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

2005 All-Russian Olympiad Regional Round, 8.6
Tags: geometry , angle bisector
In quadrilateral $ABCD$, angles $A$ and $C$ are equal. Angle bisector of $B$ intersects line $AD$ at point $P$. Perpendicular on $BP$ passing through point $A$ intersects line $BC$ at point $Q$. Prove that the lines $PQ$ and $CD$ are parallel.

1999 All-Russian Olympiad Regional Round, 8.3
Tags: inscribed , median , ratio , geometry
On sides $BC$, $CA$, $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are chosen, respectively, so that the medians $A_1A_2$, $B_1B_2$, $C_1C_2$ of the triangle $A_1B_1C_1$ are respectively parallel to straight lines $AB$, $BC$, $CA$. Determine in what ratio points $A_1$, $B_1$, $C_1$ divide the sides of the triangle $ABC$.

2018 Thailand Mathematical Olympiad, 6
Tags: diophantine , number theory , diophantine equation
Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ . a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$. b) Show that $A$ is an infinite set.

2021 Mexico National Olympiad, 6
Tags: number theory , combinatorics , algebra
Determine all non empty sets $C_1, C_2, C_3, \cdots $ such that each one of them has a finite number of elements, all their elements are positive integers, and they satisfy the following property: For any positive integers $n$ and $m$, the number of elements in the set $C_n$ plus the number of elements in the set $C_m$ equals the sum of the elements in the set $C_{m + n}$. [i]Note:[/i] We denote $\lvert C_n \lvert$ the number of elements in the set $C_n$, and $S_k$ as the sum of the elements in the set $C_n$ so the problem's condition is that for every $n$ and $m$: \[\lvert C_n \lvert + \lvert C_m \lvert = S_{n + m}\] is satisfied.

2020 Online Math Open Problems, 24
Tags:
In graph theory, a [i]triangle[/i] is a set of three vertices, every two of which are connected by an edge. For an integer $n \geq 3$, if a graph on $n$ vertices does not contain two triangles that do not share any vertices, let $f(n)$ be the maximum number of triangles it can contain. Compute $f(3) + f(4) + \cdots + f(100).$ [i]Proposed by Edward Wan[/i]

2011 Today's Calculation Of Integral, 732
Tags: trigonometry , parameterization , integration , calculus computations , calculus
Let $a$ be parameter such that $0<a<2\pi$. For $0<x<2\pi$, find the extremum of $F(x)=\int_{x}^{x+a} \sqrt{1-\cos \theta}\ d\theta$.

2000 Harvard-MIT Mathematics Tournament, 47
Tags:
Find an $n<100$ such that $n\cdot 2^n-1$ is prime. Score will be $n-5$ for correct $n$, $5-n$ for incorrect $n$ ($0$ points for answer $<5$)

1952 Moscow Mathematical Olympiad, 232
Tags: polynomial , divide , algebra
Prove that for any integer $a$ the polynomial $3x^{2n}+ax^n+2$ cannot be divided by $2x^{2m}+ax^m+3$ without a remainder.

1996 North Macedonia National Olympiad, 1
Tags: geometry , equal angles , perpendicular , parallelogram
Let $ABCD$ be a parallelogram which is not a rectangle and $E$ be the point in its plane such that $AE \perp AB$ and $CE \perp CB$. Prove that $\angle DEA = \angle CEB$.

2010 AMC 10, 1
Tags:
Mary's top book shelf holds five books with the following widths, in centimeters: $ 6$, $ \frac12$, $ 1$, $ 2.5$, and $ 10$. What is the average book width, in centimeters? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

1997 AMC 12/AHSME, 25
Tags: geometry , parallelogram
Let $ ABCD$ be a parallelogram and let $ \overrightarrow{AA^\prime}$, $ \overrightarrow{BB^\prime}$, $ \overrightarrow{CC^\prime}$, and $ \overrightarrow{DD^\prime}$ be parallel rays in space on the same side of the plane determined by $ ABCD$. If $ AA^\prime \equal{} 10$, $ BB^\prime \equal{} 8$, $ CC^\prime \equal{} 18$, $ DD^\prime \equal{} 22$, and $ M$ and $ N$ are the midpoints of $ \overline{A^{\prime}C^{\prime}}$ and $ \overline{B^{\prime}D^{\prime}}$, respectively, then $ MN \equal{}$ $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 4$

2006 MOP Homework, 6
Tags: parallel , angle , geometry
In triangle $ABC, AB \ne AC$. Circle $\omega$ passes through $A$ and meets sides $AB$ and $AC$ at $M$ and $N$, respectively, and the side $BC$ at $P$ and $Q$ such that $Q$ lies in between $B$ and $P$. Suppose that $MP // AC, NQ // AB$, and $BP \cdot AC = CQ \cdot AB$. Find $\angle BAC$.

