Found problems: 85335
2018 Moscow Mathematical Olympiad, 9
$x$ and $y$ are integer $5$-digits numbers, such that in the decimal notation, all ten digits are used exactly once. Also
$\tan{x}-\tan{y}=1+\tan{x}\tan{y}$, where $x,y$ are angles in degrees. Find maximum of $x$
2012 Junior Balkan Team Selection Tests - Moldova, 2
Let $ a,b,c $ be positive real numbers, prove the inequality:
$ (a+b+c)^2+ab+bc+ac\geq 6\sqrt{abc(a+b+c)} $
2017 Hanoi Open Mathematics Competitions, 10
Consider all words constituted by eight letters from $\{C ,H,M, O\}$. We arrange the words in an alphabet sequence.
Precisely, the first word is $CCCCCCCC$, the second one is $CCCCCCCH$, the third is $CCCCCCCM$, the fourth one is $CCCCCCCO, ...,$ and the last word is $OOOOOOOO$.
a) Determine the $2017$th word of the sequence?
b) What is the position of the word $HOMCHOMC$ in the sequence?
2010 Vietnam National Olympiad, 3
In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$
such that $BC$ not is the diameter.Consider a point $A$ varies in
$(O)$ such that $A\neq B,C$ and $AB\neq AC$.Call $D$ and $E$
respective is intersect of $BC$ and internal and external bisector
of $\widehat{BAC}$,$I$ is midpoint of $DE$.The line that pass through
orthocenter of $\triangle ABC$
and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$.
1/Prove that $MN$ pass through a fixed point
2/Determint the place of $A$ such that $S_{AMN}$ has maxium value
2004 USAMTS Problems, 2
For the equation \[ (3x^2+y^2-4y-17)^3-(2x^2+2y^2-4y-6)^3=(x^2-y^2-11)^3, \]
determine its solutions $(x, y)$ where both $x$ and $y$ are integers. Prove that your answer lists all the integer solutions.
2010 Greece Team Selection Test, 1
Solve in positive reals the system:
$x+y+z+w=4$
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=5-\frac{1}{xyzw}$
2009 VJIMC, Problem 1
A positive integer $m$ is called self-descriptive in base $b$, where $b\ge2$ is an integer, if
i) The representation of $m$ in base $b$ is of the form $(a_0a_1\ldots a_{b-1})_b$ (that is $m=a_0b^{b-1}+a_1b^{b-2}+\ldots+a_{b-2}b+a_{b-1}$, where $0\le a_i\le b-1$ are integers).
ii) $a_i$ is equal to the number of occurences of the number $i$ in the sequence $(a_0a_1\ldots a_{b-1})$.
For example, $(1210)_4$ is self-descriptive in base $4$, because it has four digits and contains one $0$, two $1$s, one $2$ and no $3$s.
1988 Balkan MO, 2
Find all polynomials of two variables $P(x,y)$ which satisfy
\[P(a,b) P(c,d) = P (ac+bd, ad+bc), \forall a,b,c,d \in \mathbb{R}.\]
2022 Bulgaria National Olympiad, 6
Let $n\geq 2$ be a positive integer. The sets $A_{1},A_{2},\ldots, A_{n}$ and $B_{1},B_{2},\ldots, B_{n}$ of positive integers are such that $A_{i}\cap B_{j}$ is non-empty $\forall i,j\in\{1,2,\ldots ,n\}$ and $A_{i}\cap A_{j}=\o$, $B_{i}\cap B_{j}=\o$ $\forall i\neq j\in \{1,2,\ldots, n\}$. We put the elements of each set in a descending order and calculate the differences between consecutive elements in this new order. Find the least possible value of the greatest of all such differences.
2011 JBMO Shortlist, 6
Let $\displaystyle {x_i> 1, \forall i \in \left \{1, 2, 3, \ldots, 2011 \right \}}$. Show that:$$\displaystyle{\frac{x^2_1}{x_2-1}+\frac{x^2_2}{x_3-1}+\frac{x^2_3}{x_4-1}+\ldots+\frac{x^2_{2010}}{x_{2011}-1}+\frac{x^2_{2011}}{x_1-1}\geq 8044}$$
When the equality holds?
2006 VJIMC, Problem 3
Two players play the following game: Let $n$ be a fixed integer greater than $1$. Starting from number $k=2$, each player has two possible moves: either replace the number $k$ by $k+1$ or by $2k$. The player who is forced to write a number greater than $n$ loses the game. Which player has a winning strategy for which $n$?
