Found problems: 85335
2012 ELMO Problems, 5
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$.
[i]Calvin Deng.[/i]
2018 Danube Mathematical Competition, 4
Let $M$ be the set of positive odd integers.
For every positive integer $n$, denote $A(n)$ the number of the subsets of $M$ whose sum of elements equals $n$.
For instance, $A(9) = 2$, because there are exactly two subsets of $M$ with the sum of their elements equal to $9$: $\{9\}$ and $\{1, 3, 5\}$.
a) Prove that $A(n) \le A(n + 1)$ for every integer $n \ge 2$.
b) Find all the integers $n \ge 2$ such that $A(n) = A(n + 1)$
2017 Saint Petersburg Mathematical Olympiad, 1
Sasha’s computer can do the following two operations: If you load the card with number $a$, it will return that card back and also prints another card with number $a+1$, and if you consecutively load the cards with numbers $a$ and $b$, it will return them back and also prints cards with all the roots of the quadratic trinomial $x^2+ax+b$ (possibly one, two, or none cards.) Initially, Sasha had only one card with number $s$. Is it true that, for any $s> 0$, Sasha can get a card with number $\sqrt{s}$?
2007 Today's Calculation Of Integral, 193
For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$.
Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin.
(1) Find the equation of $l$.
(2) Let $S(a)$ be the area of the region bounded by $OC$, $CA$ and $y=x^{2}$. Let $T(a)$ be the area of the region bounded by $AB$ and $y=x^{2}$. Find $\lim_{a \to \infty}\frac{T(a)}{S(a)}$.
2012 Putnam, 1
Let $S$ be a class of functions from $[0,\infty)$ to $[0,\infty)$ that satisfies:
(i) The functions $f_1(x)=e^x-1$ and $f_2(x)=\ln(x+1)$ are in $S;$
(ii) If $f(x)$ and $g(x)$ are in $S,$ the functions $f(x)+g(x)$ and $f(g(x))$ are in $S;$
(iii) If $f(x)$ and $g(x)$ are in $S$ and $f(x)\ge g(x)$ for all $x\ge 0,$ then the function $f(x)-g(x)$ is in $S.$
Prove that if $f(x)$ and $g(x)$ are in $S,$ then the function $f(x)g(x)$ is also in $S.$
2024 MMATHS, 7
The sum $\sum_{x=-5}^5\sum_{y=-5}^5\frac{2^x3^y}{(1+2^x)(1+3^y)}$ can be expressed as a fraction $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
1988 Austrian-Polish Competition, 8
We are given $1988$ unit cubes. Using some or all of these cubes, we form three quadratic boards $A, B,C$ of dimensions $a \times a \times 1$, $b \times b \times 1$, and $c \times c \times 1$ respectively, where $a \le b \le c$. Now we place board $B$ on board $C$ so that each cube of $B$ is precisely above a cube of $C$ and $B$ does not overlap $C$. Similarly, we place $A$ on $B$. This gives us a three-floor tower. What choice of $a, b$ and $c$ gives the maximum number of such three-floor towers?
2014 China Team Selection Test, 4
Let $k$ be a fixed odd integer, $k>3$. Prove: There exist infinitely many positive integers $n$, such that there are two positive integers $d_1, d_2$ satisfying $d_1,d_2$ each dividing $\frac{n^2+1}{2}$, and $d_1+d_2=n+k$.
1998 Harvard-MIT Mathematics Tournament, 3
Finds the sum of the infinite series $1+2\left(\dfrac{1}{1998}\right)+3\left(\dfrac{1}{1998}\right)^2+4\left(\dfrac{1}{1998}\right)^3+\cdots$.
2022 CMIMC, 5
For any integer $a$, let $f(a) = |a^4 - 36a^2 + 96a - 64|$. What is the sum of all values of $f(a)$ that are prime?
[i]Proposed by Alexander Wang[/i]
2020 June Advanced Contest, 3
Let a [i]lattice tetrahedron[/i] denote a tetrahedron whose vertices have integer coordinates. Given a lattice tetrahedron, a [i]move[/i] consists of picking some vertex and moving it parallel to one of the three edges of the face opposite the vertex so that it lands on a different point with integer coordinates. Prove that any two lattice tetrahedra with the same volume can be transformed into each other by a series of moves
1975 IMO Shortlist, 12
Consider on the first quadrant of the trigonometric circle the arcs $AM_1 = x_1,AM_2 = x_2,AM_3 = x_3, \ldots , AM_v = x_v$ , such that $x_1 < x_2 < x_3 < \cdots < x_v$. Prove that
\[\sum_{i=0}^{v-1} \sin 2x_i - \sum_{i=0}^{v-1} \sin (x_i- x_{i+1}) < \frac{\pi}{2} + \sum_{i=0}^{v-1} \sin (x_i + x_{i+1})\]
2023 Indonesia TST, 3
Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.
