Find the smallest possible $n$ for which there exist integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$ such that each integer between $1000$ and $2000$ (inclusive) can be written as the sum (without repetition), of one or more of the integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$.

Jason has n coins, among which at most one of them is counterfeit. The counterfeit coin (if there is any) is either heavier or lighter than a real coin. Jason’s grandfather also left him an old weighing balance, on which he can place any number of coins on either side and the balance will show which side is heavier. However, the old weighing balance is in fact really really old and can only be used 4 more times. What is the largest number $n$ for which it is possible for Jason to find the counterfeit coin (if it exist)?

Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$. [i]Author: Christopher Bradley, United Kingdom [/i]

Find the largest $c$ such that for any $\lambda \ge 1$ there is an a that satisfies the inequality $$\sin a + \sin (a\lambda ) \ge c.$$

17) An object is thrown directly downward from the top of a 180-meter-tall building. It takes 1.0 seconds for the object to fall the last 60 meters. With what initial downward speed was the object thrown from the roof? A) 15 m/s B) 25 m/s C) 35 m/s D) 55 m/s E) insufficient information

Find the sum of the coefficients of the polynomial $P(x) = x^4- 29x^3 + ax^2 + bx + c$, given that $P(5) = 11$, $P(11) = 17$, and $P(17) = 23$.

A castaway is building a rectangular raft $ABCD$. He fixes a mast perpendicular to the raft, with ropes passing from the top of the mast (point $S$ in the figure) to the four corners of the raft. The rope $SA$ measures $8$ meters, the rope $SB$ measures $2$ meters and the rope $SC$ measures $14$ meters. Compute the length of the rope $SD$. [asy] size(250); // Coordinates for the parallelogram ABCD pair A = (0, 0); pair B = (8, 0); pair C = (10, 5); pair D = (2, 5); // Position of point S (outside the parallelogram) pair S = (5, 8); pair T = (5, 3); // Draw the parallelogram ABCD filldraw(A--B--C--D--cycle, lightgray, black); // Draw the ropes from point S to each corner of the parallelogram draw(S--A, blue); draw(S--B, blue); draw(S--C, blue); draw(S--D, blue); draw(S--T, black); // Mark the points dot(A); dot(B); dot(C); dot(D); dot(S); dot(T); // Label the points label("A", A, SW); label("B", B, SE); label("C", C, NE); label("D", D, NW); label("S", S, N); [/asy]

(1) Show the following inequality for every natural number $ k$. \[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\] (2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$. \[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]

Let $p_1 = 2012$ and $p_n = 2012^{p_{n-1}}$ for $n > 1$. Find the largest integer $k$ such that $p_{2012}- p_{2011}$ is divisible by $2011^k$.

N numbers are marked in the set $\{1,2,...,2000\}$ so that any pair of the numbers $(1,2),(2,4),...,(1000,2000)$ contains at least one marked number. Find the least possible value of $N$. I.Gorodnin

Let $k$ be a positive integer. An arrangement of finitely many open intervals in $R$ is called [i]good [/i] if for any of the intervals the number of other intervals which intersect with it is a nonzero multiple of $k$. Find the maximum positive integer $n$ (as a function of $k$) for which there is no good arrangement with $n$ intervals

For a positive integer $m$, let $\varphi(m)$ be the number of positive integers $k \le m$ such that $k$ and $m$ are relatively prime, and let $\sigma(m)$ be the sum of the positive divisors of $m$. Find the sum of all even positive integers $n$ such that \[ \frac{n^5\sigma(n) - 2}{\varphi(n)} \] is an integer.

Let $a_0,a_1,\dots$ be a sequence of positive reals such that $$ a_{n+2} \leq \frac{2023a_n}{a_na_{n+1}+2023}$$ for all integers $n\geq 0$. Prove that either $a_{2023}<1$ or $a_{2024}<1$.

Let $m$ and $n$ be positive integers with $m\le 2000$ and $k=3-\frac{m}{n}$. Find the smallest positive value of $k$.

Consider a cylinder and a cone with a common base such that the volume of the part of the cylinder enclosed in the cone equals the volume of the part of the cylinder outside the cone. Find the ratio of the height of the cone to the height of the cylinder.

Show that if $15$ numbers lie between $2$ and $1992$ and each pair is coprime, then at least one is prime.

Are there six different positive odd numbers $a,b,c,d,e,f$ such that \[ 1/a + 1/b + 1/c + 1/d + 1/e + 1/f = 1?\]

Consider 2 non-constant polynomials $P(x),Q(x)$, with nonnegative coefficients. The coefficients of $P(x)$ is not larger than $2021$ and $Q(x)$ has at least one coefficient larger than $2021$. Assume that $P(2022)=Q(2022)$ and $P(x),Q(x)$ has a root $\frac p q \ne 0 (p,q\in \mathbb Z,(p,q)=1)$. Prove that $|p|+n|q|\le Q(n)-P(n)$ for all $n=1,2,...,2021$

1.1 Let $f(A)$ denote the difference between the maximum value and the minimum value of a set $A$. Find the sum of $f(A)$ as $A$ ranges over the subsets of $\{1, 2, \dots, n\}$. 1.2 All cells of an $8 × 8$ board are initially white. A move consists of flipping the color (white to black or vice versa) of cells in a $1\times 3$ or $3\times 1$ rectangle. Determine whether there is a finite sequence of moves resulting in the state where all $64$ cells are black. 1.3 Prove that for all positive integers $m$, there exists a positive integer $n$ such that the set $\{n, n + 1, n + 2, \dots , 3n\}$ contains exactly $m$ perfect squares.

Let $\mathbb{N}$ denote the set of all positive integers. Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$x^2-y^2+2y(f(x)+f(y))$$ is a square of an integer for all positive integers $x$ and $y$.

$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation  $b_1,b_2,\ldots,b_n$ of $a_i $'s such that all the $n$ sums $b_i+b_{i+1}$ are distinct ($1\le i\le n $ , $b_{n+1}\equiv b_1 $) [i]Proposed by Mohsen Jamali[/i]

Let $(x_1,x_2,\ldots,x_n)$ be a point chosen at random in the $n$-dimensional region defined by $0<x_1<x_2<\ldots<x_n<1$, denoting $x_0=0$ and $x_{n+1}=1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Show that the expected value of the sum $$\sum_{i=0}^n(x_{i+1}-x_i)f(x_{i+1})$$is $\int^1_0f(t)P(t)dt$., where $P$ is a polynomial of degree $n$, independent of $f$, with $0\le P(t)\le1$ for $0\le t\le1$.

Find all triples $(a, b, c)$ of positive integers with $a+b+c = 10$ such that there are $a$ red, $b$ blue and $c$ green points (all different) in the plane satisfying the following properties: $\bullet$ for each red point and each blue point we consider the distance between these two points, the sum of these distances is $37$, $\bullet$ for each green point and each red point we consider the distance between these two points, the sum of these distances is $30$, $\bullet$ for each blue point and each green point we consider the distance between these two points, the sum of these distances is $1$.

A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle? $ \textbf{(A)}\ \frac{\sqrt{\pi}}{2} \qquad\textbf{(B)}\ \sqrt{\pi} \qquad\textbf{(C)}\ \pi \qquad\textbf{(D)}\ 2\pi \qquad\textbf{(E)}\ \pi^{2}$

Let $a,b,c > 0$ and $ab+bc+ca = 1$. Prove the inequality $3\sqrt[3]{\frac{1}{abc} +6(a+b+c) }\le \frac{\sqrt[3]3}{abc}$