Found problems: 85335
2022 MOAA, 13
Determine the number of distinct positive real solutions to $$\lfloor x \rfloor ^{\{x\}} = \frac{1}{2022}x^2$$
.
Note: $\lfloor x \rfloor$ is known as the floor function, which returns the greatest integer less than or equal to $x$. Furthermore, $\{x\}$ is defined as $x - \lfloor x \rfloor$.
1991 China Team Selection Test, 2
Let $f$ be a function $f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},$ and satisfies the following conditions:
(1) $f(0) = 0, f(1) = 1,$
(2) $f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.$
Prove that for any $m \in \mathbb{N}$, there exist a $d \in \mathbb{N}$ such that $m | f(f(n)) \Leftrightarrow d | n.$
2004 Junior Balkan Team Selection Tests - Romania, 3
Let $A$ be a set of positive integers such that
a) if $a\in A$, the all the positive divisors of $a$ are also in $A$;
b) if $a,b\in A$, with $1<a<b$, then $1+ab \in A$.
Prove that if $A$ has at least 3 elements, then $A$ is the set of all positive integers.
2012 CIIM, Problem 2
A set $A\subset \mathbb{Z}$ is "padre" if whenever $x,y \in A$ with $x\leq y$ then also $2y -x \in A$. Prove that if $A$ is "padre", $0,a,b \in A$ with $0< a < b$ and $d = g.c.d(a,b)$ then \[a+b-3d, a+b-2d \in A.\]
2021 ELMO Problems, 2
Let $n > 1$ be an integer and let $a_1, a_2, \ldots, a_n$ be integers such that $n \mid a_i-i$ for all integers $1 \leq i \leq n$. Prove there exists an infinite sequence $b_1,b_2, \ldots$ such that
[list]
[*] $b_k\in\{a_1,a_2,\ldots, a_n\}$ for all positive integers $k$, and
[*] $\sum\limits_{k=1}^{\infty}\frac{b_k}{n^k}$ is an integer.
[/list]
2005 AIME Problems, 13
Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17$. Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2$, find the product $n_1\cdot n_2$.
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]