This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 85335

2019 Harvard-MIT Mathematics Tournament, 7
Tags: algebra , hmmt , summation
Find the value of \[\sum_{a = 1}^{\infty} \sum_{b = 1}^{\infty} \sum_{c = 1}^{\infty} \frac{ab(3a + c)}{4^{a+b+c} (a+b)(b+c)(c+a)}.\]

2001 AMC 10, 1
Tags:
The median of the list \[ n, n \plus{} 3, n \plus{} 4, n \plus{} 5, n \plus{} 6, n \plus{} 8, n \plus{} 10, n \plus{} 12, n \plus{} 15 \]is $ 10$. What is the mean? $ \textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }10\qquad\textbf{(E) }11$

2015 Switzerland Team Selection Test, 2
Tags: polynomial , algebra , binomial theorem , binomial coefficient , inequalities
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

1994 Iran MO (2nd round), 3
Tags: functional equation , function , induction
Find all functions $ f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}$ such that for all $ x,y \in \mathbb{Z}\setminus\{0\}$: \[ f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}\]

2016 Portugal MO, 4
Tags: geometry , equal angles , tangent , angle , parallelogram
Let $[ABCD]$ be a parallelogram with $AB <BC$ and let $E, F$ be points on the circle that passes through $A, B$ and $C$ such that $DE$ and $DF$ are tangents to this circle. Knowing that $\angle ADE = \angle CDF$ , determine $\angle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/5/e/4140b92730e9d382df49ac05ca4e8ba48332dc.png[/img]

1965 AMC 12/AHSME, 25
Tags:
Let $ ABCD$ be a quadrilateral with $ AB$ extended to $ E$ so that $ \overline{AB} \equal{} \overline{BE}$. Lines $ AC$ and $ CE$ are drawn to form angle $ ACE$. For this angle to be a right angle it is necessary that quadrilateral $ ABCD$ have: $ \textbf{(A)}\ \text{all angles equal}$ $ \textbf{(B)}\ \text{all sides equal}$ $ \textbf{(C)}\ \text{two pairs of equal sides}$ $ \textbf{(D)}\ \text{one pair of equal sides}$ $ \textbf{(E)}\ \text{one pair of equal angles}$

2025 Harvard-MIT Mathematics Tournament, 3
Tags: geometry
Point $P$ lies inside square $ABCD$ such that the areas of $\triangle{PAB}, \triangle{PBC}, \triangle{PCD},$ and $\triangle{PDA}$ are $1, 2, 3,$ and $4,$ in some order. Compute $PA \cdot PB \cdot PC \cdot PD.$

2011 Dutch IMO TST, 4
Tags: number theory , recurrence relation , sequence , prime number
Prove that there exists no in nite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$: $p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.

2021 Thailand TSTST, 3
Tags: sequence , number theory
A finite sequence of integers $a_0,,a_1,\dots,a_n$ is called [i]quadratic[/i] if for each $i\in\{1,2,\dots n\}$ we have the equality $|a_i-a_{i-1}|=i^2$. $\text{(i)}$ Prove that for any two integers $b$ and $c$, there exist a positive integer $n$ and a quadratic sequence with $a_0=b$ and $a_n = c$. $\text{(ii)}$ Find the smallest positive integer $n$ for which there exists a quadratic sequence with $a_0=0$ and $a_n=2021$.

2023-24 IOQM India, 22
Tags:
In an equilateral triangle of side length 6 , pegs are placed at the vertices and also evenly along each side at a distance of 1 from each other. Four distinct pegs are chosen from the 15 interior pegs on the sides (that is, the chosen ones are not vertices of the triangle) and each peg is joined to the respective opposite vertex by a line segment. If $N$ denotes the number of ways we can choose the pegs such that the drawn line segments divide the interior of the triangle into exactly nine regions, find the sum of the squares of the digits of $N$.

2009 Today's Calculation Of Integral, 411
Tags: geometry , calculus , integration , calculus computations
Find the area bounded by $ y\equal{}x^2\minus{}|x^2\minus{}1|\plus{}|2|x|\minus{}2|\plus{}2|x|\minus{}7$ and the $ x$ axis.

1990 IberoAmerican, 3
Tags: polynomial , number theory , algebra
Let $b$, $c$ be integer numbers, and define $f(x)=(x+b)^2-c$. i) If $p$ is a prime number such that $c$ is divisible by $p$ but not by $p^{2}$, show that for every integer $n$, $f(n)$ is not divisible by $p^{2}$. ii) Let $q \neq 2$ be a prime divisor of $c$. If $q$ divides $f(n)$ for some integer $n$, show that for every integer $r$ there exists an integer $n'$ such that $f(n')$ is divisible by $qr$.

