Found problems: 85335
1950 AMC 12/AHSME, 43
The sum to infinity of $ \frac{1}{7}\plus{}\frac {2}{7^2}\plus{}\frac{1}{7^3}\plus{}\frac{2}{7^4}\plus{}...$ is:
$\textbf{(A)}\ \frac{1}{5} \qquad
\textbf{(B)}\ \dfrac{1}{24} \qquad
\textbf{(C)}\ \dfrac{5}{48} \qquad
\textbf{(D)}\ \dfrac{1}{16} \qquad
\textbf{(E)}\ \text{None of these}$
Mexican Quarantine Mathematical Olympiad, #3
Let $\Gamma_1$ and $\Gamma_2$ be circles intersecting at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at $C$ and $D$ respectively. Let $P$ be the intersection of the lines tangent to $\Gamma_1$ at $A$ and $C$, and let $Q$ be the intersection of the lines tangent to $\Gamma_2$ at $A$ and $D$. Let $X$ be the second intersection point of the circumcircles of $BCP$ and $BDQ$, and let $Y$ be the intersection of lines $AB$ and $PQ$. Prove that $C$, $D$, $X$ and $Y$ are concyclic.
[i]Proposed by Ariel GarcÃa[/i]
2005 CentroAmerican, 4
Two players, Red and Blue, play in alternating turns on a 10x10 board. Blue goes first. In his turn, a player picks a row or column (not chosen by any player yet) and color all its squares with his own color. If any of these squares was already colored, the new color substitutes the old one.
The game ends after 20 turns, when all rows and column were chosen. Red wins if the number of red squares in the board exceeds at least by 10 the number of blue squares; otherwise Blue wins.
Determine which player has a winning strategy and describe this strategy.
2023 Thailand October Camp, 2
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
2023 Irish Math Olympiad, P8
Suppose that $a, b, c$ are positive real numbers and $a + b + c = 3$. Prove that
$$\frac{a+b}{c+2} + \frac{b+c}{a+2} + \frac{c+a}{b+2} \geq 2$$
and determine when equality holds.
Russian TST 2016, P1
Find all natural $n{}$ such that for every natural $a{}$ that is mutually prime with $n{}$, the number $a^n - 1$ is divisible by $2n^2$.
2020 AMC 8 -, 22
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.
[asy] size(300); defaultpen(linewidth(0.8)+fontsize(13)); real r = 0.05; draw((0.9,0)--(3.5,0),EndArrow(size=7)); filldraw((4,2.5)--(7,2.5)--(7,-2.5)--(4,-2.5)--cycle,gray(0.65)); fill(circle((5.5,1.25),0.8),white); fill(circle((5.5,1.25),0.5),gray(0.65)); fill((4.3,-r)--(6.7,-r)--(6.7,-1-r)--(4.3,-1-r)--cycle,white); fill((4.3,-1.25+r)--(6.7,-1.25+r)--(6.7,-2.25+r)--(4.3,-2.25+r)--cycle,white); fill((4.6,-0.25-r)--(6.4,-0.25-r)--(6.4,-0.75-r)--(4.6,-0.75-r)--cycle,gray(0.65)); fill((4.6,-1.5+r)--(6.4,-1.5+r)--(6.4,-2+r)--(4.6,-2+r)--cycle,gray(0.65)); label("$N$",(0.45,0)); draw((7.5,1.25)--(11.25,1.25),EndArrow(size=7)); draw((7.5,-1.25)--(11.25,-1.25),EndArrow(size=7)); label("if $N$ is even",(9.25,1.25),N); label("if $N$ is odd",(9.25,-1.25),N); label("$\frac N2$",(12,1.25)); label("$3N+1$",(12.6,-1.25)); [/asy]
For example, starting with an input of $N = 7$, the machine will output $3 \cdot 7 + 1 = 22$. Then if the output is repeatedly inserted into the machine five more times, the final output is $26$. $$ 7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26$$ When the same 6-step process is applied to a different starting value of $N$, the final output is $1$. What is the sum of all such integers $N$? $$ N \to \_\_ \to \_\_ \to \_\_ \to \_\_ \to \_\_ \to 1$$
$\textbf{(A)}\ 73 \qquad \textbf{(B)}\ 74 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 82 \qquad \textbf{(E)}\ 83$
1973 Bundeswettbewerb Mathematik, 3
Given $n$ digits $a_{1}, a_{2},..., a_{n}$ in that order. Does there exist a positive integer such that the first $n$ decimal digits after the dot of that number's square root are exactly those given digits¿ Give reason for your answer.
2016 Mexico National Olmypiad, 6
Let $ABCD$ a quadrilateral inscribed in a circumference, $l_1$ the parallel to $BC$ through $A$, and $l_2$ the parallel to $AD$ through $B$. The line $DC$ intersects $l_1$ and $l_2$ at $E$ and $F$, respectively. The perpendicular to $l_1$ through $A$ intersects $BC$ at $P$, and the perpendicular to $l_2$ through $B$ cuts $AD$ at $Q$. Let $\Gamma_1$ and $\Gamma_2$ be the circumferences that pass through the vertex of triangles $ADE$ and $BFC$, respectively. Prove that $\Gamma_1$ and $\Gamma_2$ are tangent to each other if and only if $DP$ is perpendicular to $CQ$.
