Find all second degree polynomial $d(x)=x^{2}+ax+b$ with integer coefficients, so that there exists an integer coefficient polynomial $p(x)$ and a non-zero integer coefficient polynomial $q(x)$ that satisfy: \[\left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1, \quad \forall x \in \mathbb R.\]

$ABC$ is a triangle and $\angle A=90^{\circ}$. Let $D$ be the meet point of the interior bisector of $\angle A$ and $BC$. And let $I_a$ be the $A-$excenter of $\triangle ABC$. Prove that: \[\frac{AD}{DI_a}\leq\sqrt{2}-1.\]

Let $ A \equal{} \{(a_1,\dots,a_8)|a_i\in\mathbb{N}$ , $ 1\leq a_i\leq i \plus{} 1$ for each $ i \equal{} 1,2\dots,8\}$.A subset $ X\subset A$ is called sparse if for each two distinct elements $ (a_1,\dots,a_8)$,$ (b_1,\dots,b_8)\in X$,there exist at least three indices $ i$,such that $ a_i\neq b_i$. Find the maximal possible number of elements in a sparse subset of set $ A$.

A circle is tangent internally by $6$ circles so that each one is tangent internally to two adjacent ones and the radii of opposite circles are pairwise equal. Prove that the sum of the radii of the inner circles is equal to the diameter of the given circle.

Do there exist infinitely many triplets of positive integers $(a, b, c)$ such that the following two conditions hold: 1. $\gcd(a, b, c) = 1$; 2. $a+b+c, a^2+b^2+c^2$ and $abc$ are all perfect squares? [i](Proposed by Ivan Chan Guan Yu)[/i]

For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$

Let $\{a_n\}$ be a sequence of nondecreasing positive integers such that $\textstyle\frac{r}{a_r} = k+1$ for some positive integers $k$ and $r$. Prove that there exists a positive integer $s$ such that $\textstyle\frac{s}{a_s} = k$.

Three angles of a non-convex, non-self-intersecting quadrilateral are equal to $45$ degrees (i.e. the last equals $225$ degrees). Prove that the midpoints of its sides are vertices of a square. (V Proizvolov)

A student wishes to determine the thickness of a rectangular piece of aluminum foil but cannot measure it directly. She can measure its density $(d)$, length $(l)$, mass $(m)$ and width $(w)$. Which relationship will give the thickness? $\textbf{(A)}\hspace{.05in}\frac{m}{d \cdot l \cdot w} \qquad\textbf{(B)}\hspace{.05in}\frac{m \cdot l \cdot w}{d} \qquad\textbf{(C)}\hspace{.05in}\frac{d \cdot l \cdot w}{m} \qquad\textbf{(D)}\hspace{.05in}\frac{d \cdot m }{l \cdot w} \qquad$

An apple orchard’s layout is a rectangular grid of unit squares. Some pairs of adjacent squares have a thick wall of grape vines between them. The orchard wants to post some robot sentries to guard its prized apple trees. Each sentry occupies a single square of the layout, and from there it can guard both its square and any square in the same row and column that it can see, where only walls and the edges of the orchard block its sight. A sample layout (not the layout of the actual orchard, which is not given) is shown below. Although a square may be guarded by multiple sentries, the sentries have not been programmed to avoid attacking other sentries. Thus, no sentry may be placed on a square guarded by another sentry. The orchard’s expert has found a way to guard all the squares of the orchard by placing 1000 sentries. However, the contractor shipped 2020 sentries. Show that it is impossible for the orchard to place all 2020 of the sentries without two of them attacking each other.

Let $ABC$ be a triangle with incenter $I$. Points $K$ and $L$ are chosen on segment $BC$ such that the incircles of $\triangle ABK$ and $\triangle ABL$ are tangent at $P$, and the incircles of $\triangle ACK$ and $\triangle ACL$ are tangent at $Q$. Prove that $IP=IQ$. [i]Ankan Bhattacharya[/i]

Let $\mathcal P$ be a finite set of squares on an infinite chessboard. Kelvin the Frog notes that $\mathcal P$ may be tiled with only $1 \times 2$ dominoes, while Alex the Kat notes that $\mathcal P$ may be tiled with only $2 \times 1$ dominoes. The dominoes cannot be rotated in each tiling. Prove that the area of $\mathcal P$ is a multiple of 4.

