Prove that there exist infinitely many integers $n$ such that $n^2+1$ is squarefree.

A circle $\omega$ has center $O$ and radius $r$. A chord $BC$ of $\omega$ also has length $r$, and the tangents to $\omega$ at $B$ and $C$ meet at $A$. Ray $AO$ meets $\omega$ at $D$ past $O$, and ray $OA$ meets the circle centered at $A$ with radius $AB$ at $E$ past $A$. Compute the degree measure of $\angle DBE$. [i]Author: Ray Li[/i]

In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$? $\textbf{(A) } 2 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 4 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } 7$

Fill in each of the ten boxes with a 3-digit number so that the following conditions are satisfied. [list=1] [*]Every number has three distinct digits that sum to $15$. $0$ may not be a leading digit. One digit of each number has been given to you. [*]No two numbers in any pair of boxes use the same three digits. For example, it is not allowed for two different boxes to have the numbers $456$ and $645$. [*]Two boxes joined by an arrow must have two numbers that share an equal hundreds digit, tens digit, or ones digit. Also, the smaller number must point to the larger.[/list] You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] size(200); defaultpen(linewidth(0.8)); path arrow; pair squares[]={(2,4),(6,4),(10,4),(0,0),(4,0),(8,0),(12,0),(2,-4),(6,-4),(10,-4)}; pair horizarrows[]={(4,4),(2,0),(6,0),(10,0),(4,-4),(8,-4)}; bool isLeft[]={false,false,true,false,false,false}; pair diagarrows[]={(1,2),(7,2),(9,2),(1,-2),(5,-2),(11,-2)}; bool isDown[]={true,false,true,false,false,true}; for(int i=0;i<=9;i=i+1) { draw(box(squares[i]-(1,1),squares[i]+(1,1))); label("$"+(string)i+"$",squares[i]); } for(int j=0;j<=5;j=j+1) { if(isLeft[j]) arrow=(horizarrows[j].x-1,horizarrows[j].y)--(horizarrows[j].x+1,horizarrows[j].y); else arrow=(horizarrows[j].x+1,horizarrows[j].y)--(horizarrows[j].x-1,horizarrows[j].y); draw(arrow,BeginArrow(size=7)); } for(int k=0;k<=5;k=k+1) { if(isDown[k]) arrow=(diagarrows[k].x-1/3,diagarrows[k].y-1)--(diagarrows[k].x+1/3,diagarrows[k].y+1); else arrow=(diagarrows[k].x-1/3,diagarrows[k].y+1)--(diagarrows[k].x+1/3,diagarrows[k].y-1); draw(arrow,BeginArrow(size=7)); } [/asy]

Let $ ABC$ be an isosceles triangle with $ AC \equal{} BC.$ Its incircle touches $ AB$ in $ D$ and $ BC$ in $ E.$ A line distinct of $ AE$ goes through $ A$ and intersects the incircle in $ F$ and $ G.$ Line $ AB$ intersects line $ EF$ and $ EG$ in $ K$ and $ L,$ respectively. Prove that $ DK \equal{} DL.$

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

Let $a, b, m$ be integers such that gcd $(a, b) = 1$ and $5 | ma^2 + b^2$ . Show that there exists an integer $n$ such that $5 | m - n^2$.

Find all pairs $(m,p)$ of natural numbers , such that $p$ is a prime and \[2^mp^2+27\] is the third power of a natural numbers

Two circles $C_1$ and $C_2$ with the respective radii $r_1$ and $r_2$ intersect in $A$ and $B.$ A variable line $r$ through $B$ meets $C_1$ and $C_2$ again at $P_r$ and $Q_r$ respectively. Prove that there exists a point $M,$ depending only on $C_1$ and $C_2,$ such that the perpendicular bisector of each segment $P_rQ_r$ passes through $M.$

Let $P_1$ and $P_2$ be two non-parallel planes in space, and $A$ a point that does not It is in none of them. For each point $X$, let $X_1$ denote its reflection with respect to $P_1$, and $X_2$ its reflection with respect to $P_2$. Determine the locus of points $X$ for the which $X_1, X_2$ and $A$ are collinear.

Prove that the number of pairs $ \left(\alpha,S\right)$ of a permutation $ \alpha$ of $ \{1,2,\dots,n\}$ and a subset $ S$ of $ \{1,2,\dots,n\}$ such that \[ \forall x\in S: \alpha(x)\not\in S\] is equal to $ n!F_{n \plus{} 1}$ in which $ F_n$ is the Fibonacci sequence such that $ F_1 \equal{} F_2 \equal{} 1$

The feet of the altitudes in the triangle $ABC$ are $A', B', C'$. Find the angles of $A'B'C'$ in terms of the angles $A, B, C$. Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$. When is it equal?

Let $\Gamma$ be the incircle of an non-isosceles triangle $ABC$, $I$ be it’s center. Let $A_1, B_1, C_1$ be the tangency points of $\Gamma$ with the sides $BC, AC, AB$, respectively. Let $A_2 = \Gamma \cap AA_1, M = C_1B_1 \cup AI$, $P$ and $Q$ be the other (different from $A_1, A_2$) intersection points of $A_1M, A_2M$ and $\Gamma$, respectively. Prove that $A, P, Q$ are collinear. (A. Voidelevich)

Consider the polynomial $$P\left(t\right)=t^3-29t^2+212t-399.$$ Find the product of all positive integers $n$ such that $P\left(n\right)$ is the sum of the digits of $n$.

