A player repeatedly throwing a die is to play until their score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n,$ and find the value of $\lim_{n \to \infty} p(n).$

Let $\omega$ denote the circumscribed circle of an acute-angled $\triangle ABC$ with $AB \neq BC$. Let $A'$ be the point symmetric to the point $A$ with respect to the line $BC$. The lines $AA'$ and $A'C$ intersect $\omega$ for the second time at points $D$ and $E$, respectively. Let the lines $AE$ and $BD$ intersect at point $P$. Prove that the line $A'P$ is tangent to the circumscribed circle of $\triangle A'BC$. [i]Proposed by Oleksii Masalitin[/i]

In the $xy$-plane, the segment with endpoints $(-5,0)$ and $(25,0)$ is the diameter of a circle. If the point $(x,15)$ is on the circle, then $x=$ $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12.5 \qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 17.5 \qquad\textbf{(E)}\ 20 $

A finite sequence of integers $ a_0, a_1, \ldots, a_n$ is called quadratic if for each $ i$ in the set $ \{1,2 \ldots, n\}$ we have the equality $ |a_i \minus{} a_{i\minus{}1}| \equal{} i^2.$ a.) Prove that any two integers $ b$ and $ c,$ there exists a natural number $ n$ and a quadratic sequence with $ a_0 \equal{} b$ and $ a_n \equal{} c.$ b.) Find the smallest natural number $ n$ for which there exists a quadratic sequence with $ a_0 \equal{} 0$ and $ a_n \equal{} 1996.$

The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$? $\textbf{(A) }20\qquad\textbf{(B) }30\qquad\textbf{(C) }60\qquad\textbf{(D) }120\qquad \textbf{(E) }180$

Farmer Boso has a busy farm with lots of animals. He tends to $5b$ cows, $5a +7$ chickens, and $b^{a-5}$ insects. Note that each insect has $6$ legs. The number of cows is equal to the number of insects. The total number of legs present amongst his animals can be expressed as $\overline{LLL }+1$, where $L$ stands for a digit. Find $L$.

Let $ABCD$ be a convex quadrilateral. Suppose that similar isosceles triangles $APB, BQC, CRD, DSA$ with the bases on the sides of $ABCD$ are constructed in the exterior of the quadrilateral such that $PQRS$ is a rectangle but not a square. Show that $ABCD$ is a rhombus.

Given an integer $ a$. Let $ p$ is prime number such that $ p|a$ and $ p \equiv \pm 3 (mod8)$. Define a sequence $ \{a_n\}_{n \equal{} 0}^\infty$ such that $ a_n \equal{} 2^n \plus{} a$. Prove that the sequence $ \{a_n\}_{n \equal{} 0}^\infty$ has finitely number of square of integer.

A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

For each integer $ n > 1,$ let $ F(n)$ be the number of solutions of the equation $ \sin x \equal{} \sin nx$ on the interval $ [0,\pi].$ What is $ \sum_{n \equal{} 2}^{2007}F(n)?$ $ \textbf{(A)}\ 2,014,524 \qquad \textbf{(B)}\ 2,015,028 \qquad \textbf{(C)}\ 2,015,033 \qquad \textbf{(D)}\ 2,016,532 \qquad \textbf{(E)}\ 2,017,033$

Six points are placed on the plane such that each three of them are the vertices of a triangle with sides of different lengths. Prove that the shortest side of one of these triangles is also the longest side of another of them.

Let $P(x)$ and $Q(x)$ be polynomials of degree $p$ and $q$ respectively such that every coefficient is $1$ or $2023$. If $P(x)$ divides $Q(x)$, prove that $p+1$ divides $q+1$.

Which of the following rationals is greater , $\frac{1995^{1994} + 1}{1995^{1995} + 1}$ or $\frac{1995^{1995} + 1}{ 1995^{1996} +1}$ ?

Find the number of finite sequences $ \{a_1,a_2,\ldots,a_{2n\plus{}1}\}$, formed with nonnegative integers, for which $ a_1\equal{}a_{2n\plus{}1}\equal{}0$ and $ |a_k \minus{}a_{k\plus{}1}|\equal{}1$, for all $ k\in\{1,2,\ldots,2n\}$.

