Find $\alpha$ such that $$\lim_{x\to0^+}x^\alpha I(x)=a\enspace\text{given}\enspace I(x)=\int^\infty_0\sqrt{1+t}\cdot e^{-xt}dt$$ where $a$ is a nonzero real number.

Lysithea and Felix each have a take-out box, and they want to select among $42$ different types of sweets to put in their boxes. They each select an even number of sweets (possibly $0$) to put in their box. In each box, there is at most one sweet of any type, although the boxes may have sweets of the same type in common. The total number of sweets they take out is $42.$ Let $N$ be the number of ways can they select sweets to take out. Compute the remainder when $N$ is divided by $42^2 - 1.$

Let $f(x) = x^3 + ax^2 + bx + c$ have solutions that are distinct negative integers. If $a+b+c = 2014$, find $c$.

Let $M$ be the circumcenter of an acute-angled triangle $ABC$. The circumcircle of triangle $BMA$ intersects $BC$ at $P$ and $AC$ at $Q$. Show that $CM \perp PQ$.

Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

Find all polynomials $P(x)$ with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.

$ABC$ is a triangle with incenter $I$. The foot of the perpendicular from $I$ to $BC$ is $D$, and the foot of the perpendicular from $I$ to $AD$ is $P$. Prove that $\angle BPD = \angle DPC$. [i]Alex Zhu.[/i]

The point $P_1, P_2,\cdots ,P_{2018} $ is placed inside or on the boundary of a given regular pentagon. Find all placement methods are made so that $$S=\sum_{1\leq i<j\leq 2018}|P_iP_j| ^2$$takes the maximum value.

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

Two cars are driving directly towards each other such that one is twice as fast as the other. The distance between their starting points is $4$ miles. When the two cars meet, how many miles is the faster car from its starting point?

$(GDR 5)$ Given a ring $G$ in the plane bounded by two concentric circles with radii $R$ and $\frac{R}{2}$, prove that we can cover this region with $8$ disks of radius $\frac{2R}{5}$. (A region is covered if each of its points is inside or on the border of some disk.)

Let $n$ points, where there are not $3$ of them on a line, and consider the segments that are formed by connecting any $2$ of the points. There are enough colors available to paint the points and the segments, coloring them with the following two rules: a) All the segments that reach the same point are painted of different colors. b) Each point is painted a different color to all the segments that reach it. Find the minimum number of colors needed to make such a coloring.

The diagonals of a quadrilateral $ ABCD$ are perpendicular: $ AC \perp BD.$ Four squares, $ ABEF,BCGH,CDIJ,DAKL,$ are erected externally on its sides. The intersection points of the pairs of straight lines $ CL, DF, AH, BJ$ are denoted by $ P_1,Q_1,R_1, S_1,$ respectively (left figure), and the intersection points of the pairs of straight lines $ AI, BK, CE DG$ are denoted by $ P_2,Q_2,R_2, S_2,$ respectively (right figure). Prove that $ P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2$ where $ P_1,Q_1,R_1, S_1$ and $ P_2,Q_2,R_2, S_2$ are the two quadrilaterals. [i]Alternative formulation:[/i] Outside a convex quadrilateral $ ABCD$ with perpendicular diagonals, four squares $ AEFB, BGHC, CIJD, DKLA,$ are constructed (vertices are given in counterclockwise order). Prove that the quadrilaterals $ Q_1$ and $ Q_2$ formed by the lines $ AG, BI, CK, DE$ and $ AJ, BL, CF, DH,$ respectively, are congruent.

What's the greatest integer $n$ for which the system $k < x^k < k + 1$ for $k = 1,2,..., n$ has a solution?

A rhombus $ABCD$ is given with $\angle BAD = 60^o$ . Point $P$ lies inside the rhombus such that $BP = 1$, $DP = 2$, $CP = 3$. Determine the length of the segment $AP$.

The circumcenter of a triangle lies on its incircle. Prove that the ratio of its greatest and smallest sides is less than two. [i]Proposed by B.Frenkin[/i]

Let $p>2$ be a prime. Prove that $\sum_{n=0}^p\binom pn\binom{p+n}n\equiv2p+1\pmod{p^2}$.

If $\frac{3}{5}=\frac{M}{45}=\frac{60}{N}$, what is $M+N$? $\textbf{(A)}\ 27\qquad \textbf{(B)}\ 29 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 105\qquad \textbf{(E)}\ 127$

Evaluate $\lim_{n\to \infty} cos\frac{x}{2}cos\frac{x}{2^2} cos\frac{x}{2^3}...cos\frac{x}{2^n}$

Let $K$ be a finite field with $q$ elements and let $n \ge q$ be an integer. Find the probability that by choosing an $n$-th degree polynomial with coefficients in $K,$ it doesn't have any root in $K.$

Positive integers $k$ and $n$ are given such that $3 \le k \le n$.Prove that among any $n$ pairwise distinct real numbers one can choose either $k$ numbers with positive sum, or $k-1$ numbers with negative sum. [i]Proposed by Mykhailo Shtandenko[/i]

Let $f:[0,1)\to (0,1)$ a continous onto (surjective) function. a) Prove that, for all $a\in(0,1)$, the function $f_a:(a,1)\to (0,1)$, given by $f_a(x) = f(x)$, for all $x\in(a,1)$ is onto; b) Give an example of such a function.

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $P$ and $Q$ be points in the half plane defined by $BC$ containing $A$, such that $BP$ and $CQ$ are tangents to $\Gamma$ and $PB = BC = CQ$. Let $K$ and $L$ be points on the external bisector of the angle $\angle CAB$ , such that $BK = BA, CL = CA$. Let $M$ be the intersection point of the lines $PK$ and $QL$. Prove that $MK=ML$.

Suppose that $a$ and $b$ are positive real numbers such that $3\log_{101}\left(\frac{1,030,301-a-b}{3ab}\right) = 3 - 2 \log_{101}(ab)$. Find $101 - \sqrt[3]{a}- \sqrt[3]{b}$.

Given acute angle triangle $ ABC$. Let $ CD$be the altitude , $ H$ be the orthocenter and $ O$ be the circumcenter of $ \triangle ABC$ The line through point $ D$ and perpendicular with $ OD$ , is intersect $ BC$ at $ E$. Prove that $ \angle DHE \equal{} \angle ABC$.