Let $ABCD$ be a convex quadrilateral such that $m \left (\widehat{ADB} \right)=15^{\circ}$, $m \left (\widehat{BCD} \right)=90^{\circ}$. The diagonals of quadrilateral are perpendicular at $E$. Let $P$ be a point on $|AE|$ such that $|EC|=4, |EA|=8$ and $|EP|=2$. What is $m \left (\widehat{PBD} \right)$? $ \textbf{(A)}\ 15^{\circ} \qquad\textbf{(B)}\ 30^{\circ} \qquad\textbf{(C)}\ 45^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 75^{\circ} $

Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$.

Let $a_n$ be the area of the part enclosed by the curve $y=x^n\ (n\geq 1)$, the line $x=\frac 12$ and the $x$ axis. Prove that : \[0\leq \ln 2-\frac 12-(a_1+a_2+\cdots\cdots+a_n)\leq \frac {1}{2^{n+1}}\]

Let $n>1$ be an even positive integer. An $2n \times 2n$ grid of unit squares is given, and it is partitioned into $n^2$ contiguous $2 \times 2$ blocks of unit squares. A subset $S$ of the unit squares satisfies the following properties: (i) For any pair of squares $A,B$ in $S$, there is a sequence of squares in $S$ that starts with $A$, ends with $B$, and has any two consecutive elements sharing a side; and (ii) In each of the $2 \times 2$ blocks of squares, at least one of the four squares is in $S$. An example for $n=2$ is shown below, with the squares of $S$ shaded and the four $2 \times 2$ blocks of squares outlined in bold. [asy] size(2.5cm); fill((0,0)--(4,0)--(4,1)--(0,1)--cycle,mediumgrey); fill((0,0)--(0,4)--(1,4)--(1,0)--cycle,mediumgrey); fill((0,3)--(4,3)--(4,4)--(0,4)--cycle,mediumgrey); fill((3,0)--(3,4)--(4,4)--(4,0)--cycle,mediumgrey); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw((1,0)--(1,4)); draw((2,0)--(2,4),linewidth(1)); draw((3,0)--(3,4)); draw((0,1)--(4,1)); draw((0,2)--(4,2),linewidth(1)); draw((0,3)--(4,3)); [/asy] In terms of $n$, what is the minimum possible number of elements in $S$?

Define a function $g :\mathbb{N} \rightarrow \mathbb{R}$ Such that $g(x)=\sqrt{4^x+\sqrt {4^{x+1}+\sqrt{4^{x+2}+...}}}$. Find the last 2 digits in the decimal representation of $g(2021)$.

There are 101 coins in a circle, each weights 10g or 11g. Prove that there exists a coin such that the total weight of the $k{}$ coins to its left is equal to the total weight of the $k{}$ coins to its right where a) $k = 50$ and b) $k = 49$. [i]Alexandr Gribalko[/i]

Given are $n$ different concentric circles on the plane.Inside the disk with the smallest radius (strictly inside it),we consider two distinct points $A,B$.We consider $k$ distinct lines passing through $A$ and $m$ distinct lines passing through $B$.There is no line passing through both $A$ and $B$ and all the lines passing through $k$ intersect with all the lines passing through $B$.The intersections do not lie on some of the circles.Determine the maximum and the minimum number of regions formed by the lines and the circles and are inside the circles.

Given a positive integer $k$, find the least integer $n_k$ for which there exist five sets $S_1, S_2, S_3, S_4, S_5$ with the following properties: \[|S_j|=k \text{ for } j=1, \cdots , 5 , \quad |\bigcup_{j=1}^{5} S_j | = n_k ;\] \[|S_i \cap S_{i+1}| = 0 = |S_5 \cap S_1|, \quad \text{for } i=1,\cdots ,4 \]

Find the sum (in base $10$) of the three greatest numbers less than $1000_{10}$ that are palindromes in both base $10$ and base $5$.

Let $n$ be given, $n\ge 4,$ and suppose that $P_1,P_2,\dots,P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i.$ What is the probability that at least one of the vertex angles of this polygon is acute.?

How many positive integers $N$ in the segment $\left[10, 10^{20} \right]$ are such that if all their digits are increased by $1$ and then multiplied, the result is $N+1$? [i](F. Bakharev)[/i]

Given positive real numbers $ {a_1},{a_2},...,{a_n} $ with $ {a_1}+{a_2}+...+{a_n}= 1 $. Prove that $ \left({\frac{1}{{a_1^2}}-1}\right)\left({\frac{1}{{a_2^2}}-1}\right)...\left({\frac{1}{{a_n^2}}-1}\right)\geqslant{({n^2}-1)^n} $.

Compute $\sum^{999}_{x=1}\gcd (x, 10x + 9)$.

Suppose that $x$, $y$, and $z$ are complex numbers of equal magnitude that satisfy \[x+y+z = -\frac{\sqrt{3}}{2}-i\sqrt{5}\] and \[xyz=\sqrt{3} + i\sqrt{5}.\] If $x=x_1+ix_2, y=y_1+iy_2,$ and $z=z_1+iz_2$ for real $x_1,x_2,y_1,y_2,z_1$ and $z_2$ then \[(x_1x_2+y_1y_2+z_1z_2)^2\] can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b.$

Let $ABC$ and $PQR$ be two triangles such that [list] [b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$. [b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$ [/list] Prove that $AB+AC=PQ+PR$.

Let $h_a$, $h_b$ and $h_c$ be the heights and $r$ the inradius of a triangle. Prove that the triangle is equilateral if and only if $h_a + h_b + h_c = 9r$.

