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

1979 Chisinau City MO, 175
Tags: inequalities , algebra
Prove that if the sum of positive numbers $a, b, c$ is equal to $1$, then $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 9.$

2015 CIIM, Problem 6
Tags: arithmetic sequence
Show that there exists a real $C > 1$ that satisfy the following property: if $n > 1$ and $a_0 < a_1 < \cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1},\dots,\frac{1}{a_n}$ are in arithmetic progression, then $a_0 > C^n.$

2014 Putnam, 3
Tags: vector , putnam matrices , pigeonhole principle
Let $A$ be an $m\times n$ matrix with rational entries. Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A.$ Show that the rank of $A$ is at least $2.$

2006 Thailand Mathematical Olympiad, 18
Tags: combinatorics
In May, the traffic police wants to select 10 days to patrol, but no two consecutive days can be selected. How many ways are there for the traffic police to select patrol days?

2010 Today's Calculation Of Integral, 592
Tags: calculus , calculus computations , logarithm , trigonometry , integration
Prove the following inequality. \[ \frac{\sqrt{2}}{4}\minus{}\frac 12\minus{}\frac 14\ln 2<\int_0^{\frac{\pi}{4}} \ln \cos x\ dx<\frac 38\pi\plus{}\frac 12\minus{}\ln \ (3\plus{}2\sqrt{2})\]

1959 Kurschak Competition, 2
Tags: angle , geometry
The angles subtended by a tower at distances $100$, $200$ and $300$ from its foot sum to $90^o$. What is its height?

2019 Nepal TST, P1
Tags: number theory
Prove that there exist infinitely many pairs of different positive integers $(m, n)$ for which $m!n!$ is a square of an integer. [i]Proposed by Anton Trygub[/i]

2010 Moldova Team Selection Test, 2
Tags: trigonometry , inequalities
Let $ x_1, x_2, \ldots, x_n$ be positive real numbers with sum $ 1$. Find the integer part of: $ E\equal{}x_1\plus{}\dfrac{x_2}{\sqrt{1\minus{}x_1^2}}\plus{}\dfrac{x_3}{\sqrt{1\minus{}(x_1\plus{}x_2)^2}}\plus{}\cdots\plus{}\dfrac{x_n}{\sqrt{1\minus{}(x_1\plus{}x_2\plus{}\cdots\plus{}x_{n\minus{}1})^2}}$

2022 JHMT HS, 9
Tags: geometry
In $\triangle{PQR}$, $PQ=4$, $PR=5$, and $QR=6$. Assume that an equilateral hexagon $ABCDEF$ is able to be drawn inside $\triangle{PQR}$ so that $\overline{AB}$ is parallel to $\overline{QR}$, $\overline{CD}$ is parallel to $\overline{PQ}$, $\overline{EF}$ is parallel to $\overline{RP}$, $\overline{BC}$ lies on $\overline{RP}$, $\overline{DE}$ lies on $\overline{QR}$, and $\overline{AF}$ lies on $\overline{PQ}$. Find the area of hexagon $ABCDEF$.

2020 USMCA, 27
Tags:
Let $\phi(n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Evaluate \[\lim_{m \rightarrow \infty}\frac{\sum_{n = 1}^m \phi(60n)}{\sum_{n = 1}^m \phi(n)}\]

1994 BMO TST – Romania, 4:
Tags: 3d geometry , geometry
Consider a tetrahedron$ A_1A_2A_3A_4$. A point $N$ is said to be a Servais point if its projections on the six edges of the tetrahedron lie in a plane $\alpha(N)$ (called Servais plane). Prove that if all the six points $Nij$ symmetric to a point $M$ with respect to the midpoints $Bij$ of the edges $A_iA_j$ are Servais points, then $M$ is contained in all Servais planes $\alpha(Nij )$

2005 China Team Selection Test, 2
Tags: number theory
Given prime number $p$. $a_1,a_2 \cdots a_k$ ($k \geq 3$) are integers not divible by $p$ and have different residuals when divided by $p$. Let \[ S_n= \{ n \mid 1 \leq n \leq p-1, (na_1)_p < \cdots < (na_k)_p \} \] Here $(b)_p$ denotes the residual when integer $b$ is divided by $p$. Prove that $|S|< \frac{2p}{k+1}$.

