For any positive integer $n$, let [list] [*]$\tau(n)$ denote the number of positive integer divisors of $n$, [*]$\sigma(n)$ denote the sum of the positive integer divisors of $n$, and [*]$\varphi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$. [/list] Let $a,b > 1$ be integers. Brandon has a calculator with three buttons that replace the integer $n$ currently displayed with $\tau(n)$, $\sigma(n)$, or $\varphi(n)$, respectively. Prove that if the calculator currently displays $a$, then Brandon can make the calculator display $b$ after a finite (possibly empty) sequence of button presses. [i]Proposed by Jaedon Whyte.[/i]

Given positive integer $n$. Prove that for any integers $a_1,a_2,\cdots,a_n,$ at least $\lceil \tfrac{n(n-6)}{19} \rceil$ numbers from the set $\{ 1,2, \cdots, \tfrac{n(n-1)}{2} \}$ cannot be represented as $a_i-a_j (1 \le i, j \le n)$.

Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$

Let $A_1A_2,B_1B_2, C_1C_2$ be three equal segments on the three sides of an equilateral triangle. Prove that in the triangle formed by the lines $B_2C_1, C_2A_1,A_2B_1$, the segments $B_2C_1, C_2A_1,A_2B_1$ are proportional to the sides in which they are contained.

There are $k$ points in the interior of a pentagon. Together with the vertices of the pentagon they form a $(k + 5)$-element set $M$. The area of the pentagon is defined by connecting lines between the points of $M$ into sub-areas in such a way that it is divided into sub-areas in such a way that no sub-areas have a point on their interior of $M$ and contains exactly three points of $M$ on the boundary of each part. None of the connecting lines has a point in common with any other connecting line or pentagon side, which does not belong to $M$. With such a decomposition of the pentagon, there can be an even number of connecting lines (including the pentagon sides) go out? The answer has to be justified.

A school organizes optional lectures for 200 students. At least 10 students have signed up for each proposed lecture, and for any two students there is at most one lecture that both of them have signed up for. Prove that it is possible to hold all these lectures over 211 days so that no one has to attend two lectures in one day.

Let $p$ and $q$ be prime numbers such that $(p-1)^{q-1}-1$ is a positive integer that divides $(2q)^{2p}-1$. Compute the sum of all possible values of $pq$. [i]Proposed by Ankit Bisain[/i]

Prove that $c=10\sqrt{24}$ is the largest constant such that if there exist positive numbers $a_1,a_2,\ldots ,a_{17}$ satisfying: \[\sum_{i=1}^{17}a_i^2=24,\ \sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i<c \] then for every $i,j,k$ such that $1\le 1<j<k\le 17$, we have that $x_i,x_j,x_k$ are sides of a triangle.

In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are integers, we call it integral point. Prove that there exists a series of concentric circles, satisfying: (1)Exery itengral point is on the concentric circles. (2)On each circle, there is exactly one itengral point.

Let $P$ be a polynomial of degree $n$ with only real zeros and real coefficients. Prove that for every real $x$ we have $(n-1)(P'(x))^2\ge nP(x)P''(x)$. When does equality occur?

If $x, y, z, w$ are nonnegative real numbers satisfying \[\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\ \end{array} \right. \] find the smallest possible value of $x$ and the values of $y, z, w$ corresponding to it.

Find the sum of all values of $a$ such that there are positive integers $a$ and $b$ satisfying $(a - b)\sqrt{ab} = 2016$.

Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.

Let $\vartriangle ABC$ be a triangle. The angle bisector of $\angle CAB$ intersects$ BC$ at $L$. On the interior of line segments $AC$ and $AB$, two points, $M$ and $N$, respectively, are chosen in such a way that the lines $AL, BM$ and $CN$ are concurrent, and such that $\angle AMN = \angle ALB$. Prove that $\angle NML = 90^o$.

