Found problems: 85335
2017 China Girls Math Olympiad, 2
Given quadrilateral $ABCD$ such that $\angle BAD+2 \angle BCD=180 ^ \circ .$
Let $E$ be the intersection of $BD$ and the internal bisector of $\angle BAD$.
The perpendicular bisector of $AE$ intersects $CB,CD$ at $X,Y,$ respectively.
Prove that $A,C,X,Y$ are concyclic.
2018 APMO, 1
Let $H$ be the orthocenter of the triangle $ABC$. Let $M$ and $N$ be the midpoints of the sides $AB$ and $AC$, respectively. Assume that $H$ lies inside the quadrilateral $BMNC$ and that the circumcircles of triangles $BMH$ and $CNH$ are tangent to each other. The line through $H$ parallel to $BC$ intersects the circumcircles of the triangles $BMH$ and $CNH$ in the points $K$ and $L$, respectively. Let $F$ be the intersection point of $MK$ and $NL$ and let $J$ be the incenter of triangle $MHN$. Prove that $F J = F A$.
1999 May Olympiad, 2
In a unit circle where $O$ is your circumcenter, let $A$ and $B$ points in the circle with $\angle BOA = 90$. In the arc $AB$(minor arc) we have the points $P$ and $Q$ such that $PQ$ is parallel to $AB$. Let $X$ and $Y$ be the points of intersections of the line $PQ$ with $OA$ and $OB$ respectively. Find the value of $PX^2 + PY^2$
1986 Kurschak Competition, 2
Let $n>2$ be a positive integer. Find the largest value $h$ and the smallest value $H$ for which
\[h<{a_1\over a_1+a_2}+{a_2\over a_2+a_3}+\cdots+{a_n\over a_n+a_1}<H\]
holds for any positive reals $a_1,\dots,a_n$.
2024 Brazil EGMO TST, 2
Let \( n, k \geq 1 \). In a school, there are \( n \) students and \( k \) clubs. Each student participates in at least one of the clubs. One day, a school uniform was established, which could be either blue or red. Each student chose only one of these colors. Every day, the principal visited one of the clubs, forcing all the students in it to switch the colors of the uniforms they wore.
Assuming that the students are distributed in clubs in such a way that any initial choice of uniforms they make, after a certain number of days, it is possible to have at most one student with one of the colors. Show that
\[
n \geq 2^{n-k-1} - 1.
\]
2017 CMIMC Algebra, 9
Define a sequence $\{a_{n}\}_{n=1}^{\infty}$ via $a_{1} = 1$ and $a_{n+1} = a_{n} + \lfloor \sqrt{a_{n}} \rfloor$ for all $n \geq 1$. What is the smallest $N$ such that $a_{N} > 2017$?
2001 AMC 12/AHSME, 25
Consider sequences of positive real numbers of the form $ x,2000,y,...,$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $ x$ does the term 2001 appear somewhere in the sequence?
$ \textbf{(A)} \ 1 \qquad \textbf{(B)} \ 2 \qquad \textbf{(C)} \ 3 \qquad \textbf{(D)} \ 4 \qquad \textbf{(E)} \ \text{more than 4}$
2020 China Team Selection Test, 5
Let $a_1,a_2,\cdots,a_n$ be a permutation of $1,2,\cdots,n$. Among all possible permutations, find the minimum of $$\sum_{i=1}^n \min \{ a_i,2i-1 \}.$$
Durer Math Competition CD Finals - geometry, 2010.C1
Dürer explains art history to his students. The following gothic window is examined.
Where the center of the arc of $BC$ is $A$, and similarly the center of the arc of $AC$ is $B$.
The question is how much is the radius of the circle (radius marked $r$ in the figure).[img]https://cdn.artofproblemsolving.com/attachments/5/c/28e5ee47005bfde7f925908b519099d5e28d91.png[/img]
2015 Swedish Mathematical Competition, 1
Given the acute triangle $ABC$. A diameter of the circumscribed circle of the triangle intersects the sides $AC$ and $BC$, dividing the side $BC$ in half. Show that the same diameter divides the side $AC$ in a ratio of $1: 3$, calculated from $A$, if and only if $\tan B = 2 \tan C$.
2015 Argentina National Olympiad, 5
Find all prime numbers $p$ such that $p^3-4p+9$ is a perfect square.
2024 Iran MO (2nd Round), 2
Find all sequences $(a_n)_{n\geq 1}$ of positive integers such that for all integers $n\geq 3$ we have
$$
\dfrac{1}{a_1 a_3} + \dfrac{1}{a_2a_4} + \cdots
+ \dfrac{1}{a_{n-2}a_n}= 1 - \dfrac{1}{a_1^2+a_2^2+\cdots +a_{n-1}^2}.
$$
2019 Philippine TST, 2
In a triangle $ABC$ with circumcircle $\Gamma$, $M$ is the midpoint of $BC$ and point $D$ lies on segment $MC$. Point $G$ lies on ray $\overrightarrow{BC}$ past $C$ such that $\frac{BC}{DC} = \frac{BG}{GC}$, and let $N$ be the midpoint of $DG$. The points $P$, $S$, and $T$ are defined as follows:
[list = i]
[*] Line $CA$ meets the circumcircle $\Gamma_1$ of triangle $AGD$ again at point $P$.
