The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

Compare the following two numbers: $2^{2^{2^{2^{2}}}}$ and $3^{3^{3^{3}}}$.

Find the smallest positive integer for which when we move the last right digit of the number to the left, the remaining number be $\frac 32$ times of the original number.

How many zeros does $100!$ have at its end in the usual decimal representation?

[b]Problem Section #1 b) Let $a, b$ be positive integers such that $b^n +n$ is a multiple of $a^n + n$ for all positive integers $n$. Prove that $a = b.$

Given $1962$ -digit number. It is divisible by $9$. Let $x$ be the sum of its digits. Let the sum of the digits of $x$ be $y$. Let the sum of the digits of $y$ be $z$. Find $z$.

Let $f: \mathbb N \to \mathbb N$ be a function such that $f(1)=1$ and \[f(n)=n - f(f(n-1)), \quad \forall n \geq 2.\] Prove that $f(n+f(n))=n $ for each positive integer $n.$

Consider an increasing sequence of real numbers $a_1<a_2<\ldots<a_{2023}$ such that all pairwise sums of the elements in the sequence are different. For such a sequence, denote by $M$ the number of pairs $(a_i,a_j)$ such that $a_i<a_j$ and $a_i+a_j<a_2+a_{2022}$. Find the minimal and the maximal possible value of $M$.

Michelle has a word with $2^n$ letters, where a word can consist of letters from any alphabet. Michelle performs a swicheroo on the word as follows: for each $k = 0, 1, \ldots, n-1$, she switches the first $2^k$ letters of the word with the next $2^k$ letters of the word. For example, for $n = 3$, Michelle changes \[ ABCDEFGH \to BACDEFGH \to CDBAEFGH \to EFGHCDBA \] in one switcheroo. In terms of $n$, what is the minimum positive integer $m$ such that after Michelle performs the switcheroo operation $m$ times on any word of length $2^n$, she will receive her original word?

[b]p22.[/b] Consider the series $\{A_n\}^{\infty}_{n=0}$, where $A_0 = 1$ and for every $n > 0$, $$A_n = A_{\left[ \frac{n}{2023}\right]} + A_{\left[ \frac{n}{2023^2}\right]}+A_{\left[ \frac{n}{2023^3}\right]},$$ where $[x]$ denotes the largest integer value smaller than or equal to $x$. Find the $(2023^{3^2}+20)$-th element of the series. [b]p23.[/b] The side lengths of triangle $\vartriangle ABC$ are $5$, $7$ and $8$. Construct equilateral triangles $\vartriangle A_1BC$, $\vartriangle B_1CA$, and $\vartriangle C_1AB$ such that $A_1$,$B_1$,$C_1$ lie outside of $\vartriangle ABC$. Let $A_2$,$B_2$, and $C_2$ be the centers of $\vartriangle A_1BC$, $\vartriangle B_1CA$, and $\vartriangle C_1AB$, respectively. What is the area of $\vartriangle A_2B_2C_2$? [b]p24. [/b]There are $20$ people participating in a random tag game around an $20$-gon. Whenever two people end up at the same vertex, if one of them is a tagger then the other also becomes a tagger. A round consists of everyone moving to a random vertex on the $20$-gon (no matter where they were at the beginning). If there are currently $10$ taggers, let $E$ be the expected number of untagged people at the end of the next round. If $E$ can be written as $\frac{a}{b}$ for $a, b$ relatively prime positive integers, compute $a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A triangle $ABC$ is inscribed in a rectangle $APQR$ so that points $B$ and $C$ lie on segments $PQ$ and $QR$, respectively. If $\alpha,\beta,\gamma$ are the angles of the triangle, prove that $$\cot\alpha\cdot S_{BCQ}=\cot\beta\cdot S_{ACR}+\cot\gamma\cdot S_{ABP}.$$

Let $ABCD$ be a cyclic quadrilateral. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ such that $KLMN$ is a rhombus with $KL \parallel AC$ and $LM \parallel BD$. Let $\omega_A, \omega_B, \omega_C, \omega_D$ be the incircles of $\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN$. Prove that the common internal tangents to $\omega_A$, and $\omega_C$ and the common internal tangents to $\omega_B$ and $\omega_D$ are concurrent.

