Determine (at least one) pair of real numbers $k,q$ such that the inequality \[2\left|\sqrt{1-x^2}-kx-q\right|\le\sqrt2-1\] holds for all $x\in[0,1].$

Call a subset of $\{1,2,\ldots, n\}$ [i]unfriendly[/i] if no two of its elements are consecutive. Show that the number of unfriendly subsets with $k$ elements is $\binom{n-k+1}{k}.$

The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. How many integers less than $2013$ but greater than $1000$ share this property? ${ \textbf{(A)}\ 33 \qquad\textbf{(B)}\ 34 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}}\ 46\qquad\textbf{(E)}\ 58 $

Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.

Let $x, y$, and $z$ be positive reals that satisfy the system $$\begin{cases} x^2 + x y + y^2 = 10 \\ x^2 + xz + z^2 = 20 \\ y^2 + yz + z^2 = 30\end{cases}$$ Find $x y + yz + xz$.

Let $ABC$ be a triangle, and let $BCDE$, $CAFG$, $ABHI$ be squares that do not overlap the triangle with centers $X$, $Y$, $Z$ respectively. Given that $AX=6$, $BY=7$, and $CA=8$, find the area of triangle $XYZ$.

Let $n_0$ be the product of the first $25$ primes. Now, choose a random divisor $n_1$ of $n_0$, where a choice $n_1$ is taken with probability proportional to $\phi(n_1)$. ($\phi(m)$ is the number of integers less than $m$ which are relatively prime to $m$.) Given this $n_1$, we let $n_2$ be a random divisor of $n_1$, again chosen with probability proportional to $\phi(n_2)$. Compute the probability that $n_2\equiv0\pmod{2310}$.

Positive integers $ a<b$ are given. Prove that among every $ b$ consecutive positive integers there are two numbers whose product is divisible by $ ab$.

Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$?

Let $ S$ be the set of odd integers greater than $ 1$. For each $ x \in S$, denote by $ \delta (x)$ the unique integer satisfying the inequality $ 2^{\delta (x)}<x<2^{\delta (x) \plus{}1}$. For $ a,b \in S$, define: $ a \ast b\equal{}2^{\delta (a)\minus{}1} (b\minus{}3)\plus{}a.$ Prove that if $ a,b,c \in S$, then: $ (a)$ $ a \ast b \in S$ and $ (b)$ $ (a \ast b)\ast c\equal{}a \ast (b \ast c)$.

Show that if \[ \frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1 \] then \[ \frac{\cos^3 y}{\cos x}+\frac{\sin^3 y}{\sin x}=1 \]

A basket is called "[i]Stuff Basket[/i]" if it includes $10$ kilograms of rice and $30$ number of eggs. A market is to distribute $100$ Stuff Baskets. We know that there is totally $1000$ kilograms of rice and $3000$ number of eggs in the baskets, but some of market's baskets include either more or less amount of rice or eggs. In each step, market workers can select two baskets and move an arbitrary amount of rice or eggs between selected baskets. Starting from an arbitrary situation, what's the minimum number of steps that workers provide $100$ Stuff Baskets?

An integer $n > 1$ is given. Two players in turns mark points on a circle. First Player uses red color while Second Player uses blue color. The game is over when each player marks $n$ points. Then each player nds the arc of maximal length with ends of his color, which does not contain any other marked points. A player wins if his arc is longer (if the lengths are equal, or both players have no such arcs, the game ends in a draw). Which player has a winning strategy?

Every pair of communities in a county are linked directly by one mode of transportation; bus, train, or airplane. All three methods of transportation are used in the county with no community being serviced by all three modes and no three communities being linked pairwise by the same mode. Determine the largest number of communities in this county.

