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

2018 CMIMC Algebra, 2
Tags: algebra
Suppose $x>1$ is a real number such that $x+\tfrac 1x = \sqrt{22}$. What is $x^2-\tfrac1{x^2}$?

1976 Miklós Schweitzer, 6
Tags: real analysis
Let $ 0 \leq c \leq 1$, and let $ \eta$ denote the order type of the set of rational numbers. Assume that with every rational number $ r$ we associate a Lebesgue-measurable subset $ H_r$ of measure $ c$ of the interval $ [0,1]$. Prove the existence of a Lebesgue-measurable set $ H \subset [0,1]$ of measure $ c$ such that for every $ x \in H$ the set \[ \{r : \;x \in H_r\ \}\] contains a subset of type $ \eta$. [i]M. Laczkovich[/i]

2022 Putnam, B3
Tags:
Assign to each positive real number a color, either red or blue. Let $D$ be the set of all distances $d>0$ such that there are two points of the same color at distance $d$ apart. Recolor the positive reals so that the numbers in $D$ are red and the numbers not in $D$ are blue. If we iterate the recoloring process, will we always end up with all the numbers red after a finite number of steps?

2018 Harvard-MIT Mathematics Tournament, 3
Tags: geometry
$HOW,BOW,$ and $DAH$ are equilateral triangles in a plane such that $WO=7$ and $AH=2$. Given that $D,A,B$ are collinear in that order, find the length of $BA$.

2024 Sharygin Geometry Olympiad, 3
Tags: geometry
Let $ABC$ be an acute-angled triangle, and $M$ be the midpoint of the minor arc $BC$ of its circumcircle. A circle $\omega$ touches the side $AB, AC$ at points $P, Q$ respectively and passes through $M$. Prove that $BP + CQ = PQ$.

2018 District Olympiad, 2
Tags: irrational number
Show that the number \[\sqrt[n]{\sqrt{2019} + \sqrt{2018}} + \sqrt[n]{\sqrt{2019} - \sqrt{2018}}\] is irrational for any $n\ge 2$.

2010 Belarus Team Selection Test, 1.1
Tags: sum , positive integer , number theory
Does there exist a subset $E$ of the set $N$ of all positive integers such that none of the elements in $E$ can be presented as a sum of at least two other (not necessarily distinct) elements from $E$ ? (E. Barabanov)

2013 ELMO Shortlist, 9
Tags: rectangle , symmetry , circumcircle , cyclic quadrilateral , projective geometry , geometry
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$. [i]Proposed by Allen Liu[/i]

1966 All Russian Mathematical Olympiad, 076
Tags: algebra
A rectangle $ABCD$ is drawn on the cross-lined paper with its sides laying on the lines, and $|AD|$ is $k$ times more than $|AB|$ ($k$ is an integer). All the shortest paths from $A$ to $C$ coming along the lines are considered. Prove that the number of those with the first link on $[AD]$ is $k$ times more then of those with the first link on $[AB]$.

2011 Irish Math Olympiad, 4
Tags: inequalities
Suppose that $x,y$ and $z$ are positive numbers such that $$1=2xyz+xy+yz+zx$$ Prove that (i) $$\frac{3}{4}\le xy+yz+zx<1$$ (ii) $$xyz\le \frac{1}{8}$$ Using (i) or otherwise, deduce that $$x+y+z\ge \frac{3}{2}$$ and derive the case of equality.

1998 Tournament Of Towns, 4
Tags: algebra , system of equations
For some positive numbers $A, B, C$ and $D$, the system of equations $$\begin{cases} x^2 + y^2 = A \\ |x| + |y| = B \end{cases}$$ has $m$ solutions, while the system of equations $$\begin{cases} x^2 + y^2 +z^2= X\\ |x| + |y| +|z| = D \end{cases}$$ has $n$ solutions. If $m > n > 1$, find $m$ and $n$. ( G Galperin)

2017 USAMO, 5
Tags:
Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $ ( x, y ) \in \mathbf{Z}^2$ with positive integers for which: [list] [*] only finitely many distinct labels occur, and [*] for each label $i$, the distance between any two points labeled $i$ is at least $c^i$. [/list] [i]Proposed by Ricky Liu[/i]

2024 HMNT, 9
Tags:
Let $ABCDEF$ be a regular hexagon with center $O$ and side length $1.$ Point $X$ is placed in the interior of the hexagon such that $\angle BXC = \angle AXE = 90^\circ.$ Compute all possible values of $OX.$

2004 Tuymaada Olympiad, 1
Tags: polynomial , calculus , integration , algebra
Do there exist a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers and a non-constant polynomial $P(x)$ such that $a_{m}+a_{n}=P(mn)$ for every positive integral $m$ and $n?$ [i]Proposed by A. Golovanov[/i]

2025 Greece National Olympiad, 1
Tags: algebra , polynomial , root , integer
Let $P(x)=x^4+5x^3+mx^2+5nx+4$ have $2$ distinct real roots, which sum up to $-5$. If $m,n \in \mathbb {Z_+}$, find the values of $m,n$ and their corresponding roots.

