A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$. What is the least number of dominoes that are entirely inside some square $2 \times 2$?

Positive integers $m$ and $n$ have no common divisor greater than one. What is the largest possible value of the greatest common divisor of $m + 2000n$ and $n + 2000m$ ? (S Zlobin)

In a tournament, $n$ players participate. Each player plays each other exactly once, with no ties. A player $A$ is said to be [i]champion [/i] if, for every other player $B$, one of the following two situations occurs: (a) $A$ beat $B$; (b) $A$ beat a player $C$ who in turn beat $B$. Prove that in such a tournament there cannot be exactly two champions.

Find $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\ _n C_k.$

[b]p1.[/b] Compute $21 \cdot 21 - 20 \cdot 20$. [b]p2.[/b] A square has side length $2$. If the square is scaled by a factor of $n$, the perimeter of the new square is equal to the area of the original square. Find $10n$. [b]p3.[/b] Kevin has $2$ red marbles and $2$ blue marbles in a box. He randomly grabs two marbles. The probability that they are the same color can be expressed as $\frac{a}{b}$ for relatively prime integers $a$ and $b$. Find $a +b$. [b]p4.[/b] In a classroom, if the teacher splits the students into groups of $3$ or $4$, there is one student left out. If the students formgroups of $5$, every student is in a group. What is the fewest possible number of students in this classroom? [b]p5.[/b] Find the sum of all positive integer values of $x$ such that $\lfloor \sqrt{x!} \rfloor = x$. [b]p6.[/b] Find the number of positive integer factors of $2021^{(2^0+2^1)} \cdot 1202^{(1^2+0^2)}$. [b]p7.[/b] Let $n$ be the number of days over a $13$ year span. Find the difference between the greatest and least possible values of $n$. Note: All years divisible by $4$ are leap years unless they are divisible by 100 but not $400$. For example, $2000$ and $2004$ are leap years, but $1900$ is not. [b]p8.[/b] In isosceles $\vartriangle ABC$, $AB = AC$, and $\angle ABC = 72^o$. The bisector of $\angle ABC$ intersects $AC$ at $D$. Given that $BC = 30$, find $AD$. [b]p9.[/b] For an arbitrary positive value of $x$, let $h$ be the area of a regular hexagon with side length $x$ and let $s$ be the area of a square with side length $x$. Find the value of $\left \lfloor \frac{10h}{s} \right \rfloor$. [b]p10.[/b] There is a half-full tub of water with a base of $4$ inches by $5$ inches and a height of $8$ inches. When an infinitely long stick with base $1$ inch by $1$ inch is inserted vertically into the bottom of the tub, the number of inches the water level rises by can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p11.[/b] Find the sum of all $4$-digit numbers with digits that are a permutation of the digits in $2021$. Note that positive integers cannot have first digit $0$. [b]p12.[/b] A $10$-digit base $8$ integer is chosen at random. The probability that it has $30$ digits when written in base $2$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p13.[/b] Call a natural number sus if it can be expressed as $k^2 +k +1$ for some positive integer $k$. Find the sum of all sus integers less than $2021$. [b]p14.[/b] In isosceles triangle $ABC$, $D$ is the intersection of $AB$ and the perpendicular to $BC$ through $C$. Given that $CD = 5$ and $AB = BC = 1$, find $\sec^2 \angle ABC$. [b]p15.[/b] Every so often, the minute and hour hands of a clock point in the same direction. The second time this happens after 1:00 is a b minutes later, where a and b are relatively prime positive integers. Find a +b. [b]p16.[/b] The $999$-digit number $N = 123123...123$ is composed of $333$ iterations of the number $123$. Find the least nonnegative integerm such that $N +m$ is a multiple of $101$. [b]p17.[/b] The sum of the reciprocals of the divisors of $2520$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p18.[/b] Duncan, Paul, and $6$ Atreides guards are boarding three helicopters. Duncan, Paul, and the guards enter the helicopters at random, with the condition that Duncan and Paul do not enter the same helicopter. Note that not all helicoptersmust be occupied. The probability that Paul has more guards with him in his helicopter than Duncan does can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p19.[/b] Let the minimum possible distance from the origin to the parabola $y = x^2 -2021$ be $d$. The value of d2 can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p20.[/b] In quadrilateral $ABCD$ with interior point $E$ and area $49 \sqrt3$, $\frac{BE}{CE}= 2 \sqrt3$, $\angle ABC = \angle BCD = 90^o$, and $\vartriangle ABC \sim \vartriangle BCD \sim \vartriangle BEC$. The length of $AD$ can be expressed aspn where $n$ is a positive integer. Find $n$. [b]p21.[/b] Find the value of $$\sum^{\infty}_{i=1}\left( \frac{i^2}{2^{i-1}}+\frac{i^2}{2^{i}}+\frac{i^2}{2^{i+1}}\right)=\left( \frac{1^2}{2^{0}}+\frac{1^2}{2^{1}}+\frac{1^2}{2^{2}}\right)+\left( \frac{2^2}{2^{1}}+\frac{2^2}{2^{2}}+\frac{2^2}{2^{3}}\right)+\left( \frac{3^2}{2^{2}}+\frac{2^2}{2^{3}}+\frac{2^2}{2^{4}}\right)+...$$ [b]p22.[/b] Five not necessarily distinct digits are randomly chosen in some order. Let the probability that they form a nondecreasing sequence be $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a +b$ is divided by$ 1000$. [b]p23.[/b] Real numbers $a$, $b$, $c$, and d satisfy $$ac -bd = 33$$ $$ad +bc = 56.$$ Given that $a^2 +b^2 = 5$, find the sum of all possible values of $c^2 +d^2$. [b]p24.[/b] Jeff has a fair tetrahedral die with sides labeled $0$, $1$, $2$, and $3$. He continuously rolls the die and record the numbers rolled in that order. For example, if he rolls a $1$, then rolls a $2$, and then rolls a $3$, he writes down $123$. He keeps rolling the die until he writes the substring $2021$. What is the expected number of times he rolls the die? [b]p25.[/b] In triangle $ABC$, $BC = 2\sqrt3$, and $AB = AC = 4\sqrt3$. Circle $\omega$ with center $O$ is tangent to segment $AB$ at $T$ , and $\omega$ is also tangent to ray $CB$ past $B$ at another point. Points $O, T$ , and $C$ are collinear. Let $r$ be the radius of $\omega$. Given that $r^2 = \frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $M$ be the set of all functions $f$ with the following properties: [b](i)[/b] $f$ is defined for all real numbers and takes only real values. [b](ii)[/b] For all $x, y \in \mathbb R$ the following equality holds: $f(x)f(y) = f(x + y) + f(x - y).$ [b](iii)[/b] $f(0) \neq 0.$ Determine all functions $f \in M$ such that [b](a)[/b] $f(1)=\frac 52$, [b](b)[/b] $f(1)= \sqrt 3$.

