Found problems: 85335
2021 MOAA, 7
Compute the number of ordered pairs $(a,b)$ of positive integers satisfying $a^b=2^{100}$.
[i]Proposed by Nathan Xiong[/i]
2002 AMC 12/AHSME, 25
Let $a$ and $b$ be real numbers such that $\sin a+\sin b=\dfrac{\sqrt2}2$ and $\cos a+\cos b=\dfrac{\sqrt6}2$. Find $\sin(a+b)$.
$\textbf{(A) }\dfrac12\qquad\textbf{(B) }\dfrac{\sqrt2}2\qquad\textbf{(C) }\dfrac{\sqrt3}2\qquad\textbf{(D) }\dfrac{\sqrt6}2\qquad\textbf{(E) }1$
2003 District Olympiad, 1
Find all functions $\displaystyle f : \mathbb N^\ast \to \mathbb N^\ast$ ($\displaystyle N^\ast = \{ 1,2,3,\ldots \}$) with the property that, for all $\displaystyle n \geq 1$, \[ f(1) + f(2) + \ldots + f(n) \] is a perfect cube $\leq n^3$.
[i]Dinu Teodorescu[/i]
2008 Tournament Of Towns, 4
Given three distinct positive integers such that one of them is the average of the two others. Can the product of these three integers be the perfect 2008th power of a positive integer?
V Soros Olympiad 1998 - 99 (Russia), 11.5
Find all values of the parameter $a$ for which the sum of all solutions (meaning real solutions) of the equation $x^4 - 5x + a = 0$ is equal to $a$
1966 IMO Longlists, 51
Consider $n$ students with numbers $1, 2, \ldots, n$ standing in the order $1, 2, \ldots, n.$ Upon a command, any of the students either remains on his place or switches his place with another student. (Actually, if student $A$ switches his place with student $B,$ then $B$ cannot switch his place with any other student $C$ any more until the next command comes.)
Is it possible to arrange the students in the order $n,1, 2, \ldots, n-1$ after two commands ?
2023-24 IOQM India, 11
A positive integer $m$ has the property that $m^2$ is expressible in the form $4n^2-5n+16$ where $n$ is an integer (of any sign). Find the maximum value of $|m-n|.$
2019 IMO Shortlist, N3
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
2016 Croatia Team Selection Test, Problem 2
Let $S$ be a set of $N \ge 3$ points in the plane. Assume that no $3$ points in $S$ are collinear. The segments with both endpoints in $S$ are colored in two colors.
Prove that there is a set of $N - 1$ segments of the same color which don't intersect except in their endpoints such that no subset of them forms a polygon with positive area.
2012 Indonesia TST, 4
Determine all integer $n > 1$ such that
\[\gcd \left( n, \dfrac{n-m}{\gcd(n,m)} \right) = 1\]
for all integer $1 \le m < n$.
2014 Bosnia and Herzegovina Junior BMO TST, 4
It is given $5$ numbers $1$, $3$, $5$, $7$, $9$. We get the new $5$ numbers such that we take arbitrary $4$ numbers(out of current $5$ numbers) $a$, $b$, $c$ and $d$ and replace them with $\frac{a+b+c-d}{2}$, $\frac{a+b-c+d}{2}$, $\frac{a-b+c+d}{2}$ and $\frac{-a+b+c+d}{2}$. Can we, with repeated iterations, get numbers:
$a)$ $0$, $2$, $4$, $6$ and $8$
$b)$ $3$, $4$, $5$, $6$ and $7$
2007 AMC 12/AHSME, 1
Isabella's house has $ 3$ bedrooms. Each bedroom is $ 12$ feet long, $ 10$ feet wide, and $ 8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $ 60$ square feet in each bedroom. How many square feet of walls must be painted?
$ \textbf{(A)}\ 678 \qquad \textbf{(B)}\ 768 \qquad \textbf{(C)}\ 786 \qquad \textbf{(D)}\ 867 \qquad \textbf{(E)}\ 876$
1972 AMC 12/AHSME, 14
A triangle has angles of $30^\circ$ and $45^\circ$. If the side opposite the $45^\circ$ angle has length $8$, then the side opposite the $30^\circ$ angle has length
$\textbf{(A) }4\qquad\textbf{(B) }4\sqrt{2}\qquad\textbf{(C) }4\sqrt{3}\qquad\textbf{(D) }4\sqrt{6}\qquad \textbf{(E) }6$
2021 Centroamerican and Caribbean Math Olympiad, 3
In a table consisting of $2021\times 2021$ unit squares, some unit squares are colored black in such a way that if we place a mouse in the center of any square on the table it can walk in a straight line (up, down, left or right along a column or row) and leave the table without walking on any black square (other than the initial one if it is black). What is the maximum number of squares that can be colored black?
