Found problems: 85335
2010 Korea Junior Math Olympiad, 5
If reals $x, y, z $ satises $tan x + tan y + tan z = 2$ and $0 < x, y,z < \frac{\pi}{2}.$ Prove that
$$sin^2 x + sin^2 y + sin^2 z < 1.$$
2019 Kyiv Mathematical Festival, 5
Is it possible to fill the cells of a table of size $2019\times2019$ with pairwise distinct positive integers in such a way that in each rectangle of size $1\times2$ or $2\times1$ the larger number is divisible by the smaller one, and the ratio of the largest number in the table to the smallest one is at most $2019?$
2005 Today's Calculation Of Integral, 10
Calculate the following indefinite integrals.
[1] $\int (2x+1)\sqrt{x+2}\ dx$
[2] $\int \frac{1+\cos x}{x+\sin x}\ dx$
[3] $\int \sin ^ 5 x \cos ^ 3 x \ dx$
[4] $\int \frac{(x-3)^2}{x^4}\ dx$
[5] $\int \frac{dx}{\tan x}\ dx$
2015 Junior Regional Olympiad - FBH, 5
In how many ways you can pay $2015\$$ using bills of $1\$$, $10\$$, $100\$$ and $200\$$
2011 Mexico National Olympiad, 5
A $(2^n - 1) \times (2^n +1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into.
2012 Iran MO (3rd Round), 8
[b]a)[/b] Does there exist an infinite subset $S$ of the natural numbers, such that $S\neq \mathbb{N}$, and such that for each natural number $n\not \in S$, exactly $n$ members of $S$ are coprime with $n$?
[b]b)[/b] Does there exist an infinite subset $S$ of the natural numbers, such that for each natural number $n\in S$, exactly $n$ members of $S$ are coprime with $n$?
[i]Proposed by Morteza Saghafian[/i]
2022 Tuymaada Olympiad, 1
Arnim and Brentano have a little vase with $1500$ candies on the table and a huge sack with spare candies under the table. They play a game taking turns, Arnim begins . At each move a player can either eat $7$ candies or take $6$ candies from under the table and add them to the vase. A player cannot go under the table in two consecutive moves. A player is declared the winner if he leaves the vase empty. In any other case, if a player cannot make a move in his turn, the game is declared a tie. Is there a winning strategy for one of the players?
PEN O Problems, 10
Let $m \ge 2$ be an integer. Find the smallest integer $n>m$ such that for any partition of the set $\{m,m+1,\cdots,n\}$ into two subsets, at least one subset contains three numbers $a, b, c$ such that $c=a^{b}$.
2012 ITAMO, 1
On the sides of a triangle $ABC$ right angled at $A$ three points $D, E$ and $F$ (respectively $BC, AC$
and $AB$) are chosen so that the quadrilateral $AFDE$ is a square. If $x$ is the length of the side of the square, show that
\[\frac{1}{x}=\frac{1}{AB}+\frac{1}{AC}\]
2020 Turkey MO (2nd round), 2
Let $P$ be an interior point of acute triangle $\Delta ABC$, which is different from the orthocenter. Let $D$ and $E$ be the feet of altitudes from $A$ to $BP$ and $CP$, and let $F$ and $G$ be the feet of the altitudes from $P$ to sides $AB$ and $AC$. Denote by $X$ the midpoint of $[AP]$, and let the second intersection of the circumcircles of triangles $\Delta DFX$ and $\Delta EGX$ lie on $BC$. Prove that $AP$ is perpendicular to $BC$ or $\angle PBA = \angle PCA$.
2008 Iran Team Selection Test, 6
Prove that in a tournament with 799 teams, there exist 14 teams, that can be partitioned into groups in a way that all of the teams in the first group have won all of the teams in the second group.
1980 IMO Longlists, 3
Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.
2008 Indonesia TST, 4
Let $a, b, c$ be positive reals. Prove that $$\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+\left(\frac{c}{c+a}\right)^2\ge \frac34$$
2006 VJIMC, Problem 1
(a) Let $u$ and $v$ be two nilpotent elements in a commutative ring (with or without unity). Prove that $u+v$ is also nilpotent.
(b) Show an example of a (non-commutative) ring $R$ and nilpotent elements $u,v\in R$ such that $u+v$ is not nilpotent.
1989 IMO Longlists, 77
Let $ a, b, c, r,$ and $ s$ be real numbers. Show that if $ r$ is a root of $ ax^2\plus{}bx\plus{}c \equal{} 0$ and s is a root of $ \minus{}ax^2\plus{}bx\plus{}c \equal{} 0,$ then \[ \frac{a}{2} x^2 \plus{} bx \plus{} c \equal{} 0\] has a root between $ r$ and $ s.$
2018 Denmark MO - Mohr Contest, 4
A sequence $a_1, a_2, a_3, . . . , a_{100}$ of $100$ (not necessarily distinct) positive numbers satisfy that the$ 99$ fractions$$\frac{a_1}{a_2},\frac{a_2}{a_3},\frac{a3}{a_4}, ... ,\frac{a_{99}}{a_{100}}$$ are all distinct. How many distinct numbers must there be, at least, in the sequence $a_1, a_2, a_3, . . . , a_{100}$?
