Lily has an unfair coin that has $\tfrac23$ probability of showing heads and $\tfrac13$ probability of showing tails. She flips the coin twice. What is the probability that the first flip is heads while the second is tails? $\textbf{(A) }0\qquad\textbf{(B) }1/9\qquad\textbf{(C) }2/9\qquad\textbf{(D) }4/9\qquad\textbf{(E) }1$

a) Prove that it is possible to choose one number out of any 39 consecutive positive integers, having the sum of its digits divisible by 11; b) Find the first 38 consecutive positive integers none of which have the sum of its digits divisible by 11.

For a positive number $x$, let $f(x)=\lim_{n\to\infty} \sum_{k=1}^n \left|\cos \left(\frac{2k+1}{2n}x\right)-\cos \left(\frac{2k-1}{2n}x\right)\right|$ Evaluate \[\lim_{x\rightarrow\infty}\frac{f(x)}{x}\]

Show that there exists a convex hexagon in the plane such that (i) all its interior angles are equal; (ii) its sides are $1,2,3,4,5,6$ in some order.

Let $ABCD$ be a trapezoid of parallel bases $ BC$ and $AD$. If $\angle CAD = 2\angle CAB, BC = CD$ and $AC = AD$, determine all the possible values of the measure of the angle $\angle CAB$.

Nine points are drawn on the surface of a regular tetrahedron with an edge of $1$ cm. Prove that among these points there are two located at a distance (in space) no greater than $0.5$ cm.

In $ \triangle{ABC}$, we have $ AB\equal{}1$ and $ AC\equal{}2$. Side $ BC$ and the median from $ A$ to $ BC$ have the same length. What is $ BC$? $ \textbf{(A)}\ \frac{1\plus{}\sqrt2}{2} \qquad \textbf{(B)}\ \frac{1\plus{}\sqrt3}{2} \qquad \textbf{(C)}\ \sqrt2 \qquad \textbf{(D)}\ \frac{3}{2} \qquad \textbf{(E)}\ \sqrt3$

If the sum $ 1 \plus{} 2 \plus{} 3 \plus{} \cdots \plus{} K$ is a perfect square $ N^2$ and if $ N$ is less than $ 100$, then the possible values for $ K$ are: $ \textbf{(A)}\ \text{only }1 \qquad\textbf{(B)}\ 1\text{ and }8 \qquad\textbf{(C)}\ \text{only }8 \qquad\textbf{(D)}\ 8\text{ and }49 \qquad\textbf{(E)}\ 1,8,\text{ and }49$

Find the number of $f:\{1,\ldots, 5\}\to \{1,\ldots, 5\}$ such that $f(f(x))=x$ (A)$26$ (B)$41$ (C)$120$ (D)$60$

Find all continuous functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all real numbers $ x, y $ $$ f(x^2+f(y))=f(f(y)-x^2)+f(xy) $$ [Extra: Can you solve this without continuity?]

Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.

Prove that \[ 1 < \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001} < 1 \frac{1}{3}. \]

Prove that the product of the first $1000$ positive even integers differs from the product of the first $1000$ positive odd integers by a multiple of $2001$.

Let $p>0$ be a given parameter. Determine all real $x$ such that \[\frac{1}{\,x+\sqrt{p-x^2\,}\,}+\frac{1}{\,x-\sqrt{p-x^2\,}\,}\ge\frac{1}{\,p\,}.\]

A circle centred at $(a, b)$ contains the origin $(0,0)$. Denote by $S^+$ the total area of the parts of the circle in the first and third quadrants, and by $S^-$ the total area of the parts of the circle in the second and the fourth quadrants. Compute $S^+ -S^-$. (G Galperin)

Find the largest positive value attained by the function \[ f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}, \qquad x\ \text{a real number} \] $ \textbf{(A)}\ \sqrt{7}-1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2\sqrt{3} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{55}-\sqrt{5} $

Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.

There are $1000$ tickets with the numbers $000, 001, ... , 999$, and $100$ boxes with the numbers $00, 01, ... , 99$. You may put a ticket in a box, if you can obtain the box number from the ticket number by deleting one digit. Prove that: a) You can put all the tickets in $50$ boxes; b) $40$ boxes is not enough for that; c) it is impossible to use less than $50$ boxes. d) Consider $10000$ $4$-digit tickets, and you are allowed to delete two digits. Prove that $34$ boxes is enough for storing all the tickets. e) What is the minimal used boxes set in the case of $k$-digit tickets?

Four consecutive even numbers are removed from the set \[A=\{ 1, 2, 3, \cdots, n \}.\] If the arithmetic mean of the remaining numbers is $51.5625$, which four numbers were removed?

Let $p$ be a prime such that $\frac{p-1}{2}$ is also prime. A pair of integers $(x, y)$ with $1\le x, y \le p-1$ is called a [i]commuter[/i] if at least one of $x^y -y^x$ or $x^y +y^x$ is divisible by $p$. Show that the number of commuters is at most $4.2p\sqrt{p}$.

[b]p1.[/b] The least prime factor of $a$ is $3$, the least prime factor of $b$ is $7$. Find the least prime factor of $a + b$. [b]p2.[/b] In a Cartesian coordinate system, the two tangent lines from $P = (39, 52)$ meet the circle defined by $x^2 + y^2 = 625$ at points $Q$ and $R$. Find the length $QR$. [b]p3.[/b] For a positive integer $n$, there is a sequence $(a_0, a_1, a_2,..., a_n)$ of real values such that $a_0 = 11$ and $(a_k + a_{k+1}) (a_k - a_{k+1}) = 5$ for every $k$ with $0 \le k \le n-1$. Find the maximum possible value of $n$. (Be careful that your answer isn’t off by one!) [b]p4.[/b] Persons $A$ and $B$ stand at point $P$ on line $\ell$. Point $Q$ lies at a distance of $10$ from point $P$ in the direction perpendicular to $\ell$. Both persons intially face towards $Q$. Person $A$ walks forward and to the left at an angle of $25^o$ with $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the right, and continues walking. Person $B$ walks forward and to the right at an angle of $55^o$ with line $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the left, and continues walking. Their paths cross at point $R$. Find the distance $PR$. [b]p5.[/b] Compute $$\frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}.$$ [b]p6.[/b] There is a unique real value $A$ such that for all $x$ with $1 < x < 3$ and $x \ne 2$, $$\left| \frac{A}{x^2-x - 2} +\frac{1}{x^2 - 6x + 8} \right|< 1999.$$ Compute $A$. [b]p7.[/b] Nine poles of height $1, 2,..., 9$ are placed in a line in random order. A pole is called [i]dominant [/i] if it is taller than the pole immediately to the left of it, or if it is the pole farthest to the left. Count the number of possible orderings in which there are exactly $2$ dominant poles. [b]p8.[/b] $\tan (11x) = \tan (34^o)$ and $\tan (19x) = \tan (21^o)$. Compute $\tan (5x)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The circle inscribed in the triangle $ABC$ touches the side $[AC]$ in the point $K$. Prove that the line connecting the midpoint of the side $[AC]$ with the centre of the circle halves the segment $[BK]$ .

A data set consists of $6$ (not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all positive values of $X$? $\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40$

The roots of the equation $x^5-180x^4+Ax^3+Bx^2+Cx+D=0$ are in geometric progression. The sum of their reciprocals is $20$. Compute $|D|$.

Show that \[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\] holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.