Let $n>2$ be a positive integer and let $p$ be a prime. Suppose that the nonzero integers are colored in $n$ colors. Let $a_1,a_2,\ldots,a_{n}$ be integers such that for all $1\le i\le n$, $p^i\nmid a_i$ and $p^{i-1}\mid a_i$. In terms of $n$, $p$, and $\{a_i\}_{i=1}^{n}$, determine if there must exist integers $x_1,x_2,\ldots,x_{n}$ of the same color such that $a_1x_1+a_2x_2+\cdots+a_{n}x_{n}=0$. [i]Ravi Jagadeesan.[/i]

Let $\vartriangle ABC$ be a triangle with circumcenter $O$. Let $ P$ and $Q$ be internal points on the sides $AB$ and $AC$ respectively such that $\angle POB = \angle ABC$ and $\angle QOC = \angle ACB$. Show that the reflection of line $BC$ over line $PQ$ is tangent to the circumcircle of triangle $\vartriangle APQ$.

Let $x>1$ ,$n$ be positive integer. Prove that$$\sum_{k=1}^{n}\frac{\{kx \}}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1}$$ Where $[kx ]$ be the integer part of $kx$ ,$\{kx \}$ be the decimal part of $kx$.

Let a positive integer $n$ be given. Determine, in terms of $n$, the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$.

If a parabola is given in the plane, find a geometric construction (ruler and compass) for the focus.

Let $p,q$ and $s{}$ be prime numbers such that $2^sq =p^y-1$ where $y > 1.$ Find all possible values of $p.$

Altitudes $BB_1$ and $CC_1$ of acute triangle $ABC$ intersect at $H$, and $\angle A = 60^{o}$, $AB < AC$. The median $AM$ intersects the circumcircle of $ABC$ at point $K$; $L$ is the midpoint of the arc $BC$ of the circumcircle that does not contain point $A$; lines $B_1C_1$ and $BC$ intersect at point $E$. Prove that $\angle EHL = \angle ABK$.

In some foreign country, there is a secret object, guarded by seven guards. Each guard has a guarding shift of 7 consecutive hours every day, in fixed hours. There is always at least one guard guarding the secret object at any given time. Prove that one of the guards can be fired, and there will still be at least one guard guarding at any given time (without changing the schedule of the other guards).

$O$ is a circle with radius $1$. $A$ and $B$ are fixed points on the circle such that $AB =\sqrt2$. Let C be any point on the circle, and let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. As $C$ travels around circle $O$, find the area of the locus of points on $MN$.

In convex quadrilateral $ABCD$, $\angle ADC = 90^\circ + \angle BAC$. Given that $AB = BC = 17$, and $CD = 16$, what is the maximum possible area of the quadrilateral? [i]Proposed by Thomas Lam[/i]

Consider an equilateral triangular grid $G$ with $20$ points on a side, where each row consists of points spaced $1$ unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and $20$ points in the last row, for a total of $210$ points. Let $S$ be a closed non-self-intersecting polygon which has $210$ vertices, using each point in $G$ exactly once. Find the sum of all possible values of the area of $S$.

The median $CM$ of the triangle $ABC$ is equal to the bisector $BL$, also $\angle BAC=2\angle ACM$. prove that the triangle is right.

Calculates the sum of all numbers that can be formed using each of the odd digits once, that is, the numbers $13579$, $13597$, ..., $97531$.

We are given an acute triangle $ABC$ and points $D$ on $BC$ and $E$ on $AC$ such that $AD$ is perpendicular to $BC$ and $BE$ is perpendicular to $AC$. The intersection of $AD$ and $BE$ is called $H$. A line through $H$ intersects line segment $BC$ in $P$, and intersects line segment $AC$ in $Q$. Furthermore, $K$ is a point on $BE$ such that $PK$ is perpendicular to $BE$, and $L$ is a point on $AD$ such that $QL$ is perpendicular to $AD$. Prove that $DK$ and $EL$ are parallel. [asy] unitsize(1 cm); pair A, B, C, D, E, H, K, L, P, Q; A = (0,0); B = (6,0); C = (3.5,4); D = (A + reflect(B,C)*(A))/2; E = (B + reflect(A,C)*(B))/2; H = extension(A, D, B, E); P = extension(H, H + dir(-10), B, C); Q = extension(H, H + dir(-10), A, C); K = (P + reflect(B,E)*(P))/2; L = (Q + reflect(A,D)*(Q))/2; draw(A--B--C--cycle); draw(A--D); draw(B--E); draw(K--P--Q--L); draw(rightanglemark(B,D,A,5)); draw(rightanglemark(B,E,A,5)); draw(rightanglemark(P,K,B,5)); draw(rightanglemark(A,L,Q,5)); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, N); dot("$D$", D, NE); dot("$E$", E, NW); dot("$H$", H, N); dot("$K$", K, SW); dot("$L$", L, SE); dot("$P$", P, NE); dot("$Q$", Q, NW); [/asy]