[*] Line $PM$ meets $\Gamma_1$ again at $S$.
[*] Line $PN$ meets the line through $A$ that is parallel to $BC$ at $Q$. Line $CQ$ meets $\Gamma$ again at $T$.
[/list]
Prove that the points $P$, $S$, $T$, and $C$ are concyclic.
1976 Miklós Schweitzer, 10
Suppose that $ \tau$ is a metrizable topology on a set $ X$ of cardinality less than or equal to continuum. Prove that there exists a separable and metrizable topology on $ X$ that is coarser that $ \tau$.
[i]L. Juhasz[/i]
2023 CCA Math Bonanza, T10
Let $ABC$ be a triangle with $AB=7, BC=8, CA=9.$ Denote by $D$ and $G$ the foot from $A$ to $BC$ and the centroid of $\triangle ABC,$ respectively. Let $M$ be the midpoint of $BC,$ and $K$ be the other intersection of the reflection of $AM$ over the angle bisector of $\angle BAC$ with $(ABC).$ Let $E$ the intersection of $DG$ and $KM.$ Find the area of $ABCE.$
[i]Team #10[/i]
2022 Brazil Team Selection Test, 2
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
2011 South East Mathematical Olympiad, 3
Find all positive integer $n$ , such that for all 35-element-subsets of $M=(1,2,3,...,50)$ ,there exists at least two different elements $a,b$ , satisfing : $a-b=n$ or $a+b=n$.
2013 NIMO Problems, 3
Jacob and Aaron are playing a game in which Aaron is trying to guess the outcome of an unfair coin which shows heads $\tfrac{2}{3}$ of the time. Aaron randomly guesses ``heads'' $\tfrac{2}{3}$ of the time, and guesses ``tails'' the other $\tfrac{1}{3}$ of the time. If the probability that Aaron guesses correctly is $p$, compute $9000p$.
[i]Proposed by Aaron Lin[/i]
2019 CCA Math Bonanza, L5.2
Suppose that a planet contains $\left(CCAMATHBONANZA_{71}\right)^{100}$ people ($100$ in decimal), where in base $71$ the digits $A,B,C,\ldots,Z$ represent the decimal numbers $10,11,12,\ldots,35$, respectively. Suppose that one person on this planet is snapping, and each time they snap, at least half of the current population disappears. Estimate the largest number of times that this person can snap without disappearing. An estimate of $E$ earns $2^{1-\frac{1}{200}\left|A-E\right|}$ points, where $A$ is the actual answer.
[i]2019 CCA Math Bonanza Lightning Round #5.2[/i]
2006 Princeton University Math Competition, 2
Professor Conway collects a total of $58$ midterms from the two sections of his introductory linear algebra course. He notices that the number of midterms from the smaller section is equal to the product of the digits of the number of midterms from his larger section. Assuming that every student handed in a midterm, how many students are there in the smaller section?
1993 Austrian-Polish Competition, 6
If $a,b \ge 0$ are real numbers, prove the inequality
$$\left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)^2\leq\frac{a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b}{4}\leq\frac{a+\sqrt{ab}+b}{3} \leq \sqrt{\left(\frac{a^{2/3}+b^{2/3}}{2}\right)^{3}}$$
For each of the inequalities, find the cases of equality.
2014 Bosnia and Herzegovina Junior BMO TST, 4
It is given $5$ numbers $1$, $3$, $5$, $7$, $9$. We get the new $5$ numbers such that we take arbitrary $4$ numbers(out of current $5$ numbers) $a$, $b$, $c$ and $d$ and replace them with $\frac{a+b+c-d}{2}$, $\frac{a+b-c+d}{2}$, $\frac{a-b+c+d}{2}$ and $\frac{-a+b+c+d}{2}$. Can we, with repeated iterations, get numbers:
$a)$ $0$, $2$, $4$, $6$ and $8$
$b)$ $3$, $4$, $5$, $6$ and $7$
2019 Kosovo Team Selection Test, 3
Prove that there exist infinitely many positive integers $n$ such that $\frac{4^n+2^n+1}{n^2+n+1}$ is a positive integer.
2018 Moldova EGMO TST, 7
Let $ABCD$ be a isosceles trapezoid with $AB \| CD $ , $AD=BC$, $ AC \cap BD = $ { $O$ }. $ M $ is the midpoint of the side $AD$ . The circumcircle of triangle $ BCM $ intersects again the side $AD$ in $K$. Prove that $OK \| AB $ .
2002 National Olympiad First Round, 34
How many positive integers $n$ are there such that $3n^2 + 3n + 7$ is a perfect cube?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 3
\qquad\textbf{d)}\ 7
\qquad\textbf{e)}\ \text{Infinitely many}
$