Found problems: 85335
Circles $k_1,k_2$ intersect at $B,C$ such that $BC$ is diameter of $k_1$.Tangent of $k_1$ at $C$ touches $k_2$ for the second time at $A$.Line $AB$ intersects $k_1$ at $E$ different from $B$, and line $CE$ intersects $k_2$ at F different from $C$. An arbitrary line through $E$ intersects segment $AF$ at $H$ and $k_1$ for the second time at $G$.If $BG$ and $AC$ intersect at $D$, prove $CH//DF$ .
There are $ n$ football teams in a round-robin competition where every 2 teams meet once. The winner of each match receives 3 points while the loser receives 0 points. In the case of a draw, both teams receive 1 point each. Let $ k$ be as follows: $ 2 \leq k \leq n \minus{} 1$. At least how many points must a certain team get in the competition so as to ensure that there are at most $ k \minus{} 1$ teams whose scores are not less than that particular team's score?
Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.
For which integers $n \geq 2$ is it possible to draw $n$ straight lines in the plane in such a way that there are at least $n - 2$ points where exactly three of the lines meet?
Points $M$ and $N$ are the midpoints of sides $AB$ and $CD$, respectively of quadrilateral $ABCD$. It is known that $BC // AD$ and $AN = CM$. Is it true that $ABCD$ is parallelogram?
A line through a vertex of a non-degenerate triangle cuts it in two similar triangles with $\sqrt{3}$ as the ratio between correspondent sides.
Find the angles of the given triangle.
At least how many moves must a knight make to get from one corner of a chessboard to the opposite corner?
Let set $\mathcal{S}$ contain all positive integers less than or equal to $2020$ that can be written in the form $n(n+1)$ for some positive integer $n$. Compute the number of ordered pairs $(a,b)$ such that $a, b\in \mathcal{S}$ and $a-b$ is a power of two.
In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} -d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$.
[i]Proposed by Eugene Chen
[/i]
In each of the $9$ small circles of the following figure we write positive integers less than $10$, without repetitions. In addition, it is true that the sum of the $5$ numbers located around each one of the $3$ circles is always equal to $S$. Find the largest possible value of $S$.
[img]https://cdn.artofproblemsolving.com/attachments/6/6/2db2c1ac7f45022606fb0099f24e6287977d10.png[/img]
Let the sequence $\{a_i\}^\infty_{i=0}$ be defined by $a_0 =\frac12$ and $a_n = 1 + (a_{n-1} - 1)^2$. Find the product $$\prod_{i=0}^\infty a_i=a_0a_1a_2\ldots$$
The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral $q$. Show that the sum of the lengths of each pair of opposite sides of $q$ is equal.
Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.
Consider the set $ M = \{1,2, \ldots ,2008\}$. Paint every number in the set $ M$ with one of the three colors blue, yellow, red such that each color is utilized to paint at least one number. Define two sets:
$ S_1=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have the same color and }2008 | (x + y + z)\}$;
$ S_2=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have three pairwisely different colors and }2008 | (x + y + z)\}$.
Prove that $ 2|S_1| > |S_2|$ (where $ |X|$ denotes the number of elements in a set $ X$).
$a, b$ are odd numbers that satisfy $(a-b)^2 \le 8\sqrt {ab}$. For $n=ab$, show that the equation
$$x^2-2([\sqrt n]+1)x+n=0$$ has two integral solutions. $[r]$ denotes the biggest integer, not strictly bigger than $r$.
2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations:
[list]
[*] Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip.
[*] Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.[/list]
Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it.
[i]Brian Hamrick.[/i]
Let $ Q$ be the centre of the inscribed circle of a triangle $ ABC.$ Prove that for any point $ P,$
\[ a(PA)^2 \plus{} b(PB)^2 \plus{} c(PC)^2 \equal{} a(QA)^2 \plus{} b(QB)^2 \plus{} c(QC)^2 \plus{} (a \plus{} b \plus{} c)(QP)^2,
\]
where $ a \equal{} BC, b \equal{} CA$ and $ c \equal{} AB.$
A positive integer has exactly $50$ divisors. Is it possible that no difference of two different divisors is divisible by $100$?
[i]Proposed by A. Chironov[/i]
Let $ a$, $ b$, and $ c$ be positive real numbers such that $ ab + bc + ca = 3$. Prove the inequality
\[ 3 + \sum_{\mathrm{\cyc}} (a - b)^2 \ge \frac {a + b^2c^2}{b + c} + \frac {b + c^2a^2}{c + a} + \frac {c + a^2b^2}{a + b} \ge 3.
\]
How many ways are there (in terms of power) to represent the number 1 as a finite number
or an infinite sum of some subset of the set:
{$\phi^{-n} | n \in Z^+$}
$\phi=\frac{1+\sqrt5}{2}$
The points $(1,1),(2,3),(4,5)$ and $(999,111)$ are marked in the coordinate system. We continue to mark points in the following way :
[list]
[*]If points $(a,b)$ are marked then $(b,a)$ and $(a-b,a+b)$ can be marked
[*]If points $(a,b)$ and $(c,d)$ are marked then so can be $(ad+bc, 4ac-4bd)$.
[/list]
Can we, after some finite number of these steps, mark a point belonging to the line $y=2x$.
Let $ a,b,c $ be positive real numbers satisfying $ a+b+c=2 $. Prove the inequality \[ \frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca} \ge \frac{27}{13} \]
Find all the real numbers $x,y,z$ which satisfy the following conditions:
$$
\begin{cases}
3(x^2+y^2+z^2)=1\\
x^2y^2+y^2z^2+z^2x^2=xyz(x+y+z)^3\\
\end{cases}
$$
Find $ a_3,a_4,...,a{}_2{}_0{}_0{}_8$, such that $ a_i =\pm1$ for $ i=3,...,2008$ and
$ \sum\limits_{i=3}^{2008} a_i2^i = 2008$
and show that the numbers $ a_3,a_4,...,a_{2008}$ are uniquely determined by these conditions.
Let $ f$ be a function satisfying $ f(xy) \equal{} f(x)/y$ for all positive real numbers $ x$ and $ y$. If $ f(500) \equal{} 3$, what is the value of $ f(600)$?
$ \textbf{(A)} \ 1 \qquad \textbf{(B)} \ 2 \qquad \textbf{(C)} \ \displaystyle \frac {5}{2} \qquad \textbf{(D)} \ 3 \qquad \textbf{(E)} \ \displaystyle \frac {18}{5}$