+p+q}{2}$, $$ \frac{m}{t+n+p+q} +\frac{n}{t+m+p+q} +\frac{p} {t+m+n+q}+\frac{q}{t+m+n+p} \geq \frac{4}{5}. $$

2008 ITest, 90
Tags: floor function , inequalities
For $a,b,c$ positive reals, let \[N=\dfrac{a^2+b^2}{c^2+ab}+\dfrac{b^2+c^2}{a^2+bc}+\dfrac{c^2+a^2}{b^2+ca}.\] Find the minimum value of $\lfloor 2008N\rfloor$.

1992 AMC 12/AHSME, 13
Tags:
How many pairs of positive integers $(a,b)$ with $a + b \le 100$ satisfy the equation $\frac{a + b^{-1}}{a^{-1} + b} = 13$? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 13 $

2013 USA TSTST, 2
Tags: floor function , inequalities , function , algebra , logarithm
A finite sequence of integers $a_1, a_2, \dots, a_n$ is called [i]regular[/i] if there exists a real number $x$ satisfying \[ \left\lfloor kx \right\rfloor = a_k \quad \text{for } 1 \le k \le n. \] Given a regular sequence $a_1, a_2, \dots, a_n$, for $1 \le k \le n$ we say that the term $a_k$ is [i]forced[/i] if the following condition is satisfied: the sequence \[ a_1, a_2, \dots, a_{k-1}, b \] is regular if and only if $b = a_k$. Find the maximum possible number of forced terms in a regular sequence with $1000$ terms.

VI Soros Olympiad 1999 - 2000 (Russia), 9.7
Tags: geometry , equilateral
In the acute-angled triangle $ABC$, the points $P$, $N$, $ M$ are the feet of the altitudes drawn from the vertices $C$, $A$, $B$, respectively. The lengths of the projections of the sides $AB$, $BC$, $CA$ on straight lines $MN$, $PM$, $NP$ respectively, are equal to each other. Prove that triangle $ABC$ is regular.

2025 Kyiv City MO Round 2, Problem 3
Tags: number theory , divisor
A positive integer \( n \), which has at least one proper divisor, is divisible by the arithmetic mean of the smallest and largest of its proper divisors (which may coincide). What can be the number of divisors of \( n \)? [i]A proper divisor of a positive integer \( n \) is any of its divisors other than \( 1 \) and \( n \).[/i] [i]Proposed by Mykhailo Shtandenko[/i]

2005 AMC 12/AHSME, 10
Tags: modular arithmetic , algebra , recursion , sequence
The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence? $ \textbf{(A)}\ 29\qquad \textbf{(B)}\ 55\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 133\qquad \textbf{(E)}\ 250$

2013 Math Prize For Girls Problems, 6
Tags: ratio , arithmetic sequence , geometric sequence , complex number , imaginary numbers
Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.

2010 Today's Calculation Of Integral, 579
Tags: calculus , integration , limit , calculus computations
Let $ a$ be a positive real number. Find $ \lim_{n\to\infty} \frac{(n\plus{}1)^a\plus{}(n\plus{}2)^a\plus{}\cdots \plus{}(n\plus{}n)^a}{1^{a}\plus{}2^{a}\plus{}\cdots \plus{}n^{a}}$

2022 Bulgaria National Olympiad, 1
Tags: coloring , parity , combinatorics
A white equilateral triangle $T$ with side length $2022$ is divided into equilateral triangles with side $1$ (cells) by lines parallel to the sides of $T$. We'll call two cells $\textit{adjacent}$ if they have a common vertex. Ivan colours some of the cells in black. Without knowing which cells are black, Peter chooses a set $S$ of cells and Ivan tells him the parity of the number of black cells in $S$. After knowing this, Peter is able to determine the parity of the number of $\textit{adjacent}$ cells of different colours. Find all possible cardinalities of $S$ such that this is always possible independent of how Ivan chooses to colour the cells.

2020/2021 Tournament of Towns, P5
Tags: geometry
Let $O{}$ be the circumcenter of an acute triangle $ABC$. Let $M{}$ be the midpoint of $AC$. The straight line $BO$ intersects the altitudes $AA_1{}$ and $CC_1{}$ at the points $H_a$ and $H_c$ respectively. The circumcircles of the triangles $BH_aA$ and $BH_cC$ have a second point of intersection $K{}$. Prove that $K{}$ lies on the straight line $BM$. [i]Mikhail Evdokimov[/i]

2015 Belarus Team Selection Test, 2
Tags: circumcircle , geometric transformation , reflection , cyclic quadrilateral , geometry
In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.

2016 Sharygin Geometry Olympiad, 3
Tags: geometry , locus , trapezoid , perpendicular
A trapezoid $ABCD$ and a line $\ell$ perpendicular to its bases $AD$ and $BC$ are given. A point $X$ moves along $\ell$. The perpendiculars from $A$ to $BX$ and from $D$ to $CX$ meet at point $Y$ . Find the locus of $Y$ . by D.Prokopenko

2020 ISI Entrance Examination, 7
Tags: number theory
Consider a right-angled triangle with integer-valued sides $a<b<c$ where $a,b,c$ are pairwise co-prime. Let $d=c-b$ . Suppose $d$ divides $a$ . Then [b](a)[/b] Prove that $d\leqslant 2$. [b](b)[/b] Find all such triangles (i.e. all possible triplets $a,b,c$) with perimeter less than $100$ .

1992 Brazil National Olympiad, 5
Tags: number theory
Let $d(n)=\sum_{0<d|n}{1}$. Show that, for any natural $n>1$, \[ \sum_{2 \leq i \leq n}{\frac{1}{i}} \leq \sum{\frac{d(i)}{n}} \leq \sum_{1 \leq i \leq n}{\frac{1}{i}} \]

2006 Korea Junior Math Olympiad, 7
Tags: geometry , tangent , perpendicular , circles
A line through point $P$ outside of circle $O$ meets the said circle at $B,C$ ($PB < PC$). Let $PO$ meet circle $O$ at $Q,D$ (with $PQ < PD$). Let the line passing $Q$ and perpendicular to $BC$ meet circle $O$ at $A$. If $BD^2 = AD\cdot CP$, prove that $PA$ is a tangent to $O$.

1986 Bulgaria National Olympiad, Problem 1
Tags: number theory
Find the smallest natural number $n$ for which the number $n^2-n+11$ has exactly four prime factors (not necessarily distinct).

2010 Spain Mathematical Olympiad, 1
Tags: number theory
A [i]pucelana[/i] sequence is an increasing sequence of $16$ consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with $3$-digit numbers only?

2014 ASDAN Math Tournament, 2
Tags:
Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$.