Two disjoint circles are inscribed in an angle with vertex $A$, whose measure is equal to $a$. The distance between their centers is $d$. A straight line tangent to both circles and not passing through $A$ intersects the sides of the angle at points $B$ and $C$. Find the radius of the circle circumscribed about triangle $ABC$.

Let $p(x, y)$ and $q(x, y)$ be polynomials in two variables such that for $x \ge 0, y \ge 0$ the following conditions hold: $(i) p(x, y)$ and $q(x, y)$ are increasing functions of $x$ for every fixed $y$. $(ii) p(x, y)$ is an increasing and $q(x)$ is a decreasing function of $y$ for every fixed $x$. $(iii) p(x, 0) = q(x, 0)$ for every $x$ and $p(0, 0) = 0$. Show that the simultaneous equations $p(x, y) = a, q(x, y) = b$ have a unique solution in the set $x \ge 0, y \ge 0$ for all $a, b$ satisfying $0 \le b \le a$ but lack a solution in the same set if $a < b$.

Sergei skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled $P$, $Q$, $R$, and $S$. What is the sorted order of the four paths from shortest to longest? [center][img]https://wiki-images.artofproblemsolving.com/9/94/2024_AMC_8_Problem_6.png[/img][/center] $\textbf{(A) }\text{P, Q, R, S}\qquad\textbf{(B) }\text{P, R, S, Q}\qquad\textbf{(C) }\text{Q, S, P, R}\qquad\textbf{(D) }\text{R, P, S, Q}\qquad\textbf{(E) }\text{R, S, P, Q}$

Let real number sequences $a_k$ and $x_k$ be defined for $1 \le k \le 7$ and suppose that $a_1=1$ and $a_{k+1}=a_k+x_k$ for $1 \le k \le 7.$ Let $x_k$ be chosen such that the quantity $S=\sum_{k=1}^7 (a_k^2+x_k^2)$ is minimized. Then $S=\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ Find $m+n.$

All sides of a convex polygon were decreased in such a way that they formed a new convex polygon. Is it possible that all diagonals were increased?

Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$ [i]Proposed by Michael Ren[/i]

$A_1, B_1, C_1$ are the midpoints of the sides $BC,CA,BA$ respectively of an equilateral triangle $ABC$. Three parallel lines, passing through $A_1, B_1, C_1$ intersect, respectively, lines $B_1C_1, C_1A_1, A_1B_1$ at points $A_2, B_2, C_2$. Prove that the lines $AA_2, BB_2, CC_2$ intersect at one point lying on the circle circumscribed around the triangle $ABC$.

Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$. [i]Proposed by Ahaan S. Rungta[/i]

Let $A,B,C,D$ be points in the space such that $|AB|=|AC|=3$, $|DB|=|DC|=5$, $|AD|=6$, and $|BC|=2$. Let $P$ be the nearest point of $BC$ to the point $D$, and $Q$ be the nearest point of the plane $ABC$ to the point $D$. What is $|PQ|$? $ \textbf{(A)}\ \frac{1}{\sqrt 2} \qquad\textbf{(B)}\ \frac{3\sqrt 7}{2} \qquad\textbf{(C)}\ \frac{57}{2\sqrt{11}} \qquad\textbf{(D)}\ \frac{9}{2\sqrt 2} \qquad\textbf{(E)}\ 2\sqrt 2 $

Let $ a,b,c$ be nonnegative real numbers. Suppose that $ a\plus{}b\plus{}c\ge abc$. Prove that: $ a^2\plus{}b^2\plus{}c^2 \ge abc.$

$n$ distinct integers are given, all of which are bigger than $-a$, where $a$ is a positive integer. It turned out that the amount of odd numbers among them is equal to the biggest even number, and the amount of even numbers is equal to the biggest odd numbers a) Find the least possible value of $n$ for all $a$ b) For each $a \geq 2$ find the maximum possible value of $n$

