Found problems: 85335
KoMaL A Problems 2022/2023, A. 852
Let $(a_i,b_i)$ be pairwise distinct pairs of positive integers for $1\le i\le n$. Prove that
\[(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n)>\frac29 n^3,\]
and show that the statement is sharp, i.e. for an arbitrary $c>\frac29$ it is possible that
\[(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n)<cn^3.\]
[i]Submitted by Péter Pál Pach, Budapest, based on an OKTV problem[/i]
1971 Miklós Schweitzer, 6
Let $ a(x)$ and $ r(x)$ be positive continuous functions defined on the interval $ [0,\infty)$, and let \[ \liminf_{x \rightarrow \infty} (x-r(x)) >0.\] Assume that $ y(x)$ is a continuous function on the whole real line, that it is differentiable on $ [0, \infty)$, and that it satisfies \[ y'(x)=a(x)y(x-r(x))\] on $ [0, \infty)$. Prove that the limit \[ \lim_{x \rightarrow \infty}y(x) \exp \left\{ -%Error. "diaplaymath" is a bad command.
\int_0^x a(u)du \right \}\] exists and is finite.
[i]I. Gyori[/i]
1982 IMO Shortlist, 9
Let $ABC$ be a triangle, and let $P$ be a point inside it such that $\angle PAC = \angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$, respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM.$
2021 Auckland Mathematical Olympiad, 4
Prove that there exist two powers of $7$ whose difference is divisible by $2021$.
2013 IPhOO, 7
A conical pendulum is formed from a rope of length $ 0.50 \, \text{m} $ and negligible mass, which is suspended from a fixed pivot attached to the ceiling. A ping-pong ball of mass $ 3.0 \, \text{g} $ is attached to the lower end of the rope. The ball moves in a circle with constant speed in the horizontal plane and the ball goes through one revolution in $ 1.0 \, \text{s} $. How high is the ceiling in comparison to the horizontal plane in which the ball revolves? Express your answer to two significant digits, in cm.
[i](Proposed by Ahaan Rungta)[/i]
[hide="Clarification"]
During the WOOT Contest, contestants wondered what exactly a conical pendulum looked like. Since contestants were not permitted to look up information during the contest, we posted this diagram: [asy]
size(6cm);
import olympiad;
draw((-1,3)--(1,3));
draw(xscale(4) * scale(0.5) * unitcircle, dotted);
draw(origin--(0,3), dashed);
label("$h$", (0,1.5), dir(180));
draw((0,3)--(2,0));
filldraw(shift(2) * scale(0.2) * unitcircle, 1.4*grey, black);
dot(origin);
dot((0,3));[/asy]The question is to find $h$.
[/hide]
1999 Singapore Team Selection Test, 1
Let $M$ and $N$ be two points on the side BC of a triangle $ABC$ such that $BM =MN = NC$. A line parallel to $AC$ meets the segments $AB, AM$ and $AN$ at the points $D, E$ and $F$ respectively. Prove that $EF = 3DE$
1976 Chisinau City MO, 119
The Serpent Gorynych has $1976$ heads. The fabulous hero can cut down $33, 21, 17$ or $1$ head with one blow of the sword, but at the same time, the Serpent grows, respectively, $48, 0, 14$ or $349$ heads. If all the heads are cut off, then no new heads will grow. Will the hero be able to defeat the Serpent?
2008 Bosnia And Herzegovina - Regional Olympiad, 3
Let $ b$ be an even positive integer. Assume that there exist integer $ n > 1$ such that $ \frac {b^{n} \minus{} 1}{b \minus{} 1}$ is perfect square.
Prove that $ b$ is divisible by 8.
1975 All Soviet Union Mathematical Olympiad, 205
a) The triangle $ABC$ was turned around the centre of the circumscribed circle by the angle less than $180$ degrees and thus was obtained the triangle $A_1B_1C_1$. The corresponding segments $[AB]$ and $[A_1B_1]$ intersect in the point $C_2, [BC]$ and $[B_1C_1]$ -- $A_2, [AC]$ and $[A_1C_1]$ -- $B_2$. Prove that the triangle $A_2B_2C_2$ is similar to the triangle $ABC$.
b) The quadrangle $ABCD$ was turned around the centre of the circumscribed circle by the angle less than $180$ degrees and thus was obtained the quadrangle $A_1B_1C_1D_1$. Prove that the points of intersection of the corresponding lines ( $(AB$) and $(A_1B_1), (BC)$ and $(B_1C_1), (CD)$ and $(C_1D_1), (DA)$ and $(D_1A_1)$ ) are the vertices of the parallelogram.
2012 NIMO Summer Contest, 8
Points $A$, $B$, and $O$ lie in the plane such that $\measuredangle AOB = 120^\circ$. Circle $\omega_0$ with radius $6$ is constructed tangent to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$. For all $i \ge 1$, circle $\omega_i$ with radius $r_i$ is constructed such that $r_i < r_{i - 1}$ and $\omega_i$ is tangent to $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\omega_{i - 1}$. If
\[
S = \sum_{i = 1}^\infty r_i,
\]
then $S$ can be expressed as $a\sqrt{b} + c$, where $a, b, c$ are integers and $b$ is not divisible by the square of any prime. Compute $100a + 10b + c$.
