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

2011 China Second Round Olympiad, 1
Tags: circumcircle , symmetry , cyclic quadrilateral , geometry
Let $P,Q$ be the midpoints of diagonals $AC,BD$ in cyclic quadrilateral $ABCD$. If $\angle BPA=\angle DPA$, prove that $\angle AQB=\angle CQB$.

2023 District Olympiad, P2
Tags: algebra , floor function
[list=a] [*]Determine all real numbers $x{}$ satisfying $\lfloor x\rfloor^2-x=-0.99$. [*]Prove that if $a\leqslant -1$, the equation $\lfloor x\rfloor^2-x=a$ does not have real solutions. [/list]

2020 MOAA, TO3
Tags: number theory , theme
Consider the addition $\begin{tabular}{cccc} & O & N & E \\ + & T & W & O \\ \hline F & O & U & R \\ \end{tabular}$ where different letters represent different nonzero digits. What is the smallest possible value of the four-digit number $FOUR$?

2018 AMC 8, 23
Tags: probability
From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon? [asy] size(3cm); pair A[]; for (int i=0; i<9; ++i) { A[i] = rotate(22.5+45*i)*(1,0); } filldraw(A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--A[6]--A[7]--cycle,gray,black); for (int i=0; i<8; ++i) { dot(A[i]); } [/asy] $\textbf{(A) } \frac{2}{7} \qquad \textbf{(B) } \frac{5}{42} \qquad \textbf{(C) } \frac{11}{14} \qquad \textbf{(D) } \frac{5}{7} \qquad \textbf{(E) } \frac{6}{7}$

1999 Israel Grosman Mathematical Olympiad, 2
Tags: inequalities , algebra
Find the smallest positive integer $n$ for which $0 <\sqrt[4]{n}- [\sqrt[4]{n}]< 10^{-5}$ .

2007 Today's Calculation Of Integral, 233
Tags: calculus , integration , trigonometry , derivative , calculus computations
Find the minimum value of the following definite integral. $ \int_0^{\pi} (a\sin x \plus{} b\sin 3x \minus{} 1)^2\ dx.$

2014 Thailand TSTST, 1
Tags: function , algebra
Find all functions $f: {\mathbb{R^\plus{}}}\to{\mathbb{R^\plus{}}}$ such that \[ f(1\plus{}xf(y))\equal{}yf(x\plus{}y)\] for all $x,y\in\mathbb{R^\plus{}}$.

2011 Gheorghe Vranceanu, 1
Tags: abstract algebra , group theory , fucntions
[b]a)[/b] Let $ B,A $ be two subsets of a finite group $ G $ such that $ |A|+|B|>|G| . $ Show that $ G=AB. $ [b]b)[/b] Show that the cyclic group of order $ n+1 $ is the product of the sets $ \{ 0,1,2,\ldots ,m \} $ and $ \{ m,m+1,m+2,\ldots ,n\} , $ where $ 0,1,2,\ldots n $ are residues modulo $ n+1 $ and $ m\le n. $

Cono Sur Shortlist - geometry, 1993.6
Tags: geometry , equilateral , equal segments
Consider in the interior of an equilateral triangle $ABC$ points $D, E$ and $F$ such that$ D$ belongs to segment $BE$, $E$ belongs to segment $CF$ and$ F$ to segment $AD$. If $AD=BE = CF$ then $DEF$ is equilateral.

2011 Rioplatense Mathematical Olympiad, Level 3, 5
Tags: combinatorial geometry , rectangle , combinatorics
A [i]form [/i] is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all the rectangles on the same side, and whose heights $m_1, ... , m_n$ satisying $m_1\ge ... \ge m_n$. An [i]angle [/i] in a [i]form [/i] consists of a box $v$ and of all the boxes to the right of $v$ and all the boxes above $v$. The size of a [i]form [/i] of an [i]angle [/i] is the number of boxes it contains. Find the maximum number of [i]angles [/i] of size $11$ in a form of size $400$. [url=http://www.oma.org.ar/enunciados/omr20.htm]source[/url]

1978 Yugoslav Team Selection Test, Problem 3
Tags: combinatorics
Let $F$ be the collection of subsets of a set with $n$ elements such that no element of $F$ is a subset of another of its elements. Prove that $$|F|\le\binom n{\lfloor n/2\rfloor}.$$

2019 239 Open Mathematical Olympiad, 8
Tags: graph theory , combinatorics , chromatic number , cycle
Given a natural number $k> 1$. Prove that if through any edge of the graph $G$ passes less than $[e(k-1)! - 1]$ simple cycles, then the vertices of this graph can be colored with $k$ colors in the correct way.

