Found problems: 85335
2015 ASDAN Math Tournament, 18
Andrew takes a square sheet of paper $ABCD$ of side length $1$ and folds a kite shape. To do this, he takes the corners at $B$ and $D$ and folds the paper such that both corners now rest at a point $E$ on $AC$. This fold results in two creases $CF$ and $CG$, respectively, where $F$ lies on $AB$ and $G$ lies on $AD$. Compute the length of $FG$.
2022 CCA Math Bonanza, I4
Burrito Bear has a white unit square. She inscribes a circle inside of the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as $\frac{a\pi+b}{c}$. Find $a+b+c$.
[i]2022 CCA Math Bonanza Individual Round #4[/i]
2008 AMC 12/AHSME, 11
Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $ 13$ visible numbers have the greatest possible sum. What is that sum?
[asy]unitsize(.8cm);
pen p = linewidth(.8pt);
draw(shift(-2,0)*unitsquare,p);
label("1",(-1.5,0.5));
draw(shift(-1,0)*unitsquare,p);
label("2",(-0.5,0.5));
label("32",(0.5,0.5));
draw(shift(1,0)*unitsquare,p);
label("16",(1.5,0.5));
draw(shift(0,1)*unitsquare,p);
label("4",(0.5,1.5));
draw(shift(0,-1)*unitsquare,p);
label("8",(0.5,-0.5));[/asy]$ \textbf{(A)}\ 154 \qquad \textbf{(B)}\ 159 \qquad \textbf{(C)}\ 164 \qquad \textbf{(D)}\ 167 \qquad \textbf{(E)}\ 189$
2019 Online Math Open Problems, 1
Compute the sum of all positive integers $n$ such that the median of the $n$ smallest prime numbers is $n$.
[i]Proposed by Luke Robitaille[/i]
2015 Caucasus Mathematical Olympiad, 1
Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.
1990 Turkey Team Selection Test, 5
Let $b_m$ be numbers of factors $2$ of the number $m!$ (that is, $2^{b_m}|m!$ and $2^{b_m+1}\nmid m!$). Find the least $m$ such that $m-b_m = 1990$.
2023 Estonia Team Selection Test, 4
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
2023 Germany Team Selection Test, 2
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
2022 Regional Competition For Advanced Students, 4
We are given the set $$M = \{-2^{2022}, -2^{2021}, . . . , -2^{2}, -2, -1, 1, 2, 2^2, . . . , 2^{2021}, 2^{2022}\}.$$
Let $T$ be a subset of $M$, such that neighbouring numbers have the same difference when the elements are ordered by size.
(a) Determine the maximum number of elements that such a set $T$ can contain.
(b) Determine all sets $T$ with the maximum number of elements.
[i](Walther Janous)[/i]
2010 China Team Selection Test, 1
Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings:
(1) $\sum_{v\in V} f(v)=|E|$;
(2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$.
Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.
1950 AMC 12/AHSME, 2
Let $R=gS-4$. When $S=8$, $R=16$. When $S=10$, $R$ is equal to:
$\textbf{(A)}\ 11 \qquad
\textbf{(B)}\ 14 \qquad
\textbf{(C)}\ 20\qquad
\textbf{(D)}\ 21 \qquad
\textbf{(E)}\ \text{None of these}$
2001 Hungary-Israel Binational, 5
In a triangle $ABC$ , $B_{1}$ and $C_{1}$ are the midpoints of $AC$ and $AB$ respectively, and $I$ is the incenter. The lines $B_{1}I$ and $C_{1}I$ meet $AB$ and $AC$ respectively at $C_{2}$ and $B_{2}$ . If the areas of $\Delta ABC$ and $\Delta AB_{2}C_{2}$ are equal, find $\angle{BAC}$ .
2015 Federal Competition For Advanced Students, 2
Let $ABC$ be an acute-angled triangle with $AC < AB$ and circumradius $R$. Furthermore, let $D$ be the foot ofthe altitude from $A$ on $BC$ and let $T$ denote the point on the line $AD$ such that $AT = 2R$ holds with $D$ lying between $A$ and $T$. Finally, let $S$ denote the mid-point of the arc $BC$ on the circumcircle that does not include $A$.
Prove: $\angle AST = 90^\circ$.
(Karl Czakler)
1967 German National Olympiad, 4
Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?
2022 VTRMC, 2
Let $A$ and $B$ be the two foci of an ellipse and let $P$ be a point on this ellipse. Prove that the focal radii of $P$ (that is, the segments $\overline{AP}$ and $\overline{BP}$ ) form equal angles with the tangent to the ellipse at $P$.
1988 AIME Problems, 14
Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form
\[ 12x^2 + bxy + cy^2 + d = 0. \]
Find the product $bc$.
1999 Federal Competition For Advanced Students, Part 2, 1
Prove that for each positive integer $n$, the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.
2018 Latvia Baltic Way TST, P8
Let natural $n \ge 2$ be given. Let Laura be a student in a class of more than $n+2$ students, all of which participated in an olympiad and solved some problems. Additionally, it is known that:
[list]
[*] for every pair of students there is exactly one problem that was solved by both students;
[*] for every pair of problems there is exactly one student who solved both of them;
[*] one specific problem was solved by Laura and exactly $n$ other students.
[/list]
Determine the number of students in Laura's class.
2003 Kazakhstan National Olympiad, 5
Prove that for all primes $p>3$, $\binom{2p}{p}-2$ is divisible by $p^3$
2005 Junior Balkan Team Selection Tests - Romania, 5
On the sides $AD$ and $BC$ of a rhombus $ABCD$ we consider the points $M$ and $N$ respectively. The line $MC$ intersects the segment $BD$ in the point $T$, and the line $MN$ intersects the segment $BD$ in the point $U$. We denote by $Q$ the intersection between the line $CU$ and the side $AB$ and with $P$ the intersection point between the line $QT$ and the side $CD$.
Prove that the triangles $QCP$ and $MCN$ have the same area.
1978 IMO Longlists, 54
Let $p, q$ and $r$ be three lines in space such that there is no plane that is parallel to all three of them. Prove that there exist three planes $\alpha, \beta$, and $\gamma$, containing $p, q$, and $r$ respectively, that are perpendicular to each other $(\alpha\perp\beta, \beta\perp\gamma, \gamma\perp \alpha).$
1978 IMO Longlists, 7
Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.
1993 IMO Shortlist, 3
Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.
2007 Stars of Mathematics, 4
Show that any subset of $ A=\{ 1,2,...,2007\} $ having $ 27 $ elements contains three distinct numbers such that the greatest common divisor of two of them divides the other one.
[i]Dan Schwarz[/i]
1997 Tournament Of Towns, (564) 5
Dima invented a secret code in which every letter is replaced by a word no longer than $10$ letters. A code is called “good” if every encoded word can be decoded in only one way. Serjozha (with the help of a computer) checked that for Dima’s code, every possible word of at most $10000$ letters can be decoded in only one way. Does it follow that Dima’s code is good? (Note that Dima and Serjozha are Russian, so they use the Cyrillic alphabet, which has $ 33$ letters! A word is any sequence of letters.)
(D Piontkovskiy, S Shalunov)