Found problems: 85335
2012 Cuba MO, 3
On a $123 \times 123$ board, each square is painted red or blue according to the following conditions:
a) Each square painted red that is not on the edge of the board has exactly $5$ blue squares among its $8$ neighboring squares.
b) Each square painted blue that is not on the edge of the board has exactly $4$ red squares among its $8$ neighboring squares.
Determine the number of red-painted squares on the board.
2020 LMT Spring, 17
Let $ABC$ be a triangle such that $AB = 26, AC = 30,$ and $BC = 28$. Let $C'$ and $B'$ be the reflections of the circumcenter $O$ over $AB$ and $AC$, respectively. The length of the portion of line segment $B'C'$ inside triangle $ABC$ can be written as $\frac{p}{q}$, where $p,q$ are relatively prime positive integers. Compute $p+q$.
1969 IMO Shortlist, 5
$(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$
$(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas.
$(b)$ Find the locus of the centers of these hyperbolas.
2022 LMT Spring, 8
The $53$-digit number
$$37,984,318,966,591,152,105,649,545,470,741,788,308,402,068,827,142,719$$
can be expressed as $n^21$ where $n$ is a positive integer. Find $n$.
2015 NIMO Problems, 2
There exists a unique strictly increasing arithmetic sequence $\{a_i\}_{i=1}^{100}$ of positive integers such that \[a_1+a_4+a_9+\cdots+a_{100}=\text{1000},\] where the summation runs over all terms of the form $a_{i^2}$ for $1\leq i\leq 10$. Find $a_{50}$.
[i]Proposed by David Altizio and Tony Kim[/i]
2008 Singapore MO Open, 5
consider a $2008 \times 2008$ chess board. let $M$ be the smallest no of rectangles that can be drawn on the chess board so that sides of every cell of the board is contained in the sides of one of the rectangles. find the value of $M$. (eg for $2\times 3$ chessboard, the value of $M$ is 3.)
2014 PUMaC Geometry A, 4
Consider the cyclic quadrilateral with side lengths $1$, $4$, $8$, $7$ in that order. What is its circumdiameter? Let the answer be of the form $a\sqrt b+c$, for $b$ squarefree. Find $a+b+c$.
Croatia MO (HMO) - geometry, 2016.3
Given a cyclic quadrilateral $ABCD$ such that the tangents at points $B$ and $D$ to its circumcircle $k$ intersect at the line $AC$. The points $E$ and $F$ lie on the circle $k$ so that the lines $AC, DE$ and $BF$ parallel. Let $M$ be the intersection of the lines $BE$ and $DF$. If $P, Q$ and $R$ are the feet of the altitides of the triangle $ABC$, prove that the points $P, Q, R$ and $M$ lie on the same circle
2019 Jozsef Wildt International Math Competition, W. 37
For real $a > 1$ find$$\lim \limits_{n \to \infty}\sqrt[n]{\prod \limits_{k=2}^n \left(a-a^{\frac{1}{k}}\right)}$$
1956 AMC 12/AHSME, 11
The expression $ 1 \minus{} \frac {1}{1 \plus{} \sqrt {3}} \plus{} \frac {1}{1 \minus{} \sqrt {3}}$ equals:
$ \textbf{(A)}\ 1 \minus{} \sqrt {3} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \minus{} \sqrt {3} \qquad\textbf{(D)}\ \sqrt {3} \qquad\textbf{(E)}\ 1 \plus{} \sqrt {3}$
2022 Cyprus JBMO TST, 1
Find all integer values of $x$ for which the value of the expression
\[x^2+6x+33\]
is a perfect square.
2010 Contests, 3
Let $n$ be a positive integer. Let $a$ be an integer such that $\gcd (a,n)=1$. Prove that
\[\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}\]
where $R$ is the reduced residue system of $n$ with each element a positive integer at most $n$.
2017 ELMO Problems, 6
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$:
(i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$
(ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$
[i]Proposed by Ashwin Sah[/i]
2015 Czech-Polish-Slovak Match, 1
On a circle of radius $r$, the distinct points $A$, $B$, $C$, $D$, and $E$ lie in this order, satisfying $AB=CD=DE>r$. Show that the triangle with vertices lying in the centroids of the triangles $ABD$, $BCD$, and $ADE$ is obtuse.
