Found problems: 85335
1952 Putnam, B6
Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincides with the center of the ellipse.
2000 Croatia National Olympiad, Problem 4
Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.
2012 Regional Olympiad of Mexico Center Zone, 6
A board of $2n$ x $2n$ is colored chess style, a movement is the changing of colors of a $2$ x $2$ square. For what integers $n$ is possible to complete the board with one color using a finite number of movements?
2020 AMC 10, 11
What is the median of the following list of $4040$ numbers$?$
$$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$
$\textbf{(A) } 1974.5 \qquad \textbf{(B) } 1975.5 \qquad \textbf{(C) } 1976.5 \qquad \textbf{(D) } 1977.5 \qquad \textbf{(E) } 1978.5$
2019 Online Math Open Problems, 2
Let $A$, $B$, $C$, and $P$ be points in the plane such that no three of them are collinear. Suppose that the areas of triangles $BPC$, $CPA$, and $APB$ are 13, 14, and 15, respectively. Compute the sum of all possible values for the area of triangle $ABC$.
[i]Proposed by Ankan Bhattacharya[/i]
2006 Iran MO (3rd Round), 2
Let $B$ be a subset of $\mathbb{Z}_{3}^{n}$ with the property that for every two distinct members $(a_{1},\ldots,a_{n})$ and $(b_{1},\ldots,b_{n})$ of $B$ there exist $1\leq i\leq n$ such that $a_{i}\equiv{b_{i}+1}\pmod{3}$. Prove that $|B| \leq 2^{n}$.
1998 All-Russian Olympiad, 8
Two distinct positive integers $a,b$ are written on the board. The smaller of them is erased and replaced with the number $\frac{ab}{|a-b|}$. This process is repeated as long as the two numbers are not equal. Prove that eventually the two numbers on the board will be equal.
2003 Iran MO (3rd Round), 1
suppose this equation: x <sup>2</sup> +y <sup>2</sup> +z <sup>2</sup> =w <sup>2</sup> . show that the solution of this equation ( if w,z have same parity) are in this form:
x=2d(XZ-YW), y=2d(XW+YZ),z=d(X <sup>2</sup> +Y <sup>2</sup> -Z <sup>2</sup> -W <sup>2</sup> ),w=d(X <sup>2</sup> +Y <sup>2</sup> +Z <sup>2</sup> +W <sup>2</sup> )
Today's calculation of integrals, 854
Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.
XMO (China) 2-15 - geometry, 15.1
As shown in the figure, in the quadrilateral $ABCD$, $AB\perp BC$, $AD\perp CD$, let $E$ be a point on line $BD$ such that $EC = CA$. The line perpendicular on line$ AC$ passing through $E$, intersects line $AB$ at point $F$, and line $AD$ at point $G$. Let $X$ and $Y$ the midpoints of line segments $AF$ and $AG$ respectively. Let $Z$ and $W$ be the midpoints of line segments $BE$ and $DE$ respectively. Prove that the circumscribed circle of $\vartriangle WBX$ is tangent to the circumscribed circle of $\vartriangle ZDY$.
[img]https://cdn.artofproblemsolving.com/attachments/0/3/1f6fca7509e6fd6cad662b42abd236fd4858ca.jpg[/img]
PEN P Problems, 12
The positive function $p(n)$ is defined as the number of ways that the positive integer $n$ can be written as a sum of positive integers. Show that, for all positive integers $n \ge 2$, \[2^{\lfloor \sqrt{n}\rfloor}< p(n) < n^{3 \lfloor\sqrt{n}\rfloor }.\]
2013 Czech-Polish-Slovak Junior Match, 2
Find all natural numbers $n$ such that the sum of the three largest divisors of $n$ is $1457$.
2007 Spain Mathematical Olympiad, Problem 6
Given a halfcircle of diameter $AB = 2R$, consider a chord $CD$ of length $c$. Let $E$ be the intersection of $AC$ with $BD$ and $F$ the inersection of $AD$ with $BC$.
Prove that the segment $EF$ has a constant length and direction when varying the chord $CD$ about the halfcircle.
