The sequence $a_1,a_2,a_3\ldots $, consisting of natural numbers, is defined by the rule: \[a_{n+1}=a_{n}+2t(n)\] for every natural number $n$, where $t(n)$ is the number of the different divisors of $n$ (including $1$ and $n$). Is it possible that two consecutive members of the sequence are squares of natural numbers?

$(MON 2)$ Given reals $x_0, x_1, \alpha, \beta$, find an expression for the solution of the system \[x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad n= 0, 1, 2, \ldots\]

If $ x$, $ y$, $ z$ are positive real numbers, prove that \[ \left(x \plus{} y \plus{} z\right)^2 \left(yz \plus{} zx \plus{} xy\right)^2 \leq 3\left(y^2 \plus{} yz \plus{} z^2\right)\left(z^2 \plus{} zx \plus{} x^2\right)\left(x^2 \plus{} xy \plus{} y^2\right) .\]

* Given an infinite cone. The measure of its unfolding’s angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the number of the curve’s self-intersections.

Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.

Consider the following $50$-term sums: $S=\frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+...+\frac{1}{99\cdot 100}$, $T=\frac{1}{51\cdot 100}+\frac{1}{52\cdot 99}+...+\frac{1}{99\cdot 52}+\frac{1}{100\cdot 51}$. Express $\frac{S}{T}$ as an irreducible fraction.

Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

Given a complete bipartite graph on $n,n$ vertices (call this $K_{n,n}$), we colour all its edges with $2$ colours , red and blue . What is the least value of $n$ such that for any colouring of the edges of the graph , there will exist at least one monochromatic $4$ cycle ?

A student adds up rational fractions incorrectly: \[\frac{a}{b}+\frac{x}{y}=\frac{a+x}{b+y}\quad (\star) \] Despite that, he sometimes obtains correct results. For a given fraction $\frac{a}{b},a,b\in\mathbb{Z},b>0$, find all fractions $\frac{x}{y},x,y\in\mathbb{Z},y>0$ such that the result obtained by $(\star)$ is correct.

Let \( \triangle ABC \) be an acute-angled triangle. Let \( D \) and \( E \) be the feet of the altitudes from \( B \) and \( C \), respectively. Let \( E' \) be the reflection of point \( E \) with respect to line \( BD \), which is assumed to lie on the circumcircle of triangle \( \triangle ABC \). Let \( C' \) be the reflection of point \( C \) with respect to line \( BD \). Prove that triangle \( C'AE \) is isosceles and determine the ratio \( AD : DC \).

Suppose $ABCD$ is a convex quadrilateral with $\angle{ABD}=105^\circ, \angle{ADB}=15^\circ, AC=7,$ and $BC=CD=5.$ Compute the sum of all possible values of $BD.$

Let $c_1,c_2,\ldots,c_n$ be real numbers such that $$c_1^k+c_2^k+\ldots+c_n^k>0\qquad\text{for all }k=1,2,\ldots$$Let us put $$f(x)=\frac1{(1-c_1x)(1-c_2x)\cdots(1-c_nx)}.$$$z\in\mathbb C$ Show that $f^{(k)}(0)>0$ for all $k=1,2,\ldots$.

Construct a right triangle with given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

A pawn is a chess piece which attacks the two squares diagonally in front if it. What is the maximum number of pawns which can be placed on an $n \times n$ chessboard such that no two pawns attack each other?

Consider two integers $n \ge m \ge 4$ and $A = \{a_1, a_2, ..., a_m\}$ a subset of the set $\{1, 2, ..., n\}$ such that: [i]for all $a, b \in A, a \ne b$, if $a + b \le n$, then $a + b \in A$.[/i] Prove that $\frac{a_1 + a_2 + ... + a_m}{m} \ge \frac{n + 1}{2}$ .