$A; B; C; D; E$ and $F$ are points in space such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, $CD$ is parallel to $FA$, but $AB \neq DE$. Prove that all six points lie in the same plane. [i](4 points)[/i]

Given an $m \times n$ table consisting of $mn$ unit cells. Alice and Bob play the following game: Alice goes first and the one who moves colors one of the empty cells with one of the given three colors. Alice wins if there is a figure, such as the ones below, having three different colors. Otherwise Bob is the winner. Determine the winner for all cases of $m$ and $n$ where $m, n \ge 3$. Proposed by [i]Toghrul Abbasov, Azerbaijan[/i]

Find a Galois extension of the field $ \mathbb{Q} $ whose Galois group is isomorphic with $ \mathbb{Z}/3\mathbb{Z} . $

Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$

A country has $100$ cities and $n$ airplane companies which take care of a total of $2018$ two-way direct flights between pairs of cities. There is a pair of cities such that one cannot reach one from the other with just one or two flights. What is the largest possible value of $n$ for which between any two cities there is a route (a sequence of flights) using only one of the airplane companies?

There are $11$ phrases written on $11$ pieces of paper (one per sheet): 1) To the left of this sheet there are no sheets with false statements. 2) Exactly one sheet to the left of this one contains a false statement. 3) Exactly $2$ sheets to the left of this one contain false statements ... 11) Exactly $10$ sheets to the left of this one contain false statements. The sheets of paper were laid out in some order in a row, going from left to right. After this, some of the written statements became true and some became false. What is the greatest possible number of true statements?

Prove that the product of four consecutive natural numbers cannot be the square of an integer.

There are $2003$ pieces of candy on a table. Two players alternately make moves. A move consists of eating one candy or half of the candies on the table (the “lesser half” if there are an odd number of candies). At least one candy must be eaten at each move. The loser is the one who eats the last candy. Which player has a winning strategy?

When a student multiplied the number $66$ by the repeating decimal, $$1. \underline{a} \underline{b} \underline{a} \underline{b} … = 1.\overline{ab},$$ where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1. \underline{a} \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$- digit integer $\underline{a} \underline{b}$? $\textbf{(A)}\ 15 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 75$

Consider the $ 12$-sided polygon $ ABCDEFGHIJKL$, as shown. Each of its sides has length $ 4$, and each two consecutive sides form a right angle. Suppose that $ \overline{AG}$ and $ \overline{CH}$ meet at $ M$. What is the area of quadrilateral $ ABCM$? [asy]unitsize(13mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(1,3), B=(2,3), C=(2,2), D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1,1), J=(0,1), K=(0,2), L=(1,2); pair M=intersectionpoints(A--G,H--C)[0]; draw(A--B--C--D--Ep--F--G--H--I--J--K--L--cycle); draw(A--G); draw(H--C); dot(M); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,NE); label("$D$",D,NE); label("$E$",Ep,SE); label("$F$",F,SE); label("$G$",G,SE); label("$H$",H,SW); label("$I$",I,SW); label("$J$",J,SW); label("$K$",K,NW); label("$L$",L,NW); label("$M$",M,W);[/asy]$ \textbf{(A)}\ \frac {44}{3}\qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ \frac {88}{5}\qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ \frac {62}{3}$

A polynomial $p(x)$ of degree $n\ge 2$ has exactly $n$ real roots, counted with multiplicity. We know that the coefficient of $x^n$ is $1$, all the roots are less than or equal to $1$, and $p(2)=3^n$. What values can $p(1)$ take?

Each side face of a regular hexagonal prism is colored in one of three colors (for example, red, yellow, blue), and the adjacent prism faces have different colors. In how many different ways can the edges of the prism be colored (using all three colors is optional)?

$ABC$ is a triangle with $\hat{A}<\hat{C}$, and $D$ is the point on $BC$ such that $B\hat{A}D=A\hat{C}B$. The perpendicular bisectors of $AD$ and $AC$ intersect in the point $E$. Prove that $B\hat{A}E=90^\circ$.

Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying $$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds. (Karl Czakler)