Found problems: 85335
1995 Tournament Of Towns, (455) 4
Prove that $\overline{a0... 09}$ (in which $a > 0$ is a digit and there is at least one zero) is not a perfect square.
(VA Senderov)
2005 Cuba MO, 1
Determine all the quadrilaterals that can be divided by a diagonal into two triangles of equal area and equal perimeter.
2013 Argentina Cono Sur TST, 5
Let $ABC$ be an equilateral triangle and $D$ a point on side $AC$. Let $E$ be a point on $BC$ such that $DE \perp BC$, $F$ on $AB$ such that $EF \perp AB$, and $G$ on $AC$ such that $FG \perp AC$. Lines $FG$ and $DE$ intersect in $P$. If $M$ is the midpoint of $BC$, show that $BP$ bisects $AM$.
2003 AMC 12-AHSME, 6
Define $ x \heartsuit y$ to be $ |x\minus{}y|$ for all real numbers $ x$ and $ y$. Which of the following statements is [b]not[/b] true?
$\textbf{(A)}\ x \heartsuit y \equal{} y \heartsuit x \text{ for all } x \text{ and } y$
$\textbf{(B)}\ 2(x \heartsuit y) \equal{} (2x) \heartsuit (2y) \text{ for all } x \text{ and } y$
$\textbf{(C)}\ x \heartsuit 0 \equal{} x \text{ for all } x$
$\textbf{(D)}\ x \heartsuit x \equal{} 0 \text{ for all } x$
$\textbf{(E)}\ x \heartsuit y > 0 \text{ if } x \ne y$
1992 IMO Longlists, 75
A sequence $\{an\}$ of positive integers is defined by
\[a_n=\left[ n +\sqrt n + \frac 12 \right] , \qquad \forall n \in \mathbb N\]
Determine the positive integers that occur in the sequence.
1998 Switzerland Team Selection Test, 8
Let $\vartriangle ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP,BP,CP$ meet the sides $BC,CA,AB$ in the points $X,Y,Z$ respectively. Prove that $XY \cdot YZ\cdot ZX \ge XB\cdot YC\cdot ZA$.
2005 AMC 10, 3
The equations $ 2x \plus{} 7 \equal{} 3$ and $ bx\minus{}10 \equal{} \minus{}\!2$ have the same solution for $ x$. What is the value of $ b$?
$ \textbf{(A)}\minus{}\!8 \qquad
\textbf{(B)}\minus{}\!4 \qquad
\textbf{(C)}\minus{}\!2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 8$
2012 ELMO Shortlist, 6
Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$.
[i]Calvin Deng.[/i]
2004 CentroAmerican, 1
In a $10\times 10$ square board, half of the squares are coloured white and half black. One side common to two squares on the board side is called a [i]border[/i] if the two squares have different colours. Determine the minimum and maximum possible number of borders that can be on the board.
2024 CCA Math Bonanza, L5.1
Michelle is birdwatching. At time $t=0$, she spots $n$ birds all standing on a power cable, in a single line. Every minute after she first spots the birds, she looks back up at the birds, counting the number of them that are left. Assume that each minute, each bird has a $50\%$ chance to fly off, and that no birds decide to perch on the cable for $t\geq0$. As $n$ approaches $\infty$, let the probability that Michelle will see exactly $1$ bird on the line at some point in time approach $p$. Estimate $\lfloor 10000p \rfloor$.
\\\\ Your score will be calculated by the function $\max(0, \lfloor20 - \frac{|A - S|}{12}\rfloor)$, where $S$ is your submission and $A$ is the true answer.
[i]Lightning 5.1[/i]
2017 China Team Selection Test, 3
Let $X$ be a set of $100$ elements. Find the smallest possible $n$ satisfying the following condition: Given a sequence of $n$ subsets of $X$, $A_1,A_2,\ldots,A_n$, there exists $1 \leq i < j < k \leq n$ such that
$$A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.$$
2019 Greece Team Selection Test, 2
Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$.( $AN$ is a diametre of the circle $\Gamma$).
2014 Tournament of Towns., 5
There are several white and black points. Every white point is connected with every black point by a segment. Each segment is equipped with a positive integer. For any closed circuit the product of the integers on the segments passed in the direction from white to black point is equal to the product of the integers on the segments passed in the opposite direction. Can one always place the integer at each point so that the integer on each segment is the product of the integers at its ends?
2023 Romania EGMO TST, P2
Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.
