Found problems: 85335
1994 Canada National Olympiad, 3
$25$ men sit around a circular table. Every hour there is a vote, and each must respond [i]yes [/i]or [i]no[/i]. Each man behaves as follows: on the $n^{\text{th}}$, vote if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the $(n+1)^{\text{th}}$ vote as on the $n^{\text{th}}$ vote; but if his response is different from that of both his neighbours on the $n^{\text{th}}$ vote, then his response on the $(n+1)^{\text{th}}$ vote will be different from his response on the $n^{\text{th}}$ vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.
2021 Harvard-MIT Mathematics Tournament., 6
A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off of the square’s perimeter $n$ times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly $2021$ regions. Compute the smallest possible value of $n$.
1985 IMO Shortlist, 15
Let $K$ and $K'$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_1, T_2, \ldots, T_p$ with mutually disjoint interiors and find translations $t_1, t_2, \ldots, t_p$ such that
\[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]
2022 Polish Junior Math Olympiad First Round, 2.
In the rectangle $ABCD$, the ratio of the lengths of sides $BC$ and $AB$ is equal to $\sqrt{2}$. Point $X$ is marked inside this rectangle so that $AB=BX=XD$. Determine the measure of angle $BXD$.
2013 Canadian Mathematical Olympiad Qualification Repechage, 5
For each positive integer $k$, let $S(k)$ be the sum of its digits. For example, $S(21) = 3$ and $S(105) = 6$. Let $n$ be the smallest integer for which $S(n) - S(5n) = 2013$. Determine the number of digits in $n$.
1997 Yugoslav Team Selection Test, Problem 3
Numbers $1,2,\ldots,1997^2$ are written in the cells of a $1997\times1997$ table. It is allowed to apply the following transformations: exchange places of any two rows or any two columns, or reverse a row or column. (When a row or column is reversed, the first and last entry exchange their positions, so do the second and second last, etc.) Is it possible that, after finitely many such transformations, arbitrary two numbers exchange their positions and no other number changes its position?
2016 Bangladesh Mathematical Olympiad, 4
Consider the set of integers $ \left \{ 1, 2, \dots , 100 \right \} $. Let $ \left \{ x_1, x_2, \dots , x_{100} \right \}$ be some arbitrary arrangement of the integers $ \left \{ 1, 2, \dots , 100 \right \}$, where all of the $x_i$ are different. Find the smallest possible value of the sum
$$S = \left | x_2 - x_1 \right | + \left | x_3 - x_2 \right | + \cdots+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | .$$
2007 Cuba MO, 1
Find all the real numbers $x, y$ such that $x^3 - y^3 = 7(x - y)$ and $x^3 + y^3 = 5(x + y).$
1994 All-Russian Olympiad Regional Round, 9.8
There are $ 16$ pupils in a class. Every month, the teacher divides the pupils into two groups. Find the smallest number of months after which it will be possible that every two pupils were in two different groups during at least one month.
2010 Contests, 1
The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$
2002 AIME Problems, 15
Polyhedron $ABCDEFG$ has six faces. Face $ABCD$ is a square with $AB=12;$ face $ABFG$ is a trapezoid with $\overline{AB}$ parallel to $\overline{GF},$ $BF=AG=8,$ and $GF=6;$ and face $CDE$ has $CE=DE=14.$ The other three faces are $ADEG, BCEF,$ and $EFG.$ The distance from $E$ to face $ABCD$ is 12. Given that $EG^2=p-q\sqrt{r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$
2024 AMC 10, 23
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
$
\textbf{(A) }212 \qquad
\textbf{(B) }247 \qquad
\textbf{(C) }258 \qquad
\textbf{(D) }276 \qquad
\textbf{(E) }284 \qquad
$
2023 HMNT, 9
Let $r_k$ denote the remainder when ${127 \choose k}$ is divided by $8$. Compute$ r_1 + 2r_2 + 3r_3 + · · · + 63r_{63}.$
1999 Turkey MO (2nd round), 6
We wish to find the sum of $40$ given numbers utilizing $40$ processors. Initially, we have the number $0$ on the screen of each processor. Each processor adds the number on its screen with a number entered directly (only the given numbers could be entered directly to the processors) or transferred from another processor in a unit time. Whenever a number is transferred from a processor to another, the former processor resets. Find the least time needed to find the desired sum.
