Found problems: 85335
PEN H Problems, 3
Does there exist a solution to the equation \[x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65\] in integers with $x, y, z, u, v$ greater than $1998$?
2015 Tournament of Towns, 7
$N$ children no two of the same height stand in a line. The following two-step procedure is applied: first, the line is split into the least possible number of groups so that in each group all children are arranged from the left to the right in ascending order of their heights (a group may consist of a single child). Second, the order of children in each group is reversed, so now in each group the children stand in descending order of their heights. Prove that in result of applying this procedure $N - 1$ times the children in the line would stand from the left to the right in descending order of their heights.
[i](12 points)[/i]
2017 Bulgaria EGMO TST, 3
Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$.
[i]Author: Christopher Bradley, United Kingdom [/i]
2016 ASDAN Math Tournament, 9
An equilateral triangle $\triangle ABC$ with side length $3$ has center $O$. A circle is drawn centered at $O$ with radius $1$. Find the area of the region contained inside both the triangle and circle.
2019 BMT Spring, 10
A [i]3-4-5 point[/i] of a triangle $ ABC $ is a point $ P $ such that the ratio $ AP : BP : CP $ is equivalent to the ratio $ 3 : 4 : 5 $. If $ \triangle ABC $ is isosceles with base $ BC = 12 $ and $ \triangle ABC $ has exactly one $ 3-4-5 $ point, compute the area of $ \triangle ABC $.
2017 AMC 12/AHSME, 22
A square is drawn in the Cartesian coordinate plane with vertices at $(2,2)$, $(-2,2)$, $(-2,-2)$, and $(2,-2)$. A particle starts at $(0,0)$. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is $\frac{1}{8}$ that the particle will move from $(x,y)$ to each of $(x,y+1)$, $(x+1,y+1)$, $(x+1,y)$, $(x+1,y-1)$, $(x,y-1)$, $(x-1,y-1)$, $(x-1,y)$, $(x-1,y+1)$. The particle will eventually hit the square for the first time, either at one of the $4$ corners of the square or one of the $12$ lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 39$
VI Soros Olympiad 1999 - 2000 (Russia), 10.2
Solve the equation
$$\frac{\pi-2}{2} + \frac{2}{1+\sin (2\sqrt{x})}+arccos(x^3-8x-1)=tg^2\sqrt{x}- \sqrt{x^4+x^3-5x^2-8x-24}$$
2011 Indonesia TST, 4
Let $a, b$, and $c$ be positive integers such that $gcd(a, b) = 1$. Sequence $\{u_k\}$, is given such that $u_0 = 0$, $u_1 = 1$, and u$_{k+2} = au_{k+1} + bu_k$ for all $k \ge 0$. Let $m$ be the least positive integer such that $c | u_m$ and $n$ be an arbitrary positive integer such that $c | u_n$. Show that $m | n$.
[hide=PS.] There was a typo in the last line, as it didn't define what n does. Wording comes from [b]tst-2011-1.pdf[/b] from [url=https://sites.google.com/site/imoidn/idntst/2011tst]here[/url]. Correction was made according to #2[/hide]
2001 ITAMO, 4
A positive integer is called [i]monotone[/i] if has at least two digits and all its digits are nonzero and appear in a strictly increasing or strictly decreasing order.
(a) Compute the sum of all monotone five-digit numbers.
(b) Find the number of final zeros in the least common multiple of all monotone numbers (with any number of digits).
2014 Polish MO Finals, 2
Let $k\ge 2$, $n\ge 1$, $a_1, a_2,\dots, a_k$ and $b_1, b_2, \dots, b_n$ be integers such that $1<a_1<a_2<\dots <a_k<b_1<b_2<\dots <b_n$. Prove that if $a_1+a_2+\dots +a_k>b_1+b_2+\dots + b_n$, then $a_1\cdot a_2\cdot \ldots \cdot a_k>b_1\cdot b_2 \cdot \ldots \cdot b_n$.
1995 India National Olympiad, 4
Let $ABC$ be a triangle and a circle $\Gamma'$ be drawn lying outside the triangle, touching its incircle $\Gamma$ externally, and also the two sides $AB$ and $AC$. Show that the ratio of the radii of the circles $\Gamma'$ and $\Gamma$ is equal to $\tan^ 2 { \left( \dfrac{ \pi - A }{4} \right) }.$
2022 CMIMC, 3
Let $ABCD$ be a rectangle with $AB=10$ and $AD=5.$ Suppose points $P$ and $Q$ are on segments $CD$ and $BC,$ respectively, such that the following conditions hold:
[list]
[*] $BD \parallel PQ$
[*] $\angle APQ=90^{\circ}.$
[/list]
What is the area of $\triangle CPQ?$
[i]Proposed by Kyle Lee[/i]
2010 Stanford Mathematics Tournament, 22
We need not restrict our number system radix to be an integer. Consider the phinary numeral system in which the radix is the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$ and the digits $0$ and $1$ are used. Compute $1010100_{\phi}-.010101_{\phi}$
1962 AMC 12/AHSME, 34
For what real values of $ K$ does $ x \equal{} K^2 (x\minus{}1)(x\minus{}2)$ have real roots?