Does there exist a polyhedron (not necessarily convex) which could have the following complete list of edges? $AB, AC, BC, BD, CD, DE, EF, EG, FG, FH, GH, AH$. [img]http://1.bp.blogspot.com/-wTdNfQHG5RU/XVk1Bf4wpqI/AAAAAAAAKhA/8kc6u9KqOgg_p1CXim2LZ1ANFXFiWgnYACK4BGAYYCw/s1600/TOT%2B1982%2BAutum%2BS2.png[/img]

We say that a graph $G$ is [i]divisive[/i], if we can write a positive integer on each of its vertices such that all the integers are distinct, and any two of these integers divide each other if and only if there is an edge running between them in $G$. Which Platonic solids form a divisive graph? [img]https://cdn.artofproblemsolving.com/attachments/1/5/7c81439ee148ccda09c429556e0740865723e0.png[/img]

Stairs are built by laying bricks as shown in the figure. If you have $2004$ bricks to build a staircase: a) How many steps (= escalones) will the ladder have? b) How many bricks will there be left over? [img]https://cdn.artofproblemsolving.com/attachments/4/b/c1b80b374daeda4e33e1bb45be1d11f4b89590.png[/img]

In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$. [i]Proposed by Andy Xu[/i]

Let $f$ be a real-valued function such that \[f(x)+2f\left(\dfrac{2002}x\right)=3x\] for all $x>0$. Find $f(2)$. $\textbf{(A) }1000\qquad\textbf{(B) }2000\qquad\textbf{(C) }3000\qquad\textbf{(D) }4000\qquad\textbf{(E) }6000$

Given a $m \times n$ rectangle where $m,n\geq 2023$. The square in the $i$-th row and $j$-th column is filled with the number $i+j$ for $1\leq i \leq m, 1\leq j \leq n$. In each move, Alice can pick a $2023 \times 2023$ subrectangle and add $1$ to each number in it. Alice wins if all the numbers are multiples of $2023$ after a finite number of moves. For which pairs $(m,n)$ can Alice win? [i]Proposed by Boon Qing Hong[/i]

Find the smallest integer $n$ satisfying the following condition: regardless of how one colour the vertices of a regular $n$-gon with either red, yellow or blue, one can always find an isosceles trapezoid whose vertices are of the same colour.

Let \( a_1 \) be an integer greater than or equal to 2. Consider the sequence such that its first term is \( a_1 \), and for \( a_n \), the \( n \)-th term of the sequence, we have \[ a_{n+1} = \frac{a_n}{p_k^{e_k - 1}} + 1, \] where \( p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \) is the prime factorization of \( a_n \), with \( 1 < p_1 < p_2 < \cdots < p_k \), and \( e_1, e_2, \dots, e_k \) positive integers. For example, if \( a_1 = 2024 = 2^3 \cdot 11 \cdot 23 \), the next two terms of the sequence are \[ a_2 = \frac{a_1}{23^{1-1}} + 1 = \frac{2024}{1} + 1 = 2025 = 3^4 \cdot 5^2; \] \[ a_3 = \frac{a_2}{5^{2-1}} + 1 = \frac{2025}{5} + 1 = 406. \] Determine for which values of \( a_1 \) the sequence is eventually periodic and what all the possible periods are. [b]Note:[/b] Let \( p \) be a positive integer. A sequence \( x_1, x_2, \dots \) is eventually periodic with period \( p \) if \( p \) is the smallest positive integer such that there exists an \( N \geq 0 \) satisfying \( x_{n+p} = x_n \) for all \( n > N \).

You are given an acute triangle $ABC$ with circumcircle $\omega$. Points $F$ on $AC$, $E$ on $AB$ and $P, Q$ on $\omega$ are chosen so that $\angle AFB = \angle AEC = \angle APE = \angle AQF = 90^\circ$. Show that lines $BC, EF, PQ$ are concurrent or parallel. [i]Proposed by Fedir Yudin[/i]

Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\tfrac{3}{5}$. If he flips heads, he adds $1$ to his score. A player wins the game if their score is higher than the other player's score. A player wins the game if their score is higher than the other player's score. The probability Dukejukem defeats the Trollner to win WASS equals $\tfrac{m}{n}$ where $m$ and $n$ are coprime positive integers. Computer $m+n$.

Find $ \lim_{a\rightarrow\infty}\int_{a}^{a\plus{}1}\frac{x}{x\plus{}\ln x}\ dx$.