CIME II 2018, 9
Let $P$ be the portion of the graph of
$$y=\frac{6x+1}{32x+8} - \frac{2x-1}{32x-8}$$
located in the first quadrant (not including the $x$ and $y$ axes). Let the shortest possible distance between the origin and a point on $P$ be $d$. Find $\lfloor 1000d \rfloor$.
[i]Proposed by [b] Th3Numb3rThr33 [/b][/i]
2019 India PRMO, 14
Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.
1993 India National Olympiad, 4
Let $ABC$ be a triangle in a plane $\pi$. Find the set of all points $P$ (distinct from $A,B,C$ ) in the plane $\pi$ such that the circumcircles of triangles $ABP$, $BCP$, $CAP$ have the same radii.
2024 Miklos Schweitzer, 11
An urn initially contains one red ball and one blue ball. In each step, we choose a uniform random ball from the urn. If it is red, then another red ball and another blue ball are placed in the urn. And when we choose a blue ball for the $k$-th time, we put a blue ball and $2k + 1$ red balls in the urn. (The chosen balls are not removed; they remain in the urn.)
Let $G_n$ denote the number of red balls in the urn after $n$ steps. Prove that there exist constants $0 < c, \alpha < \infty$ such that $\frac{G_n}{n^\alpha} \to c$ almost surely.
2007 Tournament Of Towns, 4
Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots , 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.
2016 AMC 8, 13
Two different numbers are randomly selected from the set ${ - 2, -1, 0, 3, 4, 5}$ and multiplied together. What is the probability that the product is $0$?
$\textbf{(A) }\dfrac{1}{6}\qquad\textbf{(B) }\dfrac{1}{5}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad \textbf{(E) }\dfrac{1}{2}$
2017 Singapore Junior Math Olympiad, 3
Let $ABC$ be a triangle with $AB=AC$. Let $D$ be a point on $BC$, and $E$ a point on $AD$ such that $\angle BED=\angle BAC=2\angle CED$. Prove that $BD=2CD$.
2012 AMC 10, 12
A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$) or is divisible by $4$ but not by $100$ (such as $2012$). The $200\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born?
$ \textbf{(A)}\ \text{Friday}
\qquad\textbf{(B)}\ \text{Saturday}
\qquad\textbf{(C)}\ \text{Sunday}
\qquad\textbf{(D)}\ \text{Monday}
\qquad\textbf{(E)}\ \text{Tuesday}
$
2024 Vietnam Team Selection Test, 1
Let $P(x) \in \mathbb{R}[x]$ be a monic, non-constant polynomial. Determine all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(P(x))+y+2023f(y))=P(x)+2024f(y),$$
for all reals $x,y$.
2023 Thailand TST, 1
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
2012 National Olympiad First Round, 32
How many permutations $(a_1,a_2,\dots,a_{10})$ of $1,2,3,4,5,6,7,8,9,10$ satisfy $|a_1-1|+|a_2-2|+\dots+|a_{10}-10|=4$ ?
$ \textbf{(A)}\ 60 \qquad \textbf{(B)}\ 52 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 44 \qquad \textbf{(E)}\ 36$
1993 IMO Shortlist, 2
Show that there exists a finite set $A \subset \mathbb{R}^2$ such that for every $X \in A$ there are points $Y_1, Y_2, \ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to 1, for every $i.$
2018 Brazil Undergrad MO, 21
Consider $ p (x) = x ^ n + a_ {n-1} x ^ {n-1} + ... + a_ {1} x + 1 $ a polynomial of positive real coefficients, degree $ n \geq 2 $ e with $ n $ real roots. Which of the following statements is always true?
a) $ p (2) <2 (2 ^ {n-1} +1) $ (b) $ p (1) <3 $ c) $ p (1)> 2 ^ n $ d) $ p (3 ) <3 (2 ^ {n-1} -2) $
2019 Indonesia MO, 1
Given that $n$ and $r$ are positive integers.
Suppose that
\[ 1 + 2 + \dots + (n - 1) = (n + 1) + (n + 2) + \dots + (n + r) \]
Prove that $n$ is a composite number.
1992 Spain Mathematical Olympiad, 6
For a positive integer $n$, let $S(n) $be the set of complex numbers $z = x+iy$ ($x,y \in R$) with $ |z| = 1$ satisfying
$(x+iy)^n+(x-iy)^n = 2x^n$ .
(a) Determine $S(n)$ for $n = 2,3,4$.
(b) Find an upper bound (depending on $n$) of the number of elements of $S(n)$ for $n > 5$.
2016-2017 SDML (Middle School), 2
Each term of the sequence $5, 12, 19, 26, \cdots$ is $7$ more than the term that precedes it. What is the first term of the sequence that is greater than $2017$?
$\text{(A) }2018\qquad\text{(B) }2019\qquad\text{(C) }2020\qquad\text{(D) }2021\qquad\text{(E) }2022$