How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?

Let $\triangle ABC$ be an acute angle triangle and $\omega$ be its circumcircle. Let $N$ be a point on arc $AC$ not containing $B$ and $S$ be a point on line $AB$. The line tangent to $\omega$ at $N$ intersects $BC$ at $T$, $NS$ intersects $\omega$ at $K$. Assume that $\angle NTC = \angle KSB$. Prove that $CK\parallel AN \parallel TS$.

The expression $\dfrac{\dfrac{a}{a+y}+\dfrac{y}{a-y}}{\dfrac{y}{a+y}-\dfrac{a}{a-y}}$, a real, $a\neq 0$, has the value $-1$ for: $\textbf{(A)}\ \text{all but two real values of }y \qquad \textbf{(B)}\ \text{only two real values of }y \qquad$ $\textbf{(C)}\ \text{all real values of }y \qquad \textbf{(D)}\ \text{only one real value of }y \qquad \textbf{(E)}\ \text{no real values of }y$

Find all integers $n\ge 3$ for which the following statement is true: Any arithmetic progression $a_1,\ldots ,a_n$ with $n$ terms for which $a_1+2a_2+\ldots+na_n$ is rational contains at least one rational term.

Let $\mathcal{F}$ be the set of polynomials $f(x)$ with integer coefficients for which there exists an integer root of the equation $f(x)=1$. For all $k>1$, let $m_k$ be the smallest integer greater than one for which there exists $f(x)\in \mathcal{F}$ such that $f(x)=m_k$ has exactly $k$ distinct integer roots. If the value of $\sqrt{m_{2021}-m_{2020}}$ can be written as $m\sqrt{n}$ for positive integers $m,n$ where $n$ is squarefree, compute the largest integer value of $k$ such that $2^k$ divides $\frac{m}{n}$.

Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal. [i]David Yang.[/i]

Find the smallest positive number $\lambda$ such that for an arbitrary $12$ points on the plane $P_1,P_2,...P_{12}$ (points may coincide), with distance between arbitrary two of them does not exceeds $1$, holds the inequality $\sum_{1\le i\le j\le 12} P_iP_j^2 \le \lambda$

Prove that for all integers $n > 2$, \[ 3| \sum\limits_{i=0}^{[n/3]} (-1)^i C _n ^{3i} \]

[b]p1.[/b] Consider a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$ such that the perimeter of the right triangle is numerically (ignoring units) equal to its area. Prove that there is only one possible value of $a + b - c$, and determine that value. [b]p2.[/b] Last August, Jennifer McLoud-Mann, along with her husband Casey Mann and an undergraduate David Von Derau at the University of Washington, Bothell, discovered a new tiling pattern of the plane with a pentagon. This is the fifteenth pattern of using a pentagon to cover the plane with no gaps or overlaps. It is unknown whether other pentagons tile the plane, or even if the number of patterns is finite. Below is a portion of this new tiling pattern. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS8xLzM4M2RjZDEzZTliYTlhYTJkZDU4YTA4ZGMwMTA0MzA5ODk1NjI0LnBuZw==&rn=bW1wYyAyMDE1LnBuZw==[/img] Determine the five angles (in degrees) of the pentagon $ABCDE$ used in this tiling. Explain your reasoning, and give the values you determine for the angles at the bottom. [b]p3.[/b] Let $f(x) =\sqrt{2019 + 4\sqrt{2015}} +\sqrt{2015} x$. Find all rational numbers $x$ such that $f(x)$ is a rational number. [b]p4.[/b] Alice has a whiteboard and a blackboard. The whiteboard has two positive integers on it, and the blackboard is initially blank. Alice repeats the following process. $\bullet$ Let the numbers on the whiteboard be $a$ and $b$, with $a \le b$. $\bullet$ Write $a^2$ on the blackboard. $\bullet$ Erase $b$ from the whiteboard and replace it with $b - a$. For example, if the whiteboard began with 5 and 8, Alice first writes $25$ on the blackboard and changes the whiteboard to $5$ and $3$. Her next move is to write $9$ on the blackboard and change the whiteboard to $2$ and $3$. Alice stops when one of the numbers on the whiteboard is 0. At this point the sum of the numbers on the blackboard is $2015$. a. If one of the starting numbers is $1$, what is the other? b. What are all possible starting pairs of numbers? [b]p5.[/b] Professor Beatrix Quirky has many multi-volume sets of books on her shelves. When she places a numbered set of $n$ books on her shelves, she doesn’t necessarily place them in order with book $1$ on the left and book $n$ on the right. Any volume can be placed at the far left. The only rule is that, except the leftmost volume, each volume must have a volume somewhere to its left numbered either one more or one less. For example, with a series of six volumes, Professor Quirky could place them in the order $123456$, or $324561$, or $564321$, but not $321564$ (because neither $4$ nor $6$ is to the left of $5$). Let’s call a sequence of numbers a [i]quirky [/i] sequence of length $n$ if: 1. the sequence contains each of the numbers from $1$ to $n$, once each, and 2. if $k$ is not the first term of the sequence, then either $k + 1$ or $k - 1$ occurs somewhere before $k$ in the sequence. Let $q_n$ be the number of quirky sequences of length $n$. For example, $q_3 = 4$ since the quirky sequences of length $3$ are $123$, $213$, $231$, and $321$. a. List all quirky sequences of length $4$. b. Find an explicit formula for $q_n$. Prove that your formula is correct. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\omega_1$ and $\omega_2$ be two orthogonal circles, and let the center of $\omega_1$ be $O$. Diameter $AB$ of $\omega_1$ is selected so that $B$ lies strictly inside $\omega_2$. The two circles tangent to $\omega_2$, passing through $O$ and $A$, touch $\omega_2$ at $F$ and $G$. Prove that $FGOB$ is cyclic. [i]Proposed by Eric Chen[/i]

Determine the least $n\in\mathbb{N}$ such that $n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n$ has at least $2010$ positive factors.

For any real number$m$, the equation $$x^2+(m-2)x- (m+3)=0$$ has two solutions, denoted $x_1 $and $ x_2$. Determine $m$ such that $x_1^2+x_2^2$ is the minimum possible.

Let $S_n$ be the number of subsets of the first $n$ positive integers that have the same number of even values and odd values; the empty set counts as one of these subsets. Compute the smallest positive integer $n$ such that $S_n$ is a multiple of $2020$.

Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$. What is the value of \[x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?\] $\textbf{(A)}\ 360\qquad\textbf{(B)}\ 400\qquad\textbf{(C)}\ 420\qquad\textbf{(D)}\ 440\qquad\textbf{(E)}\ 480$

A frog starts at $0$ on a number line and plays a game. On each turn the frog chooses at random to jump $1$ or $2$ integers to the right or left. It stops moving if it lands on a nonpositive number or a number on which it has already landed. If the expected number of times it will jump is $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Michael Kural[/i]

Let $a, b, c \geq 1$. Prove that $$\frac{1}{1+a^2}+\frac{1}{1+b^2}+\frac{1}{1+c^2}\geq\frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca}.$$