Found problems: 85335
2013 BMT Spring, 10
If five squares of a $3 \times 3$ board initially colored white are chosen at random and blackened, what is the expected number of edges between two squares of the same color?
2005 All-Russian Olympiad, 1
Do there exist a bounded function $f: \mathbb{R}\to\mathbb{R}$ such that $f(1)>0$ and $f(x)$ satisfies an inequality $f^2(x+y)\ge f^2(x)+2f(xy)+f^2(y)$?
2019 AMC 10, 25
For how many integers $n$ between $1$ and $50$, inclusive, is
\[
\frac{(n^2-1)!}{(n!)^n}
\]an integer? (Recall that $0! = 1$.)
$\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$
2015 Iberoamerican Math Olympiad, 1
The number $125$ can be written as a sum of some pairwise coprime integers larger than $1$. Determine the largest number of terms that the sum may have.
2019 Iran Team Selection Test, 2
In a triangle $ABC$, $\angle A$ is $60^\circ$. On sides $AB$ and $AC$ we make two equilateral triangles (outside the triangle $ABC$) $ABK$ and $ACL$. $CK$ and $AB$ intersect at $S$ , $AC$ and $BL$ intersect at $R$ , $BL$ and $CK$ intersect at $T$. Prove the radical centre of circumcircle of triangles $BSK, CLR$ and $BTC$ is on the median of vertex $A$ in triangle $ABC$.
[i]Proposed by Ali Zamani[/i]
1997 Brazil Team Selection Test, Problem 5
Consider an infinite strip, divided into unit squares. A finite number of nuts is placed in some of these squares. In a step, we choose a square $A$ which has more than one nut and take one of them and put it on the square on the right, take another nut (from $A$) and put it on the square on the left. The procedure ends when all squares has at most one nut. Prove that, given the initial configuration, any procedure one takes will end after the same number of steps and with the same final configuration.
2019 India IMO Training Camp, P1
Let the points $O$ and $H$ be the circumcenter and orthocenter of an acute angled triangle $ABC.$ Let $D$ be the midpoint of $BC.$ Let $E$ be the point on the angle bisector of $\angle BAC$ such that $AE\perp HE.$ Let $F$ be the point such that $AEHF$ is a rectangle. Prove that $D,E,F$ are collinear.
2014 CHMMC (Fall), 3
Two players play a game on a pile of $n$ beans. On each player's turn, they may take exactly $1$, $4$, or $7$ beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many $n$ between $2014$ and $2050$ (inclusive) does the second player win?
2013 Harvard-MIT Mathematics Tournament, 35
Let $P$ be the number of ways to partition $2013$ into an ordered tuple of prime numbers. What is $\log_2 (P)$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor\frac{125}2\left(\min\left(\frac CA,\frac AC\right)-\frac35\right)\right\rfloor$ or zero, whichever is larger.
2008 Danube Mathematical Competition, 4
Let $ n\geq 2$ be a positive integer. Find the [u]maximum[/u] number of segments with lenghts greater than $ 1,$ determined by $ n$ points which lie on a closed disc with radius $ 1.$
2019 CCA Math Bonanza, TB4
The number $28!$ ($28$ in decimal) has base $30$ representation \[28!=Q6T32S??OCLQJ6000000_{30}\] where the seventh and eighth digits are missing. What are the missing digits? In base $30$, we have that the digits $A=10$, $B=11$, $C=12$, $D=13$, $E=14$, $F=15$, $G=16$, $H=17$, $I=18$, $J=19$, $K=20$, $L=21$, $M=22$, $N=23$, $O=24$, $P=25$, $Q=26$, $R=27$, $S=28$, $T=29$.
[i]2019 CCA Math Bonanza Tiebreaker Round #4[/i]
2021 AMC 12/AHSME Fall, 13
The angle bisector of the acute angle formed at the origin by the graphs of the lines $y=x$ and $y=3x$ has equation $y=kx$. What is $k$?
