Found problems: 85335
2007 Junior Balkan Team Selection Tests - Romania, 3
Consider a $n$x$n$ table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a $2$x$2$ square which contains an odd number of unit squares white colored.
2002 Croatia National Olympiad, Problem 3
Let $f(x)=x^{2002}-x^{2001}+1$. Prove that for every positive integer $m$, the numbers $m,f(m),f(f(m)),\ldots$ are pairwise coprime.
2009 Harvard-MIT Mathematics Tournament, 3
A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid.
[asy]
size(150);
defaultpen(linewidth(0.8));
draw(origin--(8,0)--(8,5)--(0,5)--cycle,linewidth(1));
draw(origin--(8/3,5)^^(16/3,5)--(8,0),linetype("4 4"));
draw(origin--(4,3)--(8,0)^^(8/3,5)--(4,3)--(16/3,5),linetype("0 4"));
label("$5$",(0,5/2),W);
label("$8$",(4,0),S);
[/asy]
2016 Middle European Mathematical Olympiad, 1
Find all triples $(a, b, c)$ of real numbers such that
$$ a^2 + ab + c = 0, $$
$$b^2 + bc + a = 0, $$
$$c^2 + ca + b = 0.$$
1973 Polish MO Finals, 2
Let $p_n$ denote the probability that, in $n$ tosses, a fair coin shows the head up $100$ consecutive times. Prove that the sequence $(p_n)$ converges and determine its limit.
1989 Greece Junior Math Olympiad, 4
Simplify
i) $1+\frac{2a+\dfrac{2}{a}}{a+\dfrac{1}{a}}$
ii) $\frac{3b+\dfrac{3}{b}+\dfrac{3}{b^2}}{b+\dfrac{1}{b}+\dfrac{1}{b^2}}$
iii) $\frac{\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{ab}\right)a^6b^2-a^6-a^5b}{a^4b}$
LMT Guts Rounds, 29
Let $S$ be the set of integers that represent the number of intersections of some four distinct lines in the plane. List the elements of $S$ in ascending order.
2008 Indonesia MO, 2
Prove that for $ x,y\in\mathbb{R^ \plus{} }$,
$ \frac {1}{(1 \plus{} \sqrt {x})^{2}} \plus{} \frac {1}{(1 \plus{} \sqrt {y})^{2}} \ge \frac {2}{x \plus{} y \plus{} 2}$
1993 AMC 8, 19
$(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193) = $
$\text{(A)}\ 167,400 \qquad \text{(B)}\ 172,050 \qquad \text{(C)}\ 181,071 \qquad \text{(D)}\ 199,300 \qquad \text{(E)}\ 362,142$
2016 Serbia Additional Team Selection Test, 1
Let $P_0(x)=x^3-4x$. Sequence of polynomials is defined as following:\\
$P_{n+1}=P_n(1+x)P_n(1-x)-1$.\\
Prove that $x^{2016}|P_{2016}(x)$.
1994 Czech And Slovak Olympiad IIIA, 5
In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?
2011 Middle European Mathematical Olympiad, 7
Let $A$ and $B$ be disjoint nonempty sets with $A \cup B = \{1, 2,3, \ldots, 10\}$. Show that there exist elements $a \in A$ and $b \in B$ such that the number $a^3 + ab^2 + b^3$ is divisible by $11$.
2011 AIME Problems, 13
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers, and $r+s+t<1000$. Find $r+s+t$.
2023 Korea Junior Math Olympiad, 5
For a positive integer $n(\geq 5)$, there are $n$ white stones and $n$ black stones (total $2n$ stones) lined up in a row. The first $n$ stones from the left are white, and the next $n$ stones are black. $$\underbrace{\Circle \Circle \cdots \Circle}_n \underbrace{\CIRCLE \CIRCLE \cdots \CIRCLE}_n $$
You can swap the stones by repeating the following operation.
[b](Operation)[/b] Choose a positive integer $k (\leq 2n - 5)$, and swap $k$-th stone and $(k+5)$-th stone from the left.
Find all positive integers $n$ such that we can make first $n$ stones to be black and the next $n$ stones to be white in finite number of operations.
2004 National High School Mathematics League, 10
$p$ is a give odd prime, if $\sqrt{k^2-pk}$ is a positive integer, then the value of positive integer $k$ is________.
2007 Princeton University Math Competition, 2
A black witch's hat is in the classic shape of a cone on top of a circular brim. The cone has a slant height of $18$ inches and a base radius of $3$ inches. The brim has a radius of $5$ inches. What is the total surface area of the hat?
2004 India IMO Training Camp, 1
Prove that in any triangle $ABC$,
\[ 0 < \cot { \left( \frac{A}{4} \right)} - \tan{ \left( \frac{B}{4} \right) } - \tan{ \left( \frac{C}{4} \right) } - 1 < 2 \cot { \left( \frac{A}{2} \right) }. \]
2023 LMT Spring, 7
In $\vartriangle ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be a point on $BC$ such that $BD = 6$. Let $E$ be a point on $CA$ such that $CE = 6$. Finally, let $F$ be a point on $AB$ such that $AF = 6$. Find the area of $\vartriangle DEF$.
2019 MIG, 8
Greg plays a game in which he is given three random $1$ digit numbers, each between $0$ and $9$, inclusive, with repeats allowed. He is to put these three numbers into any order. Exactly one ordering of the three numbers is correct, and if he guesses the correct ordering, he wins $\$150$. What are Greg's expected winnings for this game, given that he randomly guesses one valid ordering when he plays?
2011 IFYM, Sozopol, 8
The fraction $\frac{1}{p}$, where $p$ is a prime number coprime with 10, is presented as an infinite periodic fraction. Prove that, if the number of digits in the period is even, then the arithmetic mean of the digits in the period is equal to $\frac{9}{2}$.
2023 CMIMC Integration Bee, 15
\[\int_0^\infty \left(1-e^{-\pi/x^2}\right)^2\,\mathrm dx\]
[i]Proposed by Vlad Oleksenko[/i]
2024 Junior Balkan Team Selection Tests - Moldova, 2
Prove that the number $ \underbrace{88\dots8}_\text{2024\; \textrm{times}}$ is divisible by 2024.
2017 Harvard-MIT Mathematics Tournament, 12
In a certain college containing $1000$ students, students may choose to major in exactly one of math, computer science, finance, or English. The [i]diversity ratio[/i] $d(s)$ of a student $s$ is the defined as number of students in a different major from $s$ divided by the number of students in the same major as $s$ (including $s$). The [i]diversity[/i] $D$ of the college is the sum of all the diversity ratios $d(s)$.
Determine all possible values of $D$.
2011 Balkan MO Shortlist, G4
Given a triangle $ABC$, the line parallel to the side $BC$ and tangent to the incircle of the triangle meets the sides $AB$ and $AC$ at the points $A_1$ and $A_2$ , the points $B_1, B_2$ and $C_1, C_2$ are de ned similarly. Show that
$$AA_1 \cdot AA_2 + BB_1 \cdot BB_2 + CC_1 \cdot CC_2 \ge \frac19 (AB^2 + BC^2 + CA^2)$$
2022 Belarusian National Olympiad, 10.7
Find all positive integers $a$ for which there exists a polynomial $p(x)$ with integer coefficients such that
$p(\sqrt{2}+1)=2-\sqrt{2}$ and $p(\sqrt{2}+2)=a$