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

2017 Caucasus Mathematical Olympiad, 6
Tags: geometry
A triangle is cut by $3$ cevians from its $3$ vertices into $7$ pieces: $4$ triangles and $3$ quadrilaterals. Determine if it is possible that all $3$ quadrilaterals are inscribed.

2014 BMT Spring, 7
Tags: algebra
If $f(x, y) = 3x^2 + 3xy + 1$ and $f(a, b) + 1 = f(b, a) = 42$, then determine $|a + b|$.

1989 Putnam, A1
Tags: algebra , difference of squares , special factorizations
How many base ten integers of the form 1010101...101 are prime?

2017 Romania National Olympiad, 3
Tags: linear algebra , matrix
Let be a natural number $ n\ge 2 $ and two $ n\times n $ complex matrices $ A,B $ that satisfy $ (AB)^3=O_n. $ Does this imply that $ (BA)^3=O_n ? $

2007 Today's Calculation Of Integral, 180
Tags: calculus , integration , geometry , trigonometry , limit , calculus computations
Let $a_{n}$ be the area surrounded by the curves $y=e^{-x}$ and the part of $y=e^{-x}|\cos x|,\ (n-1)\pi \leq x\leq n\pi \ (n=1,\ 2,\ 3,\ \cdots).$ Evaluate $\lim_{n\to\infty}(a_{1}+a_{2}+\cdots+a_{n}).$

2021 Austrian MO Beginners' Competition, 3
Tags: combinatorics
The eight points $A, B,. . ., G$ and $H$ lie on five circles as shown. Each of these letters are represented by one of the eight numbers $1, 2,. . ., 7$ and $ 8$ replaced so that the following conditions are met: (i) Each of the eight numbers is used exactly once. (ii) The sum of the numbers on each of the five circles is the same. How many ways are there to get the letters substituted through the numbers in this way? (Walther Janous) [img]https://cdn.artofproblemsolving.com/attachments/5/e/511cdd2fc31e8067f400369c4fe9cf964ef54c.png[/img]

2009 China Western Mathematical Olympiad, 2
Tags: combinatorics
Given an integer $n\ge\ 3$, find the least positive integer $k$, such that there exists a set $A$ with $k$ elements, and $n$ distinct reals $x_{1},x_{2},\ldots,x_{n}$ such that $x_{1}+x_{2}, x_{2}+x_{3},\ldots, x_{n-1}+x_{n}, x_{n}+x_{1}$ all belong to $A$.

1977 IMO Longlists, 59
Tags: analytic geometry , combinatorics , point set , centroid
Let $E$ be a set of $n$ points in the plane $(n \geq 3)$ whose coordinates are integers such that any three points from $E$ are vertices of a nondegenerate triangle whose centroid doesnt have both coordinates integers. Determine the maximal $n.$

2000 Harvard-MIT Mathematics Tournament, 1
Tags:
Find all integer solutions to $m^2=n^6+1$.

2024 ELMO Shortlist, G5
Tags: geometry
Let $ABC$ be a triangle with circumcenter $O$ and circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be points on the circumcircles of triangles $AOB$ and $AOC$, respectively, such that $A$, $P$, and $Q$ are collinear. Prove that if the circumcircle of triangle $OPQ$ is tangent to $\omega$ at $T$, then $\angle BTD=\angle CAP$. [i]Tiger Zhang[/i]