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.

AND:
OR:
NO:

Found problems: 85335

2022 Romania Team Selection Test, 3
Tags: shortlist , geometry , collinearity
Let $ABC$ be an acute triangle such that $AB < AC$. Let $\omega$ be the circumcircle of $ABC$ and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second time at $Y$, show that $A, Y$, and $M$ are collinear. [i]Proposed by Nikola Velov, North Macedonia[/i]

2013 Swedish Mathematical Competition, 2
Tags: geometry , paper , folding , nonagon , square
The paper folding art origami is usually performed with square sheets of paper. Someone folds the sheet once along a line through the center of the sheet in orde to get a nonagon. Let $p$ be the perimeter of the nonagon minus the length of the fold, i.e. the total length of the eight sides that are not folds, and denote by s the original side length of the square. Express the area of the nonagon in terms of $p$ and $s$.

2005 MOP Homework, 3
Tags: inequalities
Let $a$, $b$, $c$ be real numbers. Prove that \begin{align*}&\quad\,\,\sqrt{2(a^2+b^2)}+\sqrt{2(b^2+c^2)}+\sqrt{2(c^2+a^2)}\\&\ge \sqrt{3[(a+b)^2+(b+c)^2+(c+a)^2]}.\end{align*}

Kyiv City MO Juniors Round2 2010+ geometry, 2013.7.3
Tags: geometry , square , angle chasing , equal angles , angle
In the square $ABCD$ on the sides $AD$ and $DC$, the points $M$ and $N$ are selected so that $\angle BMA = \angle NMD = 60 { } ^ \circ $. Find the value of the angle $MBN$.

2003 Romania National Olympiad, 2
Tags: complex number , absolute value , algebra , geometry , pentagon
Let be five nonzero complex numbers having the same absolute value and such that zero is equal to their sum, which is equal to the sum of their squares. Prove that the affixes of these numbers in the complex plane form a regular pentagon. [i]Daniel Jinga[/i]

2025 Belarusian National Olympiad, 8.7
Tags: parabola , algebra
Yan and Kirill play a game. At first Kirill says 4 numbers $x_1<x_2<x_3<x_4$, and then Yan says three pairwise different non zero numbers $a_1$, $a_2$ and $a_3$. For all $i$ from $1$ to $3$ they consider the quadratic trinomial $f_i(x)$ which has roots $x_i$ and $x_{i+1}$ and leading coefficient $a_i$, and construct on the plane the graphs of that trinomials. Yan wins if in every pair $(f_1(x),f_2(x))$ and $(f_2(x),f_3(x))$ their graphs intersect at exactly one point, and if in some pair graphs do not intersect or intersect at more than one point Kirill wins. Find which player can guarantee his win regardless of the actions of his opponent. [i]V. Kamianetski[/i]

1993 All-Russian Olympiad, 4
Tags: induction , function , algebra , polynomial , ring theory , vector , binomial theorem
If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' \equal{} \frac {a_k \plus{} a_{k \plus{} 1}}2$ the [i]average sequence[/i] of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - [i]average sequence[/i] of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called [i]Good[/i]. Prove that if $ \{x_k\}$ is a [i]good[/i] sequence, then $ \{x_k^2\}$ is also [i]good[/i].

2011 Sharygin Geometry Olympiad, 18
Tags: combinatorial geometry , line , dissecting plane , geometry
On the plane, given are $n$ lines in general position, i.e. any two of them aren’t parallel and any three of them don’t concur. These lines divide the plane into several parts. What is a) the minimal, b) the maximal number of these parts that can be angles?

2016 Federal Competition For Advanced Students, P2, 3
Tags: combinatorics , chessboard
Consider arrangements of the numbers $1$ through $64$ on the squares of an $8\times 8$ chess board, where each square contains exactly one number and each number appears exactly once. A number in such an arrangement is called super-plus-good, if it is the largest number in its row and at the same time the smallest number in its column. Prove or disprove each of the following statements: (a) Each such arrangement contains at least one super-plus-good number. (b) Each such arrangement contains at most one super-plus-good number. Proposed by Gerhard J. Woeginger

2015 NIMO Summer Contest, 1
Tags: function
For all real numbers $a$ and $b$, let \[a\Join b=\dfrac{a+b}{a-b}.\] Compute $1008\Join 1007$. [i] Proposed by David Altizio [/i]

1997 Bulgaria National Olympiad, 2
Tags: ratio , circumcircle , geometry
Given a triangle $ABC$. Let $M$ and $N$ be the points where the angle bisectors of the angles $ABC$ and $BCA$ intersect the sides $CA$ and $AB$, respectively. Let $D$ be the point where the ray $MN$ intersects the circumcircle of triangle $ABC$. Prove that $\frac{1}{BD}=\frac{1}{AD}+\frac{1}{CD}$.

