[b]p1.[/b] Martin’s car insurance costed $\$6000$ before he switched to Geico, when he saved $15\%$ on car insurance. When Mayhem switched to Allstate, he, a safe driver, saved $40\%$ on car insurance. If Mayhem and Martin are now paying the same amount for car insurance, how much was Mayhem paying before he switched to Allstate? [b]p2.[/b] The $7$-digit number $N$ can be written as $\underline{A} \,\, \underline{2} \,\,\underline{0} \,\,\underline{B} \,\,\underline{2} \,\, \underline{1} \,\,\underline{5}$. How many values of $N$ are divisible by $9$? [b]p3.[/b] The solutions to the equation $x^2-18x-115 = 0$ can be represented as $a$ and $b$. What is $a^2+2ab+b^2$? [b]p4.[/b] The exterior angles of a regular polygon measure to $4$ degrees. What is a third of the number of sides of this polygon? [b]p5.[/b] Charlie Brown is having a thanksgiving party. $\bullet$ He wants one turkey, with three different sizes to choose from. $\bullet$ He wants to have two or three vegetable dishes, when he can pick from Mashed Potatoes, Saut´eed Brussels Sprouts, Roasted Butternut Squash, Buttery Green Beans, and Sweet Yams; $\bullet$ He wants two desserts out of Pumpkin Pie, Apple Pie, Carrot Cake, and Cheesecake. How many different combinations of menus are there? [b]p6.[/b] In the diagram below, $\overline{AD} \cong \overline{CD}$ and $\vartriangle DAB$ is a right triangle with $\angle DAB = 90^o$. Given that the radius of the circle is $6$ and $m \angle ADC = 30^o$, if the length of minor arc $AB$ is written as $a\pi$, what is $a$? [img]https://cdn.artofproblemsolving.com/attachments/d/9/ea57032a30c16f4402886af086064261d6828b.png[/img] [b]p7.[/b] This Halloween, Owen and his two friends dressed up as guards from Squid Game. They needed to make three masks, which were black circles with a white equilateral triangle, circle, or square inscribed in their upper halves. Resourcefully, they used black paper circles with a radius of $5$ inches and white tape to create these masks. Ignoring the width of the tape, how much tape did they use? If the length can be expressed $a\sqrt{b}+c\sqrt{d}+ \frac{e}{f} \pi$ such that $b$ and $d$ are not divisible by the square of any prime, and $e$ and $f$ are relatively prime, find $a + b + c + d + e + f$. [img]https://cdn.artofproblemsolving.com/attachments/0/c/bafe3f9939bd5767ba5cf77a51031dd32bbbec.png[/img] [b]p8.[/b] Given $LCM (10^8, 8^{10}, n) = 20^{15}$, where $n$ is a positive integer, find the total number of possible values of $n$. [b]p9.[/b] If one can represent the infinite progression $\frac{1}{11} + \frac{2}{13} + \frac{3}{121} + \frac{4}{169} + \frac{5}{1331} + \frac{6}{2197}+ ...$ as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers, what is $a$? [b]p10.[/b] Consider a tiled $3\times 3$ square without a center tile. How many ways are there to color the squares such that no two colored squares are adjacent (vertically or horizontally)? Consider rotations of an configuration to be the same, and consider the no-color configuration to be a coloring. [b]p11.[/b] Let $ABC$ be a triangle with $AB = 4$ and $AC = 7$. Let $AD$ be an angle bisector of triangle $ABC$. Point $M$ is on $AC$ such that $AD$ intersects $BM$ at point $P$, and $AP : PD = 3 : 1$. If the ratio $AM : MC$ can be expressed as $\frac{a}{b}$ such that $a$, $b$ are relatively prime positive integers, find $a + b$. [b]p12.[/b] For a positive integer $n$, define $f(n)$ as the number of positive integers less than or equal to $n$ that are coprime with $n$. For example, $f(9) = 6$ because $9$ does not have any common divisors with $1$, $2$, $4$, $5$, $7$, or $8$. Calculate: $$\sum^{100}_{i=2} \left( 29^{f(i)}\,\,\, mod \,\,i \right).$$ [b]p13.[/b] Let $ABC$ be an equilateral triangle. Let $P$ be a randomly selected point in the incircle of $ABC$. Find $a+b+c+d$ if the probability that $\angle BPC$ is acute can be expressed as $\frac{a\sqrt{b} -c\pi}{d\pi }$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, c, d) = 1$ and $b$ is not divisible by the square of any prime. [b]p14.[/b] When the following expression is simplified by expanding then combining like terms, how many terms are in the resulting expression? $$(a + b + c + d)^{100} + (a + b - c - d)^{100}$$ [b]p15.[/b] Jerry has a rectangular box with integral side lengths. If $3$ units are added to each side of the box, the volume of the box is tripled. What is the largest possible volume of this box? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We say that a four-digit number $\overline{abcd}$ , which starts at $a$ and ends at $d$, is [i]interchangeable [/i] if there is an integer $n >1$ such that $n \times \overline{abcd}$ is a four-digit number that begins with $d$ and ends with $a$. For example, $1009$ is interchangeable since $1009\times 9=9081$. Find the largest interchangeable number.

