Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.

A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? [asy] draw(Arc((0,0), 4, 0, 270)); draw((0,-4)--(0,0)--(4,0)); label("$4$", (2,0), S); [/asy] $\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7$

The number $2022$ is written on the white board. Ivan and Peter play a game, Ivan starts and they alternate. On a move, Ivan erases the number $b$, written on the board, throws a dice which shows some number $a$, and writes the residue of $(a+b) ^2$ modulo $5$. Similarly, Peter throws a dice which shows some number $a$, and changes the previously written number $b$ to the residue of $a+b$ modulo $3$. The first player to write a $0$ wins. What is the probability of Ivan winning the game?

The edges of the square in the figure have length $1$. Find the area of the marked region in terms of $a$, where $0 \le a \le 1$. [img]https://cdn.artofproblemsolving.com/attachments/2/2/f2b6ca973f66c50e39124913b3acb56feff8bb.png[/img]

Part of the graph of $ f(x) \equal{} x^3 \plus{} bx^2 \plus{} cx \plus{} d$ is shown. What is $ b$? [asy]import graph; unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; real y(real x) { return (x-1)*(x+1)*(x-2); } path bounds=(-1.5,-1)--(1.5,-1)--(1.5,2.5)--(-1.5,2.5)--cycle; pair[] points={(-1,0),(0,2),(1,0)}; draw(bounds,white); draw(graph(y,-1.5,1.5)); drawline((0,0),(1,0)); drawline((0,0),(0,1)); dot(points); label("$(-1,0)$",(-1,0),SE); label("$(1,0)$",(1,0),SW); label("$(0,2)$",(0,2),NE); clip(bounds);[/asy]$ \textbf{(A)}\minus{}\!4 \qquad \textbf{(B)}\minus{}\!2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 4$

A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$, $\text{lcm}(x, z) = 900$, and $\text{lcm}(y, z) = n$, where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property.

Let $ABC$ be a triangle with $H$ its orthocenter. The circle with diameter $[AC]$ cuts the circumcircle of triangle $ABH$ at $K$. Prove that the point of intersection of the lines $CK$ and $BH$ is the midpoint of the segment $[BH]$

In the inscribed quadrilateral $ABCD$, $P$ is the intersection point of diagonals and $M$ is the midpoint of arc $AB$. Prove that line $MP$ passes through the midpoint of segment $CD$, if and only if lines $AB, CD$ are parallel.

Parallelogram $ABCD$ is given such that $\angle ABC$ equals $30^o$ . Let $X$ be the foot of the perpendicular from $A$ onto $BC$, and $Y$ the foot of the perpendicular from $C$ to $AB$. If $AX = 20$ and $CY = 22$, find the area of the parallelogram.

Find the maximum and minimum value of $ F(x) \equal{} \frac {1}{2}x \plus{} \int_0^x (t \minus{} x)\sin t\ dt$ for $ 0\leq x\leq \pi$.

Given a triangle $A_1 A_2 A_3$, let $a_i$ denote the side opposite to $A_i$, where indices are taken modulo 3. Let $D_1 \in a_1$. For $D_i \in A_i$, let $\omega_i$ be the incircle of the triangle formed by lines $a_i, a_{i+1}, A_iD_i$, and $D_{i+1} \in a_{i+1}$ with $A_{i+1} D_{i+1}$ tangent to $\omega_i$. Show that the set $\{D_i: i \in \mathbb{N}\}$ is finite.

Let $R$ and $r$ be the radii of the circles circumscribed and inscribed, respectively, in a triangle. Prove that $R \ge 2r$, and that $R = 2r$ only for an equilateral triangle.

In the coordinate plane, given a circle $K: x^2+y^2=1,\ C: y=x^2-2$. Let $l$ be the tangent line of $K$ at $P(\cos \theta,\ \sin \theta)\ (\pi<\theta <2\pi).$ Find the minimum area of the part enclosed by $l$ and $C$.

A Pythagorean triangle is a right triangle whose three sides are integers. The best known example is the triangle with rectangular sides $3$ and $4$ and hypotenuse $5$. Determine all Pythagorean triangles whose area is twice the perimeter.

In a classroom there are $m$ students. During the month of July each of them visited the library at least once but none of them visited the library twice in the same day. It turned out that during the month of July each student visited the library a different number of times, furthermore for any two students $A$ and $B$ there was a day in which $A$ visited the library and $B$ did not and there was also a day when $B$ visited the library and $A$ did not do so. Determine the largest possible value of $m$.

A sequence of integers $a_0, a_1 …$ is called [i]kawaii[/i] if $a_0 =0, a_1=1,$ and $$(a_{n+2}-3a_{n+1}+2a_n)(a_{n+2}-4a_{n+1}+3a_n)=0$$ for all integers $n \geq 0$. An integer is called [i]kawaii[/i] if it belongs to some kawaii sequence. Suppose that two consecutive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that $m$ is divisible by $3,$ and that $m/3$ is also kawaii.

