Found problems: 85335
2006 China Team Selection Test, 2
The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that
\[ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots \]
Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.
2008 China National Olympiad, 1
Suppose $\triangle ABC$ is scalene. $O$ is the circumcenter and $A'$ is a point on the extension of segment $AO$ such that $\angle BA'A = \angle CA'A$. Let point $A_1$ and $A_2$ be foot of perpendicular from $A'$ onto $AB$ and $AC$. $H_{A}$ is the foot of perpendicular from $A$ onto $BC$. Denote $R_{A}$ to be the radius of circumcircle of $\triangle H_{A}A_1A_2$. Similiarly we can define $R_{B}$ and $R_{C}$. Show that:
\[\frac{1}{R_{A}} + \frac{1}{R_{B}} + \frac{1}{R_{C}} = \frac{2}{R}\]
where R is the radius of circumcircle of $\triangle ABC$.
1953 Moscow Mathematical Olympiad, 241
Prove that the polynomial $x^{200} y^{200} +1$ cannot be represented in the form $f(x)g(y)$, where $f$ and $g$ are polynomials of only $x$ and $y$, respectively.
2013 Balkan MO Shortlist, A3
Prove that the polynomial $P (x) = (x^2- 8x + 25) (x^2 - 16x + 100) ... (x^2 - 8nx + 25n^2)- 1$, $n \in N^*$,
cannot be written as the product of two polynomials with integer coefficients of degree greater or equal to $1$.
2010 Junior Balkan Team Selection Tests - Moldova, 5
For any strictly positive numbers $a$ and $b$ , prove the inequality $$\frac{a}{a+b} \cdot \frac{a+2b}{a+3b} < \sqrt{ \frac{a}{a+4b}}.$$
1966 IMO Longlists, 45
An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.
2017 China Western Mathematical Olympiad, 8
Let $a_1,a_2,\cdots,a_n>0$ $(n\geq 2)$. Prove that$$\sum_{i=1}^n max\{a_1,a_2,\cdots,a_i \} \cdot min
\{a_i,a_{i+1},\cdots,a_n\}\leq \frac{n}{2\sqrt{n-1}}\sum_{i=1}^n a^2_i$$
2020 Taiwan TST Round 1, 2
Let point $H$ be the orthocenter of a scalene triangle $ABC$. Line $AH$ intersects with the circumcircle $\Omega$ of triangle $ABC$ again at point $P$. Line $BH, CH$ meets with $AC,AB$ at point $E$ and $F$, respectively. Let $PE, PF$ meet $\Omega$ again at point $Q,R$, respectively. Point $Y$ lies on $\Omega$ so that lines $AY,QR$ and $EF$ are concurrent. Prove that $PY$ bisects $EF$.
KoMaL A Problems 2024/2025, A. 907
$2025$ light bulbs are operated by some switches. Each switch works on a subset of the light bulbs. When we use a switch, all the light bulbs in the subset change their state: bulbs that were on turn off, and bulbs that were off turn on. We know that every light bulb is operated by at least one of the switches. Initially, all lamps were off. Find the biggest number $k$ for which we can surely turn on at least $k$ light bulbs.
[i]Based on an OKTV problem[/i]
PEN N Problems, 8
An integer sequence $\{a_{n}\}_{n \ge 1}$ is given such that \[2^{n}=\sum^{}_{d \vert n}a_{d}\] for all $n \in \mathbb{N}$. Show that $a_{n}$ is divisible by $n$ for all $n \in \mathbb{N}$.
2012 USA Team Selection Test, 4
Find all positive integers $a,n\ge1$ such that for all primes $p$ dividing $a^n-1$, there exists a positive integer $m<n$ such that $p\mid a^m-1$.
2008 Dutch IMO TST, 4
Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer.
Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.
2020 Israel National Olympiad, 6
On a circle the numbers from 1 to 6 are written in order, as depicted in the picture. In each move, Lior picks a number $a$ on the circle whose neighbors are $b$ and $c$ and replaces it by the number $\frac{bc}{a}$. Can Lior reach a state in which the product of the numbers on the circle is greater than $10^{100}$ in
[b]a)[/b] at most 100 moves
[b]b)[/b] at most 110 moves
1987 USAMO, 5
Given a sequence $(x_1,x_2,\ldots, x_n)$ of 0's and 1's, let $A$ be the number of triples $(x_i,x_j,x_k)$ with $i<j<k$ such that $(x_i,x_j,x_k)$ equals $(0,1,0)$ or $(1,0,1)$. For $1\leq i \leq n$, let $d_i$ denote the number of $j$ for which either $j < i$ and $x_j = x_i$ or else $j > i$ and $x_j\neq x_i$.
