Found problems: 85335
2018 Rio de Janeiro Mathematical Olympiad, 2
Let $ABC$ be an equilateral triangle with side 3. A circle $C_1$ is tangent to $AB$ and $AC$.
A circle $C_2$, with a radius smaller than the radius of $C_1$, is tangent to $AB$ and $AC$ as well as externally tangent to $C_1$.
Successively, for $n$ positive integer, the circle $C_{n+1}$, with a radius smaller than the radius of $C_n$, is tangent to $AB$ and $AC$ and is externally tangent to $C_n$.
Determine the possible values for the radius of $C_1$ such that 4 circles from this sequence, but not 5, are contained on the interior of the triangle $ABC$.
1994 Turkey MO (2nd round), 5
Find the set of all ordered pairs $(s,t)$ of positive integers such that \[t^{2}+1=s(s+1).\]
1999 Gauss, 14
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between $12^2$ and $13^2$?
$\textbf{(A)}\ 105 \qquad \textbf{(B)}\ 147 \qquad \textbf{(C)}\ 156 \qquad \textbf{(D)}\ 165 \qquad \textbf{(E)}\ 175$
2010 Romanian Master of Mathematics, 2
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
2021 China National Olympiad, 2
Let $m>1$ be an integer. Find the smallest positive integer $n$, such that for any integers $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ there exists integers $x_1,x_2,\ldots ,x_n$ satisfying the following two conditions:
i) There exists $i\in \{1,2,\ldots ,n\}$ such that $x_i$ and $m$ are coprime
ii) $\sum^n_{i=1} a_ix_i \equiv \sum^n_{i=1} b_ix_i \equiv 0 \pmod m$
2022 Austrian MO National Competition, 5
Let $ABC$ be an isosceles triangle with base $AB$. We choose a point $P$ inside the triangle on altitude through $C$. The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$. Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line.
[i](Walther Janous)[/i]
2021 MOAA, 2
On Andover's campus, Graves Hall is $60$ meters west of George Washington Hall, and George Washington Hall is $80$ meters north of Paresky Commons. Jessica wants to walk from Graves Hall to Paresky Commons. If she first walks straight from Graves Hall to George Washington Hall and then walks straight from George Washington Hall to Paresky Commons, it takes her $8$ minutes and $45$ seconds while walking at a constant speed. If she walks with the same speed directly from Graves Hall to Paresky Commons, how much time does she save, in seconds?
[i]Proposed by Nathan Xiong[/i]
2019 USA TSTST, 7
Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying
$$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$
for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$.
[i]Ankan Bhattacharya[/i]
2017 China Team Selection Test, 2
In $\varDelta{ABC}$,the excircle of $A$ is tangent to segment $BC$,line $AB$ and $AC$ at $E,D,F$ respectively.$EZ$ is the diameter of the circle.$B_1$ and $C_1$ are on $DF$, and $BB_1\perp{BC}$,$CC_1\perp{BC}$.Line $ZB_1,ZC_1$ intersect $BC$ at $X,Y$ respectively.Line $EZ$ and line $DF$ intersect at $H$,$ZK$ is perpendicular to $FD$ at $K$.If $H$ is the orthocenter of $\varDelta{XYZ}$,prove that:$H,K,X,Y$ are concyclic.
1975 AMC 12/AHSME, 14
If the $ whatsis$ is $ so$ when the $ whosis$ is $ is$ and the $ so$ and $ so$ is $ is \cdot so$, what is the $ whosis \cdot whatsis$ when the $ whosis$ is $ so$, the $ so$ and $ so$ is $ so \cdot so$ and the $ is$ is two ($ whatsis$, $ whosis$, $ is$ and $ so$ are variables taking positive values)?