[i]Proposed by Aaron Lin[/i]
2007 Romania National Olympiad, 1
Show that the equation $z^{n}+z+1=0$ has a solution with $|z|=1$ if and only if $n-2$ is divisble by $3$.
1993 Polish MO Finals, 2
A circle center $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$ and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. Show that $DE = CF$ iff $AB = 2CD$.
2010 Contests, 2
On a circumference, points $A$ and $B$ are on opposite arcs of diameter $CD$. Line segments $CE$ and $DF$ are perpendicular to $AB$ such that $A-E-F-B$ (i.e., $A$, $E$, $F$ and $B$ are collinear on this order). Knowing $AE=1$, find the length of $BF$.
1990 Kurschak Competition, 3
We would like to give a present to one of $100$ children. We do this by throwing a biased coin $k$ times, after predetermining who wins in each possible outcome of this lottery.
Prove that we can choose the probability $p$ of throwing heads, and the value of $k$ such that, by distributing the $2^k$ different outcomes between the children in the right way, we can guarantee that each child has the same probability of winning.
2025 Kyiv City MO Round 1, Problem 1
Find all triples of positive integers \( a, b, c \) that satisfy the equation:
\[
a + \frac{1}{b + \frac{1}{c}} = 20.25.
\]
2005 AIME Problems, 9
Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The $27$ cubes are randomly arranged to form a $3\times 3 \times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\frac{p^a}{q^br^c},$ where $p$,$q$, and $r$ are distinct primes and $a$,$b$, and $c$ are positive integers, find $a+b+c+p+q+r$.
2013 Iran MO (3rd Round), 3
$n$ cars are racing. At first they have a particular order. At each moment a car may overtake another car. No two overtaking actions occur at the same time, and except moments a car is passing another, the cars always have an order.
A set of overtaking actions is called "small" if any car overtakes at most once.
A set of overtaking actions is called "complete" if any car overtakes exactly once.
If $F$ is the set of all possible orders of the cars after a small set of overtaking actions and $G$ is the set of all possible orders of the cars after a complete set of overtaking actions, prove that
\[\mid F\mid=2\mid G\mid\]
(20 points)
[i]Proposed by Morteza Saghafian[/i]
1997 USAMO, 1
Let $p_1, p_2, p_3, \ldots$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between 0 and 1. For positive integer $k$, define
\[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases} \]
where $\{x\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 < x_0 < 1$ for which the sequence $x_0, x_1, x_2, \ldots$ eventually becomes 0.
2015 District Olympiad, 1
[b]a)[/b] Solve the equation $ x^2-x+2\equiv 0\pmod 7. $
[b]b)[/b] Determine the natural numbers $ n\ge 2 $ for which the equation $ x^2-x+2\equiv 0\pmod n $ has an unique solution modulo $ n. $
2012 BMT Spring, 8
Let $\phi$ be the Euler totient function. Let $\phi^k (n) = (\underbrace{\phi \circ ... \circ \phi}_{k})(n)$ be $\phi$ composed with itself $k$ times. Define $\theta (n) = min \{k \in N | \phi^k (n)=1 \}$
. For example,
$\phi^1 (13) = \phi(13) = 12$
$\phi^2 (13) = \phi (\phi (13)) = 4$
$\phi^3 (13) = \phi(\phi(\phi(13))) = 2$
$\phi^4 (13) = \phi(\phi(\phi(\phi(13)))) = 1$
so $\theta (13) = 4$. Let $f(r) = \theta (13^r)$. Determine $f(2012)$.
2010 Dutch IMO TST, 1
Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if
(i) $a_n < a_{n+1}$ for all $n\ge 1$,
(ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.
1970 AMC 12/AHSME, 6
The smallest value of $x^2+8x$ for real values of $x$ is:
$\textbf{(A) }-16.25\qquad\textbf{(B) }-16\qquad\textbf{(C) }-15\qquad\textbf{(D) }-8\qquad \textbf{(E) }\text{None of these}$
2024 Turkey Team Selection Test, 9
In a scalene triangle $ABC,$ $I$ is the incenter and $O$ is the circumcenter. The line $IO$ intersects the lines $BC,CA,AB$ at points $D,E,F$ respectively. Let $A_1$ be the intersection of $BE$ and $CF$. The points $B_1$ and $C_1$ are defined similarly. The incircle of $ABC$ is tangent to sides $BC,CA,AB$ at points $X,Y,Z$ respectively. Let the lines $XA_1, YB_1$ and $ZC_1$ intersect $IO$ at points $A_2,B_2,C_2$ respectively. Prove that the circles with diameters $AA_2,BB_2$ and $CC_2$ have a common point.
2014 Online Math Open Problems, 26
Qing initially writes the ordered pair $(1,0)$ on a blackboard. Each minute, if the pair $(a,b)$ is on the board, she erases it and replaces it with one of the pairs $(2a-b,a)$, $(2a+b+2,a)$ or $(a+2b+2,b)$. Eventually, the board reads $(2014,k)$ for some nonnegative integer $k$. How many possible values of $k$ are there?
[i]Proposed by Evan Chen[/i]
2015 İberoAmerican, 5
Find all pairs of integers $(a,b)$ such that
$(b^2+7(a-b))^2=a^{3}b$.