Found problems: 85335
1994 Italy TST, 3
Find all functions $f : R \to R$ satisfying the condition $f(x- f(y)) = 1+x-y$ for all $x,y \in R$.
2010 Stanford Mathematics Tournament, 3
Bob sends a secret message to Alice using her RSA public key $n = 400000001.$ Eve wants to listen in on their conversation. But to do this, she needs Alice's private key, which is the factorization of $n.$ Eve knows that $n = pq,$ a product of two prime factors. Find $p$ and $q.$
2020 March Advanced Contest, 3
A [i]simple polygon[/i] is a polygon whose perimeter does not self-intersect. Suppose a simple polygon $\mathcal P$ can be tiled with a finite number of parallelograms. Prove that regardless of the tiling, the sum of the areas of all rectangles in the tiling is fixed.\\
[i]Note:[/i] Points will be awarded depending on the generality of the polygons for which the result is proven.
2014 Contests, 2
Let $n \ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is [i]peaceful[/i] if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.
1949-56 Chisinau City MO, 3
Prove that the number $N = 10 ...050...01$ (1, 49 zeros, 5 , 99 zeros, 1) is a not cube of an integer.
2021 CCA Math Bonanza, I9
Points $A$, $B$, $C$, $D$, and $E$ are on the same plane such that $A,E,C$ lie on a line in that order, $B,E,D$ lie on a line in that order, $AE = 1$, $BE = 4$, $CE = 3$, $DE = 2$, and $\angle AEB = 60^\circ$. Let $AB$ and $CD$ intersect at $P$. The square of the area of quadrilateral $PAED$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[i]2021 CCA Math Bonanza Individual Round #9[/i]
1996 Israel National Olympiad, 4
Eight guests arrive to a hotel with four rooms. Each guest dislikes at most three other guests and doesn’t want to share a room with any of them (this feeling is mutual). Show that the guests can reside in the four rooms, with two persons in each room
2010 China Team Selection Test, 1
Let $ABCD$ be a convex quadrilateral with $A,B,C,D$ concyclic. Assume $\angle ADC$ is acute and $\frac{AB}{BC}=\frac{DA}{CD}$. Let $\Gamma$ be a circle through $A$ and $D$, tangent to $AB$, and let $E$ be a point on $\Gamma$ and inside $ABCD$.
Prove that $AE\perp EC$ if and only if $\frac{AE}{AB}-\frac{ED}{AD}=1$.
2014 India IMO Training Camp, 2
For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold:
$x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$
$v_{1},v_{2},v_{3}$ satisfy triangle inequality
$v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality.
Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.
2013 AMC 12/AHSME, 1
Square $ ABCD $ has side length $ 10 $. Point $ E $ is on $ \overline{BC} $, and the area of $ \bigtriangleup ABE $ is $ 40 $. What is $ BE $?
$\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8 \qquad $
[asy]
pair A,B,C,D,E;
A=(0,0);
B=(0,50);
C=(50,50);
D=(50,0);
E = (30,50);
draw(A--B);
draw(B--E);
draw(E--C);
draw(C--D);
draw(D--A);
draw(A--E);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
label("A",A,SW);
label("B",B,NW);
label("C",C,NE);
label("D",D,SE);
label("E",E,N);
[/asy]
2021-2022 OMMC, 9
$12$ people stand in a row. Each person is given a red shirt or a blue shirt. Every minute, exactly one pair of two people with the same color currently standing next to each other in the row leave. After $6$ minutes, everyone has left. How many ways could the shirts have been assigned initially?
[i]Proposed by Evan Chang[/i]
2019 China Team Selection Test, 4
Does there exist a finite set $A$ of positive integers of at least two elements and an infinite set $B$ of positive integers, such that any two distinct elements in $A+B$ are coprime, and for any coprime positive integers $m,n$, there exists an element $x$ in $A+B$ satisfying $x\equiv n \pmod m$ ?
