Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ with centers $(1, 1)$ and $(4, 5)$ and radii $r_1 < r_2$, respectively, are drawn on the coordinate plane. The product of the slopes of the two common external tangents of $\mathcal{C}_1$ and $\mathcal{C}_2$ is $3$. If the value of $(r_2 - r_1)^2$ can be expressed as a common fraction in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$.

The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first $30,000$ miles the car traveled. For how many miles was each tire used? $\text{(A)}\ 6000 \qquad \text{(B)}\ 7500 \qquad \text{(C)}\ 24,000 \qquad \text{(D)}\ 30,000 \qquad \text{(E)}\ 37,500$

Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is $132.$ $\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad \textbf{(E) }12$

Evaluate \[\int_0^1 \frac{(x-1)^2(\cos x+1)-(2x-1)\sin x}{(x-1+\sqrt{\sin x})^2}\ dx\]

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$, \[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \]

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

Let $p{}$ be a fixed prime number. Juku and Miku play the following game. One of the players chooses a natural number $a$ such that $a>1$ and $a$ is not divisible by $p{}$, his opponent chooses any natural number $n{}$ such that $n>1$. Miku wins if the natural number written as $n{}$ "$1$"s in the positional numeral system with base $a$ is divisible by $p{}$, otherwise Juku wins. Which player has a winning strategy if: (a) Juku chooses the number $a$, tells it to Miku and then Miku chooses the number $n{}$; (b) Juku chooses the number $n{}$, tells it to Miku and then Miku chooses the number $a$?

Call number $A$ as interesting if $A$ is divided by every number that can be received from $A$ by crossing some last digits. Find maximum interesting number with different digits.

In the plane there is a quadrilateral $ ABCD $ and a point $ M $. Construct a parallelogram with center $ M $ and its vertices lying on the lines $ AB $, $ BC $, $ CD $, $ DA $.

Let $ S \equal{} \{1,2,3,\cdots ,280\}$. Find the smallest integer $ n$ such that each $ n$-element subset of $ S$ contains five numbers which are pairwise relatively prime.

Suppose $P,Q\in \mathbb R[x]$ that $deg\ P=deg\ Q$ and $PQ'-QP'$ has no real root. Prove that for each $\lambda \in \mathbb R$ number of real roots of $P$ and $\lambda P+(1-\lambda)Q$ are equal.

In unit cube $ABCDEFGH$ (with faces $ABCD$, $EFGH$ and connecting vertices labeled so that $\overline{AE}$, $\overline{BF}$, $\overline{CG}$, $\overline{DH}$ are edges of the cube), $L$ is the midpoint of $GH$. The area of $\vartriangle CAL$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

A circle with diameter $23$ is cut by a chord $AC$. Two different circles can be inscribed between the large circle and $AC$. Find the sum of the two radii.

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

A table $2022 \times 2022$ is divided onto the tiles of two types: $L$-tetromino and $Z$-tetromino. Determine the least amount of $Z$-tetromino one needs to use.

The sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, \ldots$ consists of 1’s separated by blocks of 2’s with n 2’s in the nth block. The sum of the first $1234$ terms of this sequence is $\text{(A)}\ 1996 \qquad \text{(B)}\ 2419 \qquad \text{(C)}\ 2429 \qquad \text{(D)}\ 2439 \qquad \text{(E)}\ 2449$

10. Michael is playing basketball. He makes $10\%$ of his shots, and gets the ball back after $90\%$ of his missed shots. If he does not get the ball back he stops playing. What is the probability that Michael eventually makes a shot? 11. How many subsets $S$ of the set $\{1, 2, \ldots , 10\}$ satisfy the property that, for all $i \in [1, 9]$, either $i$ or $i + 1$ (or both) is in S? 12. A positive integer $\overline{ABC}$, where $A, B, C$ are digits, satisfies $$\overline{ABC} = B^C - A$$ Find $\overline{ABC}$.

