[b]p1.[/b] Al usually arrives at the train station on the commuter train at $6:00$, where his wife Jane meets him and drives him home. Today Al caught the early train and arrived at $5:00$. Rather than waiting for Jane, he decided to jog along the route he knew Jane would take and hail her when he saw her. As a result, Al and Jane arrived home $12$ minutes earlier than usual. If Al was jogging at a constant speed of $5$ miles per hour, and Jane always drives at the constant speed that would put her at the station at $6:00$, what was her speed, in miles per hour? [b]p2.[/b] In the figure, points $M$ and $N$ are the respective midpoints of the sides $AB$ and $CD$ of quadrilateral $ABCD$. Diagonal $AC$ meets segment $MN$ at $P$, which is the midpoint of $MN$, and $AP$ is twice as long as $PC$. The area of triangle $ABC$ is $6$ square feet. (a) Find, with proof, the area of triangle $AMP$. (b) Find, with proof, the area of triangle $CNP$. (c) Find, with proof, the area of quadrilateral $ABCD$. [img]https://cdn.artofproblemsolving.com/attachments/a/c/4bdcd8390bae26bc90fc7eae398ace06900a67.png[/img] [b]p3.[/b] (a) Show that there is a triangle whose angles have measure $\tan^{-1}1$, $\tan^{-1}2$ and $\tan^{-1}3$. (b) Find all values of $k$ for which there is a triangle whose angles have measure $\tan^{-1}\left(\frac12 \right)$, $\tan^{-1}\left(\frac12 +k\right)$, and $\tan^{-1}\left(\frac12 +2k\right)$ [b]p4.[/b] (a) Find $19$ consecutive integers whose sum is as close to $1000$ as possible. (b) Find the longest possible sequence of consecutive odd integers whose sum is exactly $1000$, and prove that your sequence is the longest. [b]p5.[/b] Let $AB$ and $CD$ be chords of a circle which meet at a point $X$ inside the circle. (a) Suppose that $\frac{AX}{BX}=\frac{CX}{DX}$. Prove that $|AB|=|CD|$. (b) Suppose that $\frac{AX}{BX}>\frac{CX}{DX}>1$. Prove that $|AB|>|CD|$. ($|PQ|$ means the length of the segment $PQ$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Study the convergence of the sequence $$ \left( \sum_{k=2}^{n+1} \sqrt[k]{n+1} -\sum_{k=2}^{n} \sqrt[k]{n} \right)_{n\ge 2} , $$ and calculate its limit. [i]Dan Negulescu[/i]

Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$. [i]Evan O'Dorney.[/i]

