The lengths in inches of the three sides of each of four triangles I, II, III, and IV are as follows: \[ \begin{array}{rlrl} \hbox {I}& 3,\ 4,\ \hbox{and}\ 5 \qquad & \hbox{III}& 7,\ 24,\ \hbox{and}\ 25 \\ \hbox{II}& 4,\ 7\frac{1}{2},\ \hbox{and}\ 8\frac{1}{2} \qquad & \hbox{IV}& 3\frac{1}{2},\ 4\frac{1}{2},\ \hbox{and}\ 5\frac{1}{2}. \end{array} \] Of these four given triangles, the only right triangles are \[ \begin{tabular}{rlrlrl} (A) & I and II \qquad & (B) & I and III \qquad & (C) & I and IV \\ (D) & I, II, and III \qquad & (E) & I, II, and IV & \end{tabular} \]

[u]Round 5 [/u] [b]p13.[/b] A palindrome is a number that reads the same forward as backwards; for example, $121$ and $36463$ are palindromes. Suppose that $N$ is the maximal possible difference between two consecutive three-digit palindromes. Find the number of pairs of consecutive palindromes $(A, B)$ satisfying $A < B$ and $B - A = N$. [b]p14.[/b] Suppose that $x, y$, and $z$ are complex numbers satisfying $x +\frac{1}{yz} = 5$, $y +\frac{1}{zx} = 8$, and $z +\frac{1}{xy} = 6$. Find the sum of all possible values of $xyz$. [b]p15.[/b] Let $\Omega$ be a circle with radius $25\sqrt2$ centered at $O$, and let $C$ and $J$ be points on $\Omega$ such that the circle with diameter $\overline{CJ}$ passes through $O$. Let $Q$ be a point on the circle with diameter $\overline{CJ}$ satisfying $OQ = 5\sqrt2$. If the area of the region bounded by $\overline{CQ}$, $\overline{QJ}$, and minor arc $JC$ on $\Omega$ can be expressed as $\frac{a\pi-b}{c}$ for integers $a, b$, and $c$ with $gcd \,\,(a, c) = 1$, then find $a + b + c$. [u]Round 6[/u] [b]p16.[/b] Veronica writes $N$ integers between $2$ and $2020$ (inclusive) on a blackboard, and she notices that no number on the board is an integer power of another number on the board. What is the largest possible value of $N$? [b]p17.[/b] Let $ABC$ be a triangle with $AB = 12$, $AC = 16$, and $BC = 20$. Let $D$ be a point on $AC$, and suppose that $I$ and $J$ are the incenters of triangles $ABD$ and $CBD$, respectively. Suppose that $DI = DJ$. Find $IJ^2$. [b]p18.[/b] For each positive integer $a$, let $P_a = \{2a, 3a, 5a, 7a, . . .\}$ be the set of all prime multiples of $a$. Let $f(m, n) = 1$ if $P_m$ and $P_n$ have elements in common, and let $f(m, n) = 0$ if $P_m$ and $P_n$ have no elements in common. Compute $$\sum_{1\le i<j\le 50} f(i, j)$$ (i.e. compute $f(1, 2) + f(1, 3) + ,,, + f(1, 50) + f(2, 3) + f(2, 4) + ,,, + f(49, 50)$.) [u]Round 7[/u] [b]p19.[/b] How many ways are there to put the six letters in “$MMATHS$” in a two-by-three grid such that the two “$M$”s do not occupy adjacent squares and such that the letter “$A$” is not directly above the letter “$T$” in the grid? (Squares are said to be adjacent if they share a side.) [b]p20.[/b] Luke is shooting basketballs into a hoop. He makes any given shot with fixed probability $p$ with $p < 1$, and he shoots n shots in total with $n \ge 2$. Miraculously, in $n$ shots, the probability that Luke makes exactly two shots in is twice the probability that Luke makes exactly one shot in! If $p$ can be expressed as $\frac{k}{100}$ for some integer $k$ (not necessarily in lowest terms), find the sum of all possible values for $k$. [b]p21.[/b] Let $ABCD$ be a rectangle with $AB = 24$ and $BC = 72$. Call a point $P$ [i]goofy [/i] if it satisfies the following conditions: $\bullet$ $P$ lies within $ABCD$, $\bullet$ for some points $F$ and $G$ lying on sides $BC$ and $DA$ such that the circles with diameter $BF$ and $DG$ are tangent to one another, $P$ lies on their common internal tangent. Find the smallest possible area of a polygon that contains every single goofy point inside it. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2800971p24674988]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x$ be a variable that can take any positive real value. For certain positive real constants $s$ and $t$, the value of $x^2 + \frac{s}{x}$ is minimized at $x = t$, and the value of $t^2\ln(2 + tx) + \frac{1}{x^2}$ is minimized at $x = s$. Compute the ordered pair $(s, t)$.

