Found problems: 85335
2025 JBMO TST - Turkey, 1
Let $ABCD$ be a cyclic quadrilateral and let the intersection point of lines $AB$ and $CD$ be $E$. Let the points $K$ and $L$ be arbitrary points on sides $CD$ and $AB$ respectively, which satisfy the conditions
$$\angle KAD = \angle KBC \quad \text{and} \quad \angle LDA = \angle LCB.$$
Prove that $EK = EL$.
2013 IberoAmerican, 5
Let $A$ and $B$ be two sets such that $A \cup B$ is the set of the positive integers, and $A \cap B$ is the empty set. It is known that if two positive integers have a prime larger than $2013$ as their difference, then one of them is in $A$ and the other is in $B$. Find all the possibilities for the sets $A$ and $B$.
2025 Bangladesh Mathematical Olympiad, P7
Yamin and Tamim are playing a game with subsets of $\{1, 2, \ldots, n\}$ where $n \geq 3$.
[list]
[*] Tamim starts the game with the empty set.
[*] On Yamin's turn, he adds a proper non-empty subset of $\{1, 2, \ldots, n\}$ to his collection $F$ of blocked sets.
[*] On Tamim's turn, he adds or removes a positive integer less than or equal to $n$ to or from their set but Tamim can never add or remove an element so that his set becomes one of the blocked sets in $F$.
[/list]
Tamim wins if he can make his set to be $\{1, 2, \ldots, n\}$. Yamin wins if he can stop Tamim from doing so. Yamin goes first and they alternate making their moves. Does Tamim have a winning strategy?
[i]Proposed by Ahmed Ittihad Hasib[/i]
1962 Dutch Mathematical Olympiad, 2
The $n^{th}$ term of a sequence is $t_n$. For $n \ge 1$, $t_n$ is given by the relation:
$$t_n= n^3+\frac12 n^2+ \frac13 n + \frac14$$
The $n^{th}$ term of a second sequence $T_n$, where $T_n$ represents the smallest integer greater than $t_n$. Calculate: $$(T_1+T_2+...+T_{1014}) -(t_1+t_2+...+t_{1014}) $$
2005 MOP Homework, 4
Find all prime numbers $p$ and $q$ such that $3p^4+5q^4+15=13p^2q^2$.
2018 Bundeswettbewerb Mathematik, 3
Let $T$ be a point on a line segment $AB$ such that $T$ is closer to $B$ than to $A$. Show that for each point $C \ne T$ on the line through $T$ perpendicular to $AB$ there is exactly one point $D$ on the line segment $AC$ with $\angle CBD=\angle BAC$. Moreover, show that the line through $D$ perpendicular to $AC$ intersects the line $AB$ in a point $E$ which is independent of the position of $C$.
1989 Irish Math Olympiad, 1
Suppose $L$ is a fixed line, and $A$ is a fixed point not on $L$. Let $k$ be a fixed nonzero real number. For $P$ a point on $L$, let $Q$ be a point on the line $AP$ with $|AP|\cdot |AQ|=k^2$. Determine the locus of $Q$ as $P$ varies along the line $L$.
2020 Peru IMO TST, 2
Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$.
Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$.
Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$.
(Hungary)
2015 CCA Math Bonanza, I15
Let $\omega_1$ and $\omega_2$ be circles with radii $3$ and $12$ and externally tangent at point $P$. Let a common external tangent intersect $\omega_1$, $\omega_2$ at $S$, $T$ respectively and the common internal tangent at point $Q$. Define $X$ to be the point on $\overrightarrow{QP}$ such that $QX=10$. If $XS$, $XT$ intersect $\omega_1$, $\omega_2$ a second time at $A$ and $B$, determine $\tan\angle APB$.
.
[i]2015 CCA Math Bonanza Individual Round #15[/i]
1968 All Soviet Union Mathematical Olympiad, 109
Two finite sequences $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ are just rearranged sequence $1, 1/2, ... , 1/n$ with $$a_1+b_1\ge a_2+b_2\ge...\ge a_n+b_n.$$ Prove that $a_m+a_n\ge 4/m$ for every $m$ ($1\le m\le n$) .
2010 Germany Team Selection Test, 3
A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.
(a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced.
(b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.
