Found problems: 85335
2015 AMC 10, 14
The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?
[asy]
size(170);
defaultpen(linewidth(0.9)+fontsize(13pt));
draw(unitcircle^^circle((0,1.5),0.5));
path arrow = origin--(-0.13,-0.35)--(-0.06,-0.35)--(-0.06,-0.7)--(0.06,-0.7)--(0.06,-0.35)--(0.13,-0.35)--cycle;
for(int i=1;i<=12;i=i+1)
{
draw(0.9*dir(90-30*i)--dir(90-30*i));
label("$"+(string) i+"$",0.78*dir(90-30*i));
}
dot(origin);
draw(shift((0,1.87))*arrow);
draw(arc(origin,1.5,68,30),EndArrow(size=12));[/asy]
$ \textbf{(A) }\text{2 o'clock} \qquad\textbf{(B) }\text{3 o'clock} \qquad\textbf{(C) }\text{4 o'clock} \qquad\textbf{(D) }\text{6 o'clock} \qquad\textbf{(E) }\text{8 o'clock} $
2013 China Team Selection Test, 3
Let $A$ be a set consisting of 6 points in the plane. denoted $n(A)$ as the number of the unit circles which meet at least three points of $A$. Find the maximum of $n(A)$
2021 CCA Math Bonanza, T7
Find the sum of all positive integers $n$ with the following properties:
[list]
[*] $n$ is not divisible by any primes larger than $10$.
[*] For some positive integer $k$, the positive divisors of $n$ are
\[1=d_1<d_2<d_3\cdots<d_{2k}=n.\]
[*] The divisors of $n$ have the property that
\[d_1+d_2+\cdots+d_k=3k.\]
[/list]
[i]2021 CCA Math Bonanza Team Round #7[/i]
2010 Singapore Senior Math Olympiad, 3
Given $a_1\ge 1$ and $a_{k+1}\ge a_k+1$ for all $k\ge 1,2,\dots,n$, show that $a_1^3+a_2^3+\dots+a_n^3\ge (a_1+a_2+\dots+a_n)^2$
2016 Baltic Way, 8
Find all real numbers $a$ for which there exists a non-constant function $f :\Bbb R \to \Bbb R$ satisfying the following two equations for all $x\in \Bbb R:$
i) $f(ax) = a^2f(x)$ and
ii) $f(f(x)) = a f(x).$
1984 IMO Longlists, 65
A tetrahedron is inscribed in a sphere of radius $1$ such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.
2012 BAMO, 3
Two infinite rows of evenly-spaced dots are aligned as in the figure below. Arrows point from every dot in the top row to some dot in the lower row in such a way that:
[list][*]No two arrows point at the same dot.
[*]Now arrow can extend right or left by more than 1006 positions.[/list]
[img]https://cdn.artofproblemsolving.com/attachments/7/6/47abf37771176fce21bce057edf0429d0181fb.png[/img]
Show that at most 2012 dots in the lower row could have no arrow pointing to them.
2021 Hong Kong TST, 3
Let $\triangle ABC$ be an acute triangle with circumcircle $\Gamma$, and let $P$ be the midpoint of the minor arc $BC$ of $\Gamma$. Let $AP$ and $BC$ meet at $D$, and let $M$ be the midpoint of $AB$. Also, let $E$ be the point such that $AE\perp AB$ and $BE\perp MP$. Prove that $AE=DE$.
2023 All-Russian Olympiad, 6
A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$. What is the least number of dominoes that are entirely inside some square $2 \times 2$?
2016 Greece National Olympiad, 4
A square $ABCD$ is divided into $n^2$ equal small (fundamental) squares by drawing lines parallel to its sides.The vertices of the fundamental squares are called vertices of the grid.A rhombus is called [i]nice[/i] when:
$\bullet$ It is not a square
$\bullet$ Its vertices are points of the grid
$\bullet$ Its diagonals are parallel to the sides of the square $ABCD$
Find (as a function of $n$) the number of the [i]nice[/i] rhombuses ($n$ is a positive integer greater than $2$).
