A circle $S$ bisects a circle $S'$ if it cuts $S'$ at opposite ends of a diameter. $S_A$, $S_B$,$S_C$ are circles with distinct centers $A, B, C$ (respectively). Show that $A, B, C$ are collinear iff there is no unique circle $S$ which bisects each of $S_A$, $S_B$,$S_C$ . Show that if there is more than one circle $S$ which bisects each of $S_A$, $S_B$,$S_C$ , then all such circles pass through two fixed points. Find these points. [b]Original Statement:[/b] A circle $S$ is said to cut a circle $\Sigma$ [b]diametrically[/b] if and only if their common chord is a diameter of $\Sigma.$ Let $S_A, S_B, S_C$ be three circles with distinct centres $A,B,C$ respectively. Prove that $A,B,C$ are collinear if and only if there is no unique circle $S$ which cuts each of $S_A, S_B, S_C$ diametrically. Prove further that if there exists more than one circle $S$ which cuts each $S_A, S_B, S_C$ diametrically, then all such circles $S$ pass through two fixed points. Locate these points in relation to the circles $S_A, S_B, S_C.$

In the morning, $100$ students study as $50$ groups with two students in each group. In the afternoon, they study again as $50$ groups with two students in each group. No matter how the groups in the morning or groups in the afternoon are established, if it is possible to find $n$ students such that no two of them study together, what is the largest value of $n$? $ \textbf{(A)}\ 42 \qquad\textbf{(B)}\ 38 \qquad\textbf{(C)}\ 34 \qquad\textbf{(D)}\ 25 \qquad\textbf{(E)}\ \text{None of above} $

The $ABCDA_1B_1C_1D_1$ cube has unit length edges. Find the distance between two circumferences, one of those is inscribed into the $ABCD$ base, and another comes through points $A,C$ and $B_1$ .

Let $x$ be a real number such that $0<x<\frac{\pi}{2}$. Prove that \[\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1\]

For every natural number $ n $, define $ s(n) $ as the smallest natural number so that for every natural number $ a $ relatively prime to $n$, this equation holds: \[ a^{s(n)} \equiv 1 (mod n) \] Find all natural numbers $ n $ such that $ s(n) = 2010 $

Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0), (2,0), (2,1),$ and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges. [asy] size(3cm); draw((0,0)--(2,0)--(2,1)--(0,1)--cycle); draw((1,0)--(1,1)); [/asy] Compute the number of ways to choose one or more of the seven edges such that the resulting figure is traceable without lifting a pencil. (Rotations and reflections are considered distinct.)

Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.

Let $f(n) = \sum^n_{d=1} \left\lfloor \frac{n}{d} \right\rfloor$ and $g(n) = f(n) -f(n - 1)$. For how many $n$ from $1$ to $100$ inclusive is $g(n)$ even?

The lengths of the sides of triangle $ABC$ are $4,5,6$. For any point $D$ on one of the sides, draw the perpendiculars $DP, DQ$ on the other two sides. What is the minimum value of $PQ$?

Find the ratio between the sum of the areas of the circles and the area of the fourth circle that are shown in the figure Each circle passes through the center of the previous one and they are internally tangent. [img]https://cdn.artofproblemsolving.com/attachments/d/2/29d2be270f7bcf9aee793b0b01c2ef10131e06.jpg[/img]

Given seven points in the plane, some of them are connected by segments such that: [b](i)[/b] among any three of the given points, two are connected by a segment; [b](ii)[/b] the number of segments is minimal. How many segments does a figure satisfying [b](i)[/b] and [b](ii)[/b] have? Give an example of such a figure.

Chord $D E$ of the circumcircle of the triangle $A B C$ intersects sides $A B$ and $B C$ in points $P$ and $Q$ respectively. Point $P$ lies between $D$ and $Q$. Angle bisectors $D F$ and $E G$ are drawn in triangles $A D P$ and $Q E C$. It turned out that the points $D$, $F, G, E$ are concyclic. Prove that the points $A, P, Q, C$ are concyclic. Azamat Mardanov

Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? $ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 $

Find the constants $ a,\ b,\ c$ such that a function $ f(x)\equal{}a\sin x\plus{}b\cos x\plus{}c$ satisfies the following equation for any real numbers $ x$. \[ 5\sin x\plus{}3\cos x\plus{}1\plus{}\int_0^{\frac{\pi}{2}} (\sin x\plus{}\cos t)f(t)\ dt\equal{}f(x).\]

Find all non-negative integers $a,b,c,d$ such that $7^a= 4^b + 5^c + 6^d$.

