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

2010 Contests, 3
Tags: geometry , graph theory , combinatorics
Given an integer $n\ge 2$, given $n+1$ distinct points $X_0,X_1,\ldots,X_n$ in the plane, and a positive real number $A$, show that the number of triangles $X_0X_iX_j$ of area $A$ does not exceed $4n\sqrt n$.

2024 MMATHS, 12
Tags:
Let $ABC$ be a triangle with $\angle{A}=60^\circ$ and orthocenter $H.$ Let $B'$ be the reflection of $B$ over $AC,$ $C'$ be the reflection of $C$ over $AB,$ and $A'$ be the intersection of $BC'$ and $B'C.$ Let $D$ be the intersection of $A'H$ and $BC.$ If $BC=5$ and $A'D=4,$ then the area of $\triangle{ABC}$ can be expressed as $a\sqrt{b}+\sqrt{c},$ where $a,b,$ and $c$ are positive integers, and $b$ and $c$ are not divisible by the square of any prime. Find $a+b+c.$

1995 May Olympiad, 4
Tags: geometry , pyramid , minimum , distance , 3d geometry
Consider a pyramid whose base is an equilateral triangle $BCD$ and whose other faces are triangles isosceles, right at the common vertex $A$. An ant leaves the vertex $B$ arrives at a point $P$ of the $CD$ edge, from there goes to a point $Q$ of the edge $AC$ and returns to point $B$. If the path you made is minimal, how much is the angle $PQA$ ?

2023 USAMTS Problems, 4
Tags: algebra , system of equations , vieta
Prove that for any real numbers $1 \leq \sqrt{x} \leq y \leq x^2$, the following system of equations has a real solution $(a, b, c)$: \[a+b+c = \frac{x+x^2+x^4+y+y^2+y^4}{2}\] \[ab+ac+bc = \frac{x^3 + x^5 + x^6 + y^3 + y^5 + y^6}{2}\] \[abc=\frac{x^7+y^7}{2}\]

2016 HMNT, 2
Tags: geometry
What is the smallest possible perimeter of a triangle whose side lengths are all squares of distinct positive integers?

2023 JBMO Shortlist, G1
Tags: geometry
Let $ABC$ be a triangle with circumcentre $O$ and circumcircle $\Omega$. $\Gamma$ is the circle passing through $O,B$ and tangent to $AB$ at $B$. Let $\Gamma$ intersect $\Omega$ a second time at $P \neq B$. The circle passing through $P,C$ and tangent to $AC$ at $C$ intersects with $\Gamma$ at $M$. Prove that $|MP|=|MC|$.

2001 Belarusian National Olympiad, 7
Tags: geometry
The convex quadrilateral $ABCD$ is inscribed in the circle $S_1$. Let $O$ be the intersection of $AC$ and $BD$. Circle $S_2$ passes through $D$ and $ O$, intersecting $AD$ and $CD$ at $ M$ and $ N$, respectively. Lines $OM$ and $AB$ intersect at $R$, lines $ON$ and $BC$ intersect at $T$, and $R$ and $T$ lie on the same side of line $BD$ as $ A$. Prove that $O$, $R$,$T$, and $B$ are concyclic.

2023 Belarusian National Olympiad, 9.5
Tags: algebra , polynomial
The polynomial $P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\ldots+a_1x+a_0$ ($a_{2n} \neq 0$) doesn't have any real roots. Prove that the polynomial $Q(x)=a_{2n}x^{2n}+a_{2n-2}x^{2n-2}+\ldots+a_2x^2+a_0$ also doesn't have any real roots.

2003 USA Team Selection Test, 5
Tags: trigonometry , function , inequalities
Let $A, B, C$ be real numbers in the interval $\left(0,\frac{\pi}{2}\right)$. Let \begin{align*} X &= \frac{\sin A\sin (A-B)\sin (A-C)}{\sin (B+C)} \\ Y &= \frac{\sin B\sin(B-C)\sin (B-A)}{\sin (C+A)} \\ Z &= \frac{\sin C\sin (C-A)\sin (C-B)}{\sin (A+B)} . \end{align*} Prove that $X+Y+Z \geq 0$.

2015 Postal Coaching, Problem 2
Tags: algebra , functional equation , function
Find all functions $f: \mathbb{Q} \to \mathbb{R}$ such that $f(xy)=f(x)f(y)+f(x+y)-1$ for all rationals $x,y$

2002 Junior Balkan Team Selection Tests - Romania, 4
Tags: number theory , integer , divisor
Let $p, q$ be two distinct primes. Prove that there are positive integers $a, b$ such that the arithmetic mean of all positive divisors of the number $n = p^aq^b$ is an integer.

2009 Tournament Of Towns, 5
Tags: rhombus , circumcenter , geometry
In rhombus $ABCD$, angle $A$ equals $120^o$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ so that angle $NAM$ equals $30^o$. Prove that the circumcenter of triangle $NAM$ lies on a diagonal of of the rhombus.

