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 Belarus Team Selection Test, 4.2
Tags: geometry
Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.

2011 Dutch IMO TST, 5
Tags: geometry , angle bisector , bisects segment
Let $ABC$ be a triangle with $|AB|> |BC|$. Let $D$ be the midpoint of $AC$. Let $E$ be the intersection of the angular bisector of $\angle ABC$ and the line $AC$. Let $F$ be the point on $BE$ such that $CF$ is perpendicular to $BE$. Finally, let $G$ be the intersection of $CF$ and $BD$. Prove that $DF$ divides the line segment $EG$ into two equal parts.

2007 Puerto Rico Team Selection Test, 3
Tags: combinatorics
Five persons of different heights stand next to the another on numbered booths to take a picture. From how many ways can be arranged so that people in positions $ 1$ and $3$ are both taller than the person in the position $2$?

2020 SIME, 10
Tags:
Consider all $2^{20}$ paths of length $20$ units on the coordinate plane starting from point $(0, 0)$ going only up or right, each one unit at a time. Each such path has a unique [i]bubble space[/i], which is the region of points on the coordinate plane at most one unit away from some point on the path. The average area enclosed by the bubble space of each path, over all $2^{20}$ paths, can be written as $\tfrac{m + n\pi}{p}$ where $m, n, p$ are positive integers and $\gcd(m, n, p) = 1$. Find $m + n + p$.

2012 AMC 8, 21
Tags: geometry , 3d geometry
Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet? $\textbf{(A)}\hspace{.05in}5\sqrt2 \qquad \textbf{(B)}\hspace{.05in}10 \qquad \textbf{(C)}\hspace{.05in}10\sqrt2 \qquad \textbf{(D)}\hspace{.05in}50 \qquad \textbf{(E)}\hspace{.05in}50\sqrt2 $

2011 AIME Problems, 3
Tags: arithmetic sequence
The degree measures of the angles of a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.

Russian TST 2017, P2
Tags: geometry , euler line
Let $D$ be the foot of perpendicular from $A$ to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle $ABC$. A circle $\omega$ with centre $S$ passes through $A$ and $D$, and it intersects sides $AB$ and $AC$ at $X$ and $Y$ respectively. Let $P$ be the foot of altitude from $A$ to $BC$, and let $M$ be the midpoint of $BC$. Prove that the circumcentre of triangle $XSY$ is equidistant from $P$ and $M$.

2006 Germany Team Selection Test, 1
Tags: polynomial , number theory , algebra , composite number , prime number
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

TNO 2008 Senior, 2
Tags: number theory
The sequence $a_n$ for $n \in \mathbb{N}$ is defined as follows: \[ a_0 = 6, \quad a_1 = 7, \quad a_{n+2} = 3a_{n+1} - 2a_n \] Find all values of $n$ such that $n^2 = a_n$.

2019 Dürer Math Competition (First Round), P4
Tags: combinatorics , integer , coordinates
Albrecht writes numbers on the points of the first quadrant with integer coordinates in the following way: If at least one of the coordinates of a point is 0, he writes 0; in all other cases the number written on point $(a, b)$ is one greater than the average of the numbers written on points $ (a+1 , b-1) $ and $ (a-1,b+1)$ . Which numbers could he write on point $(121, 212)$? Note: The elements of the first quadrant are points where both of the coordinates are non- negative.

2000 Hungary-Israel Binational, 2
Tags: algebra , binomial theorem , number theory
Prove or disprove: For any positive integer $k$ there exists an integer $n > 1$ such that the binomial coeffcient $\binom{n}{i}$ is divisible by $k$ for any $1 \leq i \leq n-1.$

2003 AMC 12-AHSME, 25
Tags: parabola , algebra , function , inequalities , conic , domain , calculus
Let $ f(x)\equal{}\sqrt{ax^2\plus{}bx}$. For how many real values of $ a$ is there at least one positive value of $ b$ for which the domain of $ f$ and the range of $ f$ are the same set? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{infinitely many}$

2014 Albania Round 2, 5
Tags: geometry
Prove that if the angles $\alpha$ and $\beta$ satisfy $\sin(\alpha + \beta) = 2 \sin \alpha$, Then $$\alpha < \beta$$

1999 Irish Math Olympiad, 2
Tags: number theory , modular arithmetic
Show that there is a positive number in the Fibonacci sequence which is divisible by $ 1000$.