Byan Rai has 1 red cup, 4 blue cups, 1 orange cup, 2 yellow cups, 3 green cups, 3 purple cups and 8 black cups in a box. Every second, Byan will pull out a random cup from the box and magically all other cups of the same color will disappear. What is the expected number of seconds it will take for Byan to pick a blue cup? [i]Individual #13[/i]

The function $g$, with domain and real numbers, fulfills the following: $\bullet$ $g (x) \le x$, for all real $x$ $\bullet$ $g (x + y) \le g (x) + g (y)$ for all real $x,y$ Find $g (1990)$.

Find the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $a,b,c,$ and $d$ are all (not necessarily distinct) factors of $30$ and $abcd>900$.

Two lines, $l_1$ and $l_2$, are tangent to the parabola $x^2-4(x+y)+y^2=2xy+8$ such that they intersect at a point whose coordinates sum to $-32$. The minimum possible sum of the slopes of $l_1$ and $l_2$ can be written as $\frac{m}{n}$ for relatively prime integers $m$ and $n$. Find $m+n$. [I] Proposed by [b]AOPS12142015[/b][/I]

A circle and a point $P$ higher than the circle lie in the same vertical plane. A particle moves along a straight line under gravity from $P$ to a point $Q$ on the circle. Given that the distance travelled from $P$ in time $t$ is equal to $\dfrac{1}{2}gt^2 \sin{\alpha}$, where $\alpha$ is the angle of inclination of the line $PQ$ to the horizontal, give a geometrical characterization of the point $Q$ for which the time taken from $P$ to $Q$ is a minimum.

Let $ H$ be the orthocenter, $ O$ the circumcenter, and $ R$ the circumradius of an acute-angled triangle $ ABC$. Consider the circles $ k_a,k_b,k_c,k_h,k$, all with radius $ R$, centered at $ A,B,C,H,M,$ respectively. Circles $ k_a$ and $ k_b$ meet at $ M$ and $ F$; $ k_a$ and $ k_c$ meet at $ M$ and $ E$; and $ k_b$ and $ k_c$ meet at $ M$ and $ D$. $ (a)$ Prove that the points $ D,E,F$ lie on the circle $ k_h$. $ (b)$ Prove that the set of the points inside $ k_h$ that are inside exactly one of the circles $ k_a,k_b,k_c$ has the area twice the area of $ \triangle ABC$.

Prove that for any three infinite sequences of natural numbers $(a_n)_{n\ge 1}$, $(b_n)_{n\ge 1}$, $(c_n)_{n\ge 1}$, there exist numbers $p$ and $q$ such that $a_p\ge a_q$, $b_p\ge b_q$ and $c_p\ge c_q$.

Let $p$ be a prime number greater than $2$. Patricia wants $7$ not-necessarily different numbers from $\{1, 2, . . . , p\}$ to the black dots in the figure below, on such a way that the product of three numbers on a line or circle always has the same remainder when divided by $p$. [img]https://cdn.artofproblemsolving.com/attachments/3/1/ef0d63b8ff5341ffc340de0cc75b24c7229e23.png[/img] (a) Suppose Patricia uses the number $p$ at least once. How many times does she have the number $p$ then a minimum sum needed? (b) Suppose Patricia does not use the number $p$. In how many ways can she assign numbers? (Two ways are different if there is at least one black one dot different numbers are assigned. The figure is not rotated or mirrored.)

A rectangle $ABCD$ is inscribed in a circle with centre $O$. The exterior bisectors of $\angle ABD$ and $\angle ADB$ intersect at $P$; those of $\angle DAB$ and $\angle DBA$ intersect at $Q$; those of $\angle ACD$ and $\angle ADC$ intersect at $R$; and those of $\angle DAC$ and $\angle DCA$ intersect at $S$. Prove that $P,Q,R$, and $S$ are concyclic.

A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of $N$.