Found problems: 85335
2002 AMC 10, 1
The ratio $ \frac{2^{2001}\cdot3^{2003}}{6^{2002}}$ is
$ \textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{2}{3} \qquad
\textbf{(E)}\ \frac{3}{2}$
2024 Putnam, B4
Let $n$ be a positive integer. Set $a_{n,0}=1$. For $k\geq 0$, choose an integer $m_{n,k}$ uniformly at random from the set $\{1,\,\ldots,\,n\}$, and let
\[
a_{n,k+1}=
\begin{cases}
a_{n,k}+1, & \text{if $m_{n,k}>a_{n,k}$;}\\
a_{n,k}, & \text{if $m_{n,k}=a_{n,k}$;}\\
a_{n,k}-1, & \text{if $m_{n,k}<a_{n,k}$.}
\end{cases}
\]
Let $E(n)$ be the expected value of $a_{n,n}$. Determine $\lim_{n\to\infty}E(n)/n$.
2005 Junior Balkan MO, 2
Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$.
Prove that the lines $AP$ and $CS$ are parallel.
2025 Poland - Second Round, 4
Let $n\ge 2$ be an integer. Consider a $2n+1\times 2n+1$ board. All cells lying both in an even row and an even column have been removed. The remaining cells form a [i]labyrinth[/i]. An ant takes a walk in the labyrinth. A single step of the ant consists of moving to a neighbouring cell. Determine, in terms of $n$, the smallest possible number of steps so that every cell of the labirynth is visited by the ant. The ant chooses the start cell. The start cell and the end cell are considered visited. Each cell could be visited several times.
The picture depicts the labyrinth for $n=3$ and possible steps of the ant in its four locations.
1950 AMC 12/AHSME, 12
As the number of sides of a polygon increases from $3$ to $ n$, the sum of the exterior formed by extending each side in succession:
$\textbf{(A)}\ \text{Increases} \qquad
\textbf{(B)}\ \text{Decreases} \qquad
\textbf{(C)}\ \text{Remains constant} \qquad
\textbf{(D)}\ \text{Cannot be predicted} \qquad\\
\textbf{(E)}\ \text{Becomes }(n-3)\text{ straight angles}$
2018 Bosnia And Herzegovina - Regional Olympiad, 5
It is given $2018$ points in plane. Prove that it is possible to cover them with circles such that:
$i)$ sum of lengths of all diameters of all circles is not greater than $2018$
$ii)$ distance between any two circles is greater than $1$
2000 Poland - Second Round, 5
Decide whether exists function $f: \mathbb{N} \rightarrow \mathbb{N}$, such that for each $n \in \mathbb{N}$ is $f(f(n) )= 2n$.
2005 Postal Coaching, 23
Let $\Gamma$ be the incircle of an equilateral triangle $ABC$ of side length $2$ units.
(a) Show that for all points $P$ on $\Gamma$, $PA^2 +PB^2 +PC^2 = 5$.
(b) Show that for all points $P$ on $\Gamma$, it is possible to construct a triangle of sides equal to $PA,PB,PC$ and whose area is equal to $\frac{\sqrt{3}}{4}$ units.
2012 Online Math Open Problems, 12
Let $a_1,a_2,\ldots$ be a sequence defined by $a_1 = 1$ and for $n\ge1$, $a_{n+1} = \sqrt{a_n^2 -2a_n + 3} + 1$. Find $a_{513}$.
[i]Ray Li.[/i]
2007 Purple Comet Problems, 3
Square $ABCD$ has side length $36$. Point $E$ is on side $AB$ a distance $12$ from $B$, point $F$ is the midpoint of side $BC$, and point $G$ is on side $CD$ a distance $12$ from $C$. Find the area of the region that lies inside triangle $EFG$ and outside triangle $AFD$.
