Found problems: 85335
2004 Olympic Revenge, 3
$ABC$ is a triangle and $\omega$ its incircle. Let $P,Q,R$ be the intersections with $\omega$ and the sides $BC,CA,AB$ respectively. $AP$ cuts $\omega$ in $P$ and $X$. $BX,CX$ cut $\omega$ in $M,N$ respectively. Show that $MR,NQ,AP$ are parallel or concurrent.
2009 Regional Competition For Advanced Students, 4
Two infinite arithmetic progressions are called considerable different if the do not only differ by the absence of finitely many members at the beginning of one of the sequences.
How many pairwise considerable different non-constant arithmetic progressions of positive integers that contain an infinite non-constant geometric progression $ (b_n)_{n\ge0}$ with $ b_2\equal{}40 \cdot 2009$ are there?
Kvant 2025, M2830
There are coins in denominations of $a$ and $b$ doubloons, where $a$ and $b$ are given mutually prime natural numbers, with $a < b < 100$. A non-negative integer $n$ is called [i]lucky[/i] if the sum in $n$ doubloons can be scored with using no more than $1000$ coins. Find the number of lucky numbers.
[i]From the folklore[/i]
2017 Swedish Mathematical Competition, 1
Xenia and Yagve take turns in playing the following game: A coin is placed on the first box in a row of nine cells. At each turn the player may choose to move the coin forward one step, move the coin forward four steps, or move coin back two steps. For a move to be allowed, the coin must land on one of them of nine cells. The winner is one who gets to move the coin to the last ninth cell. Who wins, given that Xenia makes the first move, and both players play optimally?
2005 Brazil National Olympiad, 1
A natural number is a [i]palindrome[/i] when one obtains the same number when writing its digits in reverse order. For example, $481184$, $131$ and $2$ are palindromes.
Determine all pairs $(m,n)$ of positive integers such that $\underbrace{111\ldots 1}_{m\ {\rm ones}}\times\underbrace{111\ldots 1}_{n\ {\rm ones}}$ is a palindrome.
2013 Argentina Cono Sur TST, 3
$1390$ ants are placed near a line, such that the distance between their heads and the line is less than $1\text{cm}$ and the distance between the heads of two ants is always larger than $2\text{cm}$. Show that there is at least one pair of ants such that the distance between their heads is at least $10$ meters (consider the head of an ant as point).
LMT Speed Rounds, 2011.19
A positive six-digit integer begins and ends in $8$, and is also the product of three consecutive even numbers. What is the sum of the three even numbers?
2019 CCA Math Bonanza, L3.4
Determine the maximum possible value of \[\frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4}\] over all non-zero real numbers $x$.
[i]2019 CCA Math Bonanza Lightning Round #3.4[/i]
2024 Malaysia IMONST 2, 4
Pingu is given two positive integers $m$ and $n$ without any common factors greater than $1$.
a) Help Pingu find positive integers $p, q$ such that $$\operatorname{gcd}(pm+q, n) \cdot \operatorname{gcd}(m, pn+q) = mn$$
b) Prove to Pingu that he can never find positive integers $r, s$ such that $$\operatorname{lcm}(rm+s, n) \cdot \operatorname{lcm}(m, rn+s) = mn$$
regardless of the choice of $m$ and $n$.
2001 Manhattan Mathematical Olympiad, 2
The dates of three Sundays of a month were even numbers. What day of the week was the $20$th of the month?
1993 Tournament Of Towns, (391) 3
Each of the numbers $1, 2, 3,... 25$ is arranged in a $5$ by $5$ table. In each row they appear in increasing order (left to right). Find the maximal and minimal possible sum of the numbers in the third column.
(Folklore)
1998 AMC 8, 2
If $ \begin{tabular}{r|l}a&b\\ \hline c&d\end{tabular}=\text{a}\cdot\text{d}-\text{b}\cdot\text{c} $, what is the value of $ \begin{tabular}{r|l}3&4\\ \hline 1&2\end{tabular} $
$ \text{(A)}\ -2\qquad\text{(B)}\ -1\qquad\text{(C)}\ 0\qquad\text{(D)}\ 1\qquad\text{(E)}\ 2 $
2000 Belarus Team Selection Test, 6.3
Starting with an arbitrary pair (a,b) of vectors on the plane, we are allowed to perform the operations of the following two types:
(1) To replace $(a,b)$ with $(a+2kb,b)$ for an arbitrary integer $k \ne 0$;
(2) To replace $(a,b)$ with $(a,b+2ka)$ for an arbitrary integer $ k \ne 0$.
