Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$, and $AB$ its shortest side. Let $H$ be the orthocenter of $ABC$. Let $\Gamma$ be the circle with center $B$ and radius $BA$. Let $D$ be the second point where the line $CA$ meets $\Gamma$. Let $E$ be the second point where $\Gamma$ meets the circumcircle of the triangle $BCD$. Let $F$ be the intersection point of the lines $DE$ and $BH$. Prove that the line $BD$ is tangent to the circumcircle of the triangle $DFH$.

A 9-digit number $N$ is given, whose digits are non-zero and all different.The sums of all consecutive three-digit segments in the decimal representation of number $N$ are calculated and arranged in increasing order.Is it possible to obtain the following sequences as a result of this operation? $\text{a)}$ $11,15,16,18,19,21,22$ $\text{b)}$ $11,15,16,18,19,21,23$

Let $ABC$ be a triangle such that such that $AB=14, BC=13$, and $AC=15$. Let $X$ be a point inside triangle $ABC$. Compute the minimum possible value of $(\sqrt{2}AX+BX+CX)^2$.

Two cells in a $20 \times 20$ board are adjacent if they have a common edge (a cell is not considered adjacent to itself). What is the maximum number of cells that can be marked in a $20 \times 20$ board such that every cell is adjacent to at most one marked cell?

Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$ (1) Find the condition for which $C_1$ is inscribed in $C_2$. (2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$. (3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$. 60 point

Peter and Basil play the following game on a horizontal table $1\times{2019}$. Initially Peter chooses $n$ positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by $s$ cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by $s$ cells neither to the left, nor to the right, the coin stays in the current cell. Find the least $n$ such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays.

Define a sequence of integers by $T_1 = 2$ and for $n\ge2$, $T_n = 2^{T_{n-1}}$. Find the remainder when $T_1 + T_2 + \cdots + T_{256}$ is divided by 255. [i]Ray Li.[/i]

Let $a_1 < a_2< ...< a_n$ are real numbers, $$f(x) = \sum_{i=1}^n|x-a_i|,$$ for $n$ even. Find the minimum of this function.

Find the largest positive integer $m$ which makes it possible to color several cells of a $70\times 70$ table red such that [list] [*] There are no two red cells satisfying: the two rows in which they are have the same number of red cells, while the two columns in which they are also have the same number of red cells; [*] There are two rows with exactly $m$ red cells each. [/list]

For a prime $ p\ge 5, $ determine the number of polynomials $ X^p+pX^k+pX^l+1 $ with $ 1<k<l<p, $ that are ireducible over the integers.

Let $a,b,c,d$ be the natural numbers such that $2^a+4^b+5^c=2014^d.$ Find all $(a,b,c,d).$

Numbers $1,\ldots,2025$ are written in a circle in increasing order. For every three consecutive numbers $i,j,k$ we consider the polynomial $(x-i)(x-j)(x-k)$. Let $s(x)$ be the sum of all $2025$ these polynomials. Prove that $s(x)$ has an integral root. [i]A. Voidelevich[/i]

Prove that for all positive integers $n$ there are positive integers $a,b$ such that $$n\mid 4a^2+9b^2-1.$$

[u]Round 1 [/u] [b]p1.[/b] Ephram was born in May $2005$. How old will he turn in the first year where the product of the digits of the year number is a nonzero perfect square? [b]p2.[/b] Zhao is studying for his upcoming calculus test by reviewing each of the $13$ lectures, numbered Lecture $1$, Lecture $2$, ..., Lecture $13$. For each $n$, he spends $5n$ minutes on Lecture $n$. Which lecture is he reviewing after $4$ hours? [b]p3.[/b] Compute $$\dfrac{3^3 \div 3(3)+3}{\frac{3}{3}}+3!.$$ [u]Round 2 [/u] [b]p4.[/b] At Ingo’s shop, train tickets normally cost $\$2$, but every $5$th ticket costs only $\$1$. At Emmet’s shop, train tickets normally cost $\$3$, but every $5$th ticket is free. Both Ingo and Emmett sold $1000$ tickets. Find the absolute difference between their sales, in dollars. [b]p5.[/b] Ephram paddles his boat in a river with a $4$-mph current. Ephram travels at $10$ mph in still water. He paddles downstream and then turns around and paddles upstream back to his starting position. Find the proportion of time he spends traveling upstream, as a percentage. [b]p6.[/b] The average angle measure of a $13-14-15$ triangle is $m^o$ and the average angle measure of a $5-6-7$ triangle is $n^o$. Find $m-n$. [u]Round 3[/u] [b]p7.[/b] Let $p(x) = x^2 -10x +31$. Find the minimum value of $p(p(x))$ over all real $x$. [b]p8.[/b] Michael H. andMichael Y. are playing a game with $4$ jellybeans. Michael H starts with $3$ of the jellybeans, and Michael Y starts with the remaining $1$. Every minute, a Michael flips a coin, and if heads, Michael H takes a jellybean from Michael Y. If tails, Michael Y takes a jellybean from Michael H. WhicheverMichael gathers all $4$ jellybeans wins. The probability Michael H wins can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p9.[/b] Define the digit-product of a positive integer to be the product of its non-zero digits. Let $M$ denote the greatest five-digit number with a digit-product of $360$, and let $N$ denote the least five-digit number with a digit-product of $360$. Find the digit-product of $M-N$. [u]Round 4 [/u] [b]p10.[/b] Hannah is attending one of the three IdeaMath classes running at LHS, while Alex decides to randomly visit some combination of classes. He won’t visit all three classes, but he’s equally likely to visit any other combination. The probability Alex visits Hannah’s class can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p11.[/b] In rectangle $ABCD$, let $E$ be the intersection of diagonal $AC$ and the circle centered at $A$ passing through $D$. Angle $\angle ACD = 24^o$. Find the measure of $\angle CED$ in degrees. [b]p12.[/b] During his IdeaMath class, Zach writes the numbers $2, 3, 4, 5, 6, 7$, and $8$ on a whiteboard. Every minute, he chooses two numbers $a$ and $b$ from the board, erases them, and writes the number $ab +a +b$ on the board. He repeats this process until there’s only one number left. Find the sum of all possible remaining numbers. [u]Round 5[/u] [b]p13.[/b] In isosceles right $\vartriangle ABC$ with hypotenuse $AC$, Let $A'$ be the point on the extension of $AB$ past $A$ such that $AA' = 1$. Let $C'$ be the point on the extension of $BC$ past vertex $C$ such that $CC' = 2$. Given that the difference of the areas of triangle $A'BC'$ and $ABC$ is $10$, find the area of $ABC$. [b]p14.[/b] Compute the sumof the greatest and least values of $x$ such that $(x^2 -4x +4)^2 +x^2 -4x \le 16$. [b]p15.[/b] Ephram is starting a fan club. At the fan club’s first meeting, everyone shakes hands with everyone else exactly once, except for Ephram, who is extremely sociable and shakes hands with everyone else twice. Given that a total of $2015$ handshakes took place, how many people attended the club’s first meeting? PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167139p28823346]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that if $a$ and $b$ are positive integers such that $a^2 + b^2$ is a multiple of $7^{2009}$, then $ab$ is a multiple of $7^{2010}$.

Characterize all increasing sequences $(s_n)$ of positive reals for which there exists a set $A\subset \mathbb{R}$ with positive measure such that $\lambda(A\cap I)<\frac{s_n}{n}$ holds for every interval $I$ with length $1/n$, where $\lambda$ denotes the Lebesgue measure.