2024 India IMOTC, 7
Let $ABC$ be an acute-angled triangle with $AB<AC$, incentre $I$, and let $M$ be the midpoint of major arc $BAC$. Suppose the perpendicular line from $A$ to segment $BC$ meets lines $BI$, $CI$, and $MI$ at points $P$, $Q$, and $K$ respectively. Prove that the $A-$median line in $\triangle AIK$ passes through the circumcentre of $\triangle PIQ$.
[i]Proposed by Pranjal Srivastava and Rohan Goyal[/i]
2020 Peru Cono Sur TST., P6
Let $a_1, a_2, a_3, \ldots$ a sequence of positive integers that satisfy the following conditions:
$$a_1=1, a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor}, \forall n\ge 1$$
Prove that for every positive integer $k$ there exists a term $a_i$ that is divisible by $k$
2024 All-Russian Olympiad Regional Round, 11.4
Let $XY$ be a segment, which is a diameter of a semi-circle. Let $Z$ be a point on $XY$ and 9 rays from $Z$ are drawn that divide $\angle XZY=180^{\circ}$ into $10$ equal angles. These rays meet the semi-circle at $A_1, A_2, \ldots, A_9$ in this order in the direction from $X$ to $Y$. Prove that the sum of the areas of triangles $ZA_2A_3$ and $ZA_7A_8$ equals the area of the quadrilateral $A_2A_3A_7A_8$.
2008 District Olympiad, 1
Prove that for an integer $ n>\equal{}1$ we have $ n(1\plus{}\frac{1}{2}\plus{}\frac{1}{3}\plus{}\dots\plus{}\frac{1}{n})\geq (n\plus{}1)(\frac{1}{2}\plus{}\frac{1}{3}\plus{}\dots\frac{1}{n\plus{}1})$
2025 NCMO, 4
Let $P$ be a polynomial. Suppose that there exists a rational $q$ such that $P(m)=q^n$ for infinitely many integers $(m,n)$. Prove that $P(x)=c\cdot Q(x)^k$ for some integer constants $c$ and $k$ and irreducible polynomial $Q$ with rational coefficients.
(Here, a polynomial is $\textit{irreducible}$ if it can't be factored into the product of non-constant polynomials with rational coefficients.)
[i]Jason Lee[/i]
2020 Kazakhstan National Olympiad, 3
Let $p$ be a prime number and $k,r$ are positive integers such that $p>r$. If $pk+r$ divides $p^p+1$ then prove that $r$ divides $k$.
2012-2013 SDML (Middle School), 4
If $10\%$ of $\left(x+10\right)$ is $\left(x-10\right)$, what is $10\%$ of $x$?
$\text{(A) }\frac{11}{90}\qquad\text{(B) }\frac{9}{11}\qquad\text{(C) }1\qquad\text{(D) }\frac{11}{9}\qquad\text{(E) }\frac{110}{9}$
2006 Sharygin Geometry Olympiad, 8
The segment $AB$ divides the square into two parts, in each of which a circle can be inscribed.
The radii of these circles are equal to $r_1$ and $r_2$ respectively, where $r_1> r_2$.
Find the length of $AB$.
2023 Hong Kong Team Selection Test, Problem 4
A two digit number $s$ is special if $s$ is the common two leading digits of the decimal expansion of $4^n$ and $5^n$, where $n$ is a certain positive integer. Given that there are two special number, find these two special numbers.
1997 Greece Junior Math Olympiad, 1
Let $ABC$ be an equilateral triangle whose angle bisectors of $B$ and $C$ intersect at $D$. Perpendicular bisectors of $BD$ and $CD$ intersect $BC$ at points $E$ and $Z$ respectively.
a) Prove that $BE=EZ=ZC$.
b) Find the ratio of the areas of the triangles $BDE$ to $ABC$
2021 Bangladeshi National Mathematical Olympiad, 6
On a table near the sea, there are $N$ glass boxes where $N<2021$, each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, he can throw out all of the balls in the selected box. The player who throws out the last ball wins. Let $S$ be the sum of all values of $N$ for which Sowdha has a winning strategy and let $R$ be the sum of all values of $N$ for which Rafi has a winning strategy. What is the value of $\frac{R-S}{10}$?