[i]Proposed by Tomáš Jurík, Slovakia[/i]
1951 Moscow Mathematical Olympiad, 198
* On a plane, given points $A, B, C$ and angles $\angle D, \angle E, \angle F$ each less than $180^o$ and the sum equal to $360^o$, construct with the help of ruler and protractor a point $O$ such that $\angle AOB = \angle D, \angle BOC = \angle E$ and $\angle COA = \angle F.$
2011 Balkan MO Shortlist, N1
Given an odd number $n >1$, let
\begin{align*} S =\{ k \mid 1 \le k < n , \gcd(k,n) =1 \} \end{align*}
and let \begin{align*} T = \{ k \mid k \in S , \gcd(k+1,n) =1 \} \end{align*}
For each $k \in S$, let $r_k$ be the remainder left by $\frac{k^{|S|}-1}{n}$ upon division by $n$. Prove
\begin{align*} \prod _{k \in T} \left( r_k - r_{n-k} \right) \equiv |S| ^{|T|} \pmod{n} \end{align*}
1995 IMC, 2
Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$,
we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.
1998 Kurschak Competition, 2
Prove that for every positive integer $n$, there exists a polynomial with integer coefficients whose values at points $1,2,\dots,n$ are pairwise different powers of $2$.
2010 Polish MO Finals, 2
Positive rational number $a$ and $b$ satisfy the equality
\[a^3 + 4a^2b = 4a^2 + b^4.\]
Prove that the number $\sqrt{a}-1$ is a square of a rational number.
1998 AMC 8, 24
A rectangular board of 8 columns has squared numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result?
[asy]
unitsize(20);
for(int a = 0; a < 10; ++a)
{
draw((0,a)--(8,a));
}
for (int b = 0; b < 9; ++b)
{
draw((b,0)--(b,9));
}
draw((0,0)--(0,-.5));
draw((1,0)--(1,-1.5));
draw((.5,-1)--(1.5,-1));
draw((2,0)--(2,-.5));
draw((4,0)--(4,-.5));
draw((5,0)--(5,-1.5));
draw((4.5,-1)--(5.5,-1));
draw((6,0)--(6,-.5));
draw((8,0)--(8,-.5));
fill((0,8)--(1,8)--(1,9)--(0,9)--cycle,black);
fill((2,8)--(3,8)--(3,9)--(2,9)--cycle,black);
fill((5,8)--(6,8)--(6,9)--(5,9)--cycle,black);
fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,black);
fill((6,7)--(7,7)--(7,8)--(6,8)--cycle,black);
label("$2$",(1.5,8.2),N);
label("$4$",(3.5,8.2),N);
label("$5$",(4.5,8.2),N);
label("$7$",(6.5,8.2),N);
label("$8$",(7.5,8.2),N);
label("$9$",(0.5,7.2),N);
label("$11$",(2.5,7.2),N);
label("$12$",(3.5,7.2),N);
label("$13$",(4.5,7.2),N);
label("$14$",(5.5,7.2),N);
label("$16$",(7.5,7.2),N);
[/asy]
$\text{(A)}\ 36 \qquad \text{(B)}\ 64 \qquad \text{(C)}\ 78 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 120$
PEN F Problems, 10
The set $ S$ is a finite subset of $ [0,1]$ with the following property: for all $ s\in S$, there exist $ a,b\in S\cup\{0,1\}$ with $ a, b\neq s$ such that $ s \equal{}\frac{a\plus{}b}{2}$. Prove that all the numbers in $ S$ are rational.
2017 Bosnia And Herzegovina - Regional Olympiad, 4
Let $S$ be a set of $n$ distinct real numbers, and $A_S$ set of arithemtic means of two distinct numbers from $S$. For given $n \geq 2$ find minimal number of elements in $A_S$
1985 Traian Lălescu, 2.3
Let $ ABC $ a triangle, and $ P\neq B,C $ be a point situated upon the segment $ BC $ such that $ ABP $ and $ APC $ have the same perimeter. $ M $ represents the middle of $ BC, $ and $ I, $ the center of the incircle of $ ABC. $
Prove that $ IM\parallel AP. $
2011 Purple Comet Problems, 27
Find the smallest prime number that does not divide \[9+9^2+9^3+\cdots+9^{2010}.\]
2004 Romania Team Selection Test, 10
Prove that for all positive integers $n,m$, with $m$ odd, the following number is an integer
\[ \frac 1{3^mn}\sum^m_{k=0} { 3m \choose 3k } (3n-1)^k. \]