Let $x_1=a, x_2=a^{x_1}, ..., x_n=a^{x_{n-1}}$ where $a>1$. What is the maximum value of $a$ for which lim exists $\lim_{n\to \infty} x_n$ and what is this limit?

Show that if $x$ is positive, then \[ \log_e (1 + 1/x) > 1 / (1 + x).\]

Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board. [list] [*] $n$ is larger than any integer on the board currently. [*] $n$ cannot be written as the sum of $2$ distinct integers on the board. [/list] Find the $100$-th integer that you write on the board. Recall that at the beginning, there are already $4$ integers on the board.

[b]9.[/b] Show that if the trigonometric polynomial $f(\theta)= \sum_{v=1}^{n} a_v \cos v\theta$ monotonically decreases over the closed interval $[0,\pi]$, then the trigonometric polynomial $g(\theta)=\sum_{v=1}^{n}a_v \sin v\theta$ is non negative in the same interval. [b](S. 26)[/b]

A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction. What is the maximum value of the tension in the rod? $\textbf{(A) } mg\\ \textbf{(B) } 2mg\\ \textbf{(C) } mL\theta_0/T_0^2\\ \textbf{(D) } mg \sin \theta_0\\ \textbf{(E) } mg(3 - 2 \cos \theta_0)$

Find all sets of natural numbers $(a, b, c)$ such that $$a+1|b^2+c^2\,\, , b+1|c^2+a^2\,\,, c+1|a^2+b^2.$$

We want to construct an isosceles triangle using a compass and a straightedge. We are given two of the following four data: the length of the base of the triangle $(a),$ the length of the leg of the triangle $(b),$ the radius of the inscribed circle $(r),$ and the radius of the circumscribed circle $(R).$ In which of the six possible cases will we definitely be able to construct the triangle? [i]Proposed by György Rubóczky, Budapest[/i]

Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$.

Given real numbers $x,y,z,t\in (0,\pi /2]$ such that $$\cos^2 (x)+\cos^2 (y) +\cos^2 (z) +\cos^2 (t)=1.$$ What is the minimum possible value of $$\cot (x) +\cot (y) +\cot (z) +\cot (t)?$$

Let $ABCDEF$ be a regular hexagon. Points $P$ and $Q$ on tangents to its circumcircle at $A$ and $D$ respectively are such that $PQ$ touches the minor arc $EF$ of this circle. Find the angle between $PB$ and $QC$.

Consider the set of axis-aligned boxes in $R^d$ , $B(a, b) = \{x \in R^d: \forall i, a_i \le x_i \le b_i\}$ where $a, b \in R^d$. In terms of $d$, what is the maximum number $n$, such that there exists a set of $n$ points $S =\{x_1, ..., x_n\}$ such that no matter how one partition $S = P \cup Q$ with $P, Q$ disjoint and $P, Q$ can possibly be empty, there exists a box $B$ such that all the points in $P$ are contained in $B$, and all the points in $Q$ are outside $B$?

Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in $10$, $9$, $11$, and $12$ days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it take to write this test?

find x and y in R $\begin{array}{l} (\frac{1}{{\sqrt[3]{x}}} + \frac{1}{{\sqrt[3]{y}}})(\frac{1}{{\sqrt[3]{x}}} + 1)(\frac{1}{{\sqrt[3]{y}}} + 1) = 18 \\ \frac{1}{x} + \frac{1}{y} = 9 \\ \end{array}$

The graphs of four functions of the form $y = x^2 + ax + b$, where a and b are real coefficients, are plotted on the coordinate plane. These graphs have exactly four points of intersection, and at each one of them, exactly two graphs intersect. Prove that the sum of the largest and the smallest $x$-coordinates of the points of intersection is equal to the sum of the other two. [i](3 points)[/i]

A mouse lives in a circular cage with completely reflective walls. At the edge of this cage, a small flashlight with vertex on the circle whose beam forms an angle of $15^o$ is centered at an angle of $37.5^o$ away from the center. The mouse will die in the dark. What fraction of the total area of the cage can keep the mouse alive? [img]https://cdn.artofproblemsolving.com/attachments/1/c/283276058b7b2c85a95976743c5188ee8ee008.png[/img]

Two players play a card game. They have a deck of $n$ distinct cards. About any two cards from the deck know which of them has a different (in this case, if $A$ beats $B$, and $B$ beats $C$, then it may be that $C$ beats $A$). The deck is split between players in an arbitrary manner. In each turn the players over the top card from his deck and one whose card has a card from another player takes both cards and puts them to the bottom of your deck in any order of their discretion. Prove that for any initial distribution of cards, the players can with knowing the location agree and act so that one of the players left without a card. [i]E. Lakshtanov[/i]

Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

In an acute-angled triangle $ABC$ on the sides $AB$, $BC$, $AC$ the points $H$, $L$, $K$ so that $CH \perp AB$, $HL \parallel AC$, $HK \parallel BC$. Let $P$ and $Q$ feet of altitudes of a triangle $HBL$, drawn from the vertices $H$ and $B$ respectively. Prove that the feet of the altitudes of the triangle $AKH$, drawn from the vertices $A$ and $H$ lie on the line $PQ$.

$100$ weights, measuring $1,2, ..., 100$ grams, respectively, are placed in the two pans of a scale such that the scale is balanced. Prove that two weights can be removed from each pan such that the equilibrium is not broken.

Find all prime numbers $p$ such that there exists a positive integer $n$ where $2^n p^2 + 1$ is a square number.

We say a natural number $n$ is perfect if the sum of all the positive divisors of $n$ is equal to $2n$. For example, $6$ is perfect since its positive divisors $1,2,3,6$ add up to $12=2\times 6$. Show that an odd perfect number has at least $3$ distinct prime divisors. [i]Note: It is still not known whether odd perfect numbers exist. So assume such a number is there and prove the result.[/i]

Let $n$ be a positive integer. Determine, in terms of $n$, the greatest integer which divides every number of the form $p + 1$, where $p \equiv 2$ mod $3$ is a prime number which does not divide $n$.

In a certain game, Auntie Hall has four boxes $B_1$, $B_2$, $B_3$, $B_4$, exactly one of which contains a valuable gemstone; the other three contain cups of yogurt. You are told the probability the gemstone lies in box $B_n$ is $\frac{n}{10}$ for $n=1,2,3,4$. Initially you may select any of the four boxes; Auntie Hall then opens one of the other three boxes at random (which may contain the gemstone) and reveals its contents. Afterwards, you may change your selection to any of the four boxes, and you win if and only if your final selection contains the gemstone. Let the probability of winning assuming optimal play be $\tfrac mn$, where $m$ and $n$ are relatively prime integers. Compute $100m+n$. [i]Proposed by Evan Chen[/i]

Let $\{f_n\}_{n\ge 1}$ be the Fibonacci sequence, defined by $f_1 = f_2 = 1$, and for all positive integers $n$, $f_{n+2} = f_{n+1} + f_n$. Prove that the following inequality takes place for all positive integers $n$: $${n \choose 1}f_1 +{n \choose 2}f_2+... +{n \choose n}f_n < \frac{(2n + 2)^n}{n!}$$ .

Let $n$ and $k$ be two positive integers such that $1\leq n \leq k$. Prove that, if $d^k+k$ is a prime number for each positive divisor $d$ of $n$, then $n+k$ is a prime number.