Let n, k be positive integers. Let $f_a(x) := ||x - a||^{2n}$ , where the vectors $x = (x_1, ..., x_k) , a\in R^k$ , and ||·|| is the Euclidean norm. Let the vector space $Q_{n, k}$ be generated by the functions $f_a$ ($a\in R^k$). What is the largest integer N for which $Q_{n, k}$ contains all polynomials of $x_1, ..., x_k$ whose total degree is at most N?

Let a positive integer $n$ have at least four positive divisors. Let the least four positive divisors be $1=d_1<d_2<d_3<d_4$. Find, with proof, all solutions to $n^2=d_1^3+d_2^3+d_4^3$.

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

We can change a natural number $n$ in three ways: a) If the number $n$ has at least two digits, we erase the last digit and we subtract that digit from the remaining number (for example, from $123$ we get $12 - 3 = 9$); b) If the last digit is different from $0$, we can change the order of the digits in the opposite one (for example, from $123$ we get $321$); c) We can multiply the number $n$ by a number from the set $ \{1, 2, 3,..., 2010\}$. Can we get the number $21062011$ from the number $1012011$?

Three circles with radius $2$ are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure? [asy] filldraw((0,0)--(2,0)--(1,sqrt(3))--cycle,gray,gray); filldraw(circle((1,sqrt(3)),1),gray); filldraw(circle((0,0),1),gray); filldraw(circle((2,0),1),grey); [/asy] $ \textbf{(A)}\ 10\pi+4\sqrt3\qquad\textbf{(B)}\ 13\pi-\sqrt3\qquad\textbf{(C)}\ 12\pi+\sqrt3\qquad\textbf{(D)}\ 10\pi+9\qquad\textbf{(E)}\ 13\pi$

Let $n$, $m$ and $k$ be positive integers satisfying $(n-1)n(n+1)=m^k.$ Prove that $k=1.$

A triangle $ABC$ is given. Let $C\ensuremath{'}$ be the vertex of an isosceles triangle $ABC\ensuremath{'}$ with $\angle C\ensuremath{'} = 120^{\circ}$ constructed on the other side of $AB$ than $C$, and $B\ensuremath{'}$ be the vertex of an equilateral triangle $ACB\ensuremath{'}$ constructed on the same side of $AC$ as $ABC$. Let $K$ be the midpoint of $BB\ensuremath{'}$ Find the angles of triangle $KCC\ensuremath{'}$. [i]Proposed by A.Zaslavsky[/i]

Let $ABC$ be an acute triangle and let $k_{1},k_{2},k_{3}$ be the circles with diameters $BC,CA,AB$, respectively. Let $K$ be the radical center of these circles. Segments $AK,CK,BK$ meet $k_{1},k_{2},k_{3}$ again at $D,E,F$, respectively. If the areas of triangles $ABC,DBC,ECA,FAB$ are $u,x,y,z$, respectively, prove that \[u^{2}=x^{2}+y^{2}+z^{2}.\]

Given distinct positive integer $ a_1,a_2,…,a_{2020} $. For $ n \ge 2021 $, $a_n$ is the smallest number different from $a_1,a_2,…,a_{n-1}$ which doesn't divide $a_{n-2020}...a_{n-2}a_{n-1}$. Proof that every number large enough appears in the sequence.