[i]Proposed by Jorge Tipe, Peru[/i]
2025 Romania Team Selection Tests, P2
Let $ABC$ be a scalene acute triangle with incentre $I{}$ and circumcentre $O{}$. Let $AI$ cross $BC$ at $D$. On circle $ABC$, let $X$ and $Y$ be the mid-arc points of $ABC$ and $BCA$, respectively. Let $DX{}$ cross $CI{}$ at $E$ and let $DY{}$ cross $BI{}$ at $F{}$. Prove that the lines $FX, EY$ and $IO$ are concurrent on the external bisector of $\angle BAC$.
[i]David-Andrei Anghel[/i]
2008 Tuymaada Olympiad, 3
100 unit squares of an infinite squared plane form a $ 10\times 10$ square. Unit segments forming these squares are coloured in several colours. It is known that the border of every square with sides on grid lines contains segments of at most two colours. (Such square is not necessarily contained in the original $ 10\times 10$ square.) What maximum number of colours may appear in this colouring?
[i]Author: S. Berlov[/i]
2019 Purple Comet Problems, 20
In the diagram below, points $D, E$, and $F$ are on the inside of equilateral $\vartriangle ABC$ such that $D$ is on $\overline{AE}, E$ is on $\overline{CF}, F$ is on $\overline{BD}$, and the triangles $\vartriangle AEC, \vartriangle BDA$, and $\vartriangle CFB$ are congruent. Given that $AB = 10$ and $DE = 6$, the perimeter of $\vartriangle BDA$ is $\frac{a+b\sqrt{c}}{d}$, where $a, b, c$, and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.
[img]https://cdn.artofproblemsolving.com/attachments/8/6/98da82fc1c26fa13883a47ba6d45a015622b20.png[/img]
1962 AMC 12/AHSME, 11
The difference between the larger root and the smaller root of $ x^2 \minus{} px \plus{} (p^2 \minus{} 1)/4 \equal{} 0$ is:
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ p \qquad
\textbf{(E)}\ p\plus{}1$
2021-2022 OMMC, 24
In $\triangle ABC$, angle $B$ is obtuse, $AB = 42$ and $BC = 69$. Let $M$ and $N$ be the midpoints of $AB$ and $BC$, respectively. The angle bisectors of $\angle CAB$ and $\angle ABC$ meet $BC$ and $CA$ at $D$ and $E$ respectively. Let $X$ and $Y$ be the midpoints of $AD$ and $AN$ respectively. Let $CY$ and $BX$ meet $AB$ and $CA$ at $P$ and $Q$. If $EM$ and $PQ$ meet on $BC$, find $CA$.
[i]Proposed by Sid Doppalapudi[/i]
2005 AMC 12/AHSME, 7
Square $ EFGH$ is inside the square $ ABCD$ so that each side of $ EFGH$ can be extended to pass through a vertex of $ ABCD$. Square $ ABCD$ has side length $ \sqrt {50}$ and $ BE \equal{} 1$. What is the area of the inner square $ EFGH$?
[asy]unitsize(4cm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
pair D=(0,0), C=(1,0), B=(1,1), A=(0,1);
pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0];
pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H);
draw(A--B--C--D--cycle);
draw(D--F);
draw(C--E);
draw(B--H);
draw(A--G);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SE);
label("$D$",D,SW);
label("$E$",E,NNW);
label("$F$",F,ENE);
label("$G$",G,SSE);
label("$H$",H,WSW);[/asy]$ \textbf{(A)}\ 25\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 42$
DMM Devil Rounds, 2007
[b]p1.[/b] If
$$ \begin{cases} a^2 + b^2 + c^2 = 1000 \\
(a + b + c)^2 = 100 \\
ab + bc = 10 \end{cases}$$
what is $ac$?
[b]p2.[/b] If a and b are real numbers such that $a \ne 0$ and the numbers $1$, $a + b$, and $a$ are, in some order, the numbers $0$, $\frac{b}{a}$ , and $b$, what is $b - a$?
[b]p3.[/b] Of the first $120$ natural numbers, how many are divisible by at least one of $3$, $4$, $5$, $12$, $15$, $20$, and $60$?
[b]p4.[/b] For positive real numbers $a$, let $p_a$ and $q_a$ be the maximum and minimum values, respectively, of $\log_a(x)$ for $a \le x \le 2a$. If $p_a - q_a = \frac12$ , what is $a$?
[b]p5.[/b] Let $ABC$ be an acute triangle and let $a$, $b$, and $c$ be the sides opposite the vertices $A$, $B$, and $C$, respectively. If $a = 2b \sin A$, what is the measure of angle $B$?