Prove that for all $n\in\mathbb{Z}^+$, we have \[ \sum\limits_{p=1}^n\sum\limits_{q=1}^p\left\lfloor -\frac{1+\sqrt{8q+(2p-1)^2}}{2}\right\rfloor =-\frac{n(n+1)(n+2)}{3} \]

We have 2019 boxes. Initially, they are all empty. At one operation, we can add exactly 100 stones to some 100 boxes and exactly one stone in each of several other (perhaps none) boxes. What is the smallest possible number of moves after which all boxes will have the same (positive) number of stones. [i]Proposed by P. Kozhevnikov[/i]

Every positive integer greater than $1000$ is colored in red or blue, such that the product of any two distinct red numbers is blue. Is it possible to happen that no two blue numbers have difference $1$?

Let $ ABCD$ be a cyclic quadrilateral and let $ E$ and $ F$ be the feet of perpendiculars from the intersection of diagonals $ AC$ and $ BD$ to $ AB$ and $ CD$, respectively. Prove that $ EF$ is perpendicular to the line through the midpoints of $ AD$ and $ BC$.

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Prove that there is no positive integer $n$ such that, for $k=1, 2, \cdots, 9,$ the leftmost digit of $(n+k)!$ equals $k$.

a) Let $AB CD$ be a regular tetrahedron. On the sides $AB$, $AC$ and $AD$, the points $M$, $N$ and $P$, are considered. Determine the volume of the tetrahedron $AMNP$ in terms of $x, y, z$, where $x=AM$, $y=AN$, $z=AP$. b) Show that for any real numbers $x, y, z, t, u, v \in (0, 1)$ : $$xyz + uv(1- x) + (1- y)(1- v)t + (1- z)(1- w)(1- t) < 1.$$

Compute the number of ways to pick two rectangles in a $5 \times 5$ grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are chosen does not matter.

One million [i]bucks [/i] (i.e. one million male deer) are in different cells of a $1000 \times 1000$ grid. The left and right edges of the grid are then glued together, and the top and bottom edges of the grid are glued together, so that the grid forms a doughnut-shaped torus. Furthermore, some of the bucks are [i]honest bucks[/i], who always tell the truth, and the remaining bucks are [i]dishonest bucks[/i], who never tell the truth. Each of the million [i]bucks [/i] claims that “at most one of my neighboring bucks is an [i]honest buck[/i].” A pair of [i]neighboring bucks[/i] is said to be [i]buckaroo[/i] if exactly one of them is an [i]honest buck[/i] . What is the minimum possible number of [i]buckaroo [/i] pairs in the grid? Note: Two [i]bucks [/i] are considered to be [i]neighboring [/i] if their cells $(x_1, y_1)$ and $(x_2, y_2)$ satisfy either: $x_1 = x_2$ and $y_1 - y_2 \equiv \pm1$ (mod $1000$), or $x_1 - x_2 \equiv \pm 1$ (mod $1000$) and $y_1 = y_2$.

let $a\le b\le c \le d$ show that: $$ab^3+bc^3+cd^3+da^3\ge a^2b^2+b^2c^2+c^2d^2+d^2a^2$$

Circles $\Omega$ and $\omega$ are internally tangent at point $C$. Chord $AB$ of $\Omega$ is tangent to $\omega$ at $E$, where $E$ is the midpoint of $AB$. Another circle, $\omega_1$ is tangent to $\Omega, \omega,$ and $AB$ at $D,Z,$ and $F$ respectively. Rays $CD$ and $AB$ meet at $P$. If $M$ is the midpoint of major arc $AB$, show that $\tan \angle ZEP = \tfrac{PE}{CM}$. [i]Ray Li.[/i]