2011 National Olympiad First Round, 25
Let $S_1$ be the area of the regular pentagon $ABCDE$. And let $S_2$ be the area of the regular pentagon whose sides lie on the lines $AC, CE, EB, BD, DA$. What is values of $\frac{S_1}{S_2}$ ?
$\textbf{(A)}\ \frac{41}{6} \qquad\textbf{(B)}\ \frac{3+5\sqrt5}{2} \qquad\textbf{(C)}\ 4+\sqrt5 \qquad\textbf{(D)}\ \frac{7+3\sqrt5}2 \qquad\textbf{(E)}\ \text{None}$
2019 Saudi Arabia BMO TST, 2
Let sequences of real numbers $(x_n)$ and $(y_n)$ satisfy $x_1 = y_1 = 1$ and $x_{n+1} =\frac{x_n + 2}{x_n + 1}$ and $y_{n+1} = \frac{y_n^2 + 2}{2y_n}$ for $n = 1,2, ...$ Prove that $y_{n+1} = x_{2^n}$ holds for $n =0, 1,2, ... $
2007 China National Olympiad, 2
Show that:
1) If $2n-1$ is a prime number, then for any $n$ pairwise distinct positive integers $a_1, a_2, \ldots , a_n$, there exists $i, j \in \{1, 2, \ldots , n\}$ such that
\[\frac{a_i+a_j}{(a_i,a_j)} \geq 2n-1\]
2) If $2n-1$ is a composite number, then there exists $n$ pairwise distinct positive integers $a_1, a_2, \ldots , a_n$, such that for any $i, j \in \{1, 2, \ldots , n\}$ we have
\[\frac{a_i+a_j}{(a_i,a_j)} < 2n-1\]
Here $(x,y)$ denotes the greatest common divisor of $x,y$.
2011 Tournament of Towns, 5
On a highway, a pedestrian and a cyclist were going in the same direction, while a cart and a car were coming from the opposite direction. All were travelling at different constant speeds. The cyclist caught up with the pedestrian at $10$ o'clock. After a time interval, she met the cart, and after another time interval equal to the first, she met the car. After a third time interval, the car met the pedestrian, and after another time interval equal to the third, the car caught up with the cart. If the pedestrian met the car at $11$ o'clock, when did he meet the cart?
2007 Today's Calculation Of Integral, 174
Let $a$ be a positive number. Assume that the parameterized curve $C: \ x=t+e^{at},\ y=-t+e^{at}\ (-\infty <t< \infty)$ is touched to $x$ axis.
(1) Find the value of $a.$
(2) Find the area of the part which is surrounded by two straight lines $y=0, y=x$ and the curve $C.$
KoMaL A Problems 2019/2020, A. 779
Two circles are given in the plane, $\Omega$ and inside it $\omega$. The center of $\omega$ is $I$. $P$ is a point moving on $\Omega$. The second intersection of the tangents from $P$ to $\omega$ and circle $\Omega$ are $Q$ and $R.$ The second intersection of circle $IQR$ and lines $PI$, $PQ$ and $PR$ are $J$, $S$ and $T,$ respectively. The reflection of point $J$ across line $ST$ is $K.$
Prove that lines $PK$ are concurrent.
1989 IMO Longlists, 47
Let $ A,B$ denote two distinct fixed points in space. Let $ X, P$ denote variable points (in space), while $ K,N, n$ denote positive integers. Call $ (X,K,N,P)$ admissible if \[ (N \minus{} K) \cdot PA \plus{} K \cdot PB \geq N \cdot PX.\] Call $ (X,K,N)$ admissible if $ (X,K,N,P)$ is admissible for all choices of $ P.$ Call $ (X,N)$ admissible if $ (X,K,N)$ is admissible for some choice of $ K$ in the interval $ 0 < K < N.$ Finally, call $ X$ admissible if $ (X,N)$ is admissible for some choice of $ N, (N > 1).$ Determine:
[b](a)[/b] the set of admissible $ X;$
[b](b)[/b] the set of $ X$ for which $ (X, 1989)$ is admissible but not $ (X, n), n < 1989.$
2016 Harvard-MIT Mathematics Tournament, 4
Determine the remainder when
$$\sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor$$
is divided by 100, where $\lfloor x \rfloor$ denotes the largest integer not greater than $x$.
1961 AMC 12/AHSME, 22
If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by:
${{ \textbf{(A)}\ 3x^2-x+4 \qquad\textbf{(B)}\ 3x^2-4 \qquad\textbf{(C)}\ 3x^2+4 \qquad\textbf{(D)}\ 3x-4 }\qquad\textbf{(E)}\ 3x+4 } $