1996 Czech and Slovak Match, 2
Tags: binary operation , algebra
Let ⋆ be a binary operation on a nonempty set $M$. That is, every pair $(a,b) \in M$ is assigned an element $a$ ⋆$ b$ in $M$. Suppose that ⋆ has the additional property that $(a $ ⋆ $b) $ ⋆$ b= a$ and $a$ ⋆ $(a$ ⋆$ b)= b$ for all $a,b \in M$. (a) Show that $a$ ⋆ $b = b$ ⋆ $a$ for all $a,b \in M$. (b) On which finite sets $M$ does such a binary operation exist?

1967 Spain Mathematical Olympiad, 7
Tags: algebra
On a road a caravan of cars circulates, all at the same speed, maintaining the minimum separation between one and the other indicated by the Code of Circulation. This separation is, in meters, $\frac{u^2}{100}$, where $u$ is the speed expressed in km/h. Assuming that the length of each car is $2.89$ m, calculate the speed at which they must circulate so that the capacity of traffic is maximum, that is, so that in a fixed time the maximum number pass of vehicles at a point on the road.

1989 IMO Longlists, 81
Tags: function , algebra
A real-valued function $ f$ on $ \mathbb{Q}$ satisfies the following conditions for arbitrary $ \alpha, \beta \in \mathbb{Q}:$ [b](i)[/b] $ f(0) \equal{} 0,$ [b](ii)[/b] $ f(\alpha) > 0 \text{ if } \alpha \neq 0,$ [b](iii)[/b] $ f(\alpha \cdot \beta) \equal{} f(\alpha)f(\beta),$ [b](iv)[/b] $ f(\alpha \plus{} \beta) \leq f(\alpha) \plus{} f(\beta),$ [b](v)[/b] $ f(m) \leq 1989$ $ \forall m \in \mathbb{Z}.$ Prove that \[ f(\alpha \plus{} \beta) \equal{} \max\{f(\alpha), f(\beta)\} \text{ if } f(\alpha) \neq f(\beta).\]

2005 Today's Calculation Of Integral, 38
Tags: calculus , integration , calculus computations , logarithm
Let $a$ be a constant number such that $0<a<1$ and $V(a)$ be the volume formed by the revolution of the figure which is enclosed by the curve $y=\ln (x-a)$, the $x$-axis and two lines $x=1,x=3$ about the $x$-axis. If $a$ varies in the range of $0<a<1$, find the minimum value of $V(a)$.

2010 AMC 8, 6
Tags: symmetry , geometry , rhombus , rectangle
Which of the following has the greatest number of line of symmetry? $ \textbf{(A)}\ \text{ Equilateral Triangle}$ $\textbf{(B)}\ \text{Non-square rhombus} $ $\textbf{(C)}\ \text{Non-square rectangle}$ $\textbf{(D)}\ \text{Isosceles Triangle}$ $\textbf{(E)}\ \text{Square} $

2021 BMT, T1
Tags: algebra
The arithmetic mean of $2, 6, 8$, and $x$ is $7$. The arithmetic mean of $2, 6, 8, x$, and $y$ is $9$. What is the value of $y - x$?

2022 Indonesia TST, N
Tags: integer , quadratic , greatest common divisor , number theory , diophantine
For each natural number $n$, let $f(n)$ denote the number of ordered integer pairs $(x,y)$ satisfying the following equation: \[ x^2 - xy + y^2 = n. \] a) Determine $f(2022)$. b) Determine the largest natural number $m$ such that $m$ divides $f(n)$ for every natural number $n$.

2012 JBMO TST - Macedonia, 1
Tags: prime number , number theory
Find all prime numbers of the form $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$, where $n$ is a natural number.

2005 Sharygin Geometry Olympiad, 8
Tags: geometry , rectangle , inscribed
Around the convex quadrilateral $ABCD$, three rectangles are circumscribed . It is known that two of these rectangles are squares. Is it true that the third one is necessarily a square? (A rectangle is circumscribed around the quadrilateral $ABCD$ if there is one vertex $ABCD$ on each side of the rectangle).

1996 AMC 12/AHSME, 7
Tags:
A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges $\$4.95$ for the father and $\$0.45$ for each year of a child's age, where age is defined as the age at the most recent birthday. If the bill is $\$9.45$, which of the following could be the age of the youngest child? $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

1980 All Soviet Union Mathematical Olympiad, 301
Tags: floor function , algebra , equation
Prove that there is an infinite number of such numbers $B$ that the equation $\lfloor x^3/2\rfloor + \lfloor y^3/2 \rfloor = B$ has at least $1980$ integer solutions $(x,y)$. ($\lfloor z\rfloor$ denotes the greatest integer not exceeding $z$.)

2008 Romania National Olympiad, 2
Tags: inequalities , function , calculus , derivative , integration , geometry , real analysis
Let $ f: [0,1]\to\mathbb R$ be a derivable function, with a continuous derivative $ f'$ on $ [0,1]$. Prove that if $ f\left( \frac 12\right) \equal{} 0$, then \[ \int^1_0 \left( f'(x) \right)^2 dx \geq 12 \left( \int^1_0 f(x) dx \right)^2.\]

2019 IMC, 1
Tags:
Evaluate the product $$\prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}.$$ [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karen Keryan, Yerevan State University and American University of Armenia, Yerevan[/i]