Found problems: 85335
2005 Today's Calculation Of Integral, 58
Let $f(x)=\frac{e^x}{e^x+1}$
Prove the following equation.
\[\int_a^b f(x)dx+\int_{f(a)}^{f(b)} f^{-1}(x)dx=bf(b)-af(a)\]
2016 Junior Balkan Team Selection Test, 1
Let rightangled $\triangle ABC$ be given with right angle at vertex $C$. Let $D$ be foot of altitude from $C$ and let $k$ be circle that touches $BD$ at $E$, $CD$ at $F$ and circumcircle of $\triangle ABC$ at $G$.
$a.)$ Prove that points $A$, $F$ and $G$ are collinear.
$b.)$ Express radius of circle $k$ in terms of sides of $\triangle ABC$.
2017 Pan African, Problem 1
We consider the real sequence $(x_n)$ defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2x_n$ for $n=0,1,...$
We define the sequence $(y_n)$ by $y_n=x_n^2+2^{n+2}$ for every non negative integer $n$.
Prove that for every $n>0$, $y_n$ is the square of an odd integer
2006 Princeton University Math Competition, 8
Given that triangle $ABC$ has side lengths $a=7$, $b=8$ , $c=5$, find $$(\sin (A)+\sin (B)+\sin (C)) \cdot \left(\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}\right).$$
.
2004 Spain Mathematical Olympiad, Problem 2
${ABCD}$ is a quadrilateral, ${P}$ and ${Q}$ are midpoints of the diagonals ${BD}$ and ${AC}$, respectively. The lines parallel to the diagonals originating from ${P}$ and ${Q}$ intersect in the point ${O}$. If we join the four midpoints of the sides, ${X}$, ${Y}$, ${Z}$, and ${T}$, to ${O}$, we form four quadrilaterals: ${OXBY}$, ${OYCZ}$, ${OZDT}$, and ${OTAX}$. Prove that the four newly formed quadrilaterals have the same areas.
2024 Singapore Junior Maths Olympiad, Q1
Let $ABC$ be an isosceles right-angled triangle of area 1. Find the length of the shortest segment that divides the triangle into 2 parts of equal area.
2014 Chile TST Ibero, 3
Let $x_0 = 5$ and define the sequence recursively as $x_{n+1} = x_n + \frac{1}{x_n}$. Prove that:
\[
45 < x_{1000} < 45.1.
\]
2013 Putnam, 3
Suppose that the real numbers $a_0,a_1,\dots,a_n$ and $x,$ with $0<x<1,$ satisfy \[\frac{a_0}{1-x}+\frac{a_1}{1-x^2}+\cdots+\frac{a_n}{1-x^{n+1}}=0.\] Prove that there exists a real number $y$ with $0<y<1$ such that \[a_0+a_1y+\cdots+a_ny^n=0.\]
2011 India IMO Training Camp, 1
Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that:
$a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area.
$b) a\cdot AP=b\cdot BP=c\cdot PC.$
2024 Bundeswettbewerb Mathematik, 1
Determine all pairs $(x,y)$ of integers satisfying
\[(x+2)^4-x^4=y^3.\]
1985 AMC 8, 19
If the length and width of a rectangle are each increased by $ 10 \%$, then the perimeter of the rectangle is increased by
\[ \textbf{(A)}\ 1 \% \qquad
\textbf{(B)}\ 10 \% \qquad
\textbf{(C)}\ 20 \% \qquad
\textbf{(D)}\ 21 \% \qquad
\textbf{(E)}\ 40 \%
\]
2011 District Olympiad, 2
Let $ G $ be the set of matrices of the form $ \begin{pmatrix} a&b\\0&1 \end{pmatrix} , $ with $ a,b\in\mathbb{Z}_7,a\neq 0. $
[b]a)[/b] Verify that $ G $ is a group.
[b]b)[/b] Show that $ \text{Hom}\left( (G,\cdot) ; \left( \mathbb{Z}_7,+ \right) \right) =\{ 0\} $
2019 Philippine MO, 1
Find all functions $f : R \to R$ such that $f(2xy) + f(f(x + y)) = xf(y) + yf(x) + f(x + y)$ for all real numbers $x$ and $y$.
