There are $n\ge 3$ students in a classroom. Every day, the teacher separates the students into at least two non-empty groups, and each pair of students from the same group will shake hands once. Suppose after $k$ days, each pair of students has shaken hands exactly once, and $k$ is as minimal as possible. Prove that $$\sqrt{n} \le k-1 \le 2\sqrt{n}$$ [i]Proposed by Wong Jer Ren[/i]

Show that the number of ordered pairs $(S,T)$ of subsets of $[n]$ satisfying $s>|T|$ for all $s\in S$ and $t>|S|$ for all $t\in T$ is equal to the Fibonacci number $F_{2n+2}$. [color=#008000] Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=296007#p296007 http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=515970&hilit=Putnam+1990[/color]

Let $P_1P_2...P_n$ be a regular $n$-gon in the plane and $a_1, . . . , a_n$ be nonnegative integers. It is possible to draw $m$ circles so that for each $1 \le i \le n$, there are exactly $a_i$ circles that contain $P_i$ on their interior. Find, with proof, the minimum possible value of $m$ in terms of the $a_i$. .

Let $\{a_n\}$ be a sequence of positive terms such that $a_{n+1}=a_n+ \frac{n^2}{a_n}$ . Let $b_n =a_n-n$ . (1) Are there infinitely many $n$ such that $b_n \ge 0$ ? (2) Prove that there is a positive number $M$ such that $\sum^{\infty}_{n=3} \frac{b_n}{n+1}<M$.

Let in-circle of $ABC$ touch $AB$, $BC$, $AC$ in $C_1$, $A_1$, $B_1$ respectively. Let $H$- intersection point of altitudes in $A_1B_1C_1$, $I$ and $O$-be in-center and circumcenter of $ABC$ respectively. Prove, that $I, O, H$ lies on one line.

Let a city be in the form of a square grid which has $n \times n$ cells, each of which contain a skyscraper . At first the $m$ skyscrapers burn, but the fire spreads: everyone skyscraper that has at least two burning neighboring houses (by neighboring houses we mean only houses that border the house along a street, not just at a corner) immediately gets fire. Prove that when in the end the whole city burns down, of must have been $m \ge n$. [hide=original wording in German] Eine Stadt habe die Form eines quadratischen Gitters, welches n × n Zellen habe, von denen jede ein Hochhaus enthalte. Anfangs brennen m der Hochh¨auser, doch der Brand pflanzt sich fort: Jedes Hochhaus, das mindestens zwei brennende Nachbarh¨auser hat (unter Nachbarh¨ausern verstehen wir dabei nur H¨auser, die entlang einer Straße an das Haus angrenzen, nicht nur an einer Ecke), f¨angt sofort Feuer. Man beweise: Wenn am Ende die gesamte Stadt abgebrannt ist,muss m ≥ n gewesen sein.[/hide]

Find the number of integers $n$ that satisfy $\textit{both}$ of the following conditions: [list=i] [*]$208<n<2008$, [*]$n$ has the same remainder when divided by $24$ or by $30$.[/list]

On an infinite square grid, Gru and his $2022$ minions play a game, making moves in a cyclic order with Gru first. On any move, the current player selects $2$ adjacent cells of their choice, and paints their shared border. A border cannot be painted over more than once. Gru wins if after any move there is a $2 \times 1$ or $1 \times 2$ subgrid with its border (comprising of $6$ segments) completely colored, but the $1$ segment inside it uncolored. Can he guarantee a win? [i]Evan Chang[/i] [size=50](oops)[/size]

$2019$ students are voting on the distribution of $N$ items. For each item, each student submits a vote on who should receive that item, and the person with the most votes receives the item (in case of a tie, no one gets the item). Suppose that no student votes for the same person twice. Compute the maximum possible number of items one student can receive, over all possible values of $N$ and all possible ways of voting.

Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]

Four lighthouses are located at points $ A$, $ B$, $ C$, and $ D$. The lighthouse at $ A$ is $ 5$ kilometers from the lighthouse at $ B$, the lighthouse at $ B$ is $ 12$ kilometers from the lighthouse at $ C$, and the lighthouse at $ A$ is $ 13$ kilometers from the lighthouse at $ C$. To an observer at $ A$, the angle determined by the lights at $ B$ and $ D$ and the angle determined by the lights at $ C$ and $ D$ are equal. To an observer at $ C$, the angle determined by the lights at $ A$ and $ B$ and the angle determined by the lights at $ D$ and $ B$ are equal. The number of kilometers from $ A$ to $ D$ is given by $ \displaystyle\frac{p\sqrt{r}}{q}$, where $ p$, $ q$, and $ r$ are relatively prime positive integers, and $ r$ is not divisible by the square of any prime. Find $ p\plus{}q\plus{}r$,

Let $x$ and $y$ be two real numbers such that $2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7$. Find $\tan^2 x + 2 \tan^2 y$.

For any positive integer $n$, define $a_n$ to be the product of the digits of $n$. (a) Prove that $n \geq a(n)$ for all positive integers $n$. (b) Find all $n$ for which $n^2-17n+56 = a(n)$.

The rectangular parallelepiped (box) $P$ has some special properties. If one dimension of $P$ were doubled and another dimension were halved, then the surface area of $P$ would stay the same. If instead one dimension of $P$ were tripled and another dimension were divided by $3$, then the surface area of $P$ would still stay the same. If the middle (by length) dimension of $P$ is $1$, compute the least possible volume of $P$.

On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·

Let $ABC$ be an equilateral triangle of side length $15.$ Let $A_b$ and $B_a$ be points on side $AB,$ $A_c$ and $C_a$ be points on $AC,$ and $B_c$ and $C_b$ be points on $BC$ such that $\triangle{AA_bA_c}, \triangle{BB_cB_a},$ and $\triangle{CC_aC_b}$ are equilateral triangles with side lengths $3,4,$ and $5,$ respectively. Compute the radius of the circle tangent to segments $\overline{A_bA_c}, \overline{B_aB_c},$ and $\overline{C_aC_b}.$

Prove that for any integer $n$ greater than $5$, a square can be divided into $n$ squares.

Source: 1976 Euclid Part A Problem 9 ----- A circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. The measure of the angle subtended at the centre of the circle by the shortest side is $\textbf{(A) } 30 \qquad \textbf{(B) } 45 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } \text{none of these}$

A pile of $2020$ stones is given. Arnaldo and Bernaldo play the following game: In each move, it is allowed to remove $1, 4, 16, 64, ...$ (any power of $4$) stones from the pile. They make their moves alternately, and the player who can no longer play loses. If Arnaldo is the first to play, who has the winning strategy?

The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are $\textbf{(A) }61,63\qquad\textbf{(B) }61,65\qquad\textbf{(C) }63,65\qquad\textbf{(D) }63,67\qquad \textbf{(E) }67,69$

A tortoise walks $60$ meters per hour and a lizard walks at $240$ meters per hour. There is a rectangle $ABCD$ where $AB =60$ and $AD =120$. Both start from the vertex $A$ and in the same direction ($A \to B \to D \to A$), crossing the edge of the rectangle. The lizard has the habit of advancing two consecutive sides of the rectangle, turning to go back one, turning to go forward two, turning to go back one and so on. How many times and in what places do the tortoise and the lizard meet when the tortoise completes its third turn?

Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.

Prove that for every convex polygon, we can choose three of its consecutive vertices, such that the circle, defined by them, covers the the entire polygon. (proposed by J. Tabov)

Solve the equation, $$\sqrt{x+5}+\sqrt{16-x^2}=x^2-25$$

$p(m)$ is the number of distinct prime divisors of a positive integer $m>1$ and $f(m)$ is the $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$ th smallest prime divisor of $m$. Find all positive integers $n$ satisfying the equation: $$f(n^2+2) + f(n^2+5) = 2n-4$$