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 Mathematical Talent Reward Programme, MCQ: P4
Tags: geometry
Take a point $O$ inside $\Delta A B C$ such that $\angle B O C=90^{\circ}$, $\angle C A O=\angle A B O$, $\angle B A O=\angle B C O .$ Find the value of $\frac{A C}{O C}$ [list=1] [*] $\sqrt{2}$ [*] $\sqrt{\frac{3}{2}}$ [*] 2 [*] None of these [/list]

2014 Lusophon Mathematical Olympiad, 3
Tags: geometry
In a convex quadrilateral $ABCD$, $P$ and $Q$ are points on sides $BC$ and $DC$ such that $B\hat{A}P = D\hat{A}Q$. If the line that passes through the orthocenters of $\triangle ABP$ and $\triangle ADQ$ is perpendicular to $AC$, prove that the area of these triangles are equals.

2015 Oral Moscow Geometry Olympiad, 3
Tags: geometry , construction , equal segments
In triangle $ABC$, points $D, E$, and $F$ are marked on sides $AC, BC$, and $AB$ respectively, so that $AD = AB$, $EC = DC$, $BF = BE$. After that, they erased everything except points $E, F$ and $D$. Reconstruct the triangle $ABC$ (no study required).

2025 Harvard-MIT Mathematics Tournament, 3
Tags: team
Let $\omega_1$ and $\omega_2$ be two circles intersecting at distinct points $A$ and $B.$ Point $X$ varies along $\omega_1,$ and point $Y$ is chosen on $\omega_2$ such that $AB$ bisects angle $\angle{XAY}.$ Prove that as $X$ varies along $\omega_1,$ the circumcenter of $\triangle{AXY}$ (if it exists) varies along a fixed line.

1987 USAMO, 3
Tags: polynomial , induction , algebra
Construct a set $S$ of polynomials inductively by the rules: (i) $x\in S$; (ii) if $f(x)\in S$, then $xf(x)\in S$ and $x+(1-x)f(x)\in S$. Prove that there are no two distinct polynomials in $S$ whose graphs intersect within the region $\{0 < x < 1\}$.

1990 IMO Longlists, 7
Tags: geometry , incenter , circumcircle , inequalities
Let $S$ be the incenter of triangle $ABC$. $A_1, B_1, C_1$ are the intersections of $AS, BS, CS$ with the circumcircle of triangle $ABC$ respectively. Prove that $SA_1 + SB_1 + SC_1 \geq SA + SB + SC.$

2003 Putnam, 4
Tags: quadratic , function , calculus , derivative , inequalities , absolute value
Suppose that $a, b, c, A, B, C$ are real numbers, $a \not= 0$ and $A \not= 0$, such that \[|ax^2+ bx + c| \le |Ax^2+ Bx + C|\] for all real numbers $x$. Show that \[|b^2- 4ac| \le |B^2- 4AC|\]

2011 AMC 12/AHSME, 10
Tags: geometry , rectangle , trigonometry , trig identities , law of sines , angle
Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75 $

2012 Saint Petersburg Mathematical Olympiad, 3
Tags: geometry
$ABCD$ is inscribed. Bisector of angle between diagonals intersect $AB$ anc $CD$ at $X$ and $Y$. $M,N$ are midpoints of $AD,BC$. $XM=YM$ Prove, that $XN=YN$.

2014 Hanoi Open Mathematics Competitions, 13
Tags: algebra , inequalities
Let $a,b, c > 0$ and $abc = 1$. Prove that $\frac{a - 1}{c}+\frac{c - 1}{b}+\frac{b - 1}{a} \ge 0$

2014 Thailand Mathematical Olympiad, 1
Tags: equal segments , geometry
Let $\vartriangle ABC$ be an isosceles triangle with $\angle BAC = 100^o$. Let $D, E$ be points on ray $\overrightarrow{AB}$ so that $BC = AD = BE$. Show that $BC \cdot DE = BD \cdot CE$

2014 AMC 8, 2
Tags:
Paul owes Paula $35$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$

2003 Korea - Final Round, 2
Tags: rhombus , trigonometry , circumcircle , ratio , perpendicular bisector , angle bisector , geometry
Let $M$ be the intersection of two diagonal, $AC$ and $BD$, of a rhombus $ABCD$, where angle $A<90^\circ$. Construct $O$ on segment $MC$ so that $OB<OC$ and let $t=\frac{MA}{MO}$, provided that $O \neq M$. Construct a circle that has $O$ as centre and goes through $B$ and $D$. Let the intersections between the circle and $AB$ be $B$ and $X$. Let the intersections between the circle and $BC$ be $B$ and $Y$. Let the intersections of $AC$ with $DX$ and $DY$ be $P$ and $Q$, respectively. Express $\frac{OQ}{OP}$ in terms of $t$.

