Found problems: 85335
2024 LMT Fall, 7
Find the sum of the distinct prime factors of $512512$.
2016 Iran Team Selection Test, 1
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
2017 Math Prize for Girls Olympiad, 3
Let $ABCD$ be a cyclic quadrilateral such that $\angle BAD \le \angle ADC$. Prove that $AC + CD \le AB + BD$.
ICMC 7, 5
[list=a]
[*]Is there a non-linear integer-coefficient polynomial $P(x)$ and an integer $N{}$ such that all integers greater than $N{}$ may be written as the greatest common divisor of $P(a){}$ and $P(b){}$ for positive integers $a>b$?
[*]Is there a non-linear integer-coefficient polynomial $Q(x)$ and an integer $M{}$ such that all integers greater than $M{}$ may be written as $Q(a) - Q(b)$ for positive integers $a>b$?
[/list][i]Proposed by Dylan Toh[/i]
2016 India Regional Mathematical Olympiad, 4
There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)?
2010 Slovenia National Olympiad, 1
For a real number $t$ and positive real numbers $a,b$ we have
\[2a^2-3abt+b^2=2a^2+abt-b^2=0\]
Find $t.$
2001 USA Team Selection Test, 1
Let $\{ a_n\}_{n \ge 0}$ be a sequence of real numbers such that $a_{n+1} \ge a_n^2 + \frac{1}{5}$ for all $n \ge 0$. Prove that $\sqrt{a_{n+5}} \ge a_{n-5}^2$ for all $n \ge 5$.
2015 NIMO Summer Contest, 9
On a blackboard lies $50$ magnets in a line numbered from $1$ to $50$, with different magnets containing different numbers. David walks up to the blackboard and rearranges the magnets into some arbitrary order. He then writes underneath each pair of consecutive magnets the positive difference between the numbers on the magnets. If the expected number of times he writes the number $1$ can be written in the form $\tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i] Proposed by David Altizio [/i]
2017 ELMO Shortlist, 4
Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\omega$. Suppose a circle $\omega_B$ is tangent to $BA,BC$, and internally tangent to $\omega$ at $B_1$, while a circle $\omega_C$ is tangent to $CA, CB$, and internally tangent to $\omega$ at $C_1$. If $B_2, C_2$ are the points opposite to $B,C$ on $\omega$, respectively, and $X$ denotes the intersection of $B_1C_2, B_2C_1$, prove that $XA=XI$.
[i]Proposed by Vincent Huang and Nathan Weckwerth
2023 BMT, 8
One of Landau’s four unsolved problems asks whether there are infinitely many primes $p$ such that $p- 1$ is a perfect square. How many such primes are there less than $100$?
2018-2019 SDML (High School), 2
Given that $\frac{x}{\sqrt{x} + \sqrt{y}} = 18$ and $\frac{y}{\sqrt{x} + \sqrt{y}} = 2$, find $\sqrt{x} - \sqrt{y}$.
2019-2020 Winter SDPC, 7
Let $a,b$ be positive integers. Find, with proof, the maximum possible value of $a\lceil b\lambda \rceil - b \lfloor a \lambda \rfloor$ for irrational $\lambda$.
2000 South africa National Olympiad, 3
Let $c \geq 1$ be an integer, and define the sequence $a_1,\ a_2,\ a_3,\ \dots$ by \[ \begin{aligned} a_1 & = 2, \\ a_{n + 1} & = ca_n + \sqrt{\left(c^2 - 1\right)\left(a_n^2 - 4\right)}\textrm{ for }n = 1,2,3,\dots\ . \end{aligned} \] Prove that $a_n$ is an integer for all $n$.
1993 Turkey MO (2nd round), 5
Prove that we can draw a line (by a ruler and a compass) from a vertice of a convex quadrilateral such that, the line divides the quadrilateral to two equal areas.
2014-2015 SDML (Middle School), 3
Layna wants to paint a rectangular wall green, but she only has blue and yellow paint. She finds that a $2:1$ mix of blue paint to yellow paint produces the color green she wants, and she knows that one gallon of paint will cover $80$ square feet of wall. If the wall is $8$ feet tall and $21$ feet long, how many gallons of blue paint does Layna need? Express your answer as a fraction in simplest form.
2023 Israel TST, P2
Let $ABC$ be an isosceles triangle, $AB=AC$ inscribed in a circle $\omega$. The $B$-symmedian intersects $\omega$ again at $D$. The circle through $C,D$ and tangent to $BC$ and the circle through $A,D$ and tangent to $CD$ intersect at points $D,X$. The incenter of $ABC$ is denoted $I$. Prove that $B,C,I,X$ are concyclic.
2014 ELMO Shortlist, 11
Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define
\[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \]
Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$.
[i]Proposed by Victor Wang[/i]
2002 Tuymaada Olympiad, 1
A positive integer $c$ is given. The sequence $\{p_{k}\}$ is constructed by the following rule: $p_{1}$ is arbitrary prime and for $k\geq 1$ the number $p_{k+1}$ is any prime divisor of $p_{k}+c$ not present among the numbers $p_{1}$, $p_{2}$, $\dots$, $p_{k}$. Prove that the sequence $\{p_{k}\}$ cannot be infinite.
[i]Proposed by A. Golovanov[/i]
2010 Malaysia National Olympiad, 7
A line segment of length 1 is given on the plane. Show that a line segment of length $\sqrt{2010}$ can be constructed using only a straightedge and a compass.
2020 Thailand Mathematical Olympiad, 2
There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf $E_n$ takes $n$ minutes to travel $1 \text{ km}$ for each $n = 1,2,...,63$ , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?
2018 Austria Beginners' Competition, 4
For a positive integer $n$ we denote by $d(n)$ the number of positive divisors of $n$ and by $s(n)$ the sum of these divisors. For example, $d(2018)$ is equal to $4$ since $2018$ has four divisors $(1, 2, 1009, 2018)$ and $s(2018) = 1 + 2 + 1009 + 2018 = 3030$.
Determine all positive integers $x$ such that $s(x) \cdot d(x) = 96$.
(Richard Henner)
1979 VTRMC, 1
Show that the right circular cylinder of volume $V$ which has the least surface area is the one whose diameter is equal to its altitude. (The top and bottom are part of the surface.)
2023 Taiwan TST Round 2, 6
There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$, $B$ and $C$, respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$. On each turn, Ming chooses a two-line intersection inside $ABC$, and draws the straight line determined by the intersection and one of $A$, $B$ and $C$ of his choice. Find the maximum possible number of three-line intersections within $ABC$ after 300 turns.
[i]
Proposed by usjl[/i]
2018 Junior Balkan Team Selection Tests - Romania, 4
Consider a $ 2018\times 2018$. board. An "LC-tile" is a tile consisting of $9$ unit squares, having the shape as in the gure below. What is the maximum number of "LC-tiles" that can be placed on the board without superposing them? (Each of the $9$ unit squares of the tile must cover one of the unit squares of the board; a tile may be rotated, turned upside down, etc.)
[img]https://cdn.artofproblemsolving.com/attachments/7/4/a2f992bc0341def1a6e5e26ba8a9eb3384698a.png
[/img]
Alexandru Girban
1996 Estonia Team Selection Test, 1
Suppose that $x,y$ and $\frac{x^2+y^2+6}{xy}$ are positive integers . Prove that $\frac{x^2+y^2+6}{xy}$ is a perfect cube.