[b]p6.[/b] How many ordered triples $(x, y, z)$ of positive integers satisfy the equation $$x^3 + 2y^3 + 4z^3 = 9?$$
[b]p7.[/b] Joe has invented a robot that travels along the sides of a regular octagon. The robot starts at a vertex of the octagon and every minute chooses one of two directions (clockwise or counterclockwise) with equal probability and moves to the next vertex in that direction. What is the probability that after $8$ minutes the robot is directly opposite the vertex it started from?
[b]p8.[/b] Find the nonnegative integer $n$ such that when $$\left(x^2 -\frac{1}{x}\right)^n$$ is completely expanded the constant coefficient is $15$.
[b]p9.[/b] For each positive integer $k$, let $$f_k(x) = \frac{kx + 9}{x + 3}.$$
Compute $$f_1 \circ f_2\circ ... \circ f_{13}(2).$$
[b]p10.[/b] Exactly one of the following five integers cannot be written in the form $x^2 + y^2 + 5z^2$, where $x$, $y$, and $z$ are integers. Which one is it?
$$2003, 2004, 2005, 2006, 2007$$
[b]p11.[/b] Suppose that two circles $C_1$ and $C_2$ intersect at two distinct points $M$ and $N$. Suppose that $P$ is a point on the line $MN$ that is outside of both $C_1$ and $C_2$. Let $A$ and $B$ be the two distinct points on $C_1$ such that AP and BP are each tangent to $C_1$ and $B$ is inside $C_2$. Similarly, let $D$ and $E$ be the two distinct points on $C_2$ such that $DP$ and $EP$ are each tangent to $C_2$ and $D$ is inside $C_1$. If $AB = \frac{5\sqrt2}{2}$ , $AD = 2$, $BD = 2$, $EB = 1$, and $ED =\sqrt2$, find $AE$.
[b]p12.[/b] How many ordered pairs $(x, y)$ of positive integers satisfy the following equation? $$\sqrt{x} +\sqrt{y} =\sqrt{2007}.$$
[b]p13.[/b] The sides $BC$, $CA$, and $CB$ of triangle $ABC$ have midpoints $K$, $L$, and $M$, respectively. If
$$AB^2 + BC^2 + CA^2 = 200,$$ what is $AK^2 + BL^2 + CM^2$?
[b]p14.[/b] Let $x$ and $y$ be real numbers that satisfy: $$x + \frac{4}{x}= y +\frac{4}{y}=\frac{20}{xy}.$$ Compute the maximum value of $|x - y|$.
[b]p15.[/b] $30$ math meet teams receive different scores which are then shuffled around to lend an aura of mystery to the grading. What is the probability that no team receives their own score? Express your answer as a decimal accurate to the nearest hundredth.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2019 China Second Round Olympiad, 4
Let $V$ be a set of $2019$ points in space where any of the four points are not on the same plane, and $E$ be the set of edges connected between them. Find the smallest positive integer $n$ satisfying the following condition: if $E$ has at least $n$ elements, then there exists $908$ two-element subsets of $E$ such that
[list][*]The two edges in each subset share a common vertice,
[*]Any of the two subsets do not intersect.[/list]
2001 China National Olympiad, 2
Let $X=\{1,2,\ldots,2001\}$. Find the least positive integer $m$ such that for each subset $W\subset X$ with $m$ elements, there exist $u,v\in W$ (not necessarily distinct) such that $u+v$ is of the form $2^{k}$, where $k$ is a positive integer.
2012 HMNT, 10
Triangle $ABC$ has $AB = 4$, $BC = 5$, and $CA = 6$. Points $A'$, $B'$, $C'$ are such that $B'C'$ is tangent to the circumcircle of $ABC$ at $A$, $C'A'$ is tangent to the circumcircle at $B$, and $A'B'$ is tangent to the circumcircle at $C$. Find the length $B'C'$.
2022 IMC, 7
Let $A_1, \ldots, A_k$ be $n\times n$ idempotent complex matrices such that $A_iA_j = -A_iA_j$ for all $1 \leq i < j \leq k$. Prove that at least one of the matrices has rank not exceeding $\frac{n}{k}$.
1997 All-Russian Olympiad, 2
Given a convex polygon M invariant under a $90^\circ$ rotation, show that there exist two circles, the ratio of whose radii is $\sqrt2$, one containing M and the other contained in M.
[i]A. Khrabrov[/i]
Today's calculation of integrals, 849
Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$
2002 Germany Team Selection Test, 3
Determine all $(x,y) \in \mathbb{N}^2$ which satisfy $x^{2y} + (x+1)^{2y} = (x+2)^{2y}.$