$ \textbf{(A)}\ whosis \cdot is \cdot so \qquad
\textbf{(B)}\ whosis \qquad
\textbf{(C)}\ is \qquad
\textbf{(D)}\ so \qquad
\textbf{(E)}\ so \text{ and } so$
2019 IMEO, 5
Find all pairs of positive integers $(s, t)$, so that for any two different positive integers $a$ and $b$ there exists some positive integer $n$, for which $$a^s + b^t | a^n + b^{n+1}.$$
[i]Proposed by Oleksii Masalitin (Ukraine)[/i]
2021-2022 OMMC, 14
The corners of a $2$-dimensional room in the shape of an isosceles right triangle are labeled $A$, $B$, $C$ where $AB = BC$. Walls $BC$ and $CA$ are mirrors. A laser is shot from $A$, hits off of each of the mirrors once and lands at a point $X$ on $AB$. Let $Y$ be the point where the laser hits off $AC$. If $\tfrac{AB}{AX} = 64$, $\tfrac{CA}{AY} = \tfrac pq$ for coprime positive integers $p$, $q$. Find $p + q$.
[i]Proposed by Sid Doppalapudi[/i]
2005 Greece Junior Math Olympiad, 2
If $f(n)=\frac{2n+1+\sqrt{n(n+1)}}{\sqrt{n+1}+\sqrt{n}}$ for all positive integers $n$, evaluate
(a) $f(1)$,
(b) the sum $A=f(1)+f(2)+...+f(400)$.
1983 AIME Problems, 13
For $\{1, 2, 3, \dots, n\}$ and each of its nonempty subsets a unique [b]alternating sum[/b] is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$ and for $\{5\}$ it is simply 5.) Find the sum of all such alternating sums for $n = 7$.
2008 China Team Selection Test, 1
Let $ ABC$ be a triangle, let $ AB > AC$. Its incircle touches side $ BC$ at point $ E$. Point $ D$ is the second intersection of the incircle with segment $ AE$ (different from $ E$). Point $ F$ (different from $ E$) is taken on segment $ AE$ such that $ CE \equal{} CF$. The ray $ CF$ meets $ BD$ at point $ G$. Show that $ CF \equal{} FG$.
2016 Saint Petersburg Mathematical Olympiad, 1
Sasha multiplied all the divisors of the natural number $n$. Fedya increased each divider by $1$, and then multiplied the results. If the product found Fedya is divided by the product found by Sasha , what can $n$ be equal to ?
PEN P Problems, 24
Show that any integer can be expressed as the form $a^{2}+b^{2}-c^{2}$, where $a, b, c \in \mathbb{Z}$.
2000 Harvard-MIT Mathematics Tournament, 6
$6$ people each have a hat. If they shuffle their hats and redistribute them, what is the probability that exactly one person gets their own hat back?
1999 AIME Problems, 5
For any positive integer $x$, let $S(x)$ be the sum of the digits of $x$, and let $T(x)$ be $|S(x+2)-S(x)|.$ For example, $T(199)=|S(201)-S(199)|=|3-19|=16.$ How many values $T(x)$ do not exceed 1999?
2020 Brazil National Olympiad, 2
The following sentece is written on a board:
[center]The equation $x^2-824x+\blacksquare 143=0$ has two integer solutions.[/center]
Where $\blacksquare$ represents algarisms of a blurred number on the board. What are the possible equations originally on the board?
2017 AIME Problems, 8
Find the number of positive integers $n$ less than $2017$ such that
\[ 1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+\frac{n^4}{4!}+\frac{n^5}{5!}+\frac{n^6}{6!} \]
is an integer.
2016 JBMO TST - Turkey, 2
A and B plays a game on a pyramid whose base is a $2016$-gon. In each turn, a player colors a side (which was not colored before) of the pyramid using one of the $k$ colors such that none of the sides with a common vertex have the same color. If A starts the game, find the minimal value of $k$ for which $B$ can guarantee that all sides are colored.
2010 India IMO Training Camp, 11
Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$
2021 Brazil National Olympiad, 5
Find all triples of non-negative integers \((a, b, c)\) such that \[a^{2}+b^{2}+c^{2} = a b c+1.\]
2024 Assara - South Russian Girl's MO, 7
Find all positive integers $n$ for such the following condition holds:
"If $a$, $b$ and $c$ are positive integers such are all numbers \[ a^2+2ab+b^2,\ b^2+2bc+c^2, \ c^2+2ca+a^2 \] are divisible by $n$, then $(a+b+c)^2$ is also divisible by $n$."
[i]G.M.Sharafetdinova[/i]