Let $ABC$ be a scalene triangle with incenter $I$. The incircle of $ABC$ touches $\overline{BC},\overline{CA},\overline{AB}$ at points $D,E,F$, respectively. Let $P$ be the foot of the altitude from $D$ to $\overline{EF}$, and let $M$ be the midpoint of $\overline{BC}$. The rays $AP$ and $IP$ intersect the circumcircle of triangle $ABC$ again at points $G$ and $Q$, respectively. Show that the incenter of triangle $GQM$ coincides with $D$. [i]Zack Chroman and Daniel Liu[/i]

Let $x_1, x_2, x_3$ be positive real numbers. Prove that $$\frac{(x_1^2+x_2^2+x_3^2)^3}{(x_1^3+x_2^3+x_3^3)^2}\le 3$$

Let $ABCD$ be a parallelogram such that $\angle BAD = 60^{\circ}.$ Let $K$ and $L$ be the midpoints of $BC$ and $CD,$ respectively. Assuming that $ABKL$ is a cyclic quadrilateral, find $\angle ABD.$

The set of integers is coloured in $100$ colours in such a way that all the colours are used and the following is true. For any choice of intervals $[a, b]$ and $[c,d]$ of equal length and with integral endpoints, if a and c as well as $b$ and $d$, respectively, have the same colour, then the whole intervals $[a, b]$ and $[c,d]$ are identically coloured in that, for any integer $x$, $0 \le x \le b - a$, the numbers $a + x$ and $c + x$ are of the same colour. Prove that $-1982$ and $1982$ are of different colours

There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.

The incircle of triangle $ABC$ touches its sides in points $A', B',C'$ . It is known that the orthocenters of triangles $ABC$ and $A' B'C'$ coincide. Is triangle $ABC$ regular?

Albert and Bob and Charlie are each thinking of a number. Albert's number is one more than twice Bob's. Bob's number is one more than twice Charlie's, and Charlie's number is two more than twice Albert's. What number is Albert thinking of? $\mathrm{(A) \ } -\frac{11}{7} \qquad \mathrm{(B) \ } -2 \qquad \mathrm {(C) \ } -1 \qquad \mathrm{(D) \ } -\frac{4}{7} \qquad \mathrm{(E) \ } \frac{1}{2}$

The circle $\omega_1$ with diameter $[AB]$ and the circle $\omega_2$ with center $A$ intersects at points $C$ and $D$. Let $E$ be a point on the circle $\omega_2$, which is outside $\omega_1$ and at the same side as $C$ with respect to the line $AB$. Let the second point of intersection of the line $BE$ with $\omega_2$ be $F$. For a point $K$ on the circle $\omega_1$ which is on the same side as $A$ with respect to the diameter of $\omega_1$ passing through $C$ we have $2\cdot CK \cdot AC = CE \cdot AB$. Let the second point of intersection of the line $KF$ with $\omega_1$ be $L$. Show that the symmetric of the point $D$ with respect to the line $BE$ is on the circumcircle of the triangle $LFC$.

Trace the curve whose equation is: \[ y^4 - x^4 - 96y^2 + 100x^2 = 0. \]

Let $ n $ and $ r $ be positive integers and $ A $ be a set of lattice points in the plane such that any open disc of radius $ r $ contains a point of $ A $. Show that for any coloring of the points of $ A $ in $ n $ colors there exists four points of the same color which are the vertices of a rectangle.

Leo the fox has a $5$ by $5$ checkerboard grid with alternating red and black squares. He fills in the grid with the numbers $1, 2, 3, \dots, 25$ such that any two consecutive numbers are in adjacent squares (sharing a side) and each number is used exactly once. He then computes the sum of the numbers in the $13$ squares that are the same color as the center square. Compute the maximum possible sum Leo can obtain.

On a $1\times n$ board there are $n-1$ separating edges between neighbouring cells. Initially, none of the edges contain matches. During a move of size $0 < k < n$ a player chooses a $1\times k$ sub-board which contains no matches inside, and places a matchstick on all of the separating edges bordering the sub-board that don’t already have one. A move is considered legal if at least one matchstick can be placed and if either $k = 1$ or $k{}$ is divisible by 4. Two players take turns making moves, the player in turn must choose one of the available legal moves of the largest size $0 < k < n$ and play it. If someone does not have a legal move, the game ends and that player loses. [i]Beat the organisers twice in a row in this game! First the organisers determine the value of $n{}$, then you get to choose whether you want to play as the first or the second player.[/i]

In a triangle $ ABC$ right-angled at $ C$ , the median through $ B$ bisects the angle between $ BA$ and the bisector of $ \angle B$. Prove that \[ \frac{5}{2} < \frac{AB}{BC} < 3\]