2020 Tuymaada Olympiad, 6
Tags: geometry
$AK$ and $BL$ are altitudes of an acute triangle $ABC$. Point $P$ is chosen on the segment $AK$ so that $LK=LP$. The parallel to $BC$ through $P$ meets the parallel to $PL$ through $B$ at point $Q$. Prove that $\angle AQB = \angle ACB$. [i](S. Berlov)[/i]

2019 Junior Balkan Team Selection Tests - Romania, 4
Tags: combinatorics , parity
Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] if the number of white tokens on its left plus the number of black tokens on its right is $n$. Determine whether the number of [i]balanced[/i] tokens is even or odd.

1984 Tournament Of Towns, (063) O4
Tags: combinatorial geometry , rectangle , geometry , combinatorics , function , graph
Prove that, for any natural number $n$, the graph of any increasing function $f : [0,1] \to [0, 1]$ can be covered by $n$ rectangles each of area whose sides are parallel to the coordinate axes. Assume that a rectangle includes both its interior and boundary points. (a) Assume that $f(x)$ is continuous on $[0,1]$. (b) Do not assume that $f(x)$ is continuous on $[0,1]$. (A Andjans, Riga) PS. (a) for O Level, (b) for A Level

1989 All Soviet Union Mathematical Olympiad, 496
Tags: geometry , perimeter , inequalities , geometric inequality
A triangle with perimeter $1$ has side lengths $a, b, c$. Show that $a^2 + b^2 + c^2 + 4abc <\frac 12$.

2018 Centroamerican and Caribbean Math Olympiad, 5
Tags: polynomial , algebra
Let $n$ be a positive integer, $1<n<2018$. For each $i=1, 2, \ldots ,n$ we define the polynomial $S_i(x)=x^2-2018x+l_i$, where $l_1, l_2, \ldots, l_n$ are distinct positive integers. If the polynomial $S_1(x)+S_2(x)+\cdots+S_n(x)$ has at least an integer root, prove that at least one of the $l_i$ is greater or equal than $2018$.

2008 Iran MO (3rd Round), 4
Tags: combinatorics , function
Let $ S$ be a sequence that: \[ \left\{ \begin{array}{cc} S_0\equal{}0\hfill\\ S_1\equal{}1\hfill\\ S_n\equal{}S_{n\minus{}1}\plus{}S_{n\minus{}2}\plus{}F_n& (n>1) \end{array} \right.\] such that $ F_n$ is Fibonacci sequence such that $ F_1\equal{}F_2\equal{}1$. Find $ S_n$ in terms of Fibonacci numbers.

1970 Bulgaria National Olympiad, Problem 5
Tags: inequalities , geometry , geometric inequalities , polygon
Prove that for $n\ge5$ the side of regular inscribable $n$-gon is bigger than the side of regular $n+1$-gon circumscribed around the same circle and if $n\le4$ the opposite statement is true.

2020-2021 OMMC, 8
Tags: geometry
Let triangle $MAD$ be inscribed in circle $O$ with diameter $85$ such that $MA = 68$ and $DA = 40$. The altitudes from $M, D$ to sides $AD$ and $MA$, respectively, intersect the tangent to circle $O$ at $A$ at $X$ and $Y$ respectively. $XA \times YA$ can be expressed as $\frac{a}{b}$, where $a$ and $ b$ are relatively prime positive integers. Find $a + b$.

2017 Latvia Baltic Way TST, 14
Tags: diophantine , number theory , diophantine equation
Can you find three natural numbers $a, b, c$ whose greatest common divisor is $1$ and which satisfy the equality $$ab + bc + ac = (a + b -c)(b + c - a)(c + a - b) ?$$

2022 IFYM, Sozopol, 3
Tags: geometric inequality , geometry , tangential quadrilateral , tangential
Quadrilateral $ABCD$ is circumscribed around circle $k$. Gind the smallest possible value of $$\frac{AB + BC + CD + DA}{AC + BD}$$, as well as all quadrilaterals with the above property where it is reached.

2022 CMIMC, 13
Tags: team
Let $F_n$ denote the $n$th Fibonacci number, with $F_0=0, F_1=1$ and $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. There exists a unique two digit prime $p$ such that for all $n$, $p | F_{n+100} + F_n$. Find $p$. [i]Proposed by Sam Rosenstrauch[/i]