Here $A+B=\{a+b|a\in A, b\in B\}$.
2002 Romania Team Selection Test, 3
Let $M$ and $N$ be the midpoints of the respective sides $AB$ and $AC$ of an acute-angled triangle $ABC$. Let $P$ be the foot of the perpendicular from $N$ onto $BC$ and let $A_1$ be the midpoint of $MP$. Points $B_1$ and $C_1$ are obtained similarly. If $AA_1$, $BB_1$ and $CC_1$ are concurrent, show that the triangle $ABC$ is isosceles.
[i]Mircea Becheanu[/i]
2008 AMC 12/AHSME, 3
A semipro baseball league has teams with $ 21$ players each. League rules state that a player must be paid at least $ \$15,000$, and that the total of all players' salaries for each team cannot exceed $ \$700,000$. What is the maximum possiblle salary, in dollars, for a single player?
$ \textbf{(A)}\ 270,000 \qquad
\textbf{(B)}\ 385,000 \qquad
\textbf{(C)}\ 400,000 \qquad
\textbf{(D)}\ 430,000 \qquad
\textbf{(E)}\ 700,000$
2013 NIMO Problems, 1
Find the sum of all primes that can be written both as a sum of two primes and as a difference of two primes.
[i]Anonymous Proposal[/i]
2015 HMMT Geometry, 2
Let $ABC$ be a triangle with orthocenter $H$; suppose $AB=13$, $BC=14$, $CA=15$. Let $G_A$ be the centroid of triangle $HBC$, and define $G_B$, $G_C$ similarly. Determine the area of triangle $G_AG_BG_C$.
1987 China Team Selection Test, 2
Find all positive integer $n$ such that the equation $x^3+y^3+z^3=n \cdot x^2 \cdot y^2 \cdot z^2$ has positive integer solutions.
2012 Puerto Rico Team Selection Test, 3
$ABC$ is a triangle that is inscribed in a circle. The angle bisectors of $A, B, C$ meet the circle at $D,
E, F$, respectively. Show that $AD$ is perpendicular to $EF$.
2016 USA Team Selection Test, 2
Let $n \ge 4$ be an integer. Find all functions $W : \{1, \dots, n\}^2 \to \mathbb R$ such that for every partition $[n] = A \cup B \cup C$ into disjoint sets, \[ \sum_{a \in A} \sum_{b \in B} \sum_{c \in C} W(a,b) W(b,c) = |A| |B| |C|. \]
2012 CHMMC Fall, 1
Find the remainder when $5^{2012}$ is divided by $3$.
2014 Singapore Senior Math Olympiad, 11
Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$. Find the value of $2^{-(1+\log_23)x}$
2008 Bulgarian Autumn Math Competition, Problem 10.1
For which values of the parameter $a$ does the equation
\[(2x-a)\sqrt{ax^2-(a^2+a+2)x+2(a+1)}=0\]
has three different real roots.
2004 AMC 8, 16
Two $600$ ml pitchers contain orange juice. One pitcher is $\frac{1}{3}$ full and the other pitcher is $\frac{2}{5}$ full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice?
$\textbf{(A)}\ \frac{1}{8}\qquad
\textbf{(B)}\ \frac{3}{16}\qquad
\textbf{(C)}\ \frac{11}{30}\qquad
\textbf{(D)}\ \frac{11}{19}\qquad
\textbf{(E)}\ \frac{11}{15}$
1986 China National Olympiad, 2
In $\triangle ABC$, the length of altitude $AD$ is $12$, and the bisector $AE$ of $\angle A$ is $13$. Denote by $m$ the length of median $AF$. Find the range of $m$ when $\angle A$ is acute, orthogonal and obtuse respectively.
2018 Saudi Arabia JBMO TST, 3
Prove that in every triangle there are two sides with lengths $x$ and $y$ such that $$\frac{\sqrt{5}-1}{2}\leq\frac{x}{y}\leq\frac{\sqrt{5}+1}{2}$$