Found problems: 85335
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 $$\frac{\sqrt{5}-1}{2}\leq\frac{x}{y}\leq\frac{\sqrt{5}+1}{2}$$
2024 ELMO Shortlist, A4
The number $2024$ is written on a blackboard. Each second, if there exist positive integers $a,b,k$ such that $a^k+b^k$ is written on the blackboard, you may write $a^{k'}+b^{k'}$ on the blackboard for any positive integer $k'.$ Find all positive integers that you can eventually write on the blackboard.
[i]Srinivas Arun[/i]
2010 Tuymaada Olympiad, 1
We have a set $M$ of real numbers with $|M|>1$ such that for any $x\in M$ we have either $3x-2\in M$ or $-4x+5\in M$.
Show that $M$ is infinite.
2009 ISI B.Math Entrance Exam, 6
Let $a,b,c,d$ be integers such that $ad-bc$ is non zero. Suppose $b_1,b_2$ are integers both of which are multiples of $ad-bc$. Prove that there exist integers simultaneously satisfying both the equalities $ax+by=b_1, cx+dy=b_2$.
Kvant 2024, M2783
The sum of the digits of a natural number is $k{}.$ What is the largest possible sum of digits for[list=a]
[*] the square of this number;
[*]the fourth power of this number,
[/list] given that $k\geqslant 4.$
[i]From the folklore[/i]
1996 Romania National Olympiad, 2
$ a,b,c,d \in [0,1]$ and $ x,y,z,t \in [0, \frac{1}{2}]$ and $ a+b+c+d=x+y+z+t=1$.prove that:
$ (i)$ $ ax+by+cz+dt$ $ \geq$ $ min( {\frac{a+b}{2} , \frac{b+c}{2} , \frac{c+d}{2} , \frac{d+a}{2} , \frac{a+c}{2} , \frac{b+d}{2} )}$
$ (ii)$ $ ax+by+cz+dt$ $ \geq$ $ 54abcd$
2014 China Northern MO, 1
As shown in the figure, given $\vartriangle ABC$ with $\angle B$, $\angle C$ acute angles, $AD \perp BC$, $DE \perp AC$, $M$ midpoint of $DE$, $AM \perp BE$. Prove that $\vartriangle ABC$ is isosceles.
[img]https://cdn.artofproblemsolving.com/attachments/a/8/f553c33557979f6f7b799935c3bde743edcc3c.png[/img]
2005 Today's Calculation Of Integral, 75
A function $f(\theta)$ satisfies the following conditions $(a),(b)$.
$(a)\ f(\theta)\geq 0$
$(b)\ \int_0^{\pi} f(\theta)\sin \theta d\theta =1$
Prove the following inequality.
\[\int_0^{\pi} f(\theta)\sin n\theta \ d\theta \leq n\ (n=1,2,\cdots)\]
2017 Indonesia MO, 1
$ABCD$ is a parallelogram. $g$ is a line passing $A$. Prove that the distance from $C$ to $g$ is either the sum or the difference of the distance from $B$ to $g$, and the distance from $D$ to $g$.
2024/2025 TOURNAMENT OF TOWNS, P5
Given a circle ${\omega }_{1}$ , and a circle ${\omega }_{2}$ inside it. An arbitrary circle ${\omega }_{3}$ is chosen which is tangent to the two latter circles and both tangencies are internal. The tangency points are linked by a segment. A tangent line to ${\omega }_{2}$ is drawn through the meet point of this segment and the circle ${\omega }_{2}$ . Thus a chord of the circle ${\omega }_{3}$ is obtained. Prove that the ends of all such chords (obtained by all possible choices of ${\omega }_{3}$ ) belong to a fixed circle.
Pavel Kozhevnikov