2017 Sharygin Geometry Olympiad, P17
Tags: geometry , acute triangle , construction
Using a compass and a ruler, construct a point $K$ inside an acute-angled triangle $ABC$ so that $\angle KBA = 2\angle KAB$ and $ \angle KBC = 2\angle KCB$.

1999 All-Russian Olympiad, 4
Tags: combinatorics , search
Initially numbers from 1 to 1000000 are all colored black. A move consists of picking one number, then change the color (black to white or white to black) of itself and all other numbers NOT coprime with the chosen number. Can all numbers become white after finite numbers of moves? Edited by pbornsztein

2016 BMT Spring, 6
Tags: algebra , calculus
Amy is traveling on the $xy$-plane in a spaceship where her motion is described by the following equation $xe^y = ye^x$. Given that her $x$-component of velocity is a constant $3$ mph , the magnitude of her velocity as she approaches $(1,-1)$ can be expressed as $\sqrt{\frac{a + be^4}{ c + de^2}}$ . Find $\frac{ac}{bd}$ . (You may assume that the initial conditions do allow her to approach $(1,-1)$)

2010 Bosnia and Herzegovina Junior BMO TST, 4
Tags: circle inversion , clockwise , combinatorics
On circle in clockwise order are written positive integers from $1$ to $2010$. Let us cross out number $1$, then number $10$, then number $19$, and so on every $9$th number in that direction. Which number will be first crossed out twice? How many numbers at that moment are not crossed out?

2011 IMAC Arhimede, 3
Tags: combinatorics
Place $n$ points on a circle and draw all possible chord joining these points. If no three chord are concurent, find the number of disjoint regions created. [color=#008000]Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=260926&hilit=circle+points+segments+regions[/color]

Mathematical Minds 2024, P3
Tags: olympiad combinatorics , board , combinatorics
On the screen of a computer there is an $2^n\times 2^n$ board. On each cell of the main diagonal there is a file. At each step, we may select some files and move them to the left, on their respective rows, by the same distance. What is the minimum number of necessary moves in order to put all files on the first column? [i]Proposed by Vlad Spătaru[/i]

I Soros Olympiad 1994-95 (Rus + Ukr), 11.1
Tags: algebra , inequalities
Prove that for real $x\ge 1$, holds the inequality $$\frac{2^x +3^x }{3^x +4^x} \le \frac57$$

PEN A Problems, 62
Tags: divisibility theory
Let $p(n)$ be the greatest odd divisor of $n$. Prove that \[\frac{1}{2^{n}}\sum_{k=1}^{2^{n}}\frac{p(k)}{k}> \frac{2}{3}.\]

2018 Online Math Open Problems, 1
Tags:
Farmer James has three types of cows on his farm. A cow with zero legs is called a $\textit{ground beef}$, a cow with one leg is called a $\textit{steak}$, and a cow with two legs is called a $\textit{lean beef}$. Farmer James counts a total of $20$ cows and $18$ legs on his farm. How many more $\textit{ground beef}$s than $\textit{lean beef}$s does Farmer James have? [i]Proposed by James Lin[/i]

1997 IMO Shortlist, 5
Tags: geometry , 3d geometry , triangle , tetrahedron
Let $ ABCD$ be a regular tetrahedron and $ M,N$ distinct points in the planes $ ABC$ and $ ADC$ respectively. Show that the segments $ MN,BN,MD$ are the sides of a triangle.

2019 Dutch IMO TST, 4
Tags: functional equation , find all functions , prime number , function , number theory
Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

2008 South East Mathematical Olympiad, 1
Tags: inequalities
Let $\lambda$ be a positive real number. Inequality $|\lambda xy+yz|\le \dfrac{\sqrt5}{2}$ holds for arbitrary real numbers $x, y, z$ satisfying $x^2+y^2+z^2=1$. Find the maximal value of $\lambda$.

2001 District Olympiad, 1
Tags: arithmetic sequence , algebra , induction
Let $(a_n)_{n\ge 1}$ be a sequence of real numbers such that \[a_1\binom{n}{1}+a_2\binom{n}{2}+\ldots+a_n\binom{n}{n}=2^{n-1}a_n,\ (\forall)n\in \mathbb{N}^*\] Prove that $(a_n)_{n\ge 1}$ is an arithmetical progression. [i]Lucian Dragomir[/i]