A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that \[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]

In the triangle $ABC$, the angle - bisector $AD$ ($D \in BC$) and the median $BE$ ($E \in AC$) intersect at point $P$. Lines $AB$ and $CP$ intesect at point $F$. The parallel through $B$ to $CF$ intersects $DF$ at point $M$. Prove that $DM = BF$

A real number sequence $a_1, \cdots ,a_{2021}$ satisfies the below conditions. $$a_1=1, a_2=2, a_{n+2}=\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\leq n \leq 2019)$$ Let the minimum of $a_1, \cdots ,a_{2021}$ be $m$, and the maximum of $a_1, \cdots ,a_{2021}$ be $M$. Let a 2021 degree polynomial $$P(x):=(x-a_1)(x-a_2) \cdots (x-a_{2021})$$ $|P(x)|$ is maximum in $[m, M]$ when $x=\alpha$. Show that $1<\alpha <2$.

We consider the set of four-digit positive integers $x =\overline{abcd}$ with digits different than zero and pairwise different. We also consider the integers $y = \overline{dcba}$ and we suppose that $x > y$. Find the greatest and the lowest value of the difference $x-y$, as well as the corresponding four-digit integers $x,y$ for which these values are obtained.

Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.

Find all positive real numbers $x, y, z$, such that $$x - \frac{1}{y^2} = y - \frac{1}{z^2}= z - \frac{1}{x^2}$$

Find all primes $p,q$ such that $p^3-q^7=p-q$.

Let $ABC$ be a triangle inscribed in the circle $(O)$. Tangents at $B,C$ of the circles $(O)$ meet at $T$ . Let $M,N$ be the points on the rays $BT,CT$ respectively such that $BM = BC = CN$. The line through $M$ and $N$ intersects $CA,AB$ at $E, F$ respectively; $BE$ meets $CT$ at $P, CF$ intersects $BT$ at $Q$. Prove that $AP = AQ$. Trần Quang Hùng

Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.

A magic $3 \times 5$ board can toggle its cells between black and white. Define a \textit{pattern} to be an assignment of black or white to each of the board's $15$ cells (so there are $2^{15}$ patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than $3$ cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day $1$, compute the maximum number of days it can stay alive.

There are $n\ge3$ integers around a circle. We know that for each of these numbers the ratio between the sum of its two neighbors and the number is a positive integer. Prove that the sum of the $n$ ratios is not greater than $3n$.

For each positive integer $n$, decide whether it is possible to tile a square with exactly $n+1$ similar rectangles, each with a positive area and aspect ratio $1:n$.

Compute $\sin^4\left(7.5^\circ\right)+\sin^4\left(82.5^\circ\right)$. [i]2019 CCA Math Bonanza Lightning Round #2.3[/i]

An infinite squared sheet is given, with squares of side length $1$. The “distance” between two squares is defined as the length of the shortest path from one of these squares to the other if moving between them like a chess rook (measured along the trajectory of the centre of the rook). Determine the minimum number of colours with which it is possible to colour the sheet (each square being given a single colour) in such a way that each pair of squares with distance between them equal to $6$ units is given different colours. Give an example of such a colouring and prove that using a smaller number of colours we cannot achieve this goal. (AG Pechkovskiy, IV Itenberg)

In each square of a $4$ by $4$ grid, you put either a $+1$ or a $-1$. If any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative. What is the minimum number of $+1$'s needed to be placed to be able to satisfy the conditions

Find all integers n for which number $ \log_{2n\minus{}1}(n^2\plus{}2)$ is rational.

Find the condition of $a,b$ for which the function $f(x)\ (0\leq x\leq 2\pi)$ satisfying the following equality can be determined uniquely,then determine $f(x)$, assuming that $f(x) $ is a continuous function at $0\leq x\leq 2\pi$. \[f(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x\]

Spinners A and B are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even? [asy] defaultpen(linewidth(1)); draw(unitcircle); draw((1,0)--(-1,0)); draw((0,1)--(0,-1)); draw(shift(3,0)*unitcircle); draw(shift(3,0)*(origin--dir(90))); draw(shift(3,0)*(origin--dir(210))); draw(shift(3,0)*(origin--dir(330))); draw(0.7*dir(200)--0.7*dir(20), linewidth(0.7), EndArrow(7)); draw(shift(3,0)*(0.7*dir(180+65)--0.7*dir(65)), linewidth(0.7), EndArrow(7)); label("$1$", (-0.45,0.1), N); label("$4$", (-0.45,-0.1), S); label("$3$", (0.45,-0.1), S); label("$2$", (0.45,0.1), N); label("$1$", shift(3,0)*(-0.25,0.1), NW); label("$2$", shift(3,0)*(0.25,0.1), NE); label("$3$", shift(3,0)*(0,-0.3), S); label("$A$", (0,-1), S); label("$B$", (3,-1), S); [/asy] $ \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{4} $

Let $a,b\ge 0$. Prove that $$\sqrt{2}\left(\sqrt{a(a+b)^3}+b\sqrt{a^2+b^2}\right)\le 3(a^2+b^2)$$ with equality if and only if $a=b$.

Suppose $n>2$ and let $A_1,\dots,A_n$ be points on the plane such that no three are collinear. [b](a)[/b] Suppose $M_1,\dots,M_n$ be points on segments $A_1A_2,A_2A_3,\dots ,A_nA_1$ respectively. Prove that if $B_1,\dots,B_n$ are points in triangles $M_2A_2M_1,M_3A_3M_2,\dots ,M_1A_1M_n$ respectively then \[|B_1B_2|+|B_2B_3|+\dots+|B_nB_1| \leq |A_1A_2|+|A_2A_3|+\dots+|A_nA_1|\] Where $|XY|$ means the length of line segment between $X$ and $Y$. [b](b)[/b] If $X$, $Y$ and $Z$ are three points on the plane then by $H_{XYZ}$ we mean the half-plane that it's boundary is the exterior angle bisector of angle $\hat{XYZ}$ and doesn't contain $X$ and $Z$ ,having $Y$ crossed out. Prove that if $C_1,\dots ,C_n$ are points in ${H_{A_nA_1A_2},H_{A_1A_2A_3},\dots,H_{A_{n-1}A_nA_1}}$ then \[|A_1A_2|+|A_2A_3|+\dots +|A_nA_1| \leq |C_1C_2|+|C_2C_3|+\dots+|C_nC_1|\] Time allowed for this problem was 2 hours.

In triangle $ABC$, we draw the circle with center $A$ and radius $AB$. This circle intersects $AC$ at two points. Also we draw the circle with center $A$ and radius $AC$ and this circle intersects $AB$ at two points. Denote these four points by $A_1, A_2, A_3, A_4$. Find the points $B_1, B_2, B_3, B_4$ and $C_1, C_2, C_3, C_4$ similarly. Suppose that these $12$ points lie on two circles. Prove that the triangle $ABC$ is isosceles.

Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE =\angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$. [i]Proposed by Tony Kim[/i]

An isosceles trapezoid $ABCD$ ($AB = CD$) is given. A point $P$ on its circumcircle is such that segments $CP$ and $AD$ meet at point $Q$. Let $L$ be tha midpoint of$ QD$. Prove that the diagonal of the trapezoid is not greater than the sum of distances from the midpoints of the lateral sides to ana arbitrary point of line $PL$.