1. If $5x+3y-z=4$, $x=y$, and $z=4$, find $x+y+z$. 2. How many ways are there to pick three distinct vertices of a regular hexagon such that the triangle with those three points as its vertices shares exactly one side with the hexagon? 3. Sirena picks five distinct positive primes, $p_1 < p_2 < p_3 < p_4 < p_5$, and finds that they sum to $192$. If the product $p_1p_2p_3p_4p_5$ is as large as possible, what is $p_1 - p_2 + p_3 - p_4 + p_5$?

Consider the regular $101$-gon. A line $l$ does not contain any vertex of this polygon. Prove that line $l$ intersects even number of the diagonals of this polygon.

Given an acute triangle $ABC$, point $D$ is chosen on the side $AB$ and a point $E$ is chosen on the extension of $BC$ beyond $C$. It became known that the line through $E$ parallel to $AB$ is tangent to the circumcircle of $\triangle ADC$. Prove that one of the tangents from $E$ to the circumcircle of $\triangle BCD$ cuts the angle $\angle ABE$ in such a way that a triangle similar to $\triangle ABC$ is formed.

$x_1=\frac{1}{2}$ and $x_{k+1}=\frac{x_k}{x_1^2+...+x_k^2}$ Prove that $\sqrt{x_k^4+4\frac{x_{k-1}}{x_{k+1}}}$ is rational

Square table $5 \times 5$ is filled with numbers in a following way. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8zLzQ0Y2M1NjdiNjQ3NjhlYTAwMWQ0MTg2ZjIwZWE4NzkwYzcwYWFkLnBuZw==&rn=dGFiZWxpY2EucG5n[/img] We can change the table in a way we take two arbitrary numbers from the table and we decrease both of them with value of smaller of those two. Can we get to the table with all zeros?

For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let \[f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).\] The set of all numbers $x$ such that $1\leq x<2014$ and $f(x)\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals? $\textbf{(A) }1\qquad \textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad \textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad \textbf{(D) }\dfrac{2014}{2013}\qquad \textbf{(E) }2014^{\frac1{2014}}\qquad$

An isosceles triangle $ABC$ ($AB = BC$) is given. Circles $\omega_1$ and $\omega_2$ with centres $O_1$ and $O_2$ lie in the angle $ABC$ and touch the sides $AB$ and $CB$ at $A$ and $C$ respectively, and touch each other externally at point $X$. The side $AC$ meets the circles again at points $Y$ and $Z$. $O$ is the circumcenter of the triangle $XYZ$. Lines $O_2 O$ and $O_1 O$ intersect lines $AB$ and $BC$ at points $C_1$ and $A_1$ respectively. Prove that $B$ is the circumcentre of the triangle $A_1 OC_1$.

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21. Compute the real value of $a$ such that $$\sqrt{a(101b + 1)} - 1 = \sqrt{b(c - 1)}+ 10\sqrt{(a - c)b}.$$ [b]p20.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21. For some triangle $\vartriangle ABC$, let $\omega$ and $\omega_A$ be the incircle and $A$-excircle with centers $I$ and $I_A$, respectively. Suppose $AC$ is tangent to $\omega$ and $\omega_A$ at $E$ and $E'$, respectively, and $AB$ is tangent to $\omega$ and $\omega_A$ at $F$ and $F'$ respectively. Furthermore, let $P$ and $Q$ be the intersections of $BI$ with $EF$ and $CI$ with $EF$, respectively, and let $P'$ and $Q'$ be the intersections of $BI_A$ with $E'F'$ and $CI_A$ with $E'F'$, respectively. Given that the circumradius of $\vartriangle ABC$ is a, compute the maximum integer value of $BC$ such that the area $[P QP'Q']$ is less than or equal to $1$. [b]p21.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21. Let $c$ be a positive integer such that $gcd(b, c) = 1$. From each ordered pair $(x, y)$ such that $x$ and $y$ are both integers, we draw two lines through that point in the $x-y$ plane, one with slope $\frac{b}{c}$ and one with slope $-\frac{c}{b}$ . Given that the number of intersections of these lines in $[0, 1)^2$ is a square number, what is the smallest possible value of $ c$? Note that $[0, 1)^2$ refers to all points $(x, y)$ such that $0 \le x < 1$ and $ 0 \le y < 1$.