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

2020 Purple Comet Problems, 9
Tags: algebra
Let $a, b$, and $c$ be real numbers such that $3^a = 125$, $5^b = 49$,and $7^c = 8$1. Find the product $abc$.

2016 CHMMC (Fall), 13
Tags: abstract algebra
A sequence of numbers $a_1, a_2 , \dots a_m$ is a [i]geometric sequence modulo n of length m[/i] for $n,m \in \mathbb{Z}^+$ if for every index $i$, $a_i \in \{ 0, 1, 2, \dots , m-1\}$ and there exists an integer $k$ such that $n | a_{j+1} - ka_{j}$ for $1 \leq j \leq m-1$. How many geometric sequences modulo $14$ of length $14$ are there?

2025 AMC 8, 10
Tags:
In the figure below, $ABCD$ is a rectangle with sides of length $AB = 5$ inches and $AD = 3$ inches. Rectangle $ABCD$ is rotated $90^{\circ}$ clockwise about the midpoint of side $\overline{DC}$ to give a second rectangle. What is the total area, in square inches, covered by the two overlapping rectangles? [img]https://i.imgur.com/NyhZpL6.png[/img] $\textbf{(A) }21 \qquad\textbf{(B) }22.25 \qquad\textbf{(C) }23\qquad\textbf{(D) }23.75 \qquad\textbf{(E) }25$

2014 Belarus Team Selection Test, 2
Tags: algebra , inequalities
Prove that for all even positive integers $n$ the following inequality holds a) $\{n\sqrt6\} > \frac{1}{n}$ b)$ \{n\sqrt6\}> \frac{1}{n-1/(5n)} $ (I. Voronovich)

1994 All-Russian Olympiad Regional Round, 10.5
Tags: number theory , prime , difference
Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime).

2018 Ramnicean Hope, 1
Tags: geometry , trigonometry , algebra , inequalities
Show that $ 2/3+\sin 2018^{\circ } >0. $ [i]Costică Ambrinoc[/i]

2015 Dutch BxMO/EGMO TST, 1
Tags: number theory , divisor , divide
Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$. Prove that $m$ is a divisor of $n$.

2018 Polish Junior MO First Round, 7
Tags: 3d geometry , square , volume , plane , geometry
Square $ABCD$ with sides of length $4$ is a base of a cuboid $ABCDA'B'C'D'$. Side edges $AA'$, $BB'$, $CC'$, $DD'$ of this cuboid have length $7$. Points $K, L, M$ lie respectively on line segments $AA'$, $BB'$, $CC'$, and $AK = 3$, $BL = 2$, $CM = 5$. Plane passing through points $K, L, M$ cuts cuboid on two blocks. Calculate volumes of these blocks.

2010 JBMO Shortlist, 5
Tags: inequalities
Let $ x, y, z > 0 $ with $ x \leq 2, \;y \leq 3 \;$ and $ x+y+z = 11 $ Prove that $xyz \leq 36$

2013 National Olympiad First Round, 6
Tags:
What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation? $ \textbf{(A)}\ 760 \qquad\textbf{(B)}\ 756 \qquad\textbf{(C)}\ 755 \qquad\textbf{(D)}\ 752 \qquad\textbf{(E)}\ 750 $

2004 Turkey Junior National Olympiad, 3
Tags: inequalities
On the evening, more than $\frac 13$ of the students of a school are going to the cinema. On the same evening, More than $\frac {3}{10}$ are going to the theatre, and more than $\frac {4}{11}$ are going to the concert. At least how many students are there in this school?

2022 VIASM Summer Challenge, Problem 1
Tags: number theory
Given prime numbers $p$ and $q.$ a) Assume that $2^xq=p^y+1,$ with $x,y$ are integers greater than $1$. Can $x$ be a composite number? b) Assume that $2^uq=p^v-1,$ with $u$ is a prime number and $v$ is an integer greater than $1$. Find all possible values of $p.$

1968 Yugoslav Team Selection Test, Problem 1
Tags: inequalities , combinatorics , geometry
Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.