1992 Vietnam National Olympiad, 1
Tags: polynomial , algebra
Let $ 9 < n_{1} < n_{2} < \ldots < n_{s} < 1992$ be positive integers and \[ P(x) \equal{} 1 \plus{} x^{2} \plus{} x^{9} \plus{} x^{n_{1}} \plus{} \cdots \plus{} x^{n_{s}} \plus{} x^{1992}.\] Prove that if $ x_{0}$ is real root of $ P(x)$ then $ x_{0}\leq\frac {1 \minus{} \sqrt {5}}{2}$.

2014 Singapore Senior Math Olympiad, 13
Tags: algebra , polynomial
Suppose $a$ and $b$ are real numbers such that the polynomial $x^3+ax^2+bx+15$ has a factor of $x^2-2$. Find the value of $a^2b^2$.

1954 AMC 12/AHSME, 24
Tags:
The values of $ k$ for which the equation $ 2x^2\minus{}kx\plus{}x\plus{}8\equal{}0$ will have real and equal roots are: $ \textbf{(A)}\ 9 \text{ and }\minus{}7 \qquad \textbf{(B)}\ \text{only }\minus{}7 \qquad \textbf{(C)}\ \text{9 and 7} \\ \textbf{(D)}\ \minus{}9 \text{ and }\minus{}7 \qquad \textbf{(E)}\ \text{only 9}$

1945 Moscow Mathematical Olympiad, 098
Tags: locus , geometry
A right triangle $ABC$ moves along the plane so that the vertices $B$ and $C$ of the triangle’s acute angles slide along the sides of a given right angle. Prove that point $A$ fills in a line segment and find its length.

1981 National High School Mathematics League, 7
Tags:
The equation $x|x|+px+q=0$ is given. Which of the following is not true? $\text{(A)}$It has at most three real roots. $\text{(B)}$It has at least one real root. $\text{(C)}$Only if $p^2-4q\geq0 $,it has real roots. $\text{(D)}$If $p<0$ and $q>0$, it has three real roots.

CNCM Online Round 3, 2
Tags:
Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB, \angle BFC, \angle CGD, \angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$? [i]Proposed by Akshar Yeccherla (TopNotchMath)[/i]

2001 National Olympiad First Round, 12
Tags:
A circle with center $O$ and radius $15$ is given. Let $P$ be a point such that $|OP|=9$. How many of the chords of the circle pass through $P$ and have integer length? $ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 29 $

2023 Irish Math Olympiad, P10
Tags: combinatorics
Caitlin and Donal play a game called [i]Basketball Shoot-Out[/i]. The game consists of $10$ rounds. In each round, Caitlin and Donal both throw a ball simultaneously at each other's basket. If a player's ball falls into the basket, that player scores one point; otherwise, they score zero points. The scoreboard shows the complete sequence of points scored by each player in each of the $10$ rounds of the game. It turns out that Caitlin has scored at least as many points in total as Donal after every round of the game. Prove the number of possible scoreboards is divisible by $4$ but not by $8$.

2023 Miklós Schweitzer, 3
Tags: metric space , inequalities
Let $X =\{x_0, x_1,\ldots , x_n\}$ be the basis set of a finite metric space, where the points are inductively listed such that $x_k$ maximizes the product of the distances from the points $\{x_0, x_1,\ldots , x_{k-1}\}$ for each $1\leqslant k\leqslant n.$ Prove that if for each $x\in X$ we let $\Pi_x$ be the product of the distances from $x{}$ to every other point, then $\Pi_{x_n}\leqslant 2^{n-1}\Pi_x$ for any $x\in X.$

2009 Junior Balkan Team Selection Tests - Moldova, 6
Tags: number theory
Prove that there are no pairs of nonnegative integers $(x,y)$ that satisfy the equality $$x^3-y^3=x-y+2^{x-y}.$$