2024 China National Olympiad, 5
In acute $\triangle {ABC}$, ${K}$ is on the extention of segment $BC$. $P, Q$ are two points such that $KP \parallel AB, BK=BP$ and $KQ\parallel AC, CK=CQ$. The circumcircle of $\triangle KPQ$ intersects $AK$ again at ${T}$. Prove that:
(1) $\angle BTC+\angle APB=\angle CQA$.
(2) $AP \cdot BT \cdot CQ=AQ \cdot CT \cdot BP$.
Proposed by [i]Yijie He[/i] and [i]Yijuan Yao[/i]
2010 Postal Coaching, 2
Let $M$ be an interior point of a $\triangle ABC$ such that $\angle AM B = 150^{\circ} , \angle BM C = 120^{\circ}$. Let $P, Q, R$ be the circumcentres of the $\triangle AM B, \triangle BM C, \triangle CM A$ respectively. Prove that $[P QR] \ge [ABC]$.
2013-2014 SDML (High School), 7
Two flag poles of height $11$ and $13$ are planted vertically in level ground, and an equilateral triangle is hung as shown in the figure so that [the] lowest vertex just touches the ground. What is the length of the side of the equilateral triangle?
[asy]
pair A, B, C, D, E;
A = (0,11);
B = origin;
C = (13.8564064606,0);
D = (13.8564064606,13);
E = (8.66025403784,0);
draw(A--B--C--D--cycle);
draw(A--E--D);
label("$11$",A--B);
label("$13$",C--D);
[/asy]
2006 Turkey Team Selection Test, 2
From a point $Q$ on a circle with diameter $AB$ different from $A$ and $B$, we draw a perpendicular to $AB$, $QH$, where $H$ lies on $AB$. The intersection points of the circle of diameter $AB$ and the circle of center $Q$ and radius $QH$ are $C$ and $D$. Prove that $CD$ bisects $QH$.
2018 Azerbaijan BMO TST, 2
Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.
1974 Canada National Olympiad, 5
Given a circle with diameter $AB$ and a point $X$ on the circle different from $A$ and $B$, let $t_{a}$, $t_{b}$ and $t_{x}$ be the tangents to the circle at $A$, $B$ and $X$ respectively. Let $Z$ be the point where line $AX$ meets $t_{b}$ and $Y$ the point where line $BX$ meets $t_{a}$. Show that the three lines $YZ$, $t_{x}$ and $AB$ are either concurrent (i.e., all pass through the same point) or parallel.
[img]6762[/img]
2021 Estonia Team Selection Test, 1
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.
Proposed by United Kingdom
2011 Romania Team Selection Test, 3
The incircle of a triangle $ABC$ touches the sides $BC,CA,AB$ at points $D,E,F$, respectively. Let $X$ be a point on the incircle, different from the points $D,E,F$. The lines $XD$ and $EF,XE$ and $FD,XF$ and $DE$ meet at points $J,K,L$, respectively. Let further $M,N,P$ be points on the sides $BC,CA,AB$, respectively, such that the lines $AM,BN,CP$ are concurrent. Prove that the lines $JM,KN$ and $LP$ are concurrent.
[i]Dinu Serbanescu[/i]
1982 Vietnam National Olympiad, 1
Determine a quadric polynomial with intergral coefficients whose roots are $\cos 72^{\circ}$ and $\cos 144^{\circ}.$
2006 Putnam, A2
Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)
2014 China Team Selection Test, 2
Given a fixed positive integer $a\geq 9$. Prove: There exist finitely many positive integers $n$, satisfying:
(1)$\tau (n)=a$
(2)$n|\phi (n)+\sigma (n)$
Note: For positive integer $n$, $\tau (n)$ is the number of positive divisors of $n$, $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$, $\sigma (n)$ is the sum of positive divisors of $n$.
2008 Regional Olympiad of Mexico Center Zone, 6
In the quadrilateral $ABCD$, we have $AB = AD$ and $\angle B = \angle D = 90 ^ \circ $. The points $P$ and $Q $ lie on $BC$ and $CD$, respectively, so that $AQ$ is perpendicular on $DP$. Prove that $AP$ is perpendicular to $BQ$.