[b]p1.[/b] Given any four digit number $X = \underline{ABCD}$, consider the quantity $Y(X) = 2 \cdot \underline{AB}+\underline{CD}$. For example, if $X = 1234$, then $Y(X) = 2 \cdot 12+34 = 58$. Find the sum of all natural numbers $n \le 10000$ such that over all four digit numbers $X$, the number $n$ divides $X$ if and only if it also divides $Y(X)$. [b]p2.[/b] A sink has a red faucet, a blue faucet, and a drain. The two faucets release water into the sink at constant but different rates when turned on, and the drain removes water from the sink at a constant rate when opened. It takes $5$ minutes to fill the sink (from empty to full) when the drain is open and only the red faucet is on, it takes $10$ minutes to fill the sink when the drain is open and only the blue faucet is on, and it takes $15$ seconds to fill the sink when both faucets are on and the drain is closed. Suppose that the sink is currently one-thirds full of water, and the drain is opened. Rounded to the nearest integer, how many seconds will elapse before the sink is emptied (keeping the two faucets closed)? [b]p3.[/b] One of the bases of a right triangular prism is a triangle $XYZ$ with side lengths $XY = 13$, $YZ = 14$, $ZX = 15$. Suppose that a sphere may be positioned to touch each of the five faces of the prism at exactly one point. A plane parallel to the rectangular face of the prism containing $\overline{YZ}$ cuts the prism and the sphere, giving rise to a cross-section of area $A$ for the prism and area $15\pi$ for the sphere. Find the sum of all possible values of $A$. [b]p4.[/b] Albert, Brian, and Christine are hanging out by a magical tree. This tree gives each of them a stick, each of which have a non-negative real length. Say that Albert gets a branch of length $x$, Brian a branch of length $y$, and Christine a branch of length $z$, and the lengths follow the condition that $x+y+z = 2$. Let $m$ and $n$ be the minimum and maximum possible values of $xy+yz+xz-xyz$, respectively. What is $m+n$? [b]p5.[/b] Let $S := MATHEMATICSMATHEMATICSMATHE...$ be the sequence where $7$ copies of the word $MATHEMATICS$ are concatenated together. How many ways are there to delete all but five letters of $S$ such that the resulting subsequence is $CHMMC$? [b]p6.[/b] Consider two sequences of integers $a_n$ and $b_n$ such that $a_1 = a_2 = 1$, $b_1 = b_2 = 1$ and that the following recursive relations are satisfied for integers $n > 2$: $$a_n = a_{n-1}a_{n-2}-b_{n-1}b_{n-2},$$ $$b_n = b_{n-1}a_{n-2}+a_{n-1}b_{n-2}.$$ Determine the value of $$\sum_{1\le n\le2023,b_n \ne 0} \frac{a_n}{b_n}.$$ [b]p7.[/b] Suppose $ABC$ is a triangle with circumcenter $O$. Let $A'$ be the reflection of $A$ across $\overline{BC}$. If $BC =12$, $\angle BAC = 60^o$, and the perimeter of $ABC$ is $30$, then find $A'O$. [b]p8.[/b] A class of $10$ students wants to determine the class president by drawing slips of paper from a box. One of the students, Bob, puts a slip of paper with his name into the box. Each other student has a $\frac12$ probability of putting a slip of paper with their own name into the box and a $\frac12$ probability of not doing so. Later, one slip is randomly selected from the box. Given that Bob’s slip is selected, find the expected number of slips of paper in the box before the slip is selected. [b]p9.[/b] Let $a$ and $b$ be positive integers, $a > b$, such that $6! \cdot 11$ divides $x^a -x^b$ for all positive integers $x$. What is the minimum possible value of $a+b$? [b]p10.[/b] Find the number of pairs of positive integers $(m,n)$ such that $n < m \le 100$ and the polynomial $x^m+x^n+1$ has a root on the unit circle. [b]p11.[/b] Let $ABC$ be a triangle and let $\omega$ be the circle passing through $A$, $B$, $C$ with center $O$. Lines $\ell_A$, $\ell_B$, $\ell_C$ are drawn tangent to $\omega$ at $A$, $B$, $C$ respectively. The intersections of these lines form a triangle $XYZ$ where $X$ is the intersection of $\ell_B$ and $\ell_C$, $Y$ is the intersection of $\ell_C$ and $\ell_A$, and $Z$ is the intersection of $\ell_A$ and $\ell_B$. Let $P$ be the intersection of lines $\overline{OX}$ and $\overline{YZ}$. Given $\angle ACB = \frac32 \angle ABC$ and $\frac{AC}{AB} = \frac{15}{16}$ , find $\frac{ZP}{YP}$. [b]p12.[/b] Compute the remainder when $$\sum_{1\le a,k\le 2021} a^k$$ is divided by $2022$ (in the above summation $a,k$ are integers). [b]p13.[/b] Consider a $7\times 2$ grid of squares, each of which is equally likely to be colored either red or blue. Madeline would like to visit every square on the grid exactly once, starting on one of the top two squares and ending on one of the bottom two squares. She can move between two squares if they are adjacent or diagonally adjacent. What is the probability that Madeline may visit the squares of the grid in this way such that the sequence of colors she visits is alternating (i.e., red, blue, red,... or blue, red, blue,... )? [b]p14.[/b] Let $ABC$ be a triangle with $AB = 8$, $BC = 10$, and $CA = 12$. Denote by $\Omega_A$ the $A$-excircle of $ABC$, and suppose that $\Omega_A$ is tangent to $\overline{AB}$ and $\overline{AC}$ at $F$ and $E$, respectively. Line $\ell \ne \overline{BC}$ is tangent to $\Omega_A$ and passes through the midpoint of $\overline{BC}$. Let $T$ be the intersection of $\overline{EF}$ and $\ell$. Compute the area of triangle $ATB$. [b]p15.[/b] For any positive integer $n$, let $D_n$ be the set of ordered pairs of positive integers $(m,d)$ such that $d$ divides $n$ and gcd$(m,n) = 1$, $1 \le m \le n$. For any positive integers $a$, $b$, let $r(a,b)$ be the non-negative remainder when $a$ is divided by $b$. Denote by $S_n$ the sum $$S_n = \sum_{(m,d)\in D_n} r(m,d).$$ Determine the value of $S_{396}$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a finite sequence $a_1, a_2,...,a_n$ whose terms are natural numbers at most equal to $n$. Determine the maximum number of terms of such a sequence, if you know that every two of its neighboring terms are different and at the same time there is no quartet of terms in it such that $a_p = a_r \ne a_q = a_s$ for $p < q < r < s$.

