A deck of $52$ cards is stacked in a pile facing down. Tom takes the small pile consisting of the seven cards on the top of the deck, turns it around, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down, since the seven cards at the bottom now face up. Tom repeats this move until all cards face down again. In total, how many moves did Tom make?

Are there infinitely many natural numbers $n$ such that the sum of 2019th powers of the digits of $n$ is equal to $n$ ? [b]You don't need to find any such $n$. Just provide mathematical justification if you think there are infinitely many or finitely many such natural numbers[/b]

Suppose that $W,W_1$ and $W_2$ are subspaces of a vector space $V$ such that $V=W_1\oplus W_2$. Under what conditions we have $W=(W\cap W_1)\oplus(W\cap W_2)$?

We are given $2001$ lines in the plane, no two of which are parallel and no three of which are concurrent. These lines partition the plane into regions (not necessarily finite) bounded by segments of these lines. These segments are called [i]sides[/i], and the collection of the regions is called a [i]map[/i]. Intersection points of the lines are called [i]vertices[/i]. Two regions are [i]neighbors [/i]if they share a side, and two vertices are neighbors if they lie on the same side. A [i]legal coloring[/i] of the regions (resp. vertices) is a coloring in which each region (resp. vertex) receives one color, such that any two neighboring regions (vertices) have different colors. (a) What is the minimum number of colors required for a legal coloring of the regions? (b) What is the minimum number of colors required for a legal coloring of the vertices?

Carl has $5$ cubes each having side length $1$, and Kate has $5$ cubes each having side length $2$. What is the total volume of the $10$ cubes? $\textbf{(A) }24 \qquad \textbf{(B) }25 \qquad \textbf{(C) } 28\qquad \textbf{(D) } 40\qquad \textbf{(E) } 45$

Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?

What is the largest number of squares on $9 \times 9$ square board that can be cut along their both diagonals so that the board does not fall apart into several pieces?

The point $ I $ is the center of the inscribed circle of the triangle $ ABC $. The points $ B_1 $ and $ C_1 $ are the midpoints of the sides $ AC $ and $ AB $, respectively. It is known that $ \angle BIC_1 + \angle CIB_1 = 180^\circ $. Prove the equality $ AB + AC = 3BC $

In a circumference of a circle, $ 10000$ points are marked, and they are numbered from $ 1$ to $ 10000$ in a clockwise manner. $ 5000$ segments are drawn in such a way so that the following conditions are met: 1. Each segment joins two marked points. 2. Each marked point belongs to one and only one segment. 3. Each segment intersects exactly one of the remaining segments. 4. A number is assigned to each segment that is the product of the number assigned to each end point of the segment. Let $ S$ be the sum of the products assigned to all the segments. Show that $ S$ is a multiple of $ 4$.

Two circles with centers $O_1$ and $O_2$ meet at points $A$ and $B$. The bisector of angle $O_1AO_2$ meets the circles for the second time at points $C $and $D$. Prove that the distances from the circumcenter of triangle $CBD$ to $O_1$ and to $O_2$ are equal.

Find $\lim_{n\to\infty}\sum_{k=0}^{\infty} \int_{2k\pi}^{(2k+1)\pi} xe^{-x}\sin x\ dx.$

The average age of the $6$ people in Room A is $40$. The average age of the $4$ people in Room B is $25$. If the two groups are combined, what is the average age of all the people? $\textbf{(A)}\ 32.5 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 33.5 \qquad \textbf{(D)}\ 34\qquad \textbf{(E)}\ 35$

For every $ n$ the sum of $ n$ terms of an arithmetic progression is $ 2n \plus{} 3n^2$. The $ r$th term is: $ \textbf{(A)}\ 3r^2 \qquad \textbf{(B)}\ 3r^2 \plus{} 2r \qquad \textbf{(C)}\ 6r \minus{} 1 \qquad \textbf{(D)}\ 5r \plus{} 5 \qquad \textbf{(E)}\ 6r \plus{} 2 \qquad$

Let $ P(x)$ be the real polynomial function, $ P(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d.$ Prove that if $ |P(x)| \leq 1$ for all $ x$ such that $ |x| \leq 1,$ then \[ |a| \plus{} |b| \plus{} |c| \plus{} |d| \leq 7.\]

Let $ABCD$ be a quadrilateral and $P$ a point in the plane of the quadrilateral. Let $M,N$ be on the sides $AC,BD$ respectively such that $PM \parallel BC, PN \parallel AD$. $AC$ meets $BD$ at $E$. Prove that the orthocenter of triangles $EBC, EAD, EMN$ are collinear if and only if $P$ is on the line $AB$. Đỗ Thanh Sơn PS. Instead of the word [b]collinear[/b], it was written [b]concurrent[/b], probably a typo.

Two players in turns color the sides of an $n$-gon. The first player colors any side that has $0$ or $2$ common vertices with already colored sides. The second player colors any side that has exactly $1$ common vertex with already colored sides. The player who cannot move, loses. For which $n$ the second player has a winning strategy?

Let $x, y \in \mathbb{R}^+$. Prove that: \[ \left( 1 + \frac{1}{x} \right) \left( 1 + \frac{1}{y} \right) \geq \left( 1 + \frac{2}{x + y} \right)^2. \]

Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

Let $f : \mathbb{R}^+\to\mathbb{R}$ satisfying $f(x)=f(x+2)+2f(x^2+2x)$. Prove that if for all $x>1400^{2021}$, $xf(x) \le 2021$, then $xf(x) \le 2021$ for all $x \in \mathbb {R}^+$ Proposed by [i]Navid Safaei[/i]

Let $\mathcal{C}$ be a circle and let $P$ and $Q$ be points inside $\mathcal{C}$. Prove that there are infinitely many circle through $P$ and $Q$ that are completely contained inside of $\mathcal{C}$.

In each of the $9$ small circles of the following figure we write positive integers less than $10$, without repetitions. In addition, it is true that the sum of the $5$ numbers located around each one of the $3$ circles is always equal to $S$. Find the largest possible value of $S$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/2db2c1ac7f45022606fb0099f24e6287977d10.png[/img]

May the points of a disc of radius $1$ (including its circumference) be partitioned into three subsets in such a way that no subset contains two points separated by a distance $1$?

For a convex $n$-gon $P_n$, we say that a convex quadrangle $Q$ is a [i]diagonal-quadrangle[/i] of $P_n$, if its vertices are vertices of $P_n$ and its sides are diagonals of $P_n$. Let $d_n$ be the number of diagonal-quadrangles of a convex $n$-gon. Determine $d_n$ for all $n\geq 8$.

The expression $ \sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}}$ is equal to: $ \textbf{(A)}\ \frac{\sqrt{3}}{6} \qquad \textbf{(B)}\ \frac{-\sqrt{3}}{6} \qquad \textbf{(C)}\ \frac{\sqrt{-3}}{6} \qquad \textbf{(D)}\ \frac{5 \sqrt{3}}{6} \qquad \textbf{(E)}\ 1$

Let $a$ and $b$ be real numbers greater than 1 such that $ab=100$. The maximum possible value of $a^{(\log_{10}b)^2}$ can be written in the form $10^x$ for some real number $x$. Find $x$.