$\textbf{(A)} \: \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)} \: \frac{1+\sqrt{7}}{2} \qquad \textbf{(C)} \: \frac{2+\sqrt{3}}{2} \qquad \textbf{(D)} \: 2\qquad \textbf{(E)} \: \frac{2+\sqrt{5}}{2}$
2007 Mongolian Mathematical Olympiad, Problem 3
Let $P$ be a point outside of the triangle $ABC$ in the plane of $ABC$. Prove that by using reflections $S_{AB}$, $S_{AC}$, and $S_{BC}$ across the lines $AB$, $AC$, and $BC$ one can shift point $P$ inside the triangle $ABC$.
2010 Morocco TST, 2
Let $a$, $b$, $c$ be positive real numbers with $abc \leq a+b+c$. Show that \[ a^2 + b^2 + c^2 \geq \sqrt 3 abc. \]
[i]Cristinel Mortici, Romania[/i]
2006 Irish Math Olympiad, 4
Let $n$ be a positive integer.
Find the greatest common divisor of the numbers $\binom{2n}{1},\binom{2n}{3},\binom{2n}{5},...,\binom{2n}{2n-1}$.
2018 AMC 12/AHSME, 18
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?
$
\textbf{(A) }60 \qquad
\textbf{(B) }65 \qquad
\textbf{(C) }70 \qquad
\textbf{(D) }75 \qquad
\textbf{(E) }80 \qquad
$
2010 National Chemistry Olympiad, 3
When phenolphythalein is added to an aqueous solution containing one of the following solutes the solution turns pink. Which solute is present?
${ \textbf{(A)}\ \text{NaCl} \qquad\textbf{(B)}\ \text{KC}_2\text{H}_3\text{O}_2 \qquad\textbf{(C)}\ \text{LiBr} \qquad\textbf{(D)}\ \text{NH}_4\text{NO}_3 } $
2022 MOAA, 1
Consider the $5$ by $5$ equilateral triangular grid as shown: [img]https://cdn.artofproblemsolving.com/attachments/1/2/cac43ae24fd4464682a7992e62c99af4acaf8f.png[/img]
How many equilateral triangles are there with sides along the gridlines?
1996 VJIMC, Problem 1
Is it possible to cover the plane with the interiors of a finite number of parabolas?
1957 Moscow Mathematical Olympiad, 368
Find all real solutions of the system :
(a) $$\begin{cases}1-x_1^2=x_2 \\ 1-x_2^2=x_3\\ ...\\ 1-x_{98}^2=x_{99}\\ 1-x_{99}^2=x_1\end{cases}$$
(b)* $$\begin{cases} 1-x_1^2=x_2\\ 1-x_2^2=x_3\\ ...\\1-x_{98}^2=x_{n}\\ 1-x_{n}^2=x_1\end{cases}$$
2016 Bulgaria JBMO TST, 1
The quadrilateral $ABCD$, in which $\angle BAC < \angle DCB$ , is inscribed in a circle $c$, with center $O$. If $\angle BOD = \angle ADC = \alpha$. Find out which values of $\alpha$ the inequality $AB <AD + CD$ occurs.
1993 Bundeswettbewerb Mathematik, 2
For the real number $a$ it holds that there is exactly one square whose vertices are all on the graph with the equation $y = x^3 + ax$. Find the side length of this square.
2006 Brazil National Olympiad, 1
Let $ABC$ be a triangle. The internal bisector of $\angle B$ meets $AC$ in $P$ and $I$ is the incenter of $ABC$. Prove that if $AP+AB = CB$, then $API$ is an isosceles triangle.
2016 Thailand Mathematical Olympiad, 8
Let $\vartriangle ABC$ be an acute triangle with incenter $I$. The line passing through $I$ parallel to $AC$ intersects $AB$ at $M$, and the line passing through $I$ parallel to $AB$ intersects $AC$ at $N$. Let the line $MN$ intersect the circumcircle of $\vartriangle ABC$ at $X$ and $Y$ . Let $Z$ be the midpoint of arc $BC$ (not containing $A$). Prove that $I$ is the orthocenter of $\vartriangle XY Z$
2001 USA Team Selection Test, 9
Let $A$ be a finite set of positive integers. Prove that there exists a finite set $B$ of positive integers such that $A \subseteq B$ and
\[\prod_{x\in B} x = \sum_{x\in B} x^2.\]