Let $ABC$ be an acute-angled triangle. The tangents to the circumcircle of triangle $ABC$ at $B$ and $C$ respectively meet at $D$. The circumcircles of triangles $ABD$ and $ACD$ meet line $BC$ at additional points $E$ and $F$ respectively. Lines $DB$ and $DC$ meet the circumcircle of triangle $DEF$ at additional points $X$ and $Y$ respectively. Let $O$ be the circumcentre of triangle $DEF$. Prove that the circumcircles of triangles $ABC$ and $OXY$ are tangent to each other.

Let $n \leq 100$ be an integer. Hare puts real numbers in the cells of a $100 \times 100$ table. By asking Hare one question, Wolf can find out the sum of all numbers of a square $n \times n$, or the sum of all numbers of a rectangle $1 \times (n - 1)$ (or $(n - 1) \times 1$). Find the greatest $n{}$ such that, after several questions, Wolf can find the numbers in all cells, with guarantee.

Estimate the number of characters, excluding spaces, in the \LaTeX~source file for this Lightning Round, which includes the answer sheets and exactly one Asymptote diagram. Your score is determined by the function $max\{0, 20 - \lfloor \frac{|A - E|}{20}\rfloor\}$where $A$ is the actual answer, and $E$ is your estimate? [i]Lightning 5.3[/i]

[b]p1.[/b] While computing $7 - 2002 \cdot x$, John accidentally evaluates from left to right $((7 - 2002) \cdot x)$ instead of correctly using order of operations $(7 - (2002 \cdot x))$. If he gets the correct answer anyway, what is $x$? [b]p2.[/b] Given that $$x^2 + y^2 + z^2 = 6$$ $$ \left( \frac{x}{y} + \frac{y}{x} \right)^2 + \left( \frac{y}{z} + \frac{z}{y} \right)^2 + \left( \frac{z}{x} + \frac{x}{z} \right)^2 = 16.5,$$ what is $\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}$ ? [b]p3.[/b] Evaluate $$\frac{tan \frac{\pi}{4}}{4}+\frac{tan \frac{3\pi}{4}}{8}+\frac{tan \frac{5\pi}{4}}{16}+\frac{tan \frac{7\pi}{4}}{32}+ ...$$ [b]p4.[/b] Note that $2002 = 22 \cdot 91$, and so $2002$ is a multiple of the number obtained by removing its middle $2$ digits. Generalizing this, how many $4$-digit palindromes, $abba$, are divisible by the $2$-digit palindrome, $aa$? [b]p5.[/b] Let $ABCDE$ be a pyramid such that $BCDE$ is a square with side length $2$, and $A$ is $2$ units above the center of $BCDE$. If $F$ is the midpoint of $\overline{DE}$ and $G$ is the midpoint of $\overline{AC}$, what is the length of $\overline{DE}$? [b]p6.[/b] Suppose $a_1, a_2,..., a_{100}$ are real numbers with the property that $$i(a_1 + a_2 +... + a_i) = 1 + (a_{i+1} + a_{i+2} + ... + a_{100})$$ for all $i$. Compute $a_{10}$. [b]p7.[/b] A bug is sitting on one corner of a $3' \times 4' \times 5'$ block of wood. What is the minimum distance nit needs to travel along the block’s surface to reach the opposite corner? [b]p8.[/b] In the number game, a pair of positive integers $(n,m)$ is written on a blackboard. Two players then take turns doing the following: 1. If $n \ge m$, the player chooses a positive integer $c$ such that $n - cm \ge 0$, and replaces $(n,m)$ with $(n - cm,m)$. 2. If $m > n$, the player chooses a positive integer $c$ such that $m - cn \ge 0$, and replaces $(n,m)$ with $(n,m - cn)$. If $m$ or $n$ ever become $0$, the game ends, and the last player to have moved is declared the winner. If $(n,m)$ are originally $(20021000, 2002)$, what choices of $c$ are winning moves for the first player? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $V$ be a convex polygon. (a) Show that if $V$ has $3k$ vertices, then $V$ can be triangulated such that each vertex is in an odd number of triangles. (b) Show that if the number of vertices is not divisible with 3, then $V$ can be triangulated such that exactly 2 vertices have an even number of triangles.

Alice is thinking of a positive real number $x$, and Bob is thinking of a positive real number $y$. Given that $x^{\sqrt{y}}=27$ and $(\sqrt{x})^y=9$, compute $xy$.

Given posetive real numbers $a_1,a_2,\dots,a_n$ such that $a_1^2+2a_2^3+\dots+na_n^{n+1} <1.$ Prove that $2a_1+3a_2^2+\dots+(n+1)a_{n}^n <3.$

Planes $\alpha,\beta,\gamma,\delta$ are tangent to the circumsphere of a tetrahedron $ABCD$ at points $A,B,C,D$, respectively. Line $p$ is the intersection of $\alpha$ and $\beta$, and line $q$ is the intersection of $\gamma$ and $\delta$. Prove that if lines $p$ and $CD$ meet, then lines $q$ and $AB$ lie on a plane.

For a positive integer $n$, an $n$-configuration is a family of sets $\left\langle A_{i,j}\right\rangle_{1\le i,j\le n}.$ An $n$-configuration is called [i]sweet[/i] if for every pair of indices $(i, j)$ with $1\le i\le n -1$ and $1\le j\le n$ we have $A_{i,j}\subseteq A_{i+1,j}$ and $A_{j,i}\subseteq A_{j,i+1}.$ Let $f(n, k)$ denote the number of sweet $n$-configurations such that $A_{n,n}\subseteq \{1, 2,\ldots , k\}$. Determine which number is larger: $f\left(2024, 2024^2\right)$ or $f\left(2024^2, 2024\right).$