$ \textbf{(A)}\ \text{none} \qquad
\textbf{(B)}\ \minus{}2<K<1 \qquad
\textbf{(C)}\ \minus{}2 \sqrt{2} < K < 2 \sqrt{2} \qquad
\textbf{(D)}\ K>1 \text{ or } K<\minus{}2 \qquad
\textbf{(E)}\ \text{all}$
2018 Brazil National Olympiad, 3
Let $k$, $n$ be fixed positive integers. In a circular table, there are placed pins numbered successively with the numbers $1, 2 \dots, n$, with $1$ and $n$ neighbors. It is known that pin $1$ is golden and the others are white. Arnaldo and Bernaldo play a game, in which a ring is placed initially on one of the pins and at each step it changes position. The game begins with Bernaldo choosing a starting pin for the ring, and the first step consists of the following: Arnaldo chooses a positive integer $d$ any and Bernaldo moves the ring $d$ pins clockwise or counterclockwise (positions are considered modulo $n$, i.e., pins $x$, $y$ equal if and only if $n$ divides $x-y$). After that, the ring changes its position according to one of the following rules, to be chosen at every step by Arnaldo:
[b]Rule 1:[/b] Arnaldo chooses a positive integer $d$ and Bernaldo moves the ring $d$ pins clockwise or counterclockwise.
[b]Rule 2:[/b] Arnaldo chooses a direction (clockwise or counterclockwise), and Bernaldo moves the ring in the chosen direction in $d$ or $kd$ pins, where $d$ is the size of the last displacement performed.
Arnaldo wins if, after a finite number of steps, the ring is moved to the golden pin. Determine, as a function of $k$, the values of $n$ for which Arnaldo has a strategy that guarantees his victory, no matter how Bernaldo plays.
2014 Contests, 903
Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$.
Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$
2011 Saudi Arabia IMO TST, 2
Consider the set $S= \{(a + b)^7 - a^7 - b^7 : a,b \in Z\}$. Find the greatest common divisor of all members in $S$.
2018 VJIMC, 4
Determine all possible (finite or infinite) values of
\[\lim_{x \to -\infty} f(x)-\lim_{x \to \infty} f(x),\]
if $f:\mathbb{R} \to \mathbb{R}$ is a strictly decreasing continuous function satisfying
\[f(f(x))^4-f(f(x))+f(x)=1\]
for all $x \in \mathbb{R}$.
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 9
We draw a circle with radius 5 on a gridded paper where the grid consists of squares with sides of length 1. The center of the circle is placed in the middle of one of the squares. All the squares through which the circle passes are colored. How many squares are colored? (The figure illustrates this for a smaller circle.)
[img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1995Number9.jpg[/img]
A. 24
B. 32
C. 40
D. 64
E. None of these
2017 NIMO Problems, 3
A circle $C_0$ is inscribed in an equilateral triangle $XYZ$ of side length 112. Then, for each positive integer $n$, circle $C_n$ is inscribed in the region bounded by $XY$, $XZ$, and an arc of circle $C_{n-1}$, forming an infinite sequence of circles tangent to sides $XY$ and $XZ$ and approaching vertex $X$. If these circles collectively have area $m\pi$, find $m$.
[i]Proposed by Michael Tang[/i]
2009 QEDMO 6th, 12
Find all functions $f: R\to R$, which satisfy the equation $f (xy + f (x)) = xf (y) + f (x)$.
2024 Dutch BxMO/EGMO TST, IMO TSTST, 2
We define a sequence with $a_1=850$ and $$a_{n+1}=\frac{a_n^2}{a_n-1}$$ for $n\geq 1$. Find all values of $n$ for which $\lfloor a_n\rfloor =2024$.
1969 IMO Longlists, 49
$(NET 4)$ A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by $4.$
2024 Mathematical Talent Reward Programme, 6
Find the maximum possible length of a sequence consisting of non-zero integers, in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.
1982 IMO Longlists, 19
Show that
\[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\]
holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.