[b]p1.[/b] Let $x_1 = 0$, $x_2 = 1/2$ and for $n >2$, let $x_n$ be the average of $x_{n-1}$ and $x_{n-2}$. Find a formula for $a_n = x_{n+1} - x_{n}$, $n = 1, 2, 3, \dots$. Justify your answer. [b]p2.[/b] Given a triangle $ABC$. Let $h_a, h_b, h_c$ be the altitudes to its sides $a, b, c,$ respectively. Prove: $\frac{1}{h_a}+\frac{1}{h_b}>\frac{1}{h_c}$ Is it possible to construct a triangle with altitudes $7$, $11$, and $20$? Justify your answer. [b]p3.[/b] Does there exist a polynomial $P(x)$ with integer coefficients such that $P(0) = 1$, $P(2) = 3$ and $P(4) = 9$? Justify your answer. [b]p4.[/b] Prove that if $\cos \theta$ is rational and $n$ is an integer, then $\cos n\theta$ is rational. Let $\alpha=\frac{1}{2010}$. Is $\cos \alpha $ rational ? Justify your answer. [b]p5.[/b] Let function $f(x)$ be defined as $f(x) = x^2 + bx + c$, where $b, c$ are real numbers. (A) Evaluate $f(1) -2f(5) + f(9)$ . (B) Determine all pairs $(b, c)$ such that $|f(x)| \le 8$ for all $x$ in the interval $[1, 9]$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integer $n$ such that the equation $2x^2 + 5xy + 2y^2 = n$ has integer solution for $x$ and $y$.

Let $(G, *)$ a group of $n > 1$ elements, and let $g \in G$ be an element distinct from the identity. Ana and Bob play with the group $G$ on the following way: Starting with Ana and playing alternately, each player selects an element of $G$ that has not been selected before, until each element of $G$ have been selected or a player have selected the elements $a$ and $a * g$ for some $a \in G$. In that case it is said that the player loses and his opponent wins. $a)$ If $n$ is odd, show that, independent of element $g$, one of the two players has a winning strategy and determines which player possesses such a strategy. $b)$ If $n$ is even, show that there exists an element $g \in G$ for which none of the players has a winning strategy. Note: A group $(G, *)$ es a set $G$ together with a binary operation $* : G\times G \to G$ that satisfy the following properties $(i)$ $*$ is asociative: $\forall a, b, c \in G (a * b) * c = a * (b * c)$; $(ii)$ there exists an identity element $e \in G$ such that $\forall a \in G, a *e = e * a = a;$ $(iii)$ there exists inverse elements: $\forall a \in G \exists a^{-1} \in G$ such that $a*a^{-1} = a^{-1} *a = e.$

For which values of $ z$ does the series $ \sum_{n\equal{}1}^{\infty}c_1c_2\cdots c_n z^n$ converge, provided that $ c_k>0$ and $ \sum_{k\equal{}1}^{\infty} \frac{c_k}{k}<\infty$ ?

Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$

Let $ABC$ be an acute, non-isosceles triangle with the circumcircle $(O)$. Denote $D, E$ as the midpoints of $AB,AC$ respectively. Two circles $(ABE)$ and $(ACD)$ intersect at $K$ differs from $A$. Suppose that the ray $AK$ intersects $(O)$ at $L$. The line $LB$ meets $(ABE)$ at the second point $M$ and the line $LC$ meets $(ACD)$ at the second point $N$. a) Prove that $M, K, N$ collinear and $MN$ perpendicular to $OL$. b) Prove that $K$ is the midpoint of $MN$

Find the number of permutations $x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8$ of the integers $-3, -2, -1, 0,1,2,3,4$ that satisfy the chain of inequalities $$x_1x_2\le x_2x_3\le x_3x_4\le x_4x_5\le x_5x_6\le x_6x_7\le x_7x_8.$$

Acute triangle $ABC$ has circumcenter $O.$ The bisector of $ABC$ and the altitude from $C$ to side $AB$ intersect at $X.$ Suppose that there is a circle passing through $B, O, X,$ and $C.$ If $\angle BAC = n^\circ,$ where $n$ is a positive integer, compute the largest possible value of $n.$

Let $a, b$, and $c$ be positive real numbers. Prove that \[\frac{a^4 + 3}{b} + \frac{b^4 + 3}{c} + \frac{c^4 + 3}{a} \ge 12.\] When does equality hold? [i]Proposed by Petar Filipovski[/i]

Alice and Bob play the following game: Initially, there is a $2016\times2016$ "empty" matrix. Taking turns, with Alice playing first, each player chooses a real number and fill it into an empty entry. If the determinant of the last matrix is non-zero, then Alice wins. Otherwise, Bob wins. Who has the winning strategy?