Let be a twice-differentiable function $ f:(0,\infty )\longrightarrow\mathbb{R} $ that admits a polynomial function of degree $ 1 $ or $ 2, $ namely, $ \alpha :(0,\infty )\longrightarrow\mathbb{R} $ as its asymptote. Prove the following propositions: [b]a)[/b] $ f''>0\implies f-\alpha >0 $ [b]b)[/b] $ \text{supp} f''=(0,\infty )\wedge f-\alpha >0\implies f''=0 $

Show that \[ \frac{(b+c)(a^4-b^2c^2)}{ab+2bc+ca}+\frac{(c+a)(b^4-c^2a^2)}{bc+2ca+ab}+\frac{(a+b)(c^4-a^2b^2)}{ca+2ab+bc} \geq 0 \] for all positive real numbers $a, \: b , \: c.$

Evaluate $ \int_0^2 \frac{1}{\sqrt {1 \plus{} x^3}}\ dx$.

For $ a\in\mathbb{R}$, let $ M(a)$ be the maximum value of the function $ f(x)\equal{}\int_{0}^{\pi}\sin (x\minus{}t)\sin (2t\minus{}a)\ dt$. Evaluate $ \int_{0}^{\frac{\pi}{2}}M(a)\sin (2a)\ da$.

Let ABCD be a rectangle with $45^o < \angle ADB < 60^o$. The diagonals $AC$ and$ BD$ intersect at $O$. A line passing through $O$ and perpendicular to $BD$ meets $AD$ and $CD$ at $M$ and $N$ respectively. Let $K$ be a point on side $BC$ such that $MK \parallel AC$. Show that $\angle MKN = 90^o$. [img]https://cdn.artofproblemsolving.com/attachments/4/1/1d37b96cebaea3409ade7ce6711ac2d3fc2ef9.png[/img]

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of the elements of $S$. Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$. Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.

Ephram Chun is a senior and math captain at Lexington High School. He is well-loved by the freshmen, who seem to only listen to him. Other than being the father figure that the freshmen never had, Ephramis also part of the Science Bowl and Science Olympiad teams along with being part of the highest orchestra LHS has to offer. His many hobbies include playing soccer, volleyball, and the many forms of chess. We hope that he likes the questions that we’ve dedicated to him! [b]p1.[/b] Ephram is scared of freshmen boys. How many ways can Ephram and $4$ distinguishable freshmen boys sit together in a row of $5$ chairs if Ephram does not want to sit between $2$ freshmen boys? [b]p2.[/b] Ephram, who is a chess enthusiast, is trading chess pieces on the black market. Pawns are worth $\$100$, knights are worth $\$515$, and bishops are worth $\$396$. Thirty-four minutes ago, Ephrammade a fair trade: $5$ knights, $3$ bishops, and $9$ rooks for $8$ pawns, $2$ rooks, and $11$ bishops. Find the value of a rook, in dollars. [b]p3.[/b] Ephramis kicking a volleyball. The height of Ephram’s kick, in feet, is determined by $$h(t) = - \frac{p}{12}t^2 +\frac{p}{3}t ,$$ where $p$ is his kicking power and $t$ is the time in seconds. In order to reach the height of $8$ feet between $1$ and $2$ seconds, Ephram’s kicking power must be between reals $a$ and $b$. Find is $100a +b$. [b]p4.[/b] Disclaimer: No freshmen were harmed in the writing of this problem. Ephram has superhuman hearing: He can hear sounds up to $8$ miles away. Ephramstands in the middle of a $8$ mile by $24$ mile rectangular grass field. A freshman falls from the sky above a point chosen uniformly and randomly on the grass field. The probability Ephram hears the freshman bounce off the ground is $P\%$. Find $P$ rounded to the nearest integer. [img]https://cdn.artofproblemsolving.com/attachments/4/4/29f7a5a709523cd563f48176483536a2ae6562.png[/img] [b]p5.[/b] Ephram and Brandon are playing a version of chess, sitting on opposite sides of a $6\times 6$ board. Ephram has $6$ white pawns on the row closest to himself, and Brandon has $6$ black pawns on the row closest to himself. During each player’s turn, their only legal move is to move one pawn one square forward towards the opposing player. Pawns cannot move onto a space occupied by another pawn. Players alternate turns, and Ephram goes first (of course). Players take turns until there are no more legal moves for the active player, at which point the game ends. Find the number of possible positions the game can end in. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the volume formed by the revolution of the region satisfying $ 0\leq y\leq (x \minus{} p)(q \minus{} x)\ (0 < p < q)$ in the coordinate plane about the $ y$ -axis. You are not allowed to use the formula: $ V \equal{} \boxed{\int_a^b 2\pi x|f(x)|\ dx\ (a < b)}$ here.

A spring has an equilibrium length of $2.0$ meters and a spring constant of $10$ newtons/meter. Alice is pulling on one end of the spring with a force of $3.0$ newtons. Bob is pulling on the opposite end of the spring with a force of $3.0$ newtons, in the opposite direction. What is the resulting length of the spring? (A) $\text{1.7 m}$ (B) $\text{2.0 m}$ (C) $\text{2.3 m}$ (D) $\text{2.6 m}$ (E) $\text{8.0 m}$

Find all triples $(x,y,z)\in \mathbb{R}^{3}$ such that $x,y,z>3$ and $\frac{(x+2)^2}{y+z-2}+\frac{(y+4)^2}{z+x-4}+\frac{(z+6)^2}{x+y-6}=36$