Define a sequence by $a_0=1$, together with the rules $a_{2n+1}=a_n$ and $a_{2n+2}=a_n+a_{n+1}$ for each integer $n\ge0$. Prove that every positive rational number appears in the set $ \left\{ \tfrac {a_{n-1}}{a_n}: n \ge 1 \right\} = \left\{ \tfrac {1}{1}, \tfrac {1}{2}, \tfrac {2}{1}, \tfrac {1}{3}, \tfrac {3}{2}, \cdots \right\} $.

Let $ABCD$ be a rhombus with $\angle{ADC} = 46^{\circ}$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle{BFC}$? $\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 111 \qquad \textbf{(C)}\ 112 \qquad \textbf{(D)}\ 113 \qquad \textbf{(E)}\ 114$

There are $998$ cities in a country. Some pairs of cities are connected by two-way flights. According to the law, between any pair cities should be no more than one flight. Another law requires that for any group of cities there will be no more than $5k+10$ flights connecting two cities from this group, where $k$ is the number number of cities in the group. Prove that several new flights can be introduced so that laws still hold and the total number of flights in the country is equal to $5000$.

Circles $c_1$ and $c_2$ with centres $O_1$and $O_2$, respectively, intersect at points $A$ and $B$ so that the centre of each circle lies outside the other circle. Line $O_1A$ intersects circle $c_2$ again at point $P_2$ and line $O_2A$ intersects circle $c_1$ again at point $P_1$. Prove that the points $O_1,O_2, P_1, P_2$ and $B$ are concyclic

Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.

Prove that for every odd prime number $p{}$, the following congruence holds \[\sum_{n=1}^{p-1}n^{p-1}\equiv (p-1)!+p\pmod{p^2}.\]

The midpoint of the side $AB$ in the triangle $ABC$ is called $C'$. A point on the side $BC$ is called $D$, and $E$ is the point of intersection of $AD$ and $CC'$. Assume that $AE/ED = 2$. Show that $D$ is the midpoint of $BC$.

A triangle is said to be the Heron triangle if it has integer sides and integer area. In a Heron triangle, the sides a; b; c satisfy the equation b=a(a-c). Prove that the triangle is isosceles.

The digits $1$, $2$, $3$, $4$, and $5$ are each used once to write a five-digit number $PQRST$. The three-digit number $PQR$ is divisible by $4$, the three-digit number $QRS$ is divisible by $5$, and the three-digit number $RST$ is divisible by $3$. What is $P$? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$

The length of the longest side of a triangle is $1$. Prove that three circles of radius $\frac{1}{\sqrt3}$ with centers at the vertices cover the entire triangle.

Consider $N$ points in the plane such that the area of a triangle formed by any three of the points does not exceed $1$. Prove that there is a triangle of area not more than $4$ that contains all $N$ points.

If $a+b+c=12$ and $a^2+b^2+c^2=62$, what is $ab+bc+ac$?

A circle passing through the vertices $A$ and $B$ of a cyclic quadrilateral $ABCD$ intersects diagonals $AC$ and $BD$ at $E$ and $F$, respectively. The lines $AF$ and $BC$ meet at a point $P$, and the lines $BE$ and $AD$ meet at a point $Q$. Prove that $PQ$ is parallel to $CD$.

(a) Do there exist 2009 distinct positive integers such that their sum is divisible by each of the given numbers? (b) Do there exist 2009 distinct positive integers such that their sum is divisible by the sum of any two of the given numbers?

A triple $(a,b,c)$ of positive integers is called [b]strong[/b] if the following holds: for each integer $m>1$, the number $a+b+c$ does not divide $a^m+b^m+c^m$. The [b]sum[/b] of a strong triple $(a,b,c)$ is defined as $a+b+c$. Prove that there exists an infinite collection of strong triples, the sums of which are all pairwise coprime.

A triangle has semi-perimeter $s$, circumradius $R$ and inradius $r$. Show that it is right-angled iff $2R = s - r$.

At each step, a rectangular tile of length $1, 2$, or, $3$ is chosen at random, what is the probability that the total length is $10$ after $5$ steps?

Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA},\overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$ $\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}$

A circle is somehow divided by $3k$ points into $k$ arcs of lengths $1, 2$ and $3$ each. Prove that two of these points are always diametrically opposite.