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

2015 BMT Spring, 3
Tags: algebra , system of equations , number theory
Find all integer solutions to \begin{align*} x^2+2y^2+3z^2&=36,\\ 3x^2+2y^2+z^2&=84,\\ xy+xz+yz&=-7. \end{align*}

2024 HMNT, 26
Tags: guts
A right rectangular prism of silly powder has dimensions $20 \times 24 \times 25.$ Jerry the wizard applies $10$ bouts of highdroxylation to the box, each of which increases one dimension of the silly powder by $1$ and decreases a different dimension of the silly powder by $1,$ with every possible choice of dimensions equally likely to be chosen and independent of all previous choices. Compute the expected volume of the silly powder after Jerry’s routine.

1981 Miklós Schweitzer, 10
Tags: probability , real analysis , function , integration , complex number , probability and stats
Let $ P$ be a probability distribution defined on the Borel sets of the real line. Suppose that $ P$ is symmetric with respect to the origin, absolutely continuous with respect to the Lebesgue measure, and its density function $ p$ is zero outside the interval $ [\minus{}1,1]$ and inside this interval it is between the positive numbers $ c$ and $ d$ ($ c < d$). Prove that there is no distribution whose convolution square equals $ P$. [i]T. F. Mori, G. J. Szekely[/i]

1979 Austrian-Polish Competition, 8
Tags: midpoint , geometry , 3d geometry
Let $A,B,C,D$ be four points in space, and $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively. Show that $$AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4MN^2$$

2013 VJIMC, Problem 4
Tags: summation , algebra , binomial
Let $n$ and $k$ be positive integers. Evaluate the following sum $$\sum_{j=0}^k\binom kj^2\binom{n+2k-j}{2k}$$where $\binom nk=\frac{n!}{k!(n-k)!}$.

2021 Harvard-MIT Mathematics Tournament., 8
Tags: geometry
Two circles with radii $71$ and $100$ are externally tangent. Compute the largest possible area of a right triangle whose vertices are each on at least one of the circles.

1997 Akdeniz University MO, 5
Tags: geometry , perimeter , incenter
An $ABC$ triangle divide by a $d$ line such that, new two pieces' areas and perimeters are equal. Prove that $ABC$'s incenter lies $d$

2017 ASDAN Math Tournament, 1
Tags:
In $\triangle ABC$, we have $\angle ABC=20^\circ$. In addition, $D$ is drawn on $\overline{AB}$ such that $AC=CD=BD$. Compute $\angle ACD$ in degrees.

2023 Bulgaria National Olympiad, 3
Tags: algebra , polynomial , abstract algebra , inequalities
Let $f(x)$ be a polynomial with positive integer coefficients. For every $n\in\mathbb{N}$, let $a_{1}^{(n)}, a_{2}^{(n)}, \dots , a_{n}^{(n)}$ be fixed positive integers that give pairwise different residues modulo $n$ and let \[g(n) = \sum\limits_{i=1}^{n} f(a_{i}^{(n)}) = f(a_{1}^{(n)}) + f(a_{2}^{(n)}) + \dots + f(a_{n}^{(n)})\] Prove that there exists a constant $M$ such that for all integers $m>M$ we have $\gcd(m, g(m))>2023^{2023}$.

2010 Math Prize For Girls Problems, 10
Tags: analytic geometry , geometry , geometric transformation , congruent triangles
The triangle $ABC$ lies on the coordinate plane. The midpoint of $\overline{AB}$ has coordinates $(-16, -63)$, the midpoint of $\overline{AC}$ has coordinates $(13, 50)$, and the midpoint of $\overline{BC}$ has coordinates $(6, -85)$. What are the coordinates of point $A$?

MathLinks Contest 4th, 4.3
Tags: combinatorics
Given is a graph $G$. An [i]empty [/i] subgraph of $G$ is a subgraph of $G$ with no edges between its vertices. An edge of $G$ is called [i]important [/i] if and only if the removal of this edge will increase the size of the maximal empty subgraph. Suppose that two important edges in $G$ have a common endpoint. Prove there exists a cycle of odd length in $G$.