2005 Grigore Moisil Urziceni, 3
Define the operation $ (a,b)\circ (c,d) =(ac,ad+b). $
[b]a)[/b] Prove that $ \left( \mathbb{Q}\setminus\{ 0\}\times\mathbb{Q} ,\circ \right) $ is a group.
[b]b)[/b] Let $ H $ be an infinite subgroup of $ \left( \mathbb{Q}\setminus\{ 0\}\times\mathbb{Q} ,\circ \right) $ that is cyclic and doesn't contain any element of the form $ (1,q) , $ where $ q $ is a nonzero rational. Show that there exist two rational numbers $ a,b $ such that
$$ H=\left\{ \left.\left( a^n, b\cdot\frac{1-a^n}{1-a} \right)\right| n\in\mathbb{Z} \right\} $$
2011 India IMO Training Camp, 2
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
2013 Saudi Arabia IMO TST, 4
Determine whether it is possible to place the integers $1, 2,...,2012$ in a circle in such a way that the $2012$ products of adjacent pairs of numbers leave pairwise distinct remainders when divided by $2013$.
MOAA Gunga Bowls, 2021.1
Evaluate $2\times 0+2\times 1+ 2+0\times 2 +1$.
[i]Proposed by Nathan Xiong[/i]
2009 Kazakhstan National Olympiad, 1
Prove that for any natural $n \geq 2$, the number $ \underbrace{2^{2^{\cdots^2}}}_{n \textrm{ times}}- \underbrace{2^{2^{\cdots^2}}}_{n-1 \textrm{ times}}$ is divisible by $n$.
I know, that it is a very old problem :blush: but it is a problem from olympiad.
2006 Putnam, B5
For each continuous function $f: [0,1]\to\mathbb{R},$ let $I(f)=\int_{0}^{1}x^{2}f(x)\,dx$ and $J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$
1975 Poland - Second Round, 2
In the convex quadrilateral $ ABCD $, the corresponding points $ M $ and $ N $ are chosen on the adjacent sides $ \overline{AB} $ and $ \overline{BC} $ and the intersection point of the segments $ AN $ and $ GM $ is marked by 0. Prove that if circles can be inscribed in the quadrilaterals $ AOCD $ and $ BMON $, then a circle can also be inscribed in the quadrilateral $ ABCD $.
2016 India IMO Training Camp, 3
An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$. A [i]move[/i] consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by $1$ simultaneously. Determine all positive integers $n$ such that after performing several such moves one can obtain $9$ consecutive numbers $n,(n+1),\cdots ,(n+8)$ in some order.
[asy] size(3cm);
pair A=(0,0),D=(1,0),B,C,E,F,G,H,I;
G=rotate(60,A)*D;
B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A;
draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]
2011 Danube Mathematical Competition, 4
Given a positive integer number $n$, determine the maximum number of edges a triangle-free Hamiltonian simple graph on $n$ vertices may have.
2024 South Africa National Olympiad, 3
Each of the lattice points $(x,y)$ (where $x$ and $y$ are integers) in the plane can be coloured black or white. A single strike by an $L$-shaped punch changes the colour of the four lattice points $(a,b)$, $(a+1,b)$, $(a,b+1)$ and $(a,b+2)$. All lattice points are initially coloured white. Prove that after any number of strikes, the number of black lattice points will be either zero or greater than or equal to four.
2024 ISI Entrance UGB, P7
Consider a container of the shape obtained by revolving a segment of parabola $x = 1 + y^2$ around the $y$-axis as shown below. The container is initially empty. Water is poured at a constant rate of $1\, \text{cm}^3$ into the container. Let $h(t)$ be the height of water inside container at time $t$. Find the time $t$ when the rate of change of $h(t)$ is maximum.
1956 AMC 12/AHSME, 29
The points of intersection of $ xy \equal{} 12$ and $ x^2 \plus{} y^2 \equal{} 25$ are joined in succession. The resulting figure is:
$ \textbf{(A)}\ \text{a straight line} \qquad\textbf{(B)}\ \text{an equilateral triangle} \qquad\textbf{(C)}\ \text{a parallelogram}$
$ \textbf{(D)}\ \text{a rectangle} \qquad\textbf{(E)}\ \text{a square}$
2018 Costa Rica - Final Round, F2
Consider $f (n, m)$ the number of finite sequences of $ 1$'s and $0$'s such that each sequence that starts at $0$, has exactly n $0$'s and $m$ $ 1$'s, and there are not three consecutive $0$'s or three $ 1$'s. Show that if $m, n> 1$, then
$$f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)$$
2017 IMO Shortlist, G5
Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.