1981 AMC 12/AHSME, 2
Point $E$ is on side $AB$ of square $ABCD$. If $EB$ has length one and $EC$ has length two, then the area of the square is
$\text{(A)}\ \sqrt{3} \qquad \text{(B)}\ \sqrt{5} \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 2\sqrt{3} \qquad \text{(E)}\ 5$
2006 National Olympiad First Round, 26
For how many primes $p$, there exists an integr $m$ such that $m^3+3m-2 \equiv 0 \pmod p$ and $m^2+4m+5\equiv 0 \pmod p$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \text{Infinitely many}
$
2007 Today's Calculation Of Integral, 188
Find the volume of the solid obtained by revolving the region bounded by the graphs of $y=xe^{1-x}$ and $y=x$ around the $x$ axis.
2021/2022 Tournament of Towns, P4
Let us call a 1×3 rectangle a tromino. Alice and Bob go to different rooms, and each divides a 20 × 21 board into trominos. Then they compare the results, compute how many trominos are the same in both splittings, and Alice pays Bob that number of dollars.
What is the maximal amount Bob may guarantee to himself no matter how Alice plays?
2009 Postal Coaching, 5
Let $P$ be an interior point of a circle and $A_1,A_2...,A_{10}$ be points on the circle such that $\angle A_1PA_2 = \angle A_2PA_3 = ... = \angle A_{10}PA_1 = 36^o$. Prove that $PA_1 + PA_3 + PA_5 + PA_7 +PA_9 = PA_2 + PA_4 + PA_6 + PA_8 + PA_{10}$.
2021 Turkey Junior National Olympiad, 3
Let $x, y, z$ be real numbers such that
$$x+y+z=2, \;\;\;\; xy+yz+zx=1$$
Find the maximum possible value of $x-y$.
1979 Dutch Mathematical Olympiad, 3
Define $a_1 = 1979$ and $a_{n+1} = 9^{a_n}$ for $n = 1,2,3,...$. Determine the last two digits of $a_{1979}$.
1966 IMO Longlists, 21
Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality
\[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\]
When does equality occur?
2020 Kyiv Mathematical Festival, 4
(a) Two players take turns taking $1, 2$ or $3$ stones at random from a given set of $3$ piles, in which initially on $11, 22$ and $33$ stones. If after the move of one of the players in any two groups the same number of stones will remain, this player has won. Who will win with the right game of both players?
(b) Two players take turns taking $1$ or $2$ stones from one pile, randomly selected from a given set of $3$ ordered piles, in which at first $100, 200$ and $300$ stones, in order from left to right. Additionally it is forbidden to make a course at which, for some pair of the next handfuls, quantity of stones in the left will be more than the number of stones in the right. If after the move of one of the players of the stones in handfuls will not remain, then this player won. Who will win with the right game of both players?
[hide=original wording]
1. Два гравця по черзi беруть 1, 2 чи 3 камiнця довiльним чином з заданого набору з 3 купок, в
яких спочатку по 11, 22 i 33 камiнцiв. Якщо пiсля хода одного з гравцiв в якихось двух купках
залишиться однакова кiлькiсть камiнцiв, то цей гравець виграв. Хто виграє при правильнiй грi обох
гравцiв?
2. Два гравця по черзi беруть 1 чи 2 камiнця з одної купки, довiльної вибраної з заданого набору
з 3 впорядкованих купок, в яких спочатку по 100, 200 i 300 камiнцiв, в порядку злiва направо.
Додатково забороняется робити ход при якому, для деякої пари сусiднiх купок, кiлькiсть камiнцiв в
лiвiй стане бiльше нiж кiлькiсть камiнцiв в правiй. Якщо пiсля ходу одного з гравцiв камiнцiв в
купках не залишиться, то цей гравець виграв. Хто виграє при правильнiй грi обох гравцiв?[/hide]
1984 IMO Longlists, 15
Consider all the sums of the form
\[\displaystyle\sum_{k=1}^{1985} e_kk^5=\pm 1^5\pm 2^5\pm\cdots\pm1985^5\]
where $e_k=\pm 1$. What is the smallest nonnegative value attained by a sum of this type?
2012 IFYM, Sozopol, 3
The polynomial $p(x)$ is of degree $9$ and $p(x)-1$ is exactly divisible by $(x-1)^{5}$.
Given that $p(x) + 1$ is exactly divisible by $(x+1)^{5}$, find $p(x)$.