Find all positive integers $n$ such that the following holds. $$\tau(n)|2^{\sigma(n)}-1$$

Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(xy)=\max\{f(x+y),f(x) f(y)\} \] for all real numbers $x$ and $y$.

A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder where $m+n$ is divided by 1000.

Let $ n$ be a composite natural number and $ p$ a proper divisor of $ n.$ Find the binary representation of the smallest natural number $ N$ such that \[ \frac{(1 \plus{} 2^p \plus{} 2^{n\minus{}p})N \minus{} 1}{2^n}\] is an integer.

Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$

Let $a,b,c$ be nonnegative real numbers.Prove that $3(a^2+b^2+c^2)\ge (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2\ge (a+b+c)^2$.

9.6, 10.6 Construct for each vertex of the trapezium a symmetric point wrt to the diagonal, which doesn't contain this vertex. Prove that if four new points form a quadrilateral then it is a trapezium. ([i]L. Emel'yanov[/i])

Consider the $8\times 8\times 8$ Rubik’s cube below. Each face is painted with a different color, and it is possible to turn any layer, as you can with smaller Rubik’s cubes. Let $X$ denote the move that turns the shaded layer shown (indicated by arrows going from the top to the right of the cube) clockwise by $90$ degrees, about the axis labeled $X$. When move $X$ is performed, the only layer that moves is the shaded layer. Likewise, define move $Y$ to be a clockwise $90$-degree turn about the axis labeled Y, of just the shaded layer shown (indicated by the arrows going from the front to the top, where the front is the side pierced by the $X$ rotation axis). Let $M$ denote the move “perform $X$, then perform $Y$.” [img]https://cdn.artofproblemsolving.com/attachments/e/f/951ea75a3dbbf0ca23c45cd8da372595c2de48.png[/img] Imagine that the cube starts out in “solved” form (so each face has just one color), and we start doing move $M$ repeatedly. What is the least number of repeats of $M$ in order for the cube to be restored to its original colors?

How many seven-digit positive integers do not either start or end with $7$?

Let $p>2$ be a prime and $a,b$ positive integers such that \[\frac ab=1+\frac12+\frac13+\cdots+\frac{1}{p-1}.\] Show that $p$ is a divisor of $a.$

The parallelogram bounded by the lines $ y \equal{} ax \plus{} c,y \equal{} ax \plus{} d,y \equal{} bx \plus{} c$ and $ y \equal{} bx \plus{} d$ has area $ 18$. The parallelogram bounded by the lines $ y \equal{} ax \plus{} c,y \equal{} ax \minus{} d,y \equal{} bx \plus{} c,$ and $ y \equal{} bx \minus{} d$ has area $ 72.$ Given that $ a,b,c,$ and $ d$ are positive integers, what is the smallest possible value of $ a \plus{} b \plus{} c \plus{} d$? $ \textbf{(A)}\ 13 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17$

Jerry and Neil have a $3$-sided die that rolls the numbers $1,2,$ and $3,$ each with probability $\tfrac{1}{3}.$ Jerry rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry's. Compute the probability that Neil's final number is $3.$

Let us consider $a,b$ two integers. Prove that there exists and it is unique a pair of integers $(x,y)$ such that: \[(x+2y-a)^{2}+(2x-y-b)^{2}\leq 1.\]

How many rectangles are in this figure? [asy] pair A,B,C,D,E,F,G,H,I,J,K,L; A=(0,0); B=(20,0); C=(20,20); D=(0,20); draw(A--B--C--D--cycle); E=(-10,-5); F=(13,-5); G=(13,5); H=(-10,5); draw(E--F--G--H--cycle); I=(10,-20); J=(18,-20); K=(18,13); L=(10,13); draw(I--J--K--L--cycle);[/asy] $ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 $

Two sequences of positive reals, $ \left(x_n\right)$ and $ \left(y_n\right)$, satisfy the relations $ x_{n \plus{} 2} \equal{} x_n \plus{} x_{n \plus{} 1}^2$ and $ y_{n \plus{} 2} \equal{} y_n^2 \plus{} y_{n \plus{} 1}$ for all natural numbers $ n$. Prove that, if the numbers $ x_1$, $ x_2$, $ y_1$, $ y_2$ are all greater than $ 1$, then there exists a natural number $ k$ such that $ x_k > y_k$.

On the circle $\gamma$ the points $A$ and $B$ are selected. The circle $\omega$ touches the segment $AB$ at the point $K$ and intersects the circle $\gamma$ at the points $M$ and $N$. The points lie on the circle $\gamma$ in the following order: $A, \, \, M, \, \, N, \, \, B$. Prove that $\angle AMK = \angle KNB$. (Yuri Biletsky)

Consider a positive real number $a$ and a positive integer $m$. The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as: $x_1=1$, $x_2=a$, $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$. $a)$ Prove that the sequence is converging. $b)$ Find $\lim_{n\rightarrow \infty}{x_n}$.

There are n lilypads in a row labeled $1, 2, \dots, n$ from left to right. Fareniss the Frog picks a lilypad at random to start on, and every second she jumps to an adjacent lilypad; if there are two such lilypads, she is twice as likely to jump to the right as to the left. After some finite number of seconds, there exists two lilypads $A$ and $B$ such that Fareniss is more than $1000$ times as likely to be on $A$ as she is to be on $B$. What is the minimal number of lilypads $n$ such that this situation must occur?

Let $ABCD$ be a cyclic quadrilateral and let the intersection point of lines $AB$ and $CD$ be $E$. Let the points $K$ and $L$ be arbitrary points on sides $CD$ and $AB$ respectively, which satisfy the conditions $$\angle KAD = \angle KBC \quad \text{and} \quad \angle LDA = \angle LCB.$$ Prove that $EK = EL$.

A tamer wants to line up five lions and four tigers. We know that a tiger cannot go after another. How many ways can the beasts be distributed? The tamer cannot distinguish two animals of the same species.

A field has a shape of checkboard $\text{41x41}$ square. A tank concealed in one of the cells of the field. By one shot, a fighter airplane fires one of the cells. If a shot hits the tank, then the tank moves to a neighboring cell of the field, otherwise it stays in its cell (the cells are neighbours if they share a side). A pilot has no information about the tank , one needs to hit it twice. Find the least number of shots sufficient to destroy the tank for sure. [i](S.Berlov,A.Magazinov)[/i]

For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i=\sqrt{-1}.$ Find $b.$

Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.