Let $a_0$ be a positive integer. Define the sequence $\{a_n\}_{n \geq 0}$ as follows: if \[ a_n = \sum_{i = 0}^jc_i10^i \] where $c_i \in \{0,1,2,\cdots,9\}$, then \[ a_{n + 1} = c_0^{2005} + c_1^{2005} + \cdots + c_j^{2005}. \] Is it possible to choose $a_0$ such that all terms in the sequence are distinct?

If $i^2 = -1$, then the sum \[ \cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} \] \[ + \cdots + i^{40}\cos{3645^\circ} \] equals \[ \text{(A)}\ \frac{\sqrt{2}}{2} \qquad \text{(B)}\ -10i\sqrt{2} \qquad \text{(C)}\ \frac{21\sqrt{2}}{2} \] \[ \text{(D)}\ \frac{\sqrt{2}}{2}(21 - 20i) \qquad \text{(E)}\ \frac{\sqrt{2}}{2}(21 + 20i) \]

For an acute triangle $ABC$, let $O$ be its circumcenter, and let $O_A,O_B,O_C$ be the circumcenter of $BCO,CAO,ABO$ respectively. Show that $AO_A,BO_B,CO_C$ are concurrent.

A common external tangent to circles $\omega_1$ and $\omega_2$ touches them at points $T_1, T_2$ respectively. Let $A$ be an arbitrary point on the extension of $T_1T_2$ beyond $T_1$, and $B$ be a point on the extension of $T_1T_2$ beyond $T_2$ such that $AT_1 = BT_2$. The tangents from $A$ to $\omega_1$ and from $B$ to $\omega_2$ distinct from $T_1T_2$ meet at point $C$. Prove that all nagelians of triangles $ABC$ from $C$ have a common point.

[b]p1.[/b] Compute $45\times 45 - 6$. [b]p2.[/b] Consecutive integers have nice properties. For example, $3$, $4$, $5$ are three consecutive integers, and $8$, $9$, $10$ are three consecutive integers also. If the sum of three consecutive integers is $24$, what is the smallest of the three numbers? [b]p3.[/b] How many positive integers less than $25$ are either multiples of $2$ or multiples of $3$? [b]p4.[/b] Charlotte has $5$ positive integers. Charlotte tells you that the mean, median, and unique mode of his five numbers are all equal to $10$. What is the largest possible value of the one of Charlotte's numbers? [b]p5.[/b] Mr. Meeseeks starts with a single coin. Every day, Mr. Meeseeks goes to a magical coin converter where he can either exchange $1$ coin for $5$ coins or exchange $5$ coins for $3$ coins. What is the least number of days Mr. Meeseeks needs to end with $15$ coins? [b]p6.[/b] Twelve years ago, Violet's age was twice her sister Holo's age. In $7$ years, Holo's age will be $13$ more than a third of Violet's age. $3$ years ago, Violet and Holo's cousin Rindo's age was the sum of their ages. How old is Rindo? [b]p7.[/b] In a $2 \times 3$ rectangle composed of $6$ unit squares, let $S$ be the set of all points $P$ in the rectangle such that a unit circle centered at $P$ covers some point in exactly $3$ of the unit squares. Find the area of the region $S$. For example, the diagram below shows a valid unit circle in a $2 \times 3$ rectangle. [img]https://cdn.artofproblemsolving.com/attachments/d/9/b6e00306886249898c2bdb13f5206ced37d345.png[/img] [b]p8.[/b] What are the last four digits of $2^{1000}$? [b]p9.[/b] There is a point $X$ in the center of a $2 \times 2 \times 2$ box. Find the volume of the region of points that are closer to $X$ than to any of the vertices of the box. [b]p10.[/b] Evaluate $\sqrt{37 \cdot 41 \cdot 113 \cdot 290 - 4319^2}$ [b]p11.[/b] (Estimation) A number is abundant if the sum of all its divisors is greater than twice the number. One such number is $12$, because $1+2+3+4+6+12 = 28 > 24$: How many abundant positive integers less than $20190$ are there? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sidelengths of a polygon with $1994$ sides are $a_{i}=\sqrt{i^2 +4}$ $ \; (i=1,2,\cdots,1994)$. Prove that its vertices are not all on lattice points.

Find all natural numbers $a$ such that for every positive integer $n$ the number $n(a+n)$ is not a perfect square.

