Let $ABC$ be a triangle such that $\angle BAC = 45^{\circ}$. Let $H,O$ be the orthocenter and circumcenter of $ABC$, respectively. Let $\omega$ be the circumcircle of $ABC$ and $P$ the point on $\omega$ such that the circumcircle of $PBH$ is tangent to $BC$. Let $X$ and $Y$ be the circumcenters of $PHB$ and $PHC$ respectively. Let $O_1,O_2$ be the circumcenters of $PXO$ and $PYO$ respectively. Prove that $O_1$ and $O_2$ lie on $AB$ and $AC$, respectively.

[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].

Cole is trying to solve the [i]Collatz conjecture[/i]. She decides to make a model with a piece of wood with a hole for every natural number. For every even number there is a rope from $n$ to $\frac{n}{2}$ and for every odd number there is a rope from $n$ to $3n+1$. She wants to bring her model to a convention but in order to do that she needs to cut off the part containing the first $240$ holes. How many ropes did she break? [i]2017 CCA Math Bonanza Individual Round #4[/i]

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.

In an acute triangle $ABC$ the angle $C$ is greater than the angle $A$. Let $AE$ be a diameter of the circumcircle of the triangle. Let the intersection point of the ray $AC$ and the tangent of the circumcircle through the vertex $B$ be $K$. The perpendicular to $AE$ through $K$ intersects the circumcircle of the triangle $BCK$ for the second time at point $D$. Prove that $CE$ bisects the angle $BCD$.

Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$. Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$? $ \textbf{(A) }17\qquad\textbf{(B) }18\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21 $

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.

Inside an angle $\angle BOC$ there are three disjoint circles: $k_1,k_2$ and $k_3$, which are, each one, tangent to its sides $BO$ and $OC$. Let $r_1, r_2$ and $r_3$, respectively, be the radii of these circles, with $r_1<r_2<r_3$. The circles $k_1$ and $k_3$ are tangent to the side $OB$ at $A$ and $B$, respectively, and $k_2$ is tangent to the side $OC$ at $C$. Let $K=AC\cap k_1,L=AC\cap k_2,M=BC\cap k_2$ and $N=BC\cap k_3$. Besides that, let $P=AM\cap BK,Q=AM\cap BL,R=AN\cap BK$ and $S=AN\cap BL$. If the intersections of $CP,CQ,CR$ and $CS$ with $AB$ are $X,Y,Z$ and $T$, respectively, prove that $XZ = YT$.

When Dave walks to school, he averages 90 steps per minute, each of his steps 75cm long. It takes him 16 minutes to get to school. His brother, Jack, going to the same school by the same route, averages 100 steps per minute, but his steps are only 60 cm long. How long does it take Jack to get to school? $\textbf{(A) }14 \frac{2}{9}\qquad \textbf{(B) }15\qquad \textbf{(C) }18\qquad \textbf{(D) }20\qquad \textbf{(E) }22 \frac{2}{9}$

Let $\pi$ be the set of points in a plane and $f : \pi \to \pi$ be a mapping such that the image of any triangle (as its polygonal line) is a square. Show that $f(\pi)$ is a square.

Let $f$ be a function from positive integers to positive integers where $f(n) = \frac{n}{2}$ if $n$ is even and $f(n) = 3n+1$ if $n$ is odd. If $a$ is the smallest positive integer satisfying \[ \underbrace{f(f(\cdots f}_{2013\ f\text{'s}} (a)\cdots)) = 2013, \] find the remainder when $a$ is divided by $1000$. [i]Based on a proposal by Ivan Koswara[/i]

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]