(a) Prove that \[A = \binom n3 - \sum_{i=1}^n\binom{d_i}2.\] (Of course, $\textstyle\binom ab = \tfrac{a!}{b!(a-b)!}$.) [5 points]
(b) Given an odd number $n$, what is the maximum possible value of $A$? [15 points]
2022 BMT, Tie 3
Let $A$ be the product of all positive integers less than $1000$ whose ones or hundreds digit is $7$. Compute the remainder when $A/101$ is divided by $101$.
2009 Brazil National Olympiad, 3
There are $ 2009$ pebbles in some points $ (x,y)$ with both coordinates integer. A operation consists in choosing a point $ (a,b)$ with four or more pebbles, removing four pebbles from $ (a,b)$ and putting one pebble in each of the points
\[ (a,b\minus{}1),\ (a,b\plus{}1),\ (a\minus{}1,b),\ (a\plus{}1,b)\]
Show that after a finite number of operations each point will necessarily have at most three pebbles. Prove that the final configuration doesn't depend on the order of the operations.
The Golden Digits 2024, P2
Let $ABCD$ be a parallelogram and $P$ a point in the plane. The line $BP$ intersects the circumcircle of $ABC$ again at $X$ and the line $DP$ intersects the circumcircle of $DAC$ again at $Y$. Let $M$ be the midpoint of the side $AC$. The point $N$ lies on the circumcircle of $PXY$ so that $MN$ is a tangent to this circle. Prove that the segments $MN$ and $AM$ have the same length.
[i]Proposed by David Anghel[/i]
1985 Tournament Of Towns, (089) 5
The digits $0, 1 , 2, ..., 9$ are written in a $10 x 10$ table , each number appearing $10$ times .
(a) Is it possible to write them in such a way that in any row or column there would be not more than $4$ different digits?
(b) Prove that there must be a row or column containing more than $3$ different digits .
{ L . D . Kurlyandchik , Leningrad)
2009 Peru Iberoamerican Team Selection Test, P1
A set $P$ has the following property: “For any positive integer $k$, if $p$ is a prime factor of $k^3+6$, then $p$ belongs to $P$ ”. Prove that $P$ is infinite.
1958 AMC 12/AHSME, 9
A value of $ x$ satisfying the equation $ x^2 \plus{} b^2 \equal{} (a \minus{} x)^2$ is:
$ \textbf{(A)}\ \frac{b^2 \plus{} a^2}{2a}\qquad
\textbf{(B)}\ \frac{b^2 \minus{} a^2}{2a}\qquad
\textbf{(C)}\ \frac{a^2 \minus{} b^2}{2a}\qquad
\textbf{(D)}\ \frac{a \minus{} b}{2}\qquad
\textbf{(E)}\ \frac{a^2 \minus{} b^2}{2}$
2022 JHMT HS, 1
If three of the roots of the quartic polynomial $f(x) = x^4 + ax^3 + bx^2 + cx + d$ are $0$, $2$, and $4$, and the sum of $a$, $b$, and $c$ is at most $12$, then find the largest possible value of $f(1)$.
1998 Poland - Second Round, 2
In triangle $ABC$, the angle $\angle BCA$ is obtuse and $\angle BAC = 2\angle ABC\,.$ The line through $B$ and perpendicular to $BC$ intersects line $AC$ in $D$. Let $M$ be the midpoint of $AB$. Prove that $\angle AMC=\angle BMD$.
source : http://cage.ugent.be/~hvernaev/Olympiade/PMO982.pdf
1999 Akdeniz University MO, 4
In a sequence ,first term is $2$ and after $2.$ term all terms is equal to sum of the previous number's digits' $5.$ power. (Like this $2.$term is $2^5=32$ , $3.$term is $3^5+2^5=243+32=275\dotsm$) Prove that, this infinite sequence has at least $2$ two numbers are equal.
1999 Moldova Team Selection Test, 13
Let $N$ be a natural number. Find (with prove) the number of solutions in the segment $[1,N]$ of the equation $x^2-[x^2]=(x-[x])^2$, where $[x]$ means the floor function of $x$.
2013 China National Olympiad, 3
Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$, then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in A} x }$.