2009 Mexico National Olympiad, 2
Tags: geometry
Consider a triangle $ABC$ and a point $M$ on side $BC$. Let $P$ be the intersection of the perpendiculars from $M$ to $AB$ and from $B$ to $BC$, and let $Q$ be the intersection of the perpendiculars from $M$ to $AC$ and from $C$ to $BC$. Show that $PQ$ is perpendicular to $AM$ if and only if $M$ is the midpoint of $BC$.

2008 Princeton University Math Competition, A2
Tags: algebra
What is the polynomial of smallest degree that passes through $(-2, 2), (-1, 1), (0, 2),(1,-1)$, and $(2, 10)$?

2013 India Regional Mathematical Olympiad, 5
Tags: geometry , geometric transformation , circumcircle , incenter , homothety , reflection
In a triangle $ABC$, let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$. The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$, and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$. Prove that $H$ is the incentre of triangle $PQR$.

Kyiv City MO Juniors Round2 2010+ geometry, 2017.7.4
Tags: geometry , perimeter , rectangle , geometric inequality
On the sides $AD$ and $BC$ of a rectangle $ABCD$ select points $M, N$ and $P, Q$ respectively such that $AM = MN = ND = BP = PQ = QC$. On segment $QC$ selected point $X$, different from the ends of the segment. Prove that the perimeter of $\vartriangle ANX$ is more than the perimeter of $\vartriangle MDX$.

2007 Today's Calculation Of Integral, 208
Tags: calculus , integration , function , trigonometry , calculus computations
Find the values of real numbers $a,\ b$ for which the function $f(x)=a|\cos x|+b|\sin x|$ has local minimum at $x=-\frac{\pi}{3}$ and satisfies $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\{f(x)\}^{2}dx=2$.

PEN N Problems, 15
Tags: more sequences
In the sequence $00$, $01$, $02$, $03$, $\cdots$, $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$, $39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could remain on their places?

2018 Saint Petersburg Mathematical Olympiad, 7
Tags: combinatorics
In $10\times 10$ square we choose $n$ cells. In every chosen cell we draw one arrow from the angle to opposite angle. It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or the distance between their ends is at least 2. What is the maximum possible value of $n$?

2012 Singapore Junior Math Olympiad, 2
Tags: digit , number theory
Does there exist an integer $A$ such that each of the ten digits $0, 1, . . . , 9$ appears exactly once as a digit in exactly one of the numbers $A, A^2, A^ 3$ ?

2012 Hanoi Open Mathematics Competitions, 2
Tags: algebra , comparison
Compare the numbers $P = 2^a,Q = 3, T = 2^b$, where $a=\sqrt2 , b=1+\frac{1}{\sqrt2}$ (A) $P < Q < T$, (B) $T < P < Q$, (C) $P < T < Q$, (D) $T < Q < P$, (E) $ Q < P < T$

1992 Baltic Way, 19
Tags: angle bisector , geometry
Let $ C$ be a circle in plane. Let $ C_1$ and $ C_2$ be nonintersecting circles touching $ C$ internally at points $ A$ and $ B$ respectively. Let $ t$ be a common tangent of $ C_1$ and $ C_2$ touching them at points $ D$ and $ E$ respectively, such that both $ C_1$ and $ C_2$ are on the same side of $ t$. Let $ F$ be the point of intersection of $ AD$ and $ BE$. Show that $ F$ lies on $ C$.

2023 APMO, 1
Tags: combinatorics , combinatorial geometry
Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.

2011 National Olympiad First Round, 19
Tags: inequalities
For which inequality, there exists a line such that the region defined by the inequality and the line intersect in exactly two distinct points? $\textbf{(A)}\ x^2+y^2\leq 1 \qquad\textbf{(B)}\ |x+y|+|x-y| \leq 1 \qquad\textbf{(C)}\ |x|^3+|y|^3 \leq 1 \\ \textbf{(D)}\ |x|+|y| \leq 1 \qquad\textbf{(E)}\ |x|^{1/2} + |y|^{1/2} \leq 1$