2016 Taiwan TST Round 1, 3
Tags: binomial coefficient , function , algebra , number theory , surjective function
Let $\mathbb{Z}^+$ denote the set of all positive integers. Find all surjective functions $f:\mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ that satisfy all of the following conditions: for all $a,b,c \in \mathbb{Z}^+$, (i)$f(a,b) \leq a+b$; (ii)$f(a,f(b,c))=f(f(a,b),c)$ (iii)Both $\binom{f(a,b)}{a}$ and $\binom{f(a,b)}{b}$ are odd numbers.(where $\binom{n}{k}$ denotes the binomial coefficients)

2015 ASDAN Math Tournament, 1
Tags: algebra test
Given that $xy+x+y=5$ and $x+1=2$, compute $y+1$.

2024 Thailand Mathematical Olympiad, 4
Tags: combinatorics
In a table with $88$ rows and $253$ columns, each cell is colored either purple or yellow. Suppose that for each yellow cell $c$, $$x(c)y(c)\geq184.$$ Where $x(c)$ is the number of purple cells that lie in the same row as $c$, and $y(c)$ is the number of purple cells that lie in the same column as $c$.\\ Find the least possible number of cells that are colored purple.

2018 AMC 12/AHSME, 25
Tags:
For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$? $\textbf{(A)} \text{ 12} \qquad \textbf{(B)} \text{ 14} \qquad \textbf{(C)} \text{ 16} \qquad \textbf{(D)} \text{ 18} \qquad \textbf{(E)} \text{ 20}$

2013 China Team Selection Test, 2
Tags: circumcircle , geometric transformation , homothety , trigonometry , euler , geometry
Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

2024 Durer Math Competition Finals, 2
Tags: combinatorics , graph theory
One quadrant of the Cartesian coordinate system is tiled with $1\times 2$ dominoes. The dominoes don’t overlap with each other, they cover the entire quadrant and they all fit in the quadrant. Farringdon the flea is initially sitting at the origin and is allowed to jump from one corner of a domino to the opposite corner any number of times. Is it possible that the dominoes are arranged in such a way that Farringdon is unable to move to a distance greater than 2023 from the origin?

1998 All-Russian Olympiad, 2
Tags: trigonometry , inequalities , geometry , rectangle , function , trig identities
Two polygons are given on the plane. Assume that the distance between any two vertices of the same polygon is at most 1, and that the distance between any two vertices of different polygons is at least $ 1/\sqrt{2}$. Prove that these two polygons have no common interior points. By the way, can two sides of a polygon intersect?

I Soros Olympiad 1994-95 (Rus + Ukr), 9.10
Tags: number theory , multiple , digit
For which natural $n$ there exists a natural number multiple of $n$, whose decimal notation consists only of the digits $8$ and $9$ (possibly only from numbers $8$ or only from numbers $9$)?

2010 Turkey Team Selection Test, 3
Tags: analytic geometry , function , combinatorics
Let $\Lambda$ be the set of points in the plane whose coordinates are integers and let $F$ be the collection of all functions from $\Lambda$ to $\{1,-1\}.$ We call a function $f$ in $F$ [i]perfect[/i] if every function $g$ in $F$ that differs from $f$ at finitely many points satisfies the condition \[ \sum_{0<d(P,Q)<2010} \frac{f(P)f(Q)-g(P)g(Q)}{d(P,Q)} \geq 0 \] where $d(P,Q)$ denotes the distance between $P$ and $Q.$ Show that there exist infinitely many [i]perfect[/i] functions that are not translates of each other.

2015 Romania National Olympiad, 1
Tags: function , real analysis
Find all differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the conditions: $ \text{(i)}\quad\forall x\in\mathbb{Z} \quad f'(x) =0 $ $ \text{(ii)}\quad\forall x\in\mathbb{R}\quad f'(x)=0\implies f(x)=0 $

2013 Purple Comet Problems, 7
Tags: modular arithmetic
Find the least six-digit palindrome that is a multiple of $45$. Note that a palindrome is a number that reads the same forward and backwards such as $1441$ or $35253$.

1981 All Soviet Union Mathematical Olympiad, 320
Tags: convex polygon , perimeter , geometry , geometric inequality , inequalities
A pupil has tried to make a copy of a convex polygon, drawn inside the unit circle. He draw one side, from its end -- another, and so on. Having finished, he has noticed that the first and the last vertices do not coincide, but are situated $d$ units of length far from each other. The pupil draw angles precisely, but made relative error less than $p$ in the lengths of sides. Prove that $d < 4p$.

2024 BMT, 9
Tags: geometry
Let $\triangle{ABC}$ be a triangle with incenter $I,$ and let $M$ be the midpoint of $\overline{BC}.$ Line $AM$ intersects the circumcircle of triangle $\triangle{IBC}$ at points $P$ and $Q.$ Suppose that $AP=13, AQ=83,$ and $BC=56.$ Find the perimeter of $\triangle{ABC}.$

2005 Olympic Revenge, 1
Tags: combinatorics
Let $S=\{1,2,3,\ldots,n\}$, $n$ an odd number. Find the parity of number of permutations $\sigma : S \Rightarrow S$ such that the sequence defined by \[a(i)=|\sigma(i)-i|\] is monotonous.