1992 Rioplatense Mathematical Olympiad, Level 3, 2
Determine the integers $0 \le a \le b \le c \le d$ such that: $$2^n= a^2 + b^2 + c^2 + d^2.$$
2008 Sharygin Geometry Olympiad, 5
(A.Zaslavsky) Given two triangles $ ABC$, $ A'B'C'$. Denote by $ \alpha$ the angle between the altitude and the median from vertex $ A$ of triangle $ ABC$. Angles $ \beta$, $ \gamma$, $ \alpha'$, $ \beta'$, $ \gamma'$ are defined similarly. It is known that $ \alpha \equal{} \alpha'$, $ \beta \equal{} \beta'$, $ \gamma \equal{} \gamma'$. Can we conclude that the triangles are similar?
2021 Tuymaada Olympiad, 5
Sines of three acute angles form an arithmetic progression, while the cosines of these angles form a geometric progression. Prove that all three angles are equal.
Russian TST 2018, P3
For any finite sets $X$ and $Y$ of positive integers, denote by $f_X(k)$ the $k^{\text{th}}$ smallest positive integer not in $X$, and let $$X*Y=X\cup \{ f_X(y):y\in Y\}.$$Let $A$ be a set of $a>0$ positive integers and let $B$ be a set of $b>0$ positive integers. Prove that if $A*B=B*A$, then $$\underbrace{A*(A*\cdots (A*(A*A))\cdots )}_{\text{ A appears $b$ times}}=\underbrace{B*(B*\cdots (B*(B*B))\cdots )}_{\text{ B appears $a$ times}}.$$
[i]Proposed by Alex Zhai, United States[/i]
2012 AMC 8, 17
A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?
$\textbf{(A)}\hspace{.05in}3 \qquad \textbf{(B)}\hspace{.05in}4 \qquad \textbf{(C)}\hspace{.05in}5 \qquad \textbf{(D)}\hspace{.05in}6 \qquad \textbf{(E)}\hspace{.05in}7 $
2014 BAMO, 5
A chess tournament took place between $2n+1$ players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly $k$ games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than $n-\sqrt{2k}$ and no more than $n+\sqrt{2k}$ games.
1986 India National Olympiad, 6
Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal.
2012 Romanian Masters In Mathematics, 4
Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not.
[i](Russia) Valery Senderov[/i]
2020 LIMIT Category 1, 4
The total number of solutions of $xyz=2520$
(A)$2520$
(B)$2160$
(C)$540$
(D)None of these
2023 CUBRMC, 2
The concave decagon shown below is embedded in the Cartesian coordinate plane such that all of its vertices have integer coordinates. Two opposite edges have length $5$, whereas the remaining eight edges have length $\sqrt{10}$. Every pair of opposite edges is parallel. The sides of the decagon do not intersect each other, and the decagon has vertical and horizontal axes of symmetry. Find the area of the decagon.
[img]https://cdn.artofproblemsolving.com/attachments/1/5/daa4ab3d71af4b3274cd222f9a091eea3be705.png[/img]
2018 Latvia Baltic Way TST, P2
Find all ordered pairs $(x,y)$ of real numbers that satisfy the following system of equations:
$$\begin{cases}
y(x+y)^2=2\\
8y(x^3-y^3) = 13.
\end{cases}$$
2016 India Regional Mathematical Olympiad, 4
There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)?
2006 Baltic Way, 13
In a triangle $ABC$, points $D,E$ lie on sides $AB,AC$ respectively. The lines $BE$ and $CD$ intersect at $F$. Prove that if
$\color{white}\ .\quad\ \color{black}\ \quad BC^2=BD\cdot BA+CE\cdot CA,$
then the points $A,D,F,E$ lie on a circle.
1996 China Team Selection Test, 1
Let side $BC$ of $\bigtriangleup ABC$ be the diameter of a semicircle which cuts $AB$ and $AC$ at $D$ and $E$ respectively. $F$ and $G$ are the feet of the perpendiculars from $D$ and $E$ to $BC$ respectively. $DG$ and $EF$ intersect at $M$. Prove that $AM \perp BC$.
1983 Tournament Of Towns, (045) 2
Find all natural numbers $k$ which can be represented as the sum of two relatively prime numbers not equal to $1$.