2008 Swedish Mathematical Competition, 2
Tags: number theory , combinatorics
Determine the smallest integer $n \ge 3$ with the property that you can choose two of the numbers $1,2,\dots, n$ in such a way that their product is equal to the sum of the other $n - 2$ languages. What are the two numbers?

2008 China National Olympiad, 3
Tags: modular arithmetic , algebra , polynomial , vieta , number theory
Find all triples $(p,q,n)$ that satisfy \[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\] where $p,q$ are odd primes and $n$ is an positive integer.

2021 Latvia Baltic Way TST, P6
Tags: rectangle , rotation , domain , combinatorics
Let's call $1 \times 2$ rectangle, which can be a rotated, a domino. Prove that there exists polygon, who can be covered by dominoes in exactly $2021$ different ways.

2023 Sharygin Geometry Olympiad, 8.6
Tags: geometry
For which $n$ the plane may be paved by congruent figures bounded by $n$ arcs of circles?

2022 MIG, 1
Tags:
What is $4^0 - 3^1 - 2^2 - 1^3$? $\textbf{(A) }{-}8\qquad\textbf{(B) }{-}7\qquad\textbf{(C) }{-}5\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

2014 May Olympiad, 1
Tags: number theory , digit
A natural number $N$ is [i]good [/i] if its digits are $1, 2$, or $3$ and all $2$-digit numbers are made up of digits located in consecutive positions of $N$ are distinct numbers. Is there a good number of $10$ digits? Of $11$ digits?

2008 Danube Mathematical Competition, 2
Tags: midpoint , circles , chord , concurrency , geometry
In a triangle $ABC$ let $A_1$ be the midpoint of side $BC$. Draw circles with centers $A, A1$ and radii $AA_1, BC$ respectively and let $A'A''$ be their common chord. Similarly denote the segments $B'B''$ and $C'C''$. Show that lines $A'A'', B'B'''$ and $C'C''$ are concurrent.

1985 All Soviet Union Mathematical Olympiad, 398
Tags: coloring , polygon
You should paint all the sides and diagonals of the regular $n$-gon so, that every pair of segments, having the common point, would be painted with different colours. How many colours will you require?

2019 Belarusian National Olympiad, 10.2
Tags: circumcircle , geometry
A point $P$ is chosen in the interior of the side $BC$ of triangle $ABC$. The points $D$ and $C$ are symmetric to $P$ with respect to the vertices $B$ and $C$, respectively. The circumcircles of the triangles $ABE$ and $ACD$ intersect at the points $A$ and $X$. The ray $AB$ intersects the segment $XD$ at the point $C_1$ and the ray $AC$ intersects the segment $XE$ at the point $B_1$. Prove that the lines $BC$ and $B_1C_1$ are parallel. [i](A. Voidelevich)[/i]

1978 AMC 12/AHSME, 4
Tags:
If $a = 1,~ b = 10, ~c = 100$, and $d = 1000$, then \[(a+ b+ c-d) + (a + b- c+ d) +(a-b+ c+d)+ (-a+ b+c+d) \] is equal to $\textbf{(A) }1111\qquad\textbf{(B) }2222\qquad\textbf{(C) }3333\qquad\textbf{(D) }1212\qquad \textbf{(E) }4242$

1990 Tournament Of Towns, (270) 4
Tags: geometry , parallelogram
The sides $AB$, $BC$, $CD$ and $DA$ of the quadrilateral $ABCD$ are respectively equal to the sides $A'B'$, $B'C'$, $C'D' $ and $D'A'$ of the quadrilateral $A'B'CD$' and it is known that $AB \parallel CD$ and $B'C' \parallel D'A'$. Prove that both quadrilaterals are parallelograms. (V Proizvolov, Moscow)

2013 IMAC Arhimede, 4
Tags: number theory , greatest common divisor , floor function
Let $p,n$ be positive integers, such that $p$ is prime and $p <n$. If $p$ divides $n + 1$ and $ \left(\left[\frac{n}{p}\right], (p-1)!\right) = 1$, then prove that $p\cdot \left[\frac{n}{p}\right]^2$ divides ${n \choose p} -\left[\frac{n}{p}\right]$ . (Here $[x]$ represents the integer part of the real number $x$.)

1997 AMC 12/AHSME, 1
Tags:
If $a$ and $b$ are digits for which \begin{tabular}{ccc} & 2 & a\\ $\times$ & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{tabular} Then $a+b =$ A. 3 B. 4 C. 7 D. 9 E. 12