A pair of fair $6$-sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals $7$ at least once is greater than $\frac{1}{2}$? $\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6$

$\left(\dfrac{1}{4}\right)^{-\frac{1}{4}}=$ $\textbf{(A) }-16\qquad \textbf{(B) }-\sqrt{2}\qquad \textbf{(C) }-\dfrac{1}{16}\qquad \textbf{(D) }-\dfrac{1}{256}\qquad \textbf{(E) }\sqrt{2}$

Let $n$ distinct points in the plane be given. Prove that fewer than $2 n^{3 \slash 2}$ pairs of them are a unit distance apart.

A plane intersects a sphere of radius $10$ such that the distance from the center of the sphere to the plane is $9$. The plane moves toward the center of the bubble at such a rate that the increase in the area of the intersection of the plane and sphere is constant, and it stops once it reaches the center of the circle. Determine the distance from the center of the sphere to the plane after two-thirds of the time has passed.

Determine all pairs of positive integers $m, n,$ satisfying the equality $(2^{m}+1;2^n+1)=2^{(m;n)}+1$ , where $(a;b)$ is the greatest common divisor

Let $a,b,c\geq 0$. Prove: $$\frac{1+a+a^{2}}{1+b+c^{2}}+\frac{1+b+b^{2}}{1+c+a^{2}}+\frac{1+c+c^{2}}{1+a+b^{2}}\geq 3$$

Compute the number of ways to write the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the cells of a 3 by 3 grid such that [list] [*] each cell has exactly one number, [*] each number goes in exactly one cell, [*] the numbers in each row are increasing from left to right, [*] the numbers in each column are increasing from top to bottom, and [*]the numbers in the diagonal from the upper-right corner cell to the lower-left corner cell are increasing from upper-right to lower-left. [/list] [i]Proposed by Ankit Bisain & Luke Robitaille[/i]

A trapezoid is divided into seven strips of equal width as shown. What fraction of the trapezoid’s area is shaded?

In the quadrilateral $ABCD$ $\angle ABC = \angle CDA = 90^\circ$. Let $P = AC \cap BD$, $Q = AB\cap CD$, $R = AD \cap BC$. Let $\ell$ be the midline of the triangle $PQR$, parallel to $QR$. Show that the circumcircle of the triangle formed by lines $AB, AD, \ell$ is tangent to the circumcircle of the triangle formed by lines $CD, CB, \ell$. [i]Proposed by Fedir Yudin[/i]

Find the number of positive integer divisors of $20^{17}$ that are either perfect squares or perfect cubes.

Find the smallest positive integer $n$ with the following property: After painting black exactly $n$ cells of a $7\times 7$ board there always exists a $2\times 2$ square with at least three black cells.

Let $ a, b, c$ be integers, not all equal to $ 0$. Show that \[ \frac{1}{4a^{2}\plus{}3b^{2}\plus{}2c^{2}}\le\vert\sqrt[3]{4}a\plus{}\sqrt[3]{2}b\plus{}c\vert.\]

The checker is standing on the corner field of a $n\times n$ chess-board. Each of two players moves it in turn to the neighbour (i.e. that has the common side) field. It is forbidden to move to the field, the checker has already visited. That who cannot make a move losts. a) Prove that for even $n$ the first can always win, and if $n$ is odd, than the second can always win. b) Who wins if the checker stands initially on the neighbour to the corner field?

We are given $1999$ coins. No two coins have the same weight. A machine is provided which allows us with one operation to determine, for any three coins, which one has the middle weight. Prove that the coin that is the $1000$th by weight can be determined using no more than $1000000$ operations and that this is the only coin whose position by weight can be determined using this machine.

Given two points $p_1=(x_1, y_1)$ and $p_2=(x_2, y_2)$ on the plane, denote by $\mathcal{R}(p_1,p_2)$ the rectangle with sides parallel to the coordinate axes and with $p_1$ and $p_2$ as opposite corners, that is, \[\{(x,y)\in \mathbb{R}^2:\min\{x_1, x_2\}\leq x\leq \max\{x_1, x_2\},\min\{y_1, y_2\}\leq y\leq \max\{y_1, y_2\}\}.\] Find the largest value of $k$ for which the following statement is true: for all sets $\mathcal{S}\subset\mathbb{R}^2$ with $|\mathcal{S}|=2024$, there exist two points $p_1, p_2\in\mathcal{S}$ such that $|\mathcal{S}\cap\mathcal{R}(p_1, p_2)|\geq k$.

$A$ can run around a circular track in $40$ seconds. $B$, running in the opposite direction, meets $A$ every $15$ seconds. What is $B$'s time to run around the track, expressed in seconds? $ \textbf{(A)}\ 12\frac12 \qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 27\frac12\qquad\textbf{(E)}\ 55 $

There are $n$ cities in a country, where $n>1$. There are railroads connecting some of the cities so that you can travel between any two cities through a series of railroads (railroads run in both direction.) In addition, in this country, it is impossible to travel from a city, through a series of distinct cities, and return back to the original city. We define the [b]degree[/b] of a city as the number of cities directly connected to it by a single segment of railroad. For a city $A$ that is directly connected to $x$ cities, with $y$ of those cities having a smaller degree than city $A$, the [b]significance[/b] of city $A$ is defined as $\frac{y}{x}$. Find the smallest positive real number $t$ so that, for any $n>1$, the sum of the significance of all cities is less than $tn$, no matter how the railroads are paved. [i]Proposed by houkai[/i]

In convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at point $O$ at angle $90^{\circ}$. Let $K$, $L$, $M$ and $N$ be orthogonal projections of point $O$ to sides $AB$, $BC$, $CD$ and $DA$ of quadrilateral $ABCD$. Prove that $KLMN$ is cyclic

For each positive integer $n$, find all triples $a,b,c$ of real numbers for which \[\begin{cases}a=b^n+c^n\\ b=c^n+a^n\\ c=a^n+b^n\end{cases}\]

Let $AB$ and $CD$ be two diameters of the circle $C$. For an arbitrary point $P$ on $C$, let $R$ and $S$ be the feet of the perpendiculars from $P$ to $AB$ and $CD$, respectively. Show that the length of $RS$ is independent of the choice of $P$.

If $n$ is composite, prove that $\phi(n) \le n- \sqrt{n}$.

The positive numbers $a,b,c,A,B,C$ satisfy a condition $$a + A = b + B = c + C = k$$ Prove that $$aB + bC + cA \le k^2$$

Let $ N$ be the number of ways to write $ 2010$ in the form \[2010 \equal{} a_3 \cdot 10^3 \plus{} a_2 \cdot 10^2 \plus{} a_1 \cdot 10 \plus{} a_0,\] where the $ a_i$'s are integers, and $ 0 \le a_i \le 99$. An example of such a representation is $ 1\cdot10^3 \plus{} 3\cdot10^2 \plus{} 67\cdot10^1 \plus{} 40\cdot10^0$. Find $ N$.

Prove that \[ \sum_{n=1}^\infty {1\over n^n} = \int_0^1 x^{-x}\,dx. \]

Let $ABCD$ be a convex quadrilateral with area $202, AB = 4,$ and $\angle A = \angle B = 90^\circ$ such that there is exactly one point $E$ on line $CD$ satisfying $\angle AEB = 90^\circ.$ Compute the perimeter of $ABCD.$