2020 May Olympiad, 2
Tags: number theory
a) Determine if there are positive integers $a, b$ and $c$, not necessarily distinct, such that $a+b+c=2020$ and $2^a+2^b+2^c$ it's a perfect square. b) Determine if there are positive integers $a, b$ and $c$, not necessarily distinct, such that $a+b+c=2020$ and $3^a+3^b+3^c$ it's a perfect square.

2008 Sharygin Geometry Olympiad, 7
Tags: geometry , ratio
(F.Nilov) Given isosceles triangle $ ABC$ with base $ AC$ and $ \angle B \equal{} \alpha$. The arc $ AC$ constructed outside the triangle has angular measure equal to $ \beta$. Two lines passing through $ B$ divide the segment and the arc $ AC$ into three equal parts. Find the ratio $ \alpha / \beta$.

1986 AMC 12/AHSME, 13
Tags: quadratic , parabola , conic
A parabola $y = ax^{2} + bx + c$ has vertex $(4,2)$. If $(2,0)$ is on the parabola, then $abc$ equals $ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 12$

2022 AMC 8 -, 2
Tags:
Consider these two operations: \begin{align*} a \, \blacklozenge \, b &= a^2 - b^2\\ a \, \bigstar \, b &= (a - b)^2 \end{align*} What is the value of $(5 \, \blacklozenge \, 3) \, \bigstar \, 6?$ $\textbf{(A) } {-}20\qquad\textbf{(B) } 4\qquad\textbf{(C) } 16\qquad\textbf{(D) } 100\qquad\textbf{(E) } 220$

1962 All-Soviet Union Olympiad, 11
Tags: geometry
The triangle $ABC$ satisfies $0\le AB\le 1\le BC\le 2\le CA\le 3$. What is the maximum area it can have?

2023 Auckland Mathematical Olympiad, 7
Tags: combinatorial geometry , geometry , combinatorics
In a square of area $1$ there are situated $2024$ polygons whose total area is greater than $2023$. Prove that they have a point in common.

2007 Romania National Olympiad, 3
Tags: trigonometry
Consider the triangle $ ABC$ with $ m(\angle BAC) \equal{} 90^\circ$ and $ AB < AC$.Let a point $ D$ on the side $ AC$ such that: $ m(\angle ACB) \equal{} m(\angle DBA)$.Let $ E$ be a point on the side $ BC$ such that $ DE\perp BC$.It is known that $ BD \plus{} DE \equal{} AC$. Find the measures of the angles in the triangle $ ABC$.

2013 China Team Selection Test, 3
Tags: combinatorial geometry , parallelogram , combinatorics , geometry
Let $A$ be a set consisting of 6 points in the plane. denoted $n(A)$ as the number of the unit circles which meet at least three points of $A$. Find the maximum of $n(A)$

2017 Baltic Way, 18
Tags: number theory , abstract algebra
Let $p>3$ be a prime and let $a_1,a_2,...,a_{\frac{p-1}{2}}$ be a permutation of $1,2,...,\frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_1,a_2,...,a_{\frac{p-1}{2}}$ if it for all $i,j\in\{1,2,...,\frac{p-1}{2}\}$ with $i\not=j$ the residue of $a_ia_j$ modulo $p$ is known?

2011 AMC 8, 17
Tags:
Let $w$, $x$, $y$, and $z$ be whole numbers. If $2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588$, then what does $2w + 3x + 5y + 7z$ equal? $ \textbf{(A)}21\qquad\textbf{(B)}25\qquad\textbf{(C)}27\qquad\textbf{(D)}35\qquad\textbf{(E)}56 $

2013 Nordic, 2
Tags: maximum , arithmetic sequence , combinatorics , game
In a football tournament there are n teams, with ${n \ge 4}$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores ${1}$ more point than the following team on the scoreboard. Determine the maximum possible score of the lowest scoring team, assuming usual scoring for football games (where the winner of a game gets ${3}$ points, the loser ${0}$ points, and if there is a tie both teams get ${1}$ point).

Novosibirsk Oral Geo Oly VIII, 2020.5
Tags: midpoint , perpendicular bisector , geometry
Line $\ell$ is perpendicular to one of the medians of the triangle. The median perpendiculars to the sides of this triangle intersect the line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the other two.

2017 BMT Spring, 12
Tags: combinatorics , probability
A robot starts at the origin of the Cartesian plane. At each of $10$ steps, he decides to move $ 1$ unit in any of the following directions: left, right, up, or down, each with equal probability. After $10$ steps, the probability that the robot is at the origin is $\frac{n}{4^{10}}$ . Find$ n$