2019 Online Math Open Problems, 22
Tags:
For finite sets $A$ and $B$, call a function $f: A \rightarrow B$ an \emph{antibijection} if there does not exist a set $S \subseteq A \cap B$ such that $S$ has at least two elements and, for all $s \in S$, there exists exactly one element $s'$ of $S$ such that $f(s')=s$. Let $N$ be the number of antibijections from $\{1,2,3, \ldots 2018 \}$ to $\{1,2,3, \ldots 2019 \}$. Suppose $N$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For example, if it were true that $N=12=2\times 2\times 3$, then the answer would be $2+2+3=7$.) [i]Proposed by Ankit Bisain[/i]

2008 Iran MO (3rd Round), 3
Tags: function , induction , combinatorics
Prove that for each $ n$: \[ \sum_{k\equal{}1}^n\binom{n\plus{}k\minus{}1}{2k\minus{}1}\equal{}F_{2n}\]

2004 All-Russian Olympiad Regional Round, 8.1
Tags: algebra
On two intersecting roads with equal constant speeds Audi and BMW cars are moving fast. It turned out that as in 17.00, and at 18.00 the BMW was twice as far from the intersection, than ''Audi''. At what time could an Audi drive across the river?

1999 Turkey Team Selection Test, 3
Tags: parabola , combinatorics , conic
Prove that the plane is not a union of the inner regions of finitely many parabolas. (The outer region of a parabola is the union of the lines not intersecting the parabola. The inner region of a parabola is the set of points of the plane that do not belong to the outer region of the parabola)

Kvant 2022, M2722
Tags: geometry
Consider an acute non-isosceles triangle. In a single step it is allowed to cut any one of the available triangles into two triangles along its median. Is it possible that after a finite number of cuttings all triangles will be isosceles? [i]Proposed by E. Bakaev[/i]

2015 District Olympiad, 2
Tags: geometry , perpendicular bisector
Let $ ABC $ be an obtuse triangle with $ AB=AC, M $ the symmetric point of $ A $ with respect to $ C, $ and $ P $ the intersection of the line $ AB $ with the perpendicular bisector of the segment $ \overline{AB} . $ Knowing that $ PM $ is perpendicular to $ BC, $ show that $ APM $ is equilateral.

2024 Francophone Mathematical Olympiad, 2
Tags: combinatorics , combinatorial geometry , 3d geometry , geometry
Given a positive integer $n \ge 2$, let $\mathcal{P}$ and $\mathcal{Q}$ be two sets, each consisting of $n$ points in three-dimensional space. Suppose that these $2n$ points are distinct. Show that it is possible to label the points of $\mathcal{P}$ as $P_1,P_2,\dots,P_n$ and the points of $\mathcal{Q}$ as $Q_1,Q_2,\dots,Q_n$ such that for any indices $i$ and $j$, the balls of diameters $P_iQ_i$ and $P_jQ_j$ have at least one common point.

2002 USA Team Selection Test, 1
Tags: inequalities , trigonometry , trigonometric inequality , function
Let $ ABC$ be a triangle, and $ A$, $ B$, $ C$ its angles. Prove that \[ \sin\frac{3A}{2}+\sin\frac{3B}{2}+\sin\frac{3C}{2}\leq \cos\frac{A-B}{2}+\cos\frac{B-C}{2}+\cos\frac{C-A}{2}. \]

2016 Rioplatense Mathematical Olympiad, Level 3, 4
Tags: function , algebra , maximum , functional inequality
Let $c > 1$ be a real number. A function $f: [0 ,1 ] \to R$ is called c-friendly if $f(0) = 0, f(1) = 1$ and $|f(x) -f(y)| \le c|x - y|$ for all the numbers $x ,y \in [0,1]$. Find the maximum of the expression $|f(x) - f(y)|$ for all [i]c-friendly[/i] functions $f$ and for all the numbers $x,y \in [0,1]$.

1992 AMC 12/AHSME, 6
Tags:
If $x > y > 0$, then $\frac{x^{y}y^{x}}{y^{y}x^{x}} = $ $ \textbf{(A)}\ (x - y)^{y/x}\qquad\textbf{(B)}\ \left(\frac{x}{y}\right)^{x-y}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \left(\frac{x}{y}\right)^{y-x}\qquad\textbf{(E)}\ (x - y)^{x/y} $

2019 Harvard-MIT Mathematics Tournament, 7
Tags: hmmt , geometry , combinatorics
A convex polygon on the plane is called [i]wide[/i] if the projection of the polygon onto any line in the same plane is a segment with length at least 1. Prove that a circle of radius $\tfrac{1}{3}$ can be placed completely inside any wide polygon.

2014 Czech-Polish-Slovak Junior Match, 4
Tags: number theory , divide , divisible , digit
The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.