2018 BMT Spring, 4
Tags: number theory
What is the remainder when $201820182018... $ [$2018$ times] is divided by $15$?

2008 Sharygin Geometry Olympiad, 13
Tags: geometry , incenter
(A.Myakishev, 9--10) Given triangle $ ABC$. One of its excircles is tangent to the side $ BC$ at point $ A_1$ and to the extensions of two other sides. Another excircle is tangent to side $ AC$ at point $ B_1$. Segments $ AA_1$ and $ BB_1$ meet at point $ N$. Point $ P$ is chosen on the ray $ AA_1$ so that $ AP\equal{}NA_1$. Prove that $ P$ lies on the incircle.

2002 National High School Mathematics League, 6
Tags: rotation , geometry
Consider the area encircled by $x^2=4y,x^2=-4y,x=4,x=-4$, rotate it around $y$-axis, the volume of the revolved body is $V_1$. Then consider the figure formed by all points $(x,y)$ that $x^2+y^2\leq16,x^2+(y-2)^2\geq4,x^2+(y-2)^2\geq4$, rotate it around $y$-axis, the volume of the revolved body is $V_2$. The relationship between $V_1$ and $V_2$ is $\text{(A)}V_1=\frac{1}{2}V_2\qquad\text{(B)}V_1=\frac{2}{3}V_2\qquad\text{(C)}V_1=V_2\qquad\text{(D)}V_1=2V_2$

1977 All Soviet Union Mathematical Olympiad, 240
Tags: combinatorics
There are direct routes from every city of a certain country to every other city. The prices are known in advance. Two tourists (they do not necessary start from one city) have decided to visit all the cities, using only direct travel lines. The first always chooses the cheapest ticket to the city, he has never been before (if there are several -- he chooses arbitrary destination among the cheapests). The second -- the most expensive (they do not return to the first city). Prove that the first will spend not more money for the tickets, than the second.

2006 AIME Problems, 14
Tags: floor function , trigonometry , 3d geometry , pyramid , symmetry , geometry , tetrahedron
A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.)

1989 China Team Selection Test, 3
Tags: algebra , polynomial , combinatorial geometry , gauss , derivative , calculus
Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$

2005 India Regional Mathematical Olympiad, 7
Tags: quadratic
Let $a,b,c$ be three positive real numbers such that $a+ b +c =1$. Let $\lambda = min \{ a^3 + a^2bc , b^3 + b^2 ac , c^3 + ab c^2 \}$ Prove that the roots of $x^2 + x + 4 \lambda = 0$ are real.

1959 AMC 12/AHSME, 16
Tags: quadratic , factoring quadratics , algebra
The expression $\frac{x^2-3x+2}{x^2-5x+6}\div \frac{x^2-5x+4}{x^2-7x+12}$, when simplified is: $ \textbf{(A)}\ \frac{(x-1)(x-6)}{(x-3)(x-4)} \qquad\textbf{(B)}\ \frac{x+3}{x-3}\qquad\textbf{(C)}\ \frac{x+1}{x-1}\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2$

2015 Hanoi Open Mathematics Competitions, 9
Tags: algebra , inequalities
Let $a, b,c$ be positive numbers with $abc = 1$. Prove that $a^3 + b^3 + c^3 + 2[(ab)^3 + (bc)^3 + (ca)^3] \ge 3(a^2b + b^2c + c^2a)$.

2017 Harvard-MIT Mathematics Tournament, 8
Tags: circumcircle , inradius , geometry
Let $ABC$ be a triangle with circumradius $R=17$ and inradius $r=7$. Find the maximum possible value of $\sin \frac{A}{2}$.

1976 Spain Mathematical Olympiad, 1
Tags: construction , square , geometry
In a plane there are four fixed points $A, B, C, D$, no $3$ collinear. Construct a square with sides $a, b, c, d$ such that $A \in a$, $B \in b$, $C \in c$, $D \in d$.

2012 IFYM, Sozopol, 7
Tags: fibonacci sequence , combinatorics
Let $M=\{1,2,...,n\}$. Prove that the number of pairs $(A,a)$, where $A\subset M$ and $a$ is a permutation of $M$, for which $a(A)\cap A=\emptyset $, is equal to $n!.F_{n+1}$, where $F_{n+1}$ is the $n+1$ member of the Fibonacci sequence.

2017 HMNT, 1
Tags: number theory
Find the sum of all positive integers whose largest proper divisor is $55$. (A proper divisor of $n$ is a divisor that is strictly less than $n$.)