The intersection point $I$ of the angles bisectors of the triangle $ABC$ has reflections the points $P,Q,T$ wrt the triangle's sides . It turned out that the circle $s$ circumscribed around of the triangle $PQT$ , passes through the vertex $A$. Find the radius of the circumscribed circle of triangle $ABC$ if $BC = a$. (Gryhoriy Filippovskyi)

Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has $$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$ where $\{ x \}$ denotes the fractional part of $x$.

Find the value(s) of $ x$ such that $ 8xy\minus{}12y\plus{}2x\minus{}3\equal{}0$ is true for all values of $ y$. $ \textbf{(A)}\ \frac{2}{3} \qquad \textbf{(B)}\ \frac{3}{2}\text{ or }\minus{}\frac{1}{4} \qquad \textbf{(C)}\ \minus{}\frac{2}{3}\text{ or }\minus{}\frac{1}{4} \qquad \textbf{(D)}\ \frac{3}{2} \qquad \textbf{(E)}\ \minus{}\frac{3}{2}\text{ or }\minus{}\frac{1}{4}$

Define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. Define $\{x \} = x - \lfloor x \rfloor$. For example, $\{ 3 \} = 3-3 = 0$, $\{ \pi \} = \pi - 3$, and $\{ - \pi \} = 4-\pi$. If $\{n\} + \{ 3n\} = 1.4$, then find the sum of all possible values of $100\{n\}$. [i]Proposed by Isabella Grabski [/i]

$A, B, C, D$ are points along a circle, in that order. $AC$ intersects $BD$ at $X$. If $BC=6$, $BX=4$, $XD=5$, and $AC=11$, find $AB$

On a $16 \times 16$ torus as shown all $512$ edges are colored red or blue. A coloring is [i]good [/i]if every vertex is an endpoint of an even number of red edges. A move consists of switching the color of each of the $4$ edges of an arbitrary cell. What is the largest number of good colorings so that none of them can be converted to another by a sequence of moves?

Assume that $a,b$ are integers and $n$ is a natural number. $2^na+b$ is a perfect square for every $n$.Prove that $a=0$.

In $4$-dimensional space, a set of $1 \times 2 \times 3 \times 4$ bricks is given. Decide whether it is possible to build boxes of the following sizes using these bricks: [list]i) $2 \times 5 \times 7 \times 12$ ii) $5 \times 5 \times 10 \times 12$ iii) $6 \times 6 \times 6 \times 6$.[/list]

Let the function $f$ be defined on the set $\mathbb{N}$ such that $\text{(i)}\ \ \quad f(1)=1$ $\text{(ii)}\ \quad f(2n+1)=f(2n)+1$ $\text{(iii)}\quad f(2n)=3f(n)$ Determine the set of values taken $f$.

Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points. [b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$. [b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.

Show that it is possible to write a $n \times n$ array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.

In the lower left corner of the square $7 \times 7$ board there is a king. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different ways can the king get to the upper right corner of the board if he is prohibited from visiting the central square?

A black witch's hat is in the classic shape of a cone on top of a circular brim. The cone has a slant height of $18$ inches and a base radius of $3$ inches. The brim has a radius of $5$ inches. What is the total surface area of the hat?

In a group of $2021$ people, $1400$ of them are $\emph{saboteurs}$. Sherlock wants to find one saboteur. There are some missions that each needs exactly $3$ people to be done. A mission fails if at least one of the three participants in that mission is a saboteur! In each round, Sherlock chooses $3$ people, sends them to a mission and sees whether it fails or not. What is the minimum number of rounds he needs to accomplish his goal?

How many ordered triples of integers $(a, b, c)$ satisfy the following system? $$ \begin{cases} ab + c &= 17 \\ a + bc &= 19 \end{cases} $$ $$ \mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6 $$

Ten cars are moving at the road. There are some cities at the road. Each car is moving with some constant speed through cities and with some different constant speed outside the cities (different cars may move with different speed). There are 2011 points at the road. Cars don't overtake at the points. Prove that there are 2 points such that cars pass through these points in the same order. [i]S. Berlov[/i]

Let $f:(0;\infty) -- (0;\infty)$ such that $f(x^y)=(f(x))^{f(y)}$. Prove $f(xy)=f(x)f(y)$ and $f(x+y)=f(x)+f(y)$ for all positive real $x,y$.

For $ a,b,c\geq 0$ with $ a\plus{}b\plus{}c\equal{}1$, prove that $ \sqrt{a\plus{}\frac{(b\minus{}c)^2}{4}}\plus{}\sqrt{b}\plus{}\sqrt{c}\leq \sqrt{3}$