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$.

A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k,\thinspace 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules: $\bullet$ Any cube may be the bottom cube in the tower. $\bullet$ The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$ Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by 1000?

Add one pair of brackets to the expression \[1+2\times 3+4\times 5+6\] so that the resulting expression has a valid mathematical value, e.g., $1+2\times (3 + 4\times 5)+6=53$. What is the largest possible value that one can make? [i]Proposed by Nathan Xiong[/i]

Let us call a natural number $unlucky$ if it cannot be expressed as $\frac{x^2-1}{y^2-1} $ with natural numbers $x,y >1$. Is the number of $unlucky$ numbers finite or infinite?

The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions: [b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$ [b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\] Prove that $ c \leq \frac{1}{4n}.$

Suppose $A_1,A_2,\cdots ,A_n \subseteq \left \{ 1,2,\cdots ,2018 \right \}$ and $\left | A_i \right |=2, i=1,2,\cdots ,n$, satisfying that $$A_i + A_j, \; 1 \le i \le j \le n ,$$ are distinct from each other. $A + B = \left \{ a+b|a\in A,\,b\in B \right \}$. Determine the maximal value of $n$.

Let $I$ be an incenter of $\triangle ABC$. Denote $D, \ S \neq A$ intersections of $AI$ with $BC, \ O(ABC)$ respectively. Let $K, \ L$ be incenters of $\triangle DSB, \ \triangle DCS$. Let $P$ be a reflection of $I$ with the respect to $KL$. Prove that $BP \perp CP$.

Let H be a $G_{\delta}$ subset of $\mathbb R$ whose closure has a positive Lebesgue measure. Prove that the set $H + H + H + H = \{ x + y + z + u : x , y , z , u \in H \}$ contains an interval.

(a) Is there a polynomial $P(x)$ with real coefficients such that $P\left(\frac1k\right)=\frac{k+2}k$ for all positive integers $k$? (b) Is there a polynomial $P(x)$ with real coefficients such that $P\left(\frac1k\right)=\frac1{2k+1}$ for all positive integers $k$?

Let $O,H$ be the circumcenter and the orthocenter of triangle $ABC$. Let $F$ be the foot of the perpendicular from C onto AB, and $M$ the midpoint of $CH$. Let N be the foot of the perpendicular from C onto the parallel through H at $OM$. Let $D$ be on $AB$ such that $CA=CD$. Let $BN$ intersect $CD$ at $P$. Let $PH$ intersect $CA$ at $Q$. Prove that $QF\perp OF$.