A positive integer is said to be [b]palindromic[/b] if it remains the same when its digits are reversed. For example, $1221$ or $74847$ are both palindromic numbers. Let $k$ be a positive integer that can be expressed as an $n$-digit number $\overline{a_{n-1}a_{n-2} \cdots a_0}$. Prove that if $k$ is a palindromic number, then $k^2$ is also a palindromic number if and only if $a_0^2 + a^2_1 + \cdots + a^2_{n-1} < 10$. [i]Proposed by Ho-Chien Chen[/i]

The numbers $1,2,3,4,\ldots , 39$ are written on a blackboard. In one step we are allowed to choose two numbers $a$ and $b$ on the blackboard such that $a$ divides $b$, and replace $a$ and $b$ by the single number $\tfrac{b}{a}$. This process is continued till no number on the board divides any other number. Let $S$ be the set of numbers which is left on the board at the end. What is the smallest possible value of $|S|$? [i]Proposed by B.J. Venkatachala[/i]

Alice and Bob play a game. Bob starts by picking a set $S$ consisting of $M$ vectors of length $n$ with entries either $0$ or $1$. Alice picks a sequence of numbers $y_1\le y_2\le\dots\le y_n$ from the interval $[0,1]$, and a choice of real numbers $x_1,x_2\dots,x_n\in \mathbb{R}$. Bob wins if he can pick a vector $(z_1,z_2,\dots,z_n)\in S$ such that $$\sum_{i=1}^n x_iy_i\le \sum_{i=1}^n x_iz_i,$$otherwise Alice wins. Determine the minimum value of $M$ so that Bob can guarantee a win. [i]Proposed by DVDthe1st[/i]

Given positive numbers $a_1$ and $b_1$, consider the sequences defined by \[a_{n+1}=a_n+\frac{1}{b_n},\quad b_{n+1}=b_n+\frac{1}{a_n}\quad (n \ge 1)\] Prove that $a_{25}+b_{25} \geq 10\sqrt{2}$.

Let $ \mathcal{C}$ be the hyperbola $ y^2 \minus{} x^2 \equal{} 1$. Given a point $ P_0$ on the $ x$-axis, we construct a sequence of points $ (P_n)$ on the $ x$-axis in the following manner: let $ \ell_n$ be the line with slope $ 1$ passing passing through $ P_n$, then $ P_{n\plus{}1}$ is the orthogonal projection of the point of intersection of $ \ell_n$ and $ \mathcal C$ onto the $ x$-axis. (If $ P_n \equal{} 0$, then the sequence simply terminates.) Let $ N$ be the number of starting positions $ P_0$ on the $ x$-axis such that $ P_0 \equal{} P_{2008}$. Determine the remainder of $ N$ when divided by $ 2008$.

In triangle $\vartriangle ABC$, let $M$ be the midpoint of $\overline{AC}$. Extend $\overline{BM}$ such that it intersects the circumcircle of $\vartriangle ABC$ at a point $X$ not equal to $B$. Let $O$ be the center of the circumcircle of $\vartriangle ABC$. Given that $BM = 4MX$ and $\angle ABC = 45^o$, compute $\sin (\angle BOX)$.

Let $x_{n} = \sin^2(n^\circ)$. What is the mean of $x_{1}, x_{2}, x_{3}, \cdots, x_{90}$? $ \textbf{(A) }\frac{11}{45} \qquad \textbf{(B) }\frac{22}{45} \qquad \textbf{(C) }\frac{89}{180} \qquad \textbf{(D) }\frac{1}{2} \qquad \textbf{(E) }\frac{91}{180} \qquad $

All the roots of polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers. What is the value of $B$? $\textbf{(A)}\ -88 \qquad\textbf{(B)}\ -80 \qquad\textbf{(C)}\ -64\qquad\textbf{(D)}\ -41 \qquad\textbf{(E)}\ -40$

A cube is assembled with $27$ white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

We have an acute-angled triangle which is not isosceles. We denote the orthocenter, the circumcenter and the incenter of it by $H$, $O$, $I$ respectively. Prove that if a vertex of the triangle lies on the circle $HOI$, then there must be another vertex on this circle as well.

Let $ABC$ be a triangle. Let $D,E$ be the points outside of the triangle so that $AD=AB,AC=AE$ and $\angle DAB =\angle EAC =90^o$. Let $F$ be at the same side of the line $BC$ as $A$ such that $FB = FC$ and $\angle BFC=90^o$. Prove that the triangle $DEF$ is a right- isosceles triangle.

How many ways are there to color each of the $8$ cells below red or blue such that no two blue cells are adjacent? [asy] size(3cm); draw((0,0)--(4,0)--(4,1)--(0,1)--(0,0)); draw((1,-1)--(1,2)--(3,2)--(3,-1)--(1,-1)); draw((2,-1)--(2,2)); [/asy] $\textbf{(A) }48\qquad\textbf{(B) }50\qquad\textbf{(C) }52\qquad\textbf{(D) }54\qquad\textbf{(E) }56$