Find the least positive integer $N$ that is 50 times the number of positive integer divisors that $N$ has.

There were finitely many persons at a party among whom some were friends. Among any $4$ of them there were either $3$ who were all friends among each other or $3$ who weren't friend with each other. Prove that you can separate all the people at the party in two groups in such a way that in the first group everyone is friends with each other and that all the people in the second group are not friends to anyone else in second group. (Friendship is a mutual relation).

[i](The following problem is open in the sense that the answer to part (b) is not currently known.)[/i] [list=a] [*] Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a,b)$ of positive integers for which there exists a positive real number $r$, such that $$r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}$$ are both rational numbers. [*] Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a,b)$ of positive integers for which there exists a real number $r$, such that $$r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}$$ are both rational numbers. [/list]

A photographer took some pictures at a party with $10$ people. Each of the $45$ possible pairs of people appears together on exactly one photo, and each photo depicts two or three people. What is the smallest possible number of photos taken?

In a triangle $ ABC$ with $ \angle A\equal{}90^{\circ}$, $ D$ is the midpoint of $ BC$, $ F$ that of $ AB$, $ E$ that of $ AF$ and $ G$ that of $ FB$. Segment $ AD$ intersects $ CE,CF$ and $ CG$ in $ P,Q$ and $ R$, respectively. Determine the ratio: $ \frac{PQ}{QR}$.

A circle has a radius of $ \log_{10}(a^2)$ and a circumference of $ \log_{10}(b^4)$. What is $ \log_ab$? $ \textbf{(A)}\ \frac {1}{4\pi} \qquad \textbf{(B)}\ \frac {1}{\pi} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ 2\pi \qquad \textbf{(E)}\ 10^{2\pi}$

$ABCD$ is a convex quadrilateral inscribed in a circle with centre $O$, and with mutually perpendicular diagonals. Prove that the broken line $AOC$ divides the quadrilateral into two parts of equal area. (V Varvarkin)

Let $d(n)$ be the number of divisors of a nonnegative integer $n$ (we set $d(0)=0$). Find all positive integers $d$ such that there exists a two-variable polynomial $P(x,y)$ of degree $d$ with integer coefficients such that: [list] [*] for any positive integer $y$, there are infinitely many positive integers $x$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid x$, and [*] for any positive integer $x$, there are infinitely many positive integers $y$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid y$. [/list] [i]Allen Wang[/i]

$n$ points are marked on a plane. Each pair of these points is connected with a segment. Each segment is painted one of four different colors. Find the largest possible value of $n$ such that one can paint the segments so that for any four points there are four segments (connecting these four points) of four different colors. (E. Barabanov)

Jeffrey jogs $3$ miles at $4$ miles per hour. Then, he walks $4$ miles at $3$ miles an hour. What was Jeffrey's average speed in miles per hour for the entire trip? Express your answer as a decimal to the nearest hundredth.

Given $\triangle ABC$ with circumcircle $\Gamma$ and circumcentre $O$, let $X$ be a point on $\Gamma$. Let $XC_1$, $XB_1$ to be feet of perpendiculars from $X$ to lines $AB$ and $AC$. Define $\omega_C$ as the circle with centre the midpoint of $AB$ and passing through $C_1$ . Define $\omega_B$ similarly. Prove that $\omega_B$ and $\omega_C$ has a common point on $XO$.

Let $ABCD$ be a cyclic quadrilateral with $AB<CD$, and let $P$ be the point of intersection of the lines $AD$ and $BC$.The circumcircle of the triangle $PCD$ intersects the line $AB$ at the points $Q$ and $R$. Let $S$ and $T$ be the points where the tangents from $P$ to the circumcircle of $ABCD$ touch that circle. (a) Prove that $PQ=PR$. (b) Prove that $QRST$ is a cyclic quadrilateral.

Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.

Solve the Cauchy problem \[\frac{\partial ^2A}{\partial t^2}=9\frac{\partial^2 A}{\partial x^2}-2B,\;\frac{\partial^2 B}{\partial t^2}=6\frac{\partial^2 B}{\partial x^2}-2A\] \[A|_{t=0}=\cos x,\; B|_{t=0}=0,\; \left.\frac{\partial A}{\partial t}\right|_{t=0}=\left.\frac{\partial B}{\partial t}\right|_{t=0}=0\]

A circumscribed quadrilateral $ABCD$ is given. It is known that $\angle{ACB} = \angle{ACD}$. On the angle bisector of $\angle{C}$, a point $E$ is marked such that $AE \bot BD$. Point $F$ is the foot of the perpendicular line from point $E$ to the side $BC$. Prove that $AB = BF$.

