Given seven points on a plane, with no three points collinear. Prove that it is always possible to divide these points into the vertices of a triangle and a convex quadrilateral, with no shared parts between the two shapes.

Arthur is driving to David’s house intending to arrive at a certain time. If he drives at 60 km/h, he will arrive 5 minutes late. If he drives at 90 km/h, he will arrive 5 minutes early. If he drives at n  km/h, he will arrive exactly on time. What is the value of n?

Find the smallest $a$ for which the equation $x^2-ax +21 = 0$ has a root that is a natural number.

Let $a, b, c$, and $d$ be real numbers such that $a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018$. Evaluate $3b + 8c + 24d + 37a$.

There are $30$ children, $a_1,a_2,...,a_{30}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.

On the sides $AB,BC$ and $CA$ of the triangle $ABC$ consider the points $Z,X$ and $Y$ respectively such that \[AZ-AY=BX-BZ=CY-CX.\]Let $P,M$ and $N$ be the circumcenters of the triangles $AYZ, BZX$ and $CXY$ respectively. Prove that the incenters of the triangle $ABC$ coincides with that of the triangle $MNP$.

Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point, where his first roll does not gain him a point. Find the expected number of points that Albert receives. [i]Proposed by Nathan Ramesh

Let $n$ be a positive integer and $P(x)$ be a polynomial with integer coefficients such that $P(1)=1,P(2)=2,\ldots,P(n)=n$. Prove that $P(0)$ is divisible by $2 \cdot 3 \cdot \ldots \cdot n$. [i]A. Voidelevich[/i]

For any positive integer $n$, $A_n$ and $B_n$ are intersection of parabola $y=(n^2+n)x^2-(2n+1)x+1$ and $x$-axis. Then, the value of $|A_1B_1|+|A_2B_2|+\cdots+|A_{1992}B_{1992}|$ is $\text{(A)}\frac{1991}{1992}\qquad\text{(B)}\frac{1992}{1993}\qquad\text{(C)}\frac{1991}{1993}\qquad\text{(D)}\frac{1993}{1992}$

Pablo copied from the blackboard the problem: [list]Consider all the sequences of $2004$ real numbers $(x_0,x_1,x_2,\dots, x_{2003})$ such that: $x_0=1, 0\le x_1\le 2x_0,0\le x_2\le 2x_1\ldots ,0\le x_{2003}\le 2x_{2002}$. From all these sequences, determine the sequence which minimizes $S=\cdots$[/list] As Pablo was copying the expression, it was erased from the board. The only thing that he could remember was that $S$ was of the form $S=\pm x_1\pm x_2\pm\cdots\pm x_{2002}+x_{2003}$. Show that, even when Pablo does not have the complete statement, he can determine the solution of the problem.

Set $S$ contains exactly $36$ elements in the form of $2^m \cdot 5^n$ for integers $ 0 \le m,n \le 5$. Two distinct elements of $S$ are randomly chosen. Given that the probability that their product is divisible by $10^7$ is $a/b$, where $a$ and $b$ are relatively prime positive integers, find $a +b$. [i]Proposed by Ada Tsui[/i]

Let $ABC$ be a triangle with $\angle A<{{90}^{o}} $. Outside of a triangle we consider isosceles triangles $ABE$ and $ACZ$ with bases $AB$ and $AC$, respectively. If the midpoint $D$ of the side $BC$ is such that $DE \perp DZ$ and $EZ = 2 \cdot ED$, prove that $\angle AEB = 2 \cdot \angle AZC$ .

For $ x\geq 0,$ define a function $ f(x)\equal{}\sin \left(\frac{n\pi}{4}\right)\sin x\ (n\pi \leq x<(n\plus{}1)\pi )\ (n\equal{}0,\ 1,\ 2,\ \cdots)$. Evaluate $ \int_0^{100\pi } f(x)\ dx.$

Place parentheses in the expression $$2:2 -3:3 - 4: 4-5:5$$ so that the result is a number greater than $39$.