There are 100 positive integers written on a board. At each step, Alex composes 50 fractions using each number written on the board exactly once, brings these fractions to their irreducible form, and then replaces the 100 numbers on the board with the new numerators and denominators to create 100 new numbers. Find the smallest positive integer $n{}$ such that regardless of the values of the initial 100 numbers, after $n{}$ steps Alex can arrange to have on the board only pairwise coprime numbers.

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] Find the smallest positive whole number that ends with $17$, is divisible by $17$, and the sum of its digits is $17$. [b]p3.[/b] Three consecutive $2$-digit numbers are written next to each other. It turns out that the resulting $6$-digit number is divisible by $17$. Find all such numbers. [b]p4.[/b] Let $ABCD$ be a convex quadrilateral (a quadrilateral $ABCD$ is called convex if the diagonals $AC$ and $BD$ intersect). Suppose that $\angle CBD = \angle CAB$ and $\angle ACD = \angle BDA$ . Prove that $\angle ABC = \angle ADC$. [b]p5.[/b] A circle of radius $1$ is cut into four equal arcs, which are then arranged to make the shape shown on the picture. What is its area? [img]https://cdn.artofproblemsolving.com/attachments/f/3/49c3fe8b218ab0a5378ecc635b797a912723f9.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will $\textbf{(A)}\ \text{remain the same} \qquad \textbf{(B)}\ \text{increase by 1} \qquad \textbf{(C)}\ \text{increase by 2} \qquad \textbf{(D)}\ \text{increase by 3} \qquad \textbf{(E)}\ \text{increase by 4}$

Suppose that $m$ and $n$ are positive integers with $m < n$ such that the interval $[m, n)$ contains more multiples of $2021$ than multiples of $2000$. Compute the maximum possible value of $n - m$.

Find the smallest and largest integers with decimal representation of the form $ababa$ ($a \neq 0$) that are divisible by $11$.

Given positive integers $p$, $u$, and $v$ such that $u^2+2v^2=p$, determine, in terms of $u$ and $v$, integers $m$ and $n$ such that $3m^2-2mn+3n^2=24p$. (It is known that if $p$ is any prime number congruent to 1 or 3 modulo 8, then we can find integers $u$ and $v$ such that $u^2+2v^2=p$)

For how many pairs of primes $(p, q)$, is $p^2 + 2pq^2 + 1$ also a prime?

Points $B$,$D$ , and $J$ are midpoints of the sides of right triangle $ACG$ . Points $K$, $E$, $I$ are midpoints of the sides of triangle , etc. If the dividing and shading process is done 100 times (the first three are shown) and $ AC=CG=6 $, then the total area of the shaded triangles is nearest [asy] draw((0,0)--(6,0)--(6,6)--cycle); draw((3,0)--(3,3)--(6,3)); draw((4.5,3)--(4.5,4.5)--(6,4.5)); draw((5.25,4.5)--(5.25,5.25)--(6,5.25)); fill((3,0)--(6,0)--(6,3)--cycle,black); fill((4.5,3)--(6,3)--(6,4.5)--cycle,black); fill((5.25,4.5)--(6,4.5)--(6,5.25)--cycle,black); label("$A$",(0,0),SW); label("$B$",(3,0),S); label("$C$",(6,0),SE); label("$D$",(6,3),E); label("$E$",(6,4.5),E); label("$F$",(6,5.25),E); label("$G$",(6,6),NE); label("$H$",(5.25,5.25),NW); label("$I$",(4.5,4.5),NW); label("$J$",(3,3),NW); label("$K$",(4.5,3),S); label("$L$",(5.25,4.5),S);[/asy] $ \text{(A)}\ 6\qquad\text{(B)}\ 7\qquad\text{(C)}\ 8\qquad\text{(D)}\ 9\qquad\text{(E)}\ 10 $

In a group of $2n$ students, each student has exactly $3$ friends within the group. The friendships are mutual and for each two students $A$ and $B$ which are not friends, there is a sequence $C_1, C_2, ..., C_r$ of students such that $A$ is a friend of $C_1$, $C_1$ is a friend of $C_2$, et cetera, and $C_r$ is a friend of $B$. Every student was asked to assess each of his three friendships with: "acquaintance", "friend" and "BFF". It turned out that each student either gave the same assessment to all of his friends or gave every assessment exactly once. We say that a pair of students is in conflict if they gave each other different assessments. Let $D$ be the set of all possible values of the total number of conflicts. Prove that $|D| \geq 3n$ with equality if and only if the group can be partitioned into two subsets such that each student is separated from all of his friends.