Found problems: 85335
VI Soros Olympiad 1999 - 2000 (Russia), 10.7
The numbers $1, 2, 3, ..., 99, 100$ are randomly arranged in the cells of a square table measuring $10\times 10$ (each number is used only once). Prove that there are three cells in the table whose sum of numbers does not exceed 1$82$. The centers of these cells form an isosceles right triangle, the legs of which are parallel to the edges of the table.
2020 Czech-Austrian-Polish-Slovak Match, 3
The numbers $1, 2,..., 2020$ are written on the blackboard. Venus and Serena play the following game. First, Venus connects by a line segment two numbers such that one of them divides the other. Then Serena connects by a line segment two numbers which has not been connected and such that one of them divides the other. Then Venus again and they continue until there is a triangle with one vertex in $2020$, i.e. $2020$ is connected to two numbers that are connected with each other. The girl that has drawn the last line segment (completed the triangle) is the winner. Which of the girls has a winning strategy?
(Tomáš Bárta, Czech Republic)
2023 Stanford Mathematics Tournament, 4
Michelle is drawing segments in the plane. She begins from the origin facing up the $y$-axis and draws a segment of length $1$. Now, she rotates her direction by $120^\circ$, with equal probability clockwise or counterclockwise, and draws another segment of length $1$ beginning from the end of the previous segment. She then continues this until she hits an already drawn segment. What is the expected number of segments she has drawn when this happens?
2006 AIME Problems, 4
Let $(a_1,a_2,a_3,...,a_{12})$ be a permutation of $(1,2,3,...,12)$ for which \[ a_1>a_2>a_3>a_4>a_5>a_6 \text{ and } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}. \]
An example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12)$. Find the number of such permutations.
1994 Taiwan National Olympiad, 6
For $-1\leq x\leq 1$ and $n\in\mathbb N$ define $T_{n}(x)=\frac{1}{2^{n}}[(x+\sqrt{1-x^{2}})^{n}+(x-\sqrt{1-x^{2}})^{n}]$.
a)Prove that $T_{n}$ is a monic polynomial of degree $n$ in $x$ and that the maximum value of $|T_{n}(x)|$ is $\frac{1}{2^{n-1}}$.
b)Suppose that $p(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\in\mathbb{R}[x]$ is a monic polynomial of degree $n$ such that $p(x)>-\frac{1}{2^{n-1}}$ forall $x$, $-1\leq x\leq 1$. Prove that there exists $x_{0}$, $-1\leq x_{0}\leq 1$ such that $p(x_{0})\geq\frac{1}{2^{n-1}}$.
PEN A Problems, 4
If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.
2024 Harvard-MIT Mathematics Tournament, 31
Ash and Gary independently come up with their own lineups of $15$ fire, grass, and water monsters. Then, the first monster of both lineups will fight, with fire beating grass, grass beating water, and water beating fire. The defeated monster is then substituted with the next one from their team’s lineup; if there is a draw, both monsters get defeated.
Gary completes his lineup randomly, with each monster being equally likely to be any of the three types. Without seeing Gary’s lineup, Ash chooses a lineup that maximizes the probability p that his monsters are the last ones standing. Compute $p.$
2009 Mexico National Olympiad, 2
In boxes labeled $0$, $1$, $2$, $\dots$, we place integers according to the following rules:
$\bullet$ If $p$ is a prime number, we place it in box $1$.
$\bullet$ If $a$ is placed in box $m_a$ and $b$ is placed in box $m_b$, then $ab$ is placed in the box labeled $am_b+bm_a$.
Find all positive integers $n$ that are placed in the box labeled $n$.
2015 South Africa National Olympiad, 3
We call a divisor $d$ of a positive integer $n$ [i]special[/i] if $d + 1$ is also a divisor of $n$. Prove: at most half the positive divisors of a positive integer can be special. Determine all positive integers for which exactly half the positive divisors are special.
2023 LMT Fall, 13
Ella lays out $16$ coins heads up in a $4\times 4$ grid as shown.
[img]https://cdn.artofproblemsolving.com/attachments/3/3/a728be9c51b27f442109cc8613ef50d61182a0.png[/img]
On a move, Ella can flip all the coins in any row, column, or diagonal (including small diagonals such as $H_1$ & $H_4$). If rotations are considered distinct, how many distinct grids of coins can she create in a finite number of moves?