[b]p1.[/b] A shape made by joining four identical regular hexagons side-to-side is called a hexo. Two hexos are considered the same if one can be rotated / reflected to match the other. Find the number of different hexos. [b]p2.[/b] The sequence $1, 2, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6,... $ consists of numbers written in increasing order, where every even number $2n$ is written once, and every odd number $2n + 1$ is written $2n + 1$ times. What is the $2019^{th}$ term of this sequence? [b]p3.[/b] On planet EMCCarth, months can only have lengths of $35$, $36$, or $42$ days, and there is at least one month of each length. Victor knows that an EMCCarth year has $n$ days, but realizes that he cannot figure out how many months there are in an EMCCarth year. What is the least possible value of $n$? [b]p4.[/b] In triangle $ABC$, $AB = 5$ and $AC = 9$. If a circle centered at $A$ passing through $B$ intersects $BC$ again at $D$ and $CD = 7$, what is $BC$? [b]p5.[/b] How many nonempty subsets $S$ of the set $\{1, 2, 3,..., 11, 12\}$ are there such that the greatest common factor of all elements in $S$ is greater than $1$? [b]p6.[/b] Jasmine rolls a fair $6$-sided die, with faces labeled from $1$ to $6$, and a fair $20$-sided die, with faces labeled from $1$ to $20$. What is the probability that the product of these two rolls, added to the sum of these two rolls, is a multiple of $3$? [b]p7.[/b] Let $\{a_n\}$ be a sequence such that $a_n$ is either $2a_{n-1}$ or $a_{n-1} - 1$. Given that $a_1 = 1$ and $a_{12} = 120$, how many possible sequences $a_1$, $a_2$, $...$, $a_{12}$ are there? [b]p8.[/b] A tetrahedron has two opposite edges of length $2$ and the remaining edges have length $10$. What is the volume of this tetrahedron? [b]p9.[/b] In the garden of EMCCden, there is a tree planted at every lattice point $-10 \le x, y \le 10$ except the origin. We say that a tree is visible to an observer if the line between the tree and the observer does not intersect any other tree (assume that all trees have negligible thickness). What fraction of all the trees in the garden of EMCCden are visible to an observer standing at the origin? [b]p10.[/b] Point $P$ lies inside regular pentagon $\zeta$, which lies entirely within regular hexagon $\eta$. A point $Q$ on the boundary of pentagon $\zeta$ is called projective if there exists a point $R$ on the boundary of hexagon $\eta$ such that $P$, $Q$, $R$ are collinear and $2019 \cdot \overline{PQ} = \overline{QR}$. Given that no two sides of $\zeta$ and $\eta$ are parallel, what is the maximum possible number of projective points on $\zeta$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

4. Let $a \geq b \geq c \geq d>0$. Show that \[ \frac{b^3}{a} + \frac{c^3}{b} + \frac{d^3}{c} + \frac{a^3}{d} + 3 \left( ab+bc+cd+da \right) \geq 4 {\left( a^2 + b^2 + c^2 +d^2 \right)}. \] Other problems (in Korean) are also available at https://www.facebook.com/KoreanMathOlympiad

The complex numbers $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ are the four distinct roots of the equation $x^4+2x^3+2=0$. Determine the unordered set \[\{\alpha_1\alpha_2+\alpha_3\alpha_4,\alpha_1\alpha_3+\alpha_2\alpha_4,\alpha_1\alpha_4+\alpha_2\alpha_3\}.\]

A triangle $ABC$ is to be constructed given a side $a$ (oppisite angle $A$). angle $B$, and $h_c$, the altitude from $C$. If $N$ is the number of noncongruent solutions, then $N$ $\textbf{(A)}\ \text{is} \; 1\qquad \textbf{(B)}\ \text{is} \; 2\qquad \textbf{(C)}\ \text{must be zero}\qquad \textbf{(D)}\ \text{must be infinite}\qquad \textbf{(E)}\ \text{must be zero or infinite}$

Given is an equilateral triangle $ABC$ with circumcenter $O$. Let $D$ be a point on to minor arc $BC$ of its circumcircle such that $DB>DC$. The perpendicular bisector of $OD$ meets the circumcircle at $E, F$, with $E$ lying on the minor arc $BC$. The lines $BE$ and $CF$ meet at $P$. Prove that $PD \perp BC$.

Find all positive integers such that they have $6$ divisors (without $1$ and the number itself) and the sum of the divisors is $14133$.

Let $ S$ be the set of points $ (a,b)$ with $ 0\le a,b\le1$ such that the equation \[x^4 \plus{} ax^3 \minus{} bx^2 \plus{} ax \plus{} 1 \equal{} 0\] has at least one real root. Determine the area of the graph of $ S$.

Prove that for every real number $x$ and positive integer $n$ $$|\cos x|+|\cos2x|+|\cos2^2x|+\ldots+|\cos2^nx|\ge\frac n{2\sqrt2}.$$

Let $P(x)$ be a quadratic polynomial with real coefficients. Suppose that $P(1) = 20, P(-1) = 22,$ and $P(P(0)) = 400.$ Compute the largest possible value of $P(10).$