2016 CHMMC (Fall), 10
Let $ABC$ be a triangle with circumcircle $\omega$ such that $AB = 11$, $AC = 13$, and $\angle A = 30^o$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = 7$ and $AE = 8$. There exists a unique point $F \ne A$ on minor arc $AB$ of $\omega$ such that $\angle F DA = \angle F EA$. Compute $F A^2$.
2002 Taiwan National Olympiad, 3
Suppose $x,y,,a,b,c,d,e,f$ are real numbers satifying
i)$\max{(a,0)}+\max{(b,0)}<x+ay+bz<1+\min{(a,0)}+\min{(b,0)}$, and
ii)$\max{(c,0)}+\max{(d,0)}<cx+y+dz<1+\min{(c,0)}+\min{(d,0)}$, and
iii)$\max{(e,0)}+\max{(f,0)}<ex+fy+z<1+\min{(e,0)}+\min{(f,0)}$.
Prove that $0<x,y,z<1$.
2005 Estonia Team Selection Test, 3
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.
2025 Czech-Polish-Slovak Junior Match., 2
Find all triangles that can be divided into congruent right-angled isosceles triangles with side lengths $1, 1, \sqrt{2}$.
STEMS 2021-22 Math Cat A-B, A4 B3
Consider the starting position in a game of bughouse. Exhibit a sequence of moves
on both boards, indicating the chronology, such that at the end:
(a) The positions on both boards are the same as the original positions.
(b) It is White to play on one board, but Black to play on the other.
(c) All four players still have the right to castle subsequently (equivalently, the kings and rooks
haven’t moved).
for each of the following cases :
(a) without moving any pawns.
(b) without moving any queen.
2014 Middle European Mathematical Olympiad, 5
Let $ABC$ be a triangle with $AB < AC$. Its incircle with centre $I$ touches the sides $BC, CA,$ and $AB$ in the points $D, E,$ and $F$ respectively. The angle bisector $AI$ intersects the lines $DE$ and $DF$ in the points $X$ and $Y$ respectively. Let $Z$ be the foot of the altitude through $A$ with respect to $BC$.
Prove that $D$ is the incentre of the triangle $XYZ$.
2023 Iran MO (3rd Round), 1
Find all integers $n > 4$ st for every two subsets $A,B$ of $\{0,1,....,n-1\}$ , there exists a polynomial $f$ with integer coefficients st either $f(A) = B$ or $f(B) = A$ where the equations are considered mod n.
We say two subsets are equal mod n if they produce the same set of reminders mod n. and the set $f(X)$ is the set of reminders of $f(x)$ where $x \in X$ mod n.
2009 Singapore MO Open, 2
a palindromic number is a number which is unchanged when order of its digits is reversed.
prove that the arithmetic progression 18, 37,.. contains infinitely many palindromic numbers.
1954 Putnam, B5
Let $f(x)$ be a real-valued function, defined for $-1<x<1$ for which $f'(0)$ exists. Let $(a_n) , (b_n)$ be two sequences such that $-1 <a_n <0 <b_n <1$ for all $n$ and $\lim_{n \to \infty } a_n = 0 =\lim_{n \to \infty} b_n.$
Prove that
$$ \lim_{n \to \infty} \frac{ f(b_n )- f(a_n ) }{b_n -a_n} =f'(0).$$
2009 Sharygin Geometry Olympiad, 6
Let $M, I$ be the centroid and the incenter of triangle $ABC, A_1$ and $B_1$ be the touching points of the incircle with sides $BC$ and $AC, G$ be the common point of lines $AA_1$ and $BB_1$. Prove that angle $\angle CGI$ is right if and only if $GM // AB$.
(A.Zaslavsky)
2018 Iran MO (3rd Round), 3
Find the smallest positive integer $n$ such that we can write numbers $1,2,\dots ,n$ in a 18*18 board such that:
i)each number appears at least once
ii)In each row or column,there are no two numbers having difference 0 or 1
1978 Miklós Schweitzer, 8
Let $ X_1, \ldots ,X_n$ be $ n$ points in the unit square ($ n>1$). Let $ r_i$ be the distance of $ X_i$ from the nearest point (other than $ X_i$). Prove that the inequality \[ r_1^2\plus{} \ldots \plus{}r_n^2 \leq 4.\]
[i]L. Fejes-Toth, E. Szemeredi[/i]