1996 Vietnam Team Selection Test, 1
Given 3 non-collinear points $A,B,C$. For each point $M$ in the plane ($ABC$) let $M_1$ be the point symmetric to $M$ with respect to $AB$, $M_2$ be the point symmetric to $M_1$ with respect to $BC$ and $M'$ be the point symmetric to $M_2$ with respect to $AC$. Find all points $M$ such that $MM'$ obtains its minimum. Let this minimum value be $d$. Prove that $d$ does not depend on the order of the axes of symmetry we chose (we have 3 available axes, that is $BC$, $CA$, $AB$. In the first part the order of axes we chose $AB$, $BC$, $CA$, and the second part of the problem states that the value $d$ doesn't depend on this order).
2013 VJIMC, Problem 4
Let $\mathcal F$ be the set of all continuous functions $f:[0,1]\to\mathbb R$ with the property
$$\left|\int^x_0\frac{f(t)}{\sqrt{x-t}}\text dt\right|\le1\enspace\text{for all }x\in(0,1].$$Compute $\sup_{f\in\mathcal F}\left|\int^1_0f(x)\text dx\right|$.
2014 Online Math Open Problems, 10
Find the sum of the decimal digits of \[ \left\lfloor \frac{51525354555657\dots979899}{50} \right\rfloor. \] Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.
[i]Proposed by Evan Chen[/i]
2012 Iran MO (3rd Round), 2
Consider a set of $n$ points in plane. Prove that the number of isosceles triangles having their vertices among these $n$ points is $\mathcal O (n^{\frac{7}{3}})$. Find a configuration of $n$ points in plane such that the number of equilateral triangles with vertices among these $n$ points is $\Omega (n^2)$.
2024 Czech and Slovak Olympiad III A, 6
Find all right triangles with integer side lengths in which two congruent circles with prime radius can be inscribed such that they are externally tangent, both touch the hypotenuse, and each is tangent to another leg of the right triangle.
2016 NIMO Problems, 1
Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$.
[i] Proposed by Michael Ren [/i]
1986 Polish MO Finals, 1
A square of side $1$ is covered with $m^2$ rectangles.
Show that there is a rectangle with perimeter at least $\frac{4}{m}$.
2011 Morocco TST, 1
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
2019 IFYM, Sozopol, 3
$\Delta ABC$ is isosceles with a circumscribed circle $\omega (O)$. Let $H$ be the foot of the altitude from $C$ to $AB$ and let $M$ be the middle point of $AB$. We define a point $X$ as the second intersection point of the circle with diameter $CM$ and $\omega$ and let $XH$ intersect $\omega$ for a second time in $Y$. If $CO\cap AB=D$, then prove that the circumscribed circle of $\Delta YHD$ is tangent to $\omega$.
2018 AMC 8, 15
In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units?
[asy]
size(4cm);
filldraw(scale(2)*unitcircle,gray,black);
filldraw(shift(-1,0)*unitcircle,white,black);
filldraw(shift(1,0)*unitcircle,white,black);
[/asy]
$\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{3} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } 1 \qquad \textbf{(E) } \frac{\pi}{2}$
1951 Polish MO Finals, 2
What digits should be placed instead of zeros in the third and fifth places in the number $3000003$ to obtain a number divisible by $13$?
2002 India IMO Training Camp, 16
Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?
1991 India National Olympiad, 5
Triangle $ABC$ has an incenter $I$. Let points $X$, $Y$ be located on the line segments $AB$, $AC$ respectively, so that $BX \cdot AB = IB^2$ and $CY \cdot AC = IC^2$. Given that the points $X, I, Y$ lie on a straight line, find the possible values of the measure of angle $A$.
2017 Dutch IMO TST, 1
Let $a, b,c$ be distinct positive integers, and suppose that $p = ab+bc+ca$ is a prime number.
$(a)$ Show that $a^2,b^,c^2$ give distinct remainders after division by $p$.
(b) Show that $a^3,b^3,c^3$ give distinct remainders after division by $p$.
Novosibirsk Oral Geo Oly IX, 2023.6
Two quarter-circles touch as shown. Find the angle $x$.
[img]https://cdn.artofproblemsolving.com/attachments/b/4/e70d5d69e46d6d40368f143cb83cf10b7d6d98.png[/img]