2002 India IMO Training Camp, 7
Given two distinct circles touching each other internally, show how to construct a triangle with the inner circle as its incircle and the outer circle as its nine point circle.
2023 USA IMO Team Selection Test, 4
Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \]
[i]Carl Schildkraut[/i]
1999 Switzerland Team Selection Test, 1
Two circles intersect at points $M$ and $N$. Let $A$ be a point on the first circle, distinct from $M,N$. The lines $AM$ and $AN$ meet the second circle again at $B$ and $C$, respectively. Prove that the tangent to the first circle at $A$ is parallel to $BC$.
1984 AMC 12/AHSME, 17
A right triangle $ABC$ with hypotenuse $AB$ has side $AC = 15$. Altitude $CH$ divides $AB$ into segments $AH$ And $HB$, with $HB = 16$. The area of $\triangle ABC$ is:
[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(11pt));
pair A = origin, H = (5,0), B = (13,0), C = (5,6.5);
draw(C--A--B--C--H^^rightanglemark(C,H,B,16));
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,N);
label("$H$",H,S);
label("$15$",C/2,NW);
label("$16$",(H+B)/2,S);
[/asy]
$\textbf{(A) }120\qquad
\textbf{(B) }144\qquad
\textbf{(C) }150\qquad
\textbf{(D) }216\qquad
\textbf{(E) }144\sqrt5$
2010 AMC 12/AHSME, 15
A coin is altered so that the probability that it lands on heads is less than $ \frac {1}{2}$ and when the coin is flipped four times, the probability of an equal number of heads and tails is $ \frac {1}{6}$. What is the probability that the coin lands on heads?
$ \textbf{(A)}\ \frac {\sqrt {15} \minus{} 3}{6}\qquad
\textbf{(B)}\ \frac {6 \minus{} \sqrt {6\sqrt {6} \plus{} 2}}{12}\qquad
\textbf{(C)}\ \frac {\sqrt {2} \minus{} 1}{2}\qquad
\textbf{(D)}\ \frac {3 \minus{} \sqrt {3}}{6}\qquad
\textbf{(E)}\ \frac {\sqrt {3} \minus{} 1}{2}$
2005 Purple Comet Problems, 2
We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted?
2014 Contests, 1
Is it possible to fill a $3 \times 3$ grid with each of the numbers $1,2,\ldots,9$ once each such that the sum of any two numbers sharing a side is prime?
2003 Gheorghe Vranceanu, 2
Let be a natural number $ n $ and $ 2n $ positive real numbers $ v_1,v_2,\ldots ,v_{2n} $ such that the last $ n $ of them are greater than $ 1. $ Prove that:
$$ \sum_{i=1}^n v_iv_{n+i}\le \max_{1\le k\le n}\left( \left( -1+\prod_{l=n}^{2n} v_l \right) v_k +\sum_{m=1}^n v_m \right) $$
2006 District Olympiad, 1
Let $ a,b,c\in (0,1)$ and $ x,y,z\in (0, \plus{} \infty)$ be six real numbers such that
\[ a^x \equal{} bc , \quad b^y \equal{} ca , \quad c^z \equal{} ab .\]
Prove that
\[ \frac 1{2 \plus{} x} \plus{} \frac 1{2 \plus{} y} \plus{} \frac 1{2 \plus{} z} \leq \frac 34 .\]
[i]Cezar Lupu[/i]
1996 Putnam, 3
Let $S_n$ be the set of all permutations of $(1,2,\ldots,n)$. Then find :
\[ \max_{\sigma \in S_n} \left(\sum_{i=1}^{n} \sigma(i)\sigma(i+1)\right) \]
where $\sigma(n+1)=\sigma(1)$.
1974 IMO, 2
Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$.
[hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]