Let $u_n$ denote the least common multiple of the first $n$ terms of a strictly increasing sequence of positive integers. Prove that the series $$\sum_{n=1}^{\infty} \frac{1}{ u_n }$$ is convergent

Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1 = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$ Prove that $u_n = v_n.$

Determine whether there exists an infinite sequence $a_1, a_2, a_3, \dots$ of positive integers which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer $n$.

Find all real constants c for which there exist strictly increasing sequence $a$ of positive integers such that $(a_{2n-1}+a_{2n})/{a_n}=c$ for all positive intеgers n.

A sequence of natural numbers is written according to the following rule: [i] the first two numbers are chosen and thereafter, in order to write a new number, the sum of the last numbers is calculated using the two written numbers, we find the greatest odd divisor of their sum and the sum of this greatest odd divisor plus one is the following written number. [/i]The first numbers are $25$ and $126$ (in that order), and the sequence has $2015$ numbers. Find the last number written.

The natural numbers from $100$ to $999$ are written on separate cards. They are gathered in one pile with their numbers down in arbitrary order. Let us open them in sequence and divide into $10$ piles according to the least significant digit. The first pile will contain cards with $0$ at the end, ... , the tenth -- with $9$. Then we shall gather $10$ piles in one pile, the first -- down, then the second, ... and the tenth -- up. Let us repeat the procedure twice more, but the next time we shall divide cards according to the second digit, and the last time -- to the most significant one. What will be the order of the cards in the obtained pile?

If $a_0$ is a positive real number, consider the sequence $\{a_n\}$ defined by: \[ a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0. \] Show that there exist a real number $a > 0$ such that: [b]i.)[/b] for all $a_0 \geq a,$ the sequence $\{a_n\} \rightarrow \infty,$ [b]ii.)[/b] for all $a_0 < a,$ the sequence $\{a_n\} \rightarrow 0.$

A rational number $x$ is given. Prove that there exists a sequence $x_0, x_1, x_2, \ldots$ of rational numbers with the following properties: (a) $x_0=x$; (b) for every $n\ge1$, either $x_n = 2x_{n-1}$ or $x_n = 2x_{n-1} + \textstyle\frac{1}{n}$; (c) $x_n$ is an integer for some $n$.

We call a triple of natural numbers $(a, b, c)$ [i]square [/i] if they form an arithmetic progression (in exactly this order), the number $b$ is coprime to each of the numbers $a$ and $c$, and the number $abc$ is a perfect square. Prove that for any given a square triple, there is another square triple that has at least one common number with it.

Integers $x_0,x_1,...,x_{n-1}, x_n = x_0, x_{n+1} = x_1$ satisfy the inequality $(-1)^{x_k} x_{k-1}x_{k+1} >0$ for $k = 1,2,...,n$. Prove that the difference $\sum_{k=0}^{n-1}x_k -\sum_{k=0}^{n-1}|x_k|$ is divisible by $4$.