Prove that there exist infinitely many odd positive integers $n$ for which the number $2^n + n$ is composite. (V Senderov)

What number is directly above $142$ in this array of numbers? \[\begin{array}{cccccc} & & & 1 & & \\ & & 2 & 3 & 4 & \\ & 5 & 6 & 7 & 8 & 9 \\ 10 & 11 & 12 & \cdots & & \\ \end{array}\] $\textbf{(A)}\ 99 \qquad \textbf{(B)}\ 119 \qquad \textbf{(C)}\ 120 \qquad \textbf{(D)}\ 121 \qquad \textbf{(E)}\ 122$

Let $a_1=0$, $a_2=1$, and $a_{n+2}=a_{n+1}+a_n$ for all positive integers $n$. Show that there exists an increasing infinite arithmetic progression of integers, which has no number in common in the sequence $\{a_n\}_{n \ge 0}$.

It is known that the merchant’s $n$ clients live in locations laid along the ring road. Of these, $k$ customers have debts to the merchant for $a_1,a_2,...,a_k$ rubles, and the merchant owes the remaining $n-k$ clients, whose debts are $b_1,b_2,...,b_{n-k}$ rubles, moreover, $a_1+a_2+...+a_k=b_1+b_2+...+b_{n-k}$. Prove that a merchant who has no money can pay all his debts and have paid all the customer debts, by starting a customer walk along the road from one of points and not missing any of their customers.

Each segment whose endpoints are the vertices of a given regular $100$-gon is colored red, if the number of vertices between its endpoints is even, and blue otherwise. (For example, all sides of the $100$-gon are red.) A number is placed in every vertex so that the sum of their squares is equal to $1$. On each segment the product of the numbers at its endpoints is written. The sum of the numbers on the blue segments is subtracted from the sum of the numbers on the red segments. What is the greatest possible result? (Ilya Bogdanov)

$2024$ positive real numbers with sum $1$ are arranged on a circle. It is known that any two adjacent numbers differ at least in $2$ times. For each pair of adjacent numbers, the smaller one was subtracted from the larger one, and then all these differences were added together. What is the smallest possible value of this resulting sum? [i]Proposed by Oleksiy Masalitin[/i]

Let $ABCD$ be a convex cyclic quadrilateral with diagonals $AC$ and $BD$. Each of the four vertixes are reflected across the diagonal on which the do not lie. (a) Investigate when the four points thus obtained lie on a straight line and give as simple an equivalent condition as possible to the cyclic quadrilateral $ABCD$ for it. (b) Show that in all other cases the four points thus obtained lie on one circle. (Theresia Eisenkölbl)

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Two points $A$ and $B$ and line $\ell$ are fixed in the plane so that $\ell$ is not perpendicular to $AB$ and does not intersect the segment $AB$. We consider all circles with a centre $O$ not lying on $\ell$, passing through $A$ and $B$ and meeting $\ell$ at some points $C$ and $D$. Prove that all the circumcircles of triangles $OCD$ touch a fixed circle.

We say that a positive integer is an [i]almost square[/i], if it is equal to the product of two consecutive positive integers. Prove that every almost square can be expressed as a quotient of two almost squares. V. Senderov

Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$, \[\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),\]where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$. [i]Calvin Deng.[/i]

Given an $\triangle ABC$ isoceles with base $BC$ we note with $M$ the midpoint of said base. Let $X$ be any point on the shortest arc $AM$ of the circumcircle of $\triangle ABM$ and let $T$ be a point on the inside $\angle BMA$ such that $\angle TMX = 90^o$ and $TX = BX$. Show that $\angle MTB - \angle CTM$ does not depend on $X$.

If $x$ and $y$ are real numbers such that $x^{2013}+y^{2013}>x^{2012}+y^{2012}$, prove that $x^{2014}+y^{2014}>x^{2013}+y^{2013}$

The average of all ten-digit base-ten positive integers $\underline{d_9} \ \underline{d_8} \ldots \underline{d_1} \ \underline{d_0}$ that satisfy the property $|d_i - i| \leq 1$ for all $i \in \{0, 1, \ldots, 9\}$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Compute the remainder when $p + q$ is divided by $10^6.$

Let $ABC$ be an acute-angled triangle, $O$ its circumcenter and $H$ its orthocenter. The orthogonal projection of the vertex $A$ to the line $BC$ lies on the perpendicular bisector of the segment $AC$. Compute $\frac{CH}{BO}$ . (J. Willemson)