Find the product of all positive real solutions to the equation $x^{-x} + x^{\frac{1}{x}} = \frac{2021}{2020}.$ [i]Proposed by Gabriel Wu[/i]

The equation $ x^3 - x^2 + ax - 2^n = 0$ has three integer roots. Determine $a$ and $n$.

An equiangular octagon has four sides of length $ 1$ and four sides of length $ \frac{\sqrt{2}}{2}$, arranged so that no two consecutive sides have the same length. What is the area of the octagon? $ \textbf{(A)}\ \frac{7}{2}\qquad \textbf{(B)}\ \frac{7\sqrt{2}}{2}\qquad \textbf{(C)}\ \frac{5 \plus{} 4\sqrt{2}}{2}\qquad \textbf{(D)}\ \frac{4 \plus{} 5\sqrt{2}}{2}\qquad \textbf{(E)}\ 7$

In $\triangle ABC$, a point $D$ lies on line $BC$. The circumcircle of $ABD$ meets $AC$ at $F$ (other than $A$), and the circumcircle of $ADC$ meets $AB$ at $E$ (other than $A$). Prove that as $D$ varies, the circumcircle of $AEF$ always passes through a fixed point other than $A$, and that this point lies on the median from $A$ to $BC$. [i]Proposed by Allen Liu[/i]

Suppose a box contains $28$ balls: $1$ red, $2$ blue, $3$ yellow, $4$ orange, $5$ purple, $6$ green, and $7$ pink. One by one, each ball is removed uniformly at random and without replacement until all $28$ balls have been removed. Determine the probability that the most likely “scenario of exhaustion” occurs; that is, determine the probability that the first color to have all such balls removed from the box is red, that the second is blue, the third is yellow, the fourth is orange, the fifth is purple, the sixth is green, and the seventh is pink.

Suppose two polygons may be glued together at an edge if and only if corresponding edges of the same length are made to coincide. A $3\times 4$ rectangle is cut into $n$ pieces by making straight line cuts. What is the minimum value of $n$ so that it’s possible to cut the pieces in such a way that they may be glued together two at a time into a polygon with perimeter at least $2021$?

Real numbers $a,b$ and $c$ have arithmetic mean $0$. The arithmetic mean of $a^2, b^2$ and $c^2$ is $10$. What is the arithmetic mean of $ab, ac$ and $bc$? $ \textbf{(A) }-5 \qquad \textbf{(B) }-\frac{10}{3} \qquad \textbf{(C) }-\frac{10}{9} \qquad \textbf{(D) }0 \qquad \textbf{(E) }\frac{10}{9} \qquad $

A quota of diplomas at the All-Russian Olympiad should be strictly less than $45\%$. More than $20$ students took part in the olympiad. After the olympiad the Authorities declared the results low because the quota of diplomas was significantly less than $45\%$. The Jury responded that the quota was already maximum possible on this olympiad or any other olympiad with smaller number of participants. Then the Authorities ordered to increase the number of participants for the next olympiad so that the quota of diplomas became at least two times closer to $45\%$. Prove that the number of participants should be at least doubled.

Compute the number of (not necessarily convex) polygons in the coordinate plane with the following properties: [list] [*] If the coordinates of a vertex are $(x,y)$, then $x,y$ are integers and $1\leq |x|+|y|\leq 3$ [*] Every side of the polygon is parallel to either the x or y axis [*] The point $(0,0)$ is contained in the interior of the polygon. [/list] [i]2021 CCA Math Bonanza Lightning Round #4.2[/i]

Three piles of stones are given, initially containing 2000, 4000, and 4899 stones respectively. Ali and Baba play the following game, taking turns, with Ali starting first. In one move, a player can choose two piles and transfer some stones from one pile to the other, provided that at the end of the move, the pile from which the stones are moved has no fewer stones than the pile to which the stones are moved. The player who cannot make a move loses. Does either player have a winning strategy, and if so, who?

A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? [asy] draw((0,8)--(0,0)--(4,0)--(4,8)--(0,8)--(3.5,8.5)--(3.5,8)); draw((2,-1)--(2,9),dashed);[/asy] $ \text{(A)}\ \frac{1}{3}\qquad\text{(B)}\ \frac{1}{2}\qquad\text{(C)}\ \frac{3}{4}\qquad\text{(D)}\ \frac{4}{5}\qquad\text{(E)}\ \frac{5}{6} $

Let $ABC$ be an acute triangle with incenter $O$, orthocenter $H$, altitude $AD. AO$ meets $BC$ at $E$. Line through $D$ parallel to $OH$ meet $AB,AC$ at $M,N$, respectively. Let $I$ be the midpoint of $AE$, and $DI$ intersect $AB,AC$ at $P,Q$ respectively. $MQ$ meets $NP$ at $T$. Prove that $D,O, T$ are collinear. Trần Quang Hùng

Let $ 0 < x <y$ be real numbers. Let $ H\equal{}\frac{2xy}{x\plus{}y}$ , $ G\equal{}\sqrt{xy}$ , $ A\equal{}\frac{x\plus{}y}{2}$ , $ Q\equal{}\sqrt{\frac{x^2\plus{}y^2}{2}}$ be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of $ x$ and $ y$. As generally known $ H<G<A<Q$. Arrange the intervals $ [H,G]$ , $ [G,A]$ and $ [A,Q]$ in ascending order by their length.

Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC=8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI=EJ=EK=2$. A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE}$, and containing the edges, $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $S$, including the walls of the tunnel, is $m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p$.

In a convex quadrilateral $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. If $MP$ and $NQ$ divide $ABCD$ in four quadrilaterals with the same area, prove that $ABCD$ is a parallelogram.

Show that if an infinite arithmetic progression of positive integers contains a square and a cube, it must contain a sixth power.

Work base 3. (so each digit is 0,1,2) A good number of size $n$ is a number in which there are no consecutive $1$'s and no consecutive $2$'s. How many good 10-digit numbers are there?

A tetrahedron has three edges of length $2$ and three edges of length $4$, and one of its faces is an equilateral triangle. Compute the radius of the sphere that is tangent to every edge of this tetrahedron.

Andrew generates a finite random sequence $\{a_n\}$ of distinct integers according to the following criteria: [list] [*] $a_0 = 1$, $0 < |a_n| < 7$ for all $n$, and $a_i \neq a_j$ for all $i < j$. [*] $a_{n+1}$ is selected uniformly at random from the set $\{a_n - 1, a_n + 1, -a_n\}$, conditioned on the above rule. The sequence terminates if no element of the set satisfies the first condition. [/list] For example, if $(a_0, a_1) = (1, 2)$, then $a_2$ would be chosen from the set $\{-2,3\}$, each with probability $\tfrac12$. Determine the probability that there exists an integer $k$ such that $a_k = 6$.

If $m,n\in\mathbb N_{\ge2}$, find the best constant $k\in\mathbb R$ for which $$\sum_{j=2}^m\sum_{i=2}^n\frac1{i^j}<k$$ [i]Proposed by Dorin Mărghidanu[/i]

Let $f:S\to\mathbb{R}$ be the function from the set of all right triangles into the set of real numbers, defined by $f(\Delta ABC)=\frac{h}{r}$, where $h$ is the height with respect to the hypotenuse and $r$ is the inscribed circle's radius. Find the image, $Im(f)$, of the function.

Let $n$ be a posititve integer. On a $n \times n$ grid there are $n^2$ unit squares and on these we color the sides with blue such that every unit square has exactly one side with blue. [b]a)[/b] Find the maximun number of blue unit sides we can have on the $n \times n$ grid. [b]b)[/b] Find the minimun number of blue unit sides we can have on the $n \times n$ grid.

For each polynomial $p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ we define it's derivative as this and we show it by $p'(x)$: \[p'(x)=na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+...+2a_2x+a_1\] a) For each two polynomials $p(x)$ and $q(x)$ prove that:(3 points) \[(p(x)q(x))'=p'(x)q(x)+p(x)q'(x)\] b) Suppose that $p(x)$ is a polynomial with degree $n$ and $x_1,x_2,...,x_n$ are it's zeros. prove that:(3 points) \[\frac{p'(x)}{p(x)}=\sum_{i=1}^{n}\frac{1}{x-x_i}\] c) $p(x)$ is a monic polynomial with degree $n$ and $z_1,z_2,...,z_n$ are it's zeros such that: \[|z_1|=1, \quad \forall i\in\{2,..,n\}:|z_i|\le1\] Prove that $p'(x)$ has at least one zero in the disc with length one with the center $z_1$ in complex plane. (disc with length one with the center $z_1$ in complex plane: $D=\{z \in \mathbb C: |z-z_1|\le1\}$)(20 points)

Call a $ 7$-digit telephone number $ d_1d_2d_3 \minus{} d_4d_5d_6d_7$ [i]memorable[/i] if the prefix sequence $ d_1d_2d_3$ is exactly the same as either of the sequences $ d_4d_5d_6$ or $ d_5d_6d_7$ (possibly both). Assuming that each $ d_i$ can be any of the ten decimal digits $ 0,1,2,\ldots9$, the number of different memorable telephone numbers is $ \textbf{(A)}\ 19,\!810 \qquad \textbf{(B)}\ 19,\!910 \qquad \textbf{(C)}\ 19,\!990 \qquad \textbf{(D)}\ 20,\!000 \qquad \textbf{(E)}\ 20,\!100$

For how many integers $x$ is the point $(x,-x)$ inside or on the circle of radius $10$ centered at $(5,5)$? $\textbf{(A) } 11 \qquad\textbf{(B) } 12 \qquad\textbf{(C) } 13 \qquad\textbf{(D) } 14 \qquad\textbf{(E) } 15 $