Found problems: 85335
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 $BQ$.
2009 Saint Petersburg Mathematical Olympiad, 2
$[x,y]-[x,z]=y-z$ and $x \neq y \neq z \neq x$
Prove, that $x|y,x|z$
1994 AMC 12/AHSME, 22
Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose their chairs?
$ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 630 $
1984 National High School Mathematics League, 3
For any integers $1\leq n\leq m\leq5$, how many hyperbolas does the equation $\rho=\frac{1}{1-\text{C}_m^n \cos\theta}$ represent?
Note: $\text{C}_m^n=\frac{m!}{n!(m-n)!}$.
$\text{(A)}15\qquad\text{(B)}10\qquad\text{(C)}7\qquad\text{(D)}6$
2020 MBMT, 14
Mr. Schwartz has been hired to paint a row of 7 houses. Each house must be painted red, blue, or green. However, to make it aesthetically pleasing, he doesn't want any three consecutive houses to be the same color. Find the number of ways he can fulfill his task.
[i]Proposed by Daniel Monroe[/i]
2001 Brazil National Olympiad, 1
Show that for any $a,b,c$ positive reals,
\[ (a+b)(a+c) \geq 2 \sqrt{abc(a+b+c)} \]