Find the smallest $n$ for which we can find $15$ distinct elements $a_{1},a_{2},...,a_{15}$ of $\{16,17,...,n\}$ such that $a_{k}$ is a multiple of $k$.

Two perfect squares are [i]friends[/i] if one is obtained from the other adding the digit $1$ at the left. For instance, $1225 = 35^2$ and $225 = 15^2$ are friends. Prove that there are infinite pairs of odd perfect squares that are friends.

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

[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 $z$.

[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 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]

Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$? $\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}$

Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\] Proposed by Moldova

$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?

For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a [i]stump[/i] of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$.

Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$: (i) move the last digit of $a$ to the first position to obtain the numb er $b$; (ii) square $b$ to obtain the number $c$; (iii) move the first digit of $c$ to the end to obtain the number $d$. (All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.) Find all numbers $a$ for which $d\left( a\right) =a^2$. [i]Proposed by Zoran Sunic, USA[/i]

Find all positive integers $n$ such that there exists integers $n_1,\ldots,n_k\ge 3$, for some integer $k$, satisfying \[n=n_1n_2\cdots n_k=2^{\frac{1}{2^k}(n_1-1)\cdots (n_k-1)}-1.\]

For every non-negative integer $n$ , let $s_n$ be the sum of digits in the decimal expansion of $2^n$. Is the sequence $(s_n)_{n \in \mathbb{N}}$ eventually increasing ?

[b]а)[/b] Prove that for any positive integer $n$ there exist a pair of positive integers $(m, k)$ such that \[{k + m^k + n^{m^k}} = 2009^n.\] [b]b)[/b] Prove that there are infinitely many positive integers $n$ for which there is only one such pair.

Determine all pairs $(m, n)$ of non-negative integers that satisfy the equation $$ 20^m - 10m^2 + 1 = 19^n. $$

Let $ABCD$ be a square and consider the points $K\in AB, L\in BC,$ and $M\in CD$ such that $\Delta KLM$ is a right isosceles triangle, with the right angle at $L$. Prove that the lines $AL$ and $DK$ are perpendicular to each other.

Suppose that $p$ is a prime number. Find all natural numbers $n$ such that $p|\varphi(n)$ and for all $a$ such that $(a,n)=1$ we have \[ n|a^{\frac{\varphi(n)}{p}}-1 \]

The measure of a given angle is $\frac{180^{\circ}}{n}$ where $n$ is a positive integer not divisible by $3$. Prove that the angle can be trisected by Euclidean means (straightedge and compasses).

Let $f{}$ and $g{}$ be polynomials with integers coefficients. The leading coefficient of $g{}$ is equal to 1. It is known that for infinitely many natural numbers $n{}$ the number $f(n)$ is divisible by $g(n)$ . Prove that $f(n)$ is divisible by $g(n)$ for all positive integers $n{}$ such that $g(n)\neq 0$. [i]From the folklore[/i]

A bus ticket is considered to be lucky if the sum of the first three digits equals to the sum of the last three ($6$ digits in Russian buses). Prove that the sum of all the lucky numbers is divisible by $13$.

Let $a_1, a_2, a_3,\ldots$ and $b_1, b_2, b_3,\ldots$ be infinite increasing arithmetic progressions. Their terms are positive numbers. It is known that the ratio $a_k/b_k$ is an integer for all k. Is it true that this ratio does not depend on $k{}$? [i]Boris Frenkin[/i]