LMT Team Rounds 2010-20, 2020.S9
A function $f(x)$ is such that for any integer $x$, $f(x)+xf(2-x)=6$. Compute $-2019f(2020)$.
2021 239 Open Mathematical Olympiad, 7
Given $n$ lines on the plane, they divide the plane onto several
bounded or bounded polygonal regions. Define the rank of a region as
the number of vertices on its boundary (a vertex is a point which
belongs to at least two lines). Prove that the sum of squares of
ranks of all regions does not exceed $10n^2$.
(D. Fomin)
2002 Italy TST, 2
Prove that for each prime number $p$ and positive integer $n$, $p^n$ divides
\[\binom{p^n}{p}-p^{n-1}. \]
2014 Contests, 2
Let $ AB$ be the diameter of semicircle $O$ ,
$C, D $ be points on the arc $AB$,
$P, Q$ be respectively the circumcenter of $\triangle OAC $ and $\triangle OBD $ .
Prove that:$CP\cdot CQ=DP \cdot DQ$.[asy]
import cse5; import olympiad; unitsize(3.5cm); dotfactor=4; pathpen=black;
real h=sqrt(55/64);
pair A=(-1,0), O=origin, B=(1,0),C=shift(-3/8,h)*O,D=shift(4/5,3/5)*O,P=circumcenter(O,A,C), Q=circumcenter(O,D,B);
D(arc(O,1,0,180),darkgreen);
D(MP("A",A,W)--MP("C",C,N)--MP("P",P,SE)--MP("D",D,E)--MP("Q",Q,E)--C--MP("O",O,S)--D--MP("B",B,E)--cycle,deepblue);
D(O);
[/asy]
2016 Purple Comet Problems, 22
In $\triangle{ABC}$, $cos\angle{A} =\frac{2}{3}$, $cos\angle{B} =\frac{1}{9}$, and $BC = 24$. Find the length $AC$.
2019 Belarusian National Olympiad, 11.1
[b]a)[/b] Find all real numbers $a$ such that the parabola $y=x^2-a$ and the hyperbola $y=1/x$ intersect each other in three different points.
[b]b)[/b] Find the locus of centers of circumcircles of such triples of intersection points when $a$ takes all possible values.
[i](I. Gorodnin)[/i]
2020 ISI Entrance Examination, 3
Let $A$ and $B$ be variable points on $x-$axis and $y-$axis respectively such that the line segment $AB$ is in the first quadrant and of a fixed length $2d$ . Let $C$ be the mid-point of $AB$ and $P$ be a point such that
[b](a)[/b] $P$ and the origin are on the opposite sides of $AB$ and,
[b](b)[/b] $PC$ is a line segment of length $d$ which is perpendicular to $AB$ .
Find the locus of $P$ .
2011 National Olympiad First Round, 10
How many interger tuples $(x,y,z)$ are there satisfying $0\leq x,y,z < 2011$, $xy+yz+zx \equiv 0 \pmod{2011}$, and $x+y+z \equiv 0 \pmod{2011}$ ?
$\textbf{(A)}\ 2010 \qquad\textbf{(B)}\ 2011 \qquad\textbf{(C)}\ 2012 \qquad\textbf{(D)}\ 4021 \qquad\textbf{(E)}\ 4023$
2019 Simon Marais Mathematical Competition, A3
For some positive integer $n$, a coin will be flipped $n$ times to obtain a sequence of $n$ heads and tails. For each flip of the coin, there is probability $p$ of obtaining a head and probability $1-p$ of obtaining a tail, where $0<p<1$ is a rational number.
Kim writes all $2^n$ possible sequences of $n$ heads and tails in two columns, with some sequences in the left column and the remaining sequences in the right column. Kim would like the sequence produced by the coin flips to appear in the left column with probability $1/2$.
Determine all pairs $(n,p)$ for which this is possible.
2017 Junior Balkan Team Selection Tests - Romania, 2
Given $x_1,x_2,...,x_n$ real numbers, prove that there exists a real number $y$, such that,
$$\{y-x_1\}+\{y-x_2\}+...+\{y-x_n\} \leq \frac{n-1}{2}$$
2022 VJIMC, 2
Let $n\ge1$. Assume that $A$ is a real $n\times n$ matrix which satisfies the equality
$$A^7+A^5+A^3+A-I=0.$$
Show that $\det(A)>0$.
2011 Junior Balkan Team Selection Tests - Moldova, 5
The real numbers $a, b$ satisfy $| a | \ne | b |$ and $$ \frac{a + b}{a - b}+\frac{a - b}{a + b}= -\frac52.$$
Determine the value of the expression $$E= \frac{a^4 - b^4}{a^4 + b^4} - \frac{a^4 + b^4}{a^4- b^4}.$$