2013 Oral Moscow Geometry Olympiad, 4
Tags: diagonal , geometry , similar triangles , perpendicular , equal segments
Similar triangles $ABM, CBP, CDL$ and $ADK$ are built on the sides of the quadrilateral $ABCD$ with perpendicular diagonals in the outer side (the neighboring ones are oriented differently). Prove that $PK = ML$.

2021 CCA Math Bonanza, L5.1
Tags:
Estimate the number of distinct submissions to this problem. Your submission must be a positive integer less than or equal to $50$. If you submit $E$, and the actual number of distinct submissions is $D$, you will receive a score of $\frac{2}{0.5|E-D|+1}$. [i]2021 CCA Math Bonanza Lightning Round #5.1[/i]

2023 Brazil Team Selection Test, 2
Tags: number theory
Find all integers $n$ satisfying $n \geq 2$ and $\dfrac{\sigma(n)}{p(n)-1} = n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$.

2018 China Western Mathematical Olympiad, 2
Tags: algebra , inequalities , fractional part , n-variable inequality
Let $n \geq 2$ be an integer. Positive reals $x_1, x_2, \cdots, x_n$ satisfy $x_1x_2 \cdots x_n = 1$. Show: $$\{x_1\} + \{x_2\} + \cdots + \{x_n\} < \frac{2n-1}{2}$$ Where $\{x\}$ denotes the fractional part of $x$.

2024 Macedonian TST, Problem 5
Tags: combinatorics
Let \(P\) be a convex polyhedron with the following properties: [b]1)[/b] \(P\) has exactly \(666\) edges. [b]2)[/b] The degrees of all vertices of \(P\) differ by at most \(1\). [b]3)[/b] There is an edge‐coloring of \(P\) with \(k\) colors such that for each color \(c\) and any two distinct vertices \(V_1,V_2\), there exists a path from \(V_1\) to \(V_2\) all of whose edges have color \(c\). Determine the largest positive integer \(k\) for which such a polyhedron \(P\) exists.

JBMO Geometry Collection, 2006
Tags: geometry , incenter
The triangle $ABC$ is isosceles with $AB=AC$, and $\angle{BAC}<60^{\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\angle{ABD}\equiv\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\angle{BDC}$ and $\angle{ACB}$. Compute $\angle{COD}$.

PEN D Problems, 19
Tags: modular arithmetic , congruence
Let $a_{1}$, $\cdots$, $a_{k}$ and $m_{1}$, $\cdots$, $m_{k}$ be integers with $2 \le m_{1}$ and $2m_{i}\le m_{i+1}$ for $1 \le i \le k-1$. Show that there are infinitely many integers $x$ which do not satisfy any of congruences \[x \equiv a_{1}\; \pmod{m_{1}}, x \equiv a_{2}\; \pmod{m_{2}}, \cdots, x \equiv a_{k}\; \pmod{m_{k}}.\]

2003 Tuymaada Olympiad, 3
Tags: combinatorics
Alphabet $A$ contains $n$ letters. $S$ is a set of words of finite length composed of letters of $A$. It is known that every infinite sequence of letters of $A$ begins with one and only one word of $S$. Prove that the set $S$ is finite. [i]Proposed by F. Bakharev[/i]

1999 Bundeswettbewerb Mathematik, 1
Tags: combinatorial geometry , combinatorics
The vertices of a regular $2n$-gon (with $n > 2$ an integer) are labelled with the numbers $1,2,...,2n$ in some order. Assume that the sum of the labels at any two adjacent vertices equals the sum of the labels at the two diametrically opposite vertices. Prove that this is possible if and only if $n$ is odd.

2005 iTest, 40
Tags: number theory
$ITEST + AHSIMC = 6666CS$. Each letter represents a unique digit from $0$ to $9$. How many solutions of the form $(C,A,S,H)$ exist?

2004 India IMO Training Camp, 2
Tags: induction , number theory
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

2003 District Olympiad, 3
Tags: combinatorics
Consider an array $n \times n$ ($n\ge 2$) with $n^2$ integers. In how many ways one can complete the array if the product of the numbers on any row and column is $5$ or $-5$?

2003 AMC 8, 2
Tags:
Which of the following numbers has the smallest prime factor? $\textbf{(A)}\ 55 \qquad \textbf{(B)}\ 57 \qquad \textbf{(C)}\ 58 \qquad \textbf{(D)}\ 59\qquad \textbf{(E)}\ 61$

2007 Thailand Mathematical Olympiad, 5
Tags: geometry , equal segments
A triangle $\vartriangle ABC$ has $\angle A = 90^o$, and a point $D$ is chosen on $AC$. Point $F$ is the foot of altitude from $A$ to $BC$. Suppose that $BD = DC = CF = 2$. Compute $AC$.