Farmer John has a $47 \times 53$ rectangular square grid. He labels the first row $1, 2, \cdots, 47$, the second row $48, 49, \cdots, 94$, and so on. He plants corn on any square of the form $47x + 53y$, for non-negative integers $x, y$. Given that the unplanted squares form a contiguous region $R$, find the perimeter of $R$.

Prove that there exist infinitely many non prime positive integers $n$ such that $7^{n-1} - 3^{n-1}$ is divisible by $n$. Lê Anh Vinh

Given a triangle $ABC$ with the unit area. The first player chooses a point $X$ on the side $[AB]$, than the second -- $Y$ on $[BC]$ side, and, finally, the first chooses a point $Z$ on $[AC]$ side. The first tries to obtain the greatest possible area of the $XYZ$ triangle, the second -- the smallest. What area can obtain the first for sure and how?

In rhombus ABCD, $\angle A = 60^\circ$. Rhombus $BEFG$ is constructed, where $E$ and $G$ are the midpoints of $BC$ and $AB$, respectively. Rhombus $BHIJ$ is constructed, where $H$ and $J$ are the midpoints of $BE$ and $BG$, respectively. This process is repeated forever. If the area of $ABCD$ and the sum of the areas of all of the rhombi are both integers, compute the smallest possible value of $AB$.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x + y) + y \le f(f(f(x)))\] holds for all $x, y \in \mathbb{R}$.

In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$. [i]Ray Li.[/i]

Find, with proof, a positive integer $n$ such that \[\frac{(n + 1)(n + 2)
\cdots
(n + 500)}{500!}\] is an integer with no prime factors less than $500$.

Prove the following two inequalities: 1) If $n > 49$, then exist positive integers $a, b > 1$ such that $a+b=n$ and $$\frac{\phi (a)}{a}+\frac{\phi (b)}{b}<1$$ 2) If $n > 4$, then exist integer integers $a, b > 1$ such that $a+b=n$ and $$\frac{\phi (a)}{a}+\frac{\phi (b)}{b}>1$$

Let $ABC$ is acute angled triangle and $K,L$ is points on $AC,BC$ respectively such that $\angle{AKB}=\angle{ALB}$. $P$ is intersection of $AL$ and $BK$ and $Q$ is the midpoint of segment $KL$. Let $T,S$ are the intersection $AL,BK$ with $(ABC)$ respectively. Prove that $TK,SL,PQ$ are concurrent.

Let $ S$ be a semigroup without proper two-sided ideals and suppose that for every $ a,b \in S$ at least one of the products $ ab$ and $ ba$ is equal to one of the elements $ a,b$. Prove that either $ ab\equal{}a$ for all $ a,b \in S$ or $ ab\equal{}b$ for all $ a,b \in S$. [i]L. Megyesi[/i]

For a real number $x$, the minimum value of the expression $$\frac{2x^2 + x - 3}{x^2 - 2x + 3}$$ can be written in the form $\frac{a-\sqrt{b}}{c}$, where $a, b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$

Find the maximal possible value of $n$ such that there exist points $P_1,P_2,P_3,\ldots,P_n$ in the plane and real numbers $r_1,r_2,\ldots,r_n$ such that the distance between any two different points $P_i$ and $P_j$ is $r_i+r_j$.

In convex hexagon $ABCDEF$, $\angle A \cong \angle B$, $\angle C \cong \angle D$, and $\angle E \cong \angle F$. Prove that the perpendicular bisectors of $\overline{AB}$, $\overline{CD}$, and $\overline{EF}$ pass through a common point. [i]Proposed by Lewis Chen[/i]