Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

Three concentric circles centered at $O$ have radii of $1$, $2$, and $3$. Points $B$ and $C$ lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle $BOC$, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of $\angle{BOC}$ in degrees? [asy] size(100); import graph; draw(circle((0,0),3)); real radius = 3; real angleStart = -54; // starting angle of the sector real angleEnd = 54; // ending angle of the sector label("$O$",(0,0),W); pair O = (0, 0); filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, lightgray); filldraw(circle((0,0),2),lightgray); filldraw(circle((0,0),1),white); draw((1.763,2.427)--(0,0)--(1.763,-2.427)); label("$B$",(1.763,2.427),NE); label("$C$",(1.763,-2.427),SE); [/asy] $\textbf{(A)}\ 108 \qquad \textbf{(B)}\ 120 \qquad \textbf{(C)}\ 135 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 150$

Segment $AD$ is the diameter of the circumscribed circle of an acute-angled triangle $ABC$. Through the intersection of the altitudes of this triangle, a straight line was drawn parallel to the side $BC$, which intersects sides $AB$ and $AC$ at points $E$ and $F$, respectively. Prove that the perimeter of the triangle $DEF$ is two times larger than the side $BC$.

We have $n > 2$ nonzero integers such that everyone of them is divisible by the sum of the other $n - 1$ numbers, Show that the sum of the $n$ numbers is precisely $0$.

Let $ABC$ be a triangle. On its sides $AB$ and $BC$ are fixed points $C_1$ and $A_1$, respectively. Find a point $ P$ on the circumscribed circle of triangle $ABC$ such that the distance between the centers of the circumscribed circles of the triangles $APC_1$ and $CPA_1$ is minimal.

Consider the triangle $ABC$, with $\angle A= 90^o, \angle B = 30^o$, and $D$ is the foot of the altitude from $A$. Let the point $E \in (AD)$ such that $DE = 3AE$ and $F$ the foot of the perpendicular from $D$ to the line $BE$. a) Prove that $AF \perp FC$. b) Determine the measure of the angle $AFB$.

Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$. Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$

Let $ABC$ be a triangle such that $AB=6,BC=5,AC=7.$ Let the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at $X.$ Let $Z$ be a point on the circumcircle of $ABC.$ Let $Y$ be the foot of the perpendicular from $X$ to $CZ.$ Let $K$ be the intersection of the circumcircle of $BCY$ with line $AB.$ Given that $Y$ is on the interior of segment $CZ$ and $YZ=3CY,$ compute $AK.$

Let $A_k$ be the number of pairs $(a_1, a_2, ..., a_{2k})$ for $k\leq 50$, where $a_1, a_2, ..., a_{2k}$ are all different positive integers that satisfy the following. [b]$\cdot$[/b] $a_1, a_2, ..., a_{2k} \leq 100$ [b]$\cdot$[/b] For an odd number less or equal than $2k-1$, we have $a_i > a_{i+1}$ [b]$\cdot$[/b] For an even number less or equal than $2k-2$, we have $a_i < a_{i+1}$ Prove that $A_1 \leq A_2 \leq \cdots \leq A_{49}$.

Let $T$ be the triangle in the $xy$-plane with vertices $(0, 0)$, $(3, 0)$, and $\left(0, \frac32\right)$. Let $E$ be the ellipse inscribed in $T$ which meets each side of $T$ at its midpoint. Find the distance from the center of $E$ to $(0, 0)$.

$n > 1$ is an odd number and $a_1, a_2, . . . , a_n$ are positive integers such that $gcd(a_1, a_2, . . . , a_n) = 1$. If $d = gcd (a_1^n + a_1.a_2. . . a_n, a_2^n + a_1.a_2. . . a_n, . . . , a_n^n + a_1.a_2. . . a_n) $ find all possible values of $d$.

There are $6$ different bus lines in a city, each stopping at exactly $5$ stations and running in both directions. Nevertheless, for every two different stations there is always a bus line connecting these two stations. Determine the maximum number of stations in this city. [i](Karl Czakler)[/i]

Let $ABC$ be a triangle and $E$ and $F$ two arbitrary points on sides $AB$ and $AC$, respectively. The circumcircle of triangle $AEF$ meets the circumcircle of triangle $ABC$ again at point $M$. The point $D$ is such that $EF$ bisects the segment $MD$ . Finally, $O$ is the circumcenter of triangle $ABC$. Prove that $D$ lies on line $BC$ if and only if $O$ lies on the circumcircle of triangle $AEF$.