Determine all pairs of positive integers $(x, y)$ such that: $$x^4 - 6x^2 + 1 = 7\cdot 2^y$$

Let $k$ be a positive integer. Show that there exist infinitely many positive integers $n$ such that $\frac{n^n-1}{n-1}$ has at least $k$ distinct prime divisors. [i]Proposed by Adnan Sadik[/i]

Diagonal $ DB$ of rectangle $ ABCD$ is divided into $ 3$ segments of length $ 1$ by parallel lines $ L$ and $ L'$ that pass through $ A$ and $ C$ and are perpendicular to $ DB$. The area of $ ABCD$, rounded to the nearest tenth, is [asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); real x=sqrt(6), y=sqrt(3), a=0.4; pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D); real r=degrees(B); pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90); markscalefactor=0.02; draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B,6)^^rightanglemark(N1,E,B,6)); pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135); label("A", A, NE); label("B", B, NE); label("C", C, dir(0)); label("D", D, dir(180)); label("$L$", (x/2,0), SW); label("$L^\prime$", C, SW); label("1", D--F, NW); label("1", F--E, SE); label("1", E--B, SE); clip(W--X--Y--Z--cycle); [/asy] $ \textbf{(A)}\ 4.1 \qquad \textbf{(B)}\ 4.2 \qquad \textbf{(C)}\ 4.3 \qquad \textbf{(D)}\ 4.4 \qquad \textbf{(E)}\ 4.5$

A triangle $ABC$ is given. $N$ and $M$ are the midpoints of $AB$ and $BC$, respectively. The bisector of angle $B$ meets the segment $MN$ at $E$. $H$ is the base of the altitude drawn from $B$ in the triangle $ABC$. The point $T$ on the circumcircle of $ABC$ is such that the circumcircles of $TMN$ and $ABC$ are tangent. Prove that points $T, H, E, B$ are concyclic. [i]Proposed by M. Yumatov[/i]

An engineer said he could finish a highway section in $ 3$ days with his present supply of a certain type of machine. However, with $ 3$ more of these machines the job could be done in $ 2$ days. If the machines all work at the same rate, how many days would it take to do the job with one machine? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 36$

In triangle $ABC$ point $O$ is circumcenter. Point $T$ is centroid of $ABC$, and points $D$, $E$ and $F$ are circumcenters of triangles $TBC$, $TCA$ and $TAB$. Prove that $O$ is centroid of $DEF$

Let $A'$ be a point on one of the sides of the trapezoid $ABCD$ such that line $AA'$ divides the area of the trapezoid in half. Points $B'$, $C'$, $D'$ are defined similarly. Prove that the intersection points of the diagonals of quadrilaterals $ABCD$ and $A'B'C'D'$ are symmetrical wrt the midpoint of midline of trapezoid $ABCD$.

Two squares with side $12$ lie exactly on top of each other. One square is rotated around a corner point through an angle of $30$ degrees relative to the other square. Determine the area of the common piece of the two squares. [asy] unitsize (2 cm); pair A, B, C, D, Bp, Cp, Dp, P; A = (0,0); B = (-1,0); C = (-1,1); D = (0,1); Bp = rotate(-30)*(B); Cp = rotate(-30)*(C); Dp = rotate(-30)*(D); P = extension(C, D, Bp, Cp); fill(A--Bp--P--D--cycle, gray(0.8)); draw(A--B--C--D--cycle); draw(A--Bp--Cp--Dp--cycle); label("$30^\circ$", (-0.5,0.1), fontsize(10)); [/asy]

Find the maximum positive integer $k$ such that there exist $k$ positive integers which do not exceed $2017$ and have the property that every number among them cannot be a power of any of the remaining $k-1$ numbers.

On a circle there're $1000$ marked points, each colored in one of $k$ colors. It's known that among any $5$ pairwise intersecting segments, endpoints of which are $10$ distinct marked points, there're at least $3$ segments, each of which has its endpoints colored in different colors. Determine the smallest possible value of $k$ for which it's possible.