However, we must change the type of operetion in any step.
(a) Is it possible to obtain $((1,0), (2,1))$ from $((1,0), (0,1))$, if the first operation is of the type (1)?
(b) Find all pairs of vectors that can be obtained from $((1,0), (0,1))$ (the type of first operation can be selected arbitrarily).
2010 Singapore Senior Math Olympiad, 2
The numbers $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$, $y$, erases them and then writes down $x + y + xy$. He continues to do this until only one number is left on the blackboard. What is this number?
2018 Regional Olympiad of Mexico Northeast, 1
$N$ different positive integers are arranged around a circle , in such a way that the sum of every $5$ consecutive numbers in the circle is a multiple of $13$. Let $A $ be the smallest possible sum of the $n$ numbers. Calculate the value of $A$ for
$\bullet$ $n = 99$,
$\bullet$ $n = 100$.
1995 Poland - Second Round, 1
For a polynomial $P$ with integer coefficients, $P(5)$ is divisible by $2$ and $P(2)$ is divisible by $5$. Prove that $P(7)$ is divisible by $10$.
2006 Kyiv Mathematical Festival, 2
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
2006 equilateral triangles are located in the square with side 1. The sum of their perimeters is equal to 300. Prove that at least three of them have a common point.
2019 AIME Problems, 15
Let $\overline{AB}$ be a chord of a circle $\omega$, and let $P$ be a point on the chord $\overline{AB}$. Circle $\omega_1$ passes through $A$ and $P$ and is internally tangent to $\omega$. Circle $\omega_2$ passes through $B$ and $P$ and is internally tangent to $\omega$. Circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$. Line $PQ$ intersects $\omega$ at $X$ and $Y$. Assume that $AP=5$, $PB=3$, $XY=11$, and $PQ^2 = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1976 Putnam, 4
Let $r$ be a root of $P(x)=x^3+ax^2+bx-1=0$ and $r+1$ be a root of $y^3+cy^2+dy+1=0,$ where $a,b,c$ and $d$ are integers. Also let $P(x)$ be irreducible over the rational numbers. Express another root $s$ of $P(x)=0$ as a function of $r$ which does not explicitly involve $a,b,c$ or $d.$
2008 Tournament Of Towns, 5
Standing in a circle are $99$ girls, each with a candy. In each move, each girl gives her candy to either neighbour. If a girl receives two candies in the same move, she eats one of them. What is the minimum number of moves after which only one candy remains?
1993 Chile National Olympiad, 2
Given a rectangle, circumscribe a rectangle of maximum area.
2018 Sharygin Geometry Olympiad, 5
Let $ABCD$ be a cyclic quadrilateral, $BL$ and $CN$ be the internal angle bisectors in triangles $ABD$ and $ACD$ respectively. The circumcircles of triangles $ABL$ and $CDN$ meet at points $P$ and $Q$. Prove that the line $PQ$ passes through the midpoint of the arc $AD$ not containing $B$.
2011 F = Ma, 16
What magnitude force does Jonathan need to exert on the physics book to keep the rope from slipping?
(A) $Mg$
(B) $\mu_k Mg$
(C) $\mu_k Mg/\mu_s$
(D) $(\mu_s + \mu_k)Mg$
(E) $Mg/\mu_s$
2025 Greece National Olympiad, 3
Let $f(x):\mathbb {Q} \rightarrow \mathbb {Q}$ be a function satisfying $f(x+2y)+f(2x-y)=5f(x)+5f(y)$
Find all such functions.
1982 Czech and Slovak Olympiad III A, 5
Given is a sequence of real numbers $\{a_n\}^{\infty}_{n=1}$ such that $a_n \ne a_m$ for $n\ne m,$ given is a natural number $k$. Construct an injective map $P:\{1,2,\ldots,20k\}\to\mathbb Z^+$ such that the following inequalities hold:
$$a_{p(1)}<a_{p(2)}<...<a_{p(10)}$$
$$ a_{p(10)}>a_{p(11)}>...>a_{p(20)}$$
$$a_{p(20)}<a_{p(21)}<...<a_{p(30)}$$
$$...$$
$$a_{p(20k-10)}>a_{p(20k-9)}>...>a_{p(20k)}$$
$$a_{p(10)}>a_{p(30)}>...>a_{p((20k-10))} $$
$$a_{p(1)}<a_{p(20)}<...<a_{p(20k)},$$