Let $a_1,a_2,...a_n$ be positive numbers. And let $s=a_1+a_2+...+a_n$,and $p=a_1*a_2*...*a_n$.Prove that $2^{n}*\sqrt{p} \leq 1+\frac{s}{1!}+\frac{s^2}{2!}+...+\frac{s^n}{n!}$

Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$

We know that $R^3 = \{(x_1, x_2, x_3) | x_i \in R, i = 1, 2, 3\}$ is a vector space regarding the laws of composition $(x_1, x_2, x_3) + (y_1, y_2, y_3) = (x_1 + y_1, x_2 + y_2, x_3 + y_3)$, $\lambda (x_1, x_2, x_3) = (\lambda x_1, \lambda x_2, \lambda x_3)$, $\lambda \in R$. We consider the following subset of $R^3$ : $L =\{(x_1, x2, x_3) \in R^3 | x_1 + x_2 + x_3 = 0\}$. a) Prove that $L$ is a vector subspace of $R^3$ . b) In $R^3$ the following relation is defined $\overline{x} R \overline{y} \Leftrightarrow \overline{x} -\overline{y} \in L, \overline{x} , \overline{y} \in R^3$. Prove that it is an equivalence relation. c) Find two vectors of $R^3$ that belong to the same class as the vector $(-1, 3, 2)$.

At a turnament between $n$ persons, everyone playes exactly one time against everyone else, and at one game there is everytime a winner and a looser. Prove that one can arrange the participants in a chain$P_1 \to P_2 \to ... \to P_n$ such that the $i$-th person has won against the $(i+1)$-th person.

Given that $\sqrt{x+2y}-\sqrt{x-2y}=2,$ compute the minimum value of $x+y.$ [i]Proposed by Alex Li[/i]

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard? [hide=Solution] With probability $1$ the number of corners covered is $0$, $1$, or $2$ for example by the diameter of a square being $\sqrt{2}$ so it suffices to compute the probability that the square covers $2$ corners. This is due to the fact that density implies the mean number of captured corners is $1$. For the lattice with offset angle $\theta \in \left[0,\frac{\pi}{2}\right]$ consider placing a lattice uniformly randomly on to it and in particular say without loss of generality consider the square which covers the horizontal lattice midpoint $\left(\frac{1}{2},0 \right)$. The locus of such midpoint locations so that the square captures the $2$ points $(0,0),(1,0)$, is a rectangle. As capturing horizontally adjacent points does not occur when capturing vertically adjacent points one computes twice that probability as $\frac{4}{\pi} \int_0^{\frac{\pi}{2}} (1-\sin(\theta))(1-\cos(\theta)) d\theta=\boxed{\frac{2(\pi-3)}{\pi}}$ \\ [asy] draw((0,0)--(80,40)); draw((0,0)--(-40,80)); draw((80,40)--(40,120)); draw((-40,80)--(40,120)); draw((80,40)--(-20,40)); draw((-40,80)--(60,80)); draw((32*sqrt(5),16*sqrt(5))--(-8*sqrt(5),16*sqrt(5))); draw((40+8*sqrt(5),120-16*sqrt(5))--(40-32*sqrt(5),120-16*sqrt(5))); draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)+2*(40-16*sqrt(5)),16*sqrt(5)+(40-16*sqrt(5)))); draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)-(80-16*sqrt(5))/2,16*sqrt(5)+(80-16*sqrt(5)))); draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)+(120-16*sqrt(5)-40)/2,120-16*sqrt(5)-(120-16*sqrt(5)-40))); draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)-2*(120-16*sqrt(5)-80),120-16*sqrt(5)-(120-16*sqrt(5)-80))); [/asy] [/hide]

Let $n\ge 2$ be a positive integer. Find whether there exist $n$ pairwise nonintersecting nonempty subsets of $\{1, 2, 3, \ldots \}$ such that each positive integer can be expressed in a unique way as a sum of at most $n$ integers, all from different subsets.

A point in the plane with a cartesian coordinate system is called a [i]mixed point[/i] if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coefficients such that their graphs do not contain any mixed point.

assume that A is a finite subset of prime numbers, and a is an positive integer. prove that there are only finitely many positive integers m s.t: prime divisors of a^m-1 are contained in A.

The numbers $1,2,...,64 $ are written on the unit squares of a table $8 \times 8$. The two smallest numbers of every row are marked black and the two smaller numbers of every comlumn are marked white. Prove or disprove that there are at least k numbers on the table that are marked both black and white when: a) $k=3$ b) $k=4$ c) $k=5$ .