Father moves $3$ steps forward just as son moves $5$ steps, but this while the father takes $6$ steps, the son does $7$ steps. At first, father and son are together, then the son begins to walk away from his father in a straight line. When the son has done $30$ steps, the father starts to follow him. In a few steps, Dad arrives after the son?

Given $n$ a positive integer, define $T_n$ the number of quadruples of positive integers $(a,b,x,y)$ such that $a>b$ and $n= ax+by$. Prove that $T_{2023}$ is odd.

Bobo the clown was juggling his spherical cows again when he realized that when he drops a cow is related to how many cows he started off juggling. If he juggles $1$, he drops it after $64$ seconds. When juggling $2$, he drops one after $55$ seconds, and the other $55$ seconds later. In fact, he was able to create the following table: [img]https://cdn.artofproblemsolving.com/attachments/1/0/554a9bace83b4b3595c6012dfdb42409465829.png[/img] He can only juggle up to $22$ cows. To juggle the cows the longest, what number of cows should he start off juggling? How long (in minutes) can he juggle for?

There are 2012 lamps arranged on a table. Two persons play the following game. In each move the player flips the switch of one lamp, but he must never get back an arrangement of the lit lamps that has already been on the table. A player who cannot move loses. Which player has a winning strategy?

Find all natural numbers $n$ such that there exist $ x_1,x_2,\ldots,x_n,y\in\mathbb{Z}$ where $x_1,x_2,\ldots,x_n,y\neq 0$ satisfying: \[x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n \equal{} 0\] \[ny^2 \equal{} x_1^2 \plus{} x_2^2 \plus{} \ldots \plus{} x_n^2\]

Let $ABCD$ be a convex quadrilateral.Let diagonals $AC$ and $BD$ intersect at $P$. Let $PE,PF,PG$ and $PH$ are altitudes from $P$ on the side $AB,BC,CD$ and $DA$ respectively. Show that $ABCD$ has a incircle if and only if $\frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH}.$

Let $n$ be a positive integer. Prove that all roots of the equation $$x(x + 2) (x + 4 )... (x + 2n) + (x +1) (x + 3 )... (x + 2n - 1) = 0$$ are real and irrational.

Let be two nonnegative real numbers $ a,b $ with $ b>a, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ of real numbers such that the sequence $ \left( \frac{x_1+x_2+\cdots +x_n}{n^a} \right)_{n\ge 1} $ is bounded. Show that the sequence $ \left( x_1+\frac{x_2}{2^b} +\frac{x_3}{3^b} +\cdots +\frac{x_n}{n^b} \right)_{n\ge 1} $ is convergent.

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Let $f:\mathbb N\to \mathbb N$. Show that $f(m)+n\mid f(n)+m$ for all positive integers $m\le n$ if and only if $f(m)+n\mid f(n)+m$ for all positive integers $m\ge n$. [i]Proposed by Carl Schildkraut[/i]

Two circles of radius $ 2$ are centered at $ (2,0)$ and at $ (0,2)$. What is the area of the intersection of the interiors of the two circles? $ \textbf{(A)}\ \pi \minus{} 2\qquad \textbf{(B)}\ \frac {\pi}{2}\qquad \textbf{(C)}\ \frac {\pi\sqrt {3}}{3}\qquad \textbf{(D)}\ 2(\pi \minus{} 2)\qquad \textbf{(E)}\ \pi$

For non-negative integers $n$ and $k$, let $P_{n, k}(x)$ denote the rational function \[\frac{(x^{n}-1)(x^{n}-x) \cdots (x^{n}-x^{k-1})}{(x^{k}-1)(x^{k}-x) \cdots (x^{k}-x^{k-1})}.\] Show that $P_{n, k}(x)$ is actually a polynomial for all $n, k \in \mathbb{N}$.

$ \lim_{n\to\infty } n\left( \sqrt[3]{n^3-6n^2+6n+1}-\sqrt{n^2-an+5} \right) $

Let $c$ be a constant. The simultaneous equations \begin{align*} x-y = &\ 2 \\ cx+y = &\ 3 \\ \end{align*} have a solution $(x, y)$ inside Quadrant I if and only if $ \textbf{(A)}\ c=-1 \qquad\textbf{(B)}\ c>-1 \qquad\textbf{(C)}\ c<\frac{3}{2} \qquad\textbf{(D)}\ 0<c<\frac{3}{2}\\ \qquad\textbf{(E)}\ -1<c<\frac{3}{2} $

In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.

Given a positive integer $n,$ let $M(n)$ be the largest integer $m$ such that \[\binom{m}{n-1}>\binom{m-1}{n}.\] Evaluate \[\lim_{n\to\infty}\frac{M(n)}{n}.\]

There are several triangles. From them a new triangle is obtained according to the following rule. The largest side of the new triangle is equal to the sum of the large sides of the data, the middle one is equal to the sum of the middle sides, and the smallest one is the sum of the smaller ones. Prove that if all the angles of these triangles were less than $a$, and $\phi$, where $\phi$ is the largest angle