Prove that for any integer $n$ greater than $5$, a square can be divided into $n$ squares.

Call a rational number $r$ [i]powerful[/i] if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p, q$ and some integer $k >1$. Let $a, b, c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x, y, z$ such that $a^x + b^y + c^z$ is an integer. Prove that $a, b, c$ are all [i]powerful[/i]. [i]Jeck Lim, Singapore[/i]

A teacher gives a multiple choice test to $15$ students and that each student answered each question. Each question had $5$ choices, but remarkably, no pair of students had more than $2$ answers in common. What is the maximum number of questions that could have been on the quiz?

Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$ Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$ [i]Proposed by Ma Zhao Yu

Let $ABC$ be triangle with $H$ is the orthocenter and $I$ is incenter. Denote $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ be the points on the rays $AB, AC, BC, CA, CB$, respectively such that $$AA_{1} = AA_{2} = BC, BB_{1} = BB_{2} = CA, CC_{1} = CC_{2} = AB.$$ Suppose that $B_{1}B_{2}$ cuts $C_{1}C_{2}$ at $A'$, $C_{1}C_{2}$ cuts $A_{1}A_{2}$ at $B'$ and $A_{1}A_{2}$ cuts $B_{1}B_{2}$ at $C'$. a) Prove that area of triangle $A'B'C'$ is smaller than or equal to the area of triangle $ABC$. b) Let $J$ be circumcenter of triangle $A'B'C'$. $AJ$ cuts $BC$ at $R$, $BJ$ cuts $CA$ at $S$ and $CJ$ cuts $AB$ at $T$. Suppose that $(AST), (BTR), (CRS)$ intersect at $K$. Prove that if triangle $ABC$ is not isosceles then $HIJK$ is a parallelogram.

a) Let $MNP$ be a triangle such that $\angle MNP> 60^o$. Show that the side $MP$ cannot be the smallest side of the triangle $MNP$. b) In a plane the equilateral triangle $ABC$ is considered. The point $V$ that does not belong to the plane $(ABC)$ is chosen so that $\angle VAB = \angle VBC = \angle VCA$. Show that if $VA = AB$, the tetrahedron $VABC$ is regular. Valentin Vornicu

Let $ABC$ be a triangle and $\ell$ be a line parallel to $BC$ that passes through vertex $A$. Draw two circles congruent to the circle inscribed in triangle $ABC$ and tangent to line $\ell$, $AB$ and $BC$ (see picture). Lines $DE$ and $FG$ intersect at point $P$. Prove that $P$ lies on $BC$ if and only if $P$ is the midpoint of $BC$. (Mykhailo Plotnikov) [img]https://cdn.artofproblemsolving.com/attachments/8/b/2dacf9a6d94a490511a2dc06fbd36f79f25eec.png[/img]

The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. The line through $A$ parallel to $DF$ meets the line through $C$ parallel to $EF$ at $G$. $(a)$ Prove that the quadrilateral $AICG$ is cyclic. $(b)$ Prove that the points $B,I,G$ are collinear.

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

Let x; y; z be real numbers, satisfying the relations $x \ge 20$ $y \ge 40$ $z \ge 1675$ x + y + z = 2015 Find the greatest value of the product P = $x
y z$

$n$ is a given positive integer, such that it’s possible to weigh out the mass of any product weighing $1,2,3,\cdots ,ng$ with a counter balance without sliding poise and $k$ counterweights, which weigh $x_ig(i=1,2,\cdots ,k),$ respectively, where $x_i\in \mathbb{N}^*$ for any $i \in \{ 1,2,\cdots ,k\}$ and $x_1\leq x_2\leq\cdots \leq x_k.$ $(1)$Let $f(n)$ be the least possible number of $k$. Find $f(n)$ in terms of $n.$ $(2)$Find all possible number of $n,$ such that sequence $x_1,x_2,\cdots ,x_{f(n)}$ is uniquely determined.