2007 IMO Shortlist, 1
Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define
\[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \}
\]
and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$.
(a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$,
\[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*)
\]
(b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*).
[i]Author: Michael Albert, New Zealand[/i]
2002 All-Russian Olympiad Regional Round, 8.4
Given a triangle $ABC$ with pairwise distinct sides. on his on the sides, regular triangles $ABC_1$, $BCA_1$, $CAB_1$. are constructed externally. Prove that triangle $A_1B_1C_1$ cannot be regular.
2020 Online Math Open Problems, 4
Let $ABCD$ be a square with side length $16$ and center $O$. Let $\mathcal S$ be the semicircle with diameter $AB$ that lies outside of $ABCD$, and let $P$ be a point on $\mathcal S$ so that $OP = 12$. Compute the area of triangle $CDP$.
[i]Proposed by Brandon Wang[/i]
2012 Dutch IMO TST, 4
Let $n$ be a positive integer divisible by $4$. We consider the permutations $(a_1, a_2,...,a_n)$ of $(1,2,..., n)$ having the following property: for each j we have $a_i + j = n + 1$ where $i = a_j$ . Prove that there are exactly $\frac{ (\frac12 n)!}{(\frac14 n)!}$ such permutations.
2011 All-Russian Olympiad, 3
Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$.
[i]A. Golovanov[/i]
1999 May Olympiad, 2
In a parallelogram $ABCD$ , $BD$ is the largest diagonal. By matching $B$ with $D$ by a bend, a regular pentagon is formed. Calculate the measures of the angles formed by the diagonal $BD$ with each of the sides of the parallelogram.
1970 IMO Longlists, 55
A turtle runs away from an UFO with a speed of $0.2 \ m/s$. The UFO flies $5$ meters above the ground, with a speed of $20 \ m/s$. The UFO's path is a broken line, where after flying in a straight path of length $\ell$ (in meters) it may turn through for any acute angle $\alpha$ such that $\tan \alpha < \frac{\ell}{1000}$. When the UFO's center approaches within $13$ meters of the turtle, it catches the turtle. Prove that for any initial position the UFO can catch the turtle.
2013 NIMO Problems, 8
The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.
[i]Proposed by Evan Chen[/i]
2000 All-Russian Olympiad Regional Round, 10.7
In a convex quadrilateral $ABCD$ we draw the bisectors $\ell_a$, $\ell_b$, $\ell_c$, $\ell_d$ of external angles $A$, $B$, $C$, $D$ respectively. The intersection points of the lines $\ell_a$ and $\ell_b$, $\ell_b$ and $\ell_c$, $\ell_c$ and $\ell_d$, $\ell_d$ and $\ell_a$ are designated by $K$, $L$, $M$, $N$. It is known that $3$ perpendiculars drawn from $K$ on $AB$, from $L$ om $BC$, from $M$ on $CD$ intersect at one point. Prove that the quadrilateral $ABCD$ is cyclic.
2003 Greece JBMO TST, 2
Calculate if $n\in N$ with $n>2$ the value of
$$B=\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+...+\sqrt{1+\frac{1}{(n-1)^2}+\frac{1}{n^2}} $$
2022 JHMT HS, 5
Consider an array of white unit squares arranged in a rectangular grid with $59$ rows of unit squares and $c$ columns of unit squares, for some positive integer $c$. What is the smallest possible value of $c$ such that, if we shade exactly $25$ unit squares in each column black, then there must necessarily be some row with at least $18$ black unit squares?
1949-56 Chisinau City MO, 44
Determine the locus of points, for each of which the difference between the squares of the distances to two given points is a constant value.
1991 National High School Mathematics League, 1
Set $S=\{1,2,\cdots,n\}$. $A$ is an increasing arithmetic sequence (at least two numbers), and all numbers are in $S$. Also, we can't add any number in $S$ to $A$ without changing its tolerance. Find the number of such sequence $A$.
2017 Peru IMO TST, 4
The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?
2022 Malaysia IMONST 2, 2
It is known that there are $n$ integers $a_1, a_2, \cdots, a_n$ such that
$$a_1 + a_2 + \cdots + a_n = 0 \qquad \text{and} \qquad a_1 \times a_2 \times \cdots \times a_n = n.$$
Determine all possible values of $n$.