Prove that the cube of any positive integer greater than 1 can be represented as a difference of the squares of two positive integers.

Every phone number in an area consists of eight digits and starts with digit $ 8$. Mr Edy, who has just moved to the area, apply for a new phone number. What is the chance that Mr Edy gets a phone number which consists of at most five different digits?

Let $f(n)$ be the number of points of intersection of diagonals of a $n$-dimensional hypercube that is not the vertex of the cube. For example, $f(3) = 7$ because the intersection points of a cube’s diagonals are at the centers of each face and the center of the cube. Find $f(5)$.

Let $p,q,r$ be primes such that for all positive integer $n$, $$n^{pqr}\equiv n (\mod{pqr})$$ Prove that this happens if and only if $p,q,r$ are pairwise distinct and $LCM(p-1,q-1,r-1)|pqr-1$

$f(x)=a\sin x+b\sqrt[3]{x}+4$. If $f(\lg\log_{3}10)=5$, then the value of $f(\lg\lg 3)$ is $\text{(A)}-5\qquad\text{(B)}-3\qquad\text{(C)}3\qquad\text{(D)}$ not sure

Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.

The sidelengths of a triangle are $a,b,c$. (a) Prove that there is a triangle which has the sidelengths $\sqrt a,\sqrt b,\sqrt c$. (b) Prove that $\displaystyle \sqrt{ab}+\sqrt{bc}+\sqrt{ca} \leq a+b+c < 2 \sqrt{ab} + 2 \sqrt{bc} + 2 \sqrt{ca}$.

An ant is currently located in the center (vertex $S$) of the adjoined hexagonal configuration, as shown in the figure below. Each minute, it walks along $1$ of the $15$ possible edges, traveling from one vertex to another. How many ways are there for the ant to be back to its original position after $2020$ minutes?

For each positive integer, define a function \[ f(n)=\begin{cases}0, &\text{if n is the square of an integer}\\ \\ \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. \] Find the value of $\sum_{k=1}^{200} f(k)$.

Let $n$ be an integer greater than or equal to $3$, and let $P_n$ be the collection of all planar (simple) $n$-gons no two distinct sides of which are parallel or lie along some line. For each member $P$ of $P_n$, let $f_n(P)$ be the least cardinal a cover of $P$ by triangles formed by lines of support of sides of $P$ may have. Determine the largest value $f_n(P)$ may achieve, as $P$ runs through $P_n$.

Let $ A_1,A_2,...,A_{3n} $ be $ 3n\ge 3 $ planar points such that $ A_1A_2A_3 $ is an equilateral triangle and $ A_{3k+1} ,A_{3k+2} ,A_{3k+3} $ are the midpoints of the sides of $ A_{3k-2}A_{3k-1}A_{3k} , $ for all $ 1\le k<n. $ Of two different colors, each one of these points are colored, either with one, either with another. [b]a)[/b] Prove that, if $ n\ge 7, $ then some of these points form a monochromatic (only one color) isosceles trapezoid. [b]b)[/b] What about $ n=6? $

Prove that the equation $ x^{5} \plus{} 31 \equal{} y^{2}$ has no integer solution.

A number is called [i]capicua[/i] if when it is written in decimal notation, it can be read equal from left to right as from right to left; for example: $8, 23432, 6446$. Let $x_1<x_2<\cdots<x_i<x_{i+1},\cdots$ be the sequence of all capicua numbers. For each $i$ define $y_i=x_{i+1}-x_i$. How many distinct primes contains the set $\{y_1,y_2, \ldots\}$?

Find positive reals $a, b, c$ which maximizes the value of $abc$ subject to the constraint that $b(a^2 + 2) + c(a + 2) = 12$.

Let $n>3$ be a positive integer. Consider $n$ sets, each having two elements, such that the intersection of any two of them is a set with one element. Prove that the intersection of all sets is non-empty. [i]Sever Moldoveanu[/i]

$ABCD$ is a fixed rhombus. Segment $PQ$ is tangent to the inscribed circle of $ABCD$, where $P$ is on side $AB$, $Q$ is on side $AD$. Show that, when segment $PQ$ is moving, the area of $\Delta CPQ$ is a constant.

Let $ABC$ be a triangle. Let $I$ be the Incenter of $ABC$ and $S$ be the midpoint of arc $BAC$. Define $IA$ as the $A$-excenter wrt $ABC$. Define $\omega$ to be the circle centred at $S$ with radius $SB$. Let $AI_A \cap \omega = X$, $Y$. Show that $\angle BCX = \angle ACY$.

[i]Greedy Algorithms, Inc.[/i] offers the following string-processing service. Each string submitted for processing has a starting price of $1$ dollar. The customer can then ask for any two adjacent characters in the string to be swapped. This may be done an arbitrary number of times, but each swap doubles the price for processing the string. Then the company returns the modified string and charges the customer $2^S$ dollars, where $S$ is the number of swaps executed. If a customer submits [b]all [/b]permutations of the string $\text{PUMAC}$ for processing and wants all of the strings to be identical after processing, what is the lowest price, in dollars, she could pay?

A triangular pyramid $ABCD$ is given. A sphere $\omega_A$ is tangent to the face $BCD$ and to the planes of other faces in points don't lying on faces. Similarly, sphere $\omega_B$ is tangent to the face $ACD$ and to the planes of other faces in points don't lying on faces. Let $K$ be the point where $\omega_A$ is tangent to $ACD$, and let $L$ be the point where $\omega_B$ is tangent to $BCD$. The points $X$ and $Y$ are chosen on the prolongations of $AK$ and $BL$ over $K$ and $L$ such that $\angle CKD = \angle CXD + \angle CBD$ and $\angle CLD = \angle CYD +\angle CAD$. Prove that the distances from the points $X$, $Y$ to the midpoint of $CD$ are the same. [hide=thanks ]Thanks to the user Vlados021 for translating the problem.[/hide]

Consider a triangular grid of equilateral triangles with unit sides. Assume that $\mathcal{P}$ is a non-self-intersecting polygon with perimeter 1399 and sides from the grid. Prove that $\mathcal{P}$ has either an internal or an external 120-degree angle. [i]Proposed by Seyed Hessam Firouzi[/i]

Does there exist a function $f\colon \mathbb{R}\to\mathbb{R}$ such that for any $x>y,$ it satisfies $f(x)-f(y)>\sqrt{x-y}.$

Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. [i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$. [i]Floor van Lamoen[/i]

The equation $$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$$ is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

Let $M$ be the point of intersection of the diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$. Let$ K$ be the intersection point of the extension of side $AB$ (from $A$) with the bisector of the $\angle ACD$. If $MA \cdot MC + MA \cdot CD = MB\cdot MD$ , prove that $\angle BKC = \angle CDB$.

A semicircle $(O)$ is drawn with the center $O$, where $O$ lies on a line $\ell$. $C$ and $D$ lie on the circle $(O)$, and the tangent lines of $(O)$ at points $C$ and $D$ intersects the line $\ell$ at points $B$ and $A$, respectively, such that $O$ lies between points $B$ and $A$. Let $E$ be the intersection point between $AC$ and $BD$, and $F$ the point on $\ell$ so that $EF $ is perpendicular to line $\ell$. Prove that $EF$ bisects the angle $\angle CFD$.