A numerical sequence is called lusophone if it satisfies the following three conditions: i) The first term of the sequence is number $1$. ii) To obtain the next term of the sequence we can multiply the previous term by a positive prime number ($2,3,5,7,11, ...$) or add $1$. (iii) The last term of the sequence is the number $2016$. For example: $1\overset{{\times 11}}{\to}11 \overset{{\times 61}}{\to} 671 \overset{{+1}}{\to}672 \overset{{\times 3}}{\to}2016$ How many Lusophone sequences exist in which (as in the example above) the add $1$ operation was used exactly once and not multiplied twice by the same prime number?

Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}.$

Points $P$ and $Q$ lie on the sides $AB$, $BC$ of the triangle $ABC$, such that $AC=CP =PQ=QB$ and $A \neq P$ and $C \neq Q$. If $\sphericalangle ACB = 104^{\circ}$, determine the measures of all angles of the triangle $ABC$.

On a large wooden block there are four twelve-hour analog clocks of varying accuracy. At 7PM on April 3, 2015, they all correctly displayed the time. The first clock is accurate, the second clock is two times as fast as the first clock, the third clock is three times as fast as the first clock, and the last clock doesn't move at all. How many hours must elapse (from 7PM) before the times displayed on the clocks coincide again? (The clocks do not distinguish between AM and PM.) [asy] import olympiad; import cse5; size(12cm); defaultpen(linewidth(0.9)+fontsize(11pt)); picture clock(real hh, real mm, string nn) { picture p; draw(p, unitcircle); for(int i=1;i<=12;i=i+1) { // draw(p, 0.9*dir(90-30*i)--dir(90-30*i)); label(p, "$"+(string) i+"$",0.84*dir(90-30*i), fontsize(9pt)); } dot(p, origin); pair hpoint = 0.5 * dir(90 - 30 * (hh + mm/60)); pair mpoint = 0.75 * dir(90 - 6 * mm); draw(p, origin--hpoint, EndArrow(HookHead, 3)); draw(p, origin--mpoint, EndArrow(HookHead, 5)); string tlabel; if (mm > 10) { tlabel = (string) hh + ":" + (string) mm; } else { tlabel = (string) hh + ":0" + (string) mm; } label(p, tlabel, dir(90)*1.2, dir(90)); label(p, tlabel, dir(90)*1.2, dir(90)); label(p, nn, dir(-90)*1.1, dir(-90)); return p; } // The block real h = 1; filldraw( (-1.2,-1)--(8.4,-1)--(8.4,-1-h)--(-1.2,-1-h)--cycle, 0.7*lightgrey, black); add(shift((0.0,0)) * clock(10,22, "I")); add(shift((2.4,0)) * clock( 1,44, "II")); add(shift((4.8,0)) * clock( 5,06, "III")); add(shift((7.2,0)) * clock( 7,00, "IV")); label("\emph{Omnes vulnerant, postuma necat}", (3.6, -1.8), origin); [/asy] [i]Proposed by Evan Chen[/i]

each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 2 block , at least one of the 4 squares is white. let P be the number of ways of colouring the grid. find the largest k so that $3^k$ divides P.

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$. Denotes by $S_1,S_2$ and $S$ the areas of the triangles $ABE$, $CDE$ and the quadrilateral $ABCD$ respectively. Prove that $\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S}$ . When equality is reached?

If $p=2^{16}+1$ is a prime, find the maximum possible number of elements in a set $S$ of positive integers less than $p$ so no two distinct $a,b$ in $S$ satisfy $$a^2\equiv b\pmod{p}.$$

Determine all pairs $(m,n)$ of positive integers such that $m^2+n^2=2018(m-n)$

Let $A,B,C$ and $D$ be non-coplanar points such that $\angle ABC=\angle ADC$ and $\angle BAD=\angle BCD$. Show that $AB=CD$ and $AD=BC$.

Let $a,b,c>0$ and $a^2+b^2+c^2=3$ . Prove that $$ \frac{a(a-b^2)}{a+b^2}+\frac{b(b-c^2)}{b+c^2}+\frac{c(c-a^2)}{c+a^2}\ge 0.$$

Find all functions $f: (0,\infty)\rightarrow(0,\infty)$ with the following properties: $f(x+1)=f(x)+1$ and $f\left(\frac{1}{f(x)}\right)=\frac{1}{x}$. [i]Proposed by P. Volkmann[/i]

Find all positive rational $(x,y)$ that satisfy the equation : $$yx^y=y+1$$