The smallest positive integer that does not divide $1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$ is:

In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are rational numbers, we call it rational point. If $a$ is an irrational number, then in all lines that passes $(a,0)$, $\text{(A)}$There are infinitely many lines, on which there are at least two rational points. $\text{(B)}$There are exactly $n(n\geq2)$ lines, on which there are at least two rational points. $\text{(C)}$There are exactly 1 line, on which there are at least two rational points. $\text{(D)}$Every line passes at least one rational point.

When a bucket is two-thirds full of water, the bucket and water weigh $ a$ kilograms. When the bucket is one-half full of water the total weight is $ b$ kilograms. In terms of $ a$ and $ b$, what is the total weight in kilograms when the bucket is full of water? $ \textbf{(A)}\ \frac23a\plus{}\frac13b\qquad \textbf{(B)}\ \frac32a\minus{}\frac12b\qquad \textbf{(C)}\ \frac32a\plus{}b$ $ \textbf{(D)}\ \frac32a\plus{}2b\qquad \textbf{(E)}\ 3a\minus{}2b$

Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled: 1) For every prime number $p$ and every natural number $n$, the numbers $p^n,p^{n+1}$ and $p^{n+2}$ do not have the same colour. 2) There does not exist an infinite geometric sequence of natural numbers of the same colour.

The $12$-sided polygon below was created by placing three $3$ × $3$ squares with their sides parallel so that vertices of two of the squares are at the center of the third square. Find the perimeter of this $12$-sided polygon.

How many polynomials $P$ of integer coefficients and degree at most $4$ satisfy $0 \le P(x) < 72$ for all $x\in \{0, 1, 2, 3, 4\}$? Harvard-MIT Mathematics Tournament 2011

The fraction $ \frac{\sqrt{a^2\plus{}x^2}\minus{}(x^2\minus{}a^2)/\sqrt{a^2\plus{}x^2}}{a^2\plus{}x^2}$ reduces to: $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac{2a^2}{a^2\plus{}x^2} \qquad\textbf{(C)}\ \frac{2x^2}{(a^2\plus{}x^2)^{\frac{3}{2}}} \qquad\textbf{(D)}\ \frac{2a^2}{(a^2\plus{}x^2)^{\frac{3}{2}}} \qquad\textbf{(E)}\ \frac{2x^2}{a^2\plus{}x^2}$

Let $n{}$ be a positive integer and let $a{}$ and $b{}$ be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer $k{}$ such that at least one of the numbers $a^k-b$ and $b^k-a$ is divisible by $2^n.$ [i]Cătălin Liviu Gherghe[/i]

13. How many functions $f : \{0, 1\}^3 \to \{0, 1\}$ satisfy the property that, for all ordered triples $(a_1, a_2, a_3)$ and $(b_1, b_2, b_3)$ such that $a_i \ge b_i$ for all $i$, $f(a_1, a_2, a_3) \ge f(b_1, b_2, b_3)$? 14. The very hungry caterpillar lives on the number line. For each non-zero integer $i$, a fruit sits on the point with coordinate $i$. The caterpillar moves back and forth; whenever he reaches a point with food, he eats the food, increasing his weight by one pound, and turns around. The caterpillar moves at a speed of $2^{-w}$ units per day, where $w$ is his weight. If the caterpillar starts off at the origin, weighing zero pounds, and initially moves in the positive $x$ direction, after how many days will he weigh $10$ pounds? 15. Let $ABCD$ be an isosceles trapezoid with parallel bases $AB = 1$ and $CD = 2$ and height $1$. Find the area of the region containing all points inside $ABCD$ whose projections onto the four sides of the trapezoid lie on the segments formed by $AB,BC,CD$ and $DA$.

Prove there exists a constant $k_0$ such that for any $k\ge k_0$, the equation \[a^{2n}+b^{4n}+2013=ka^nb^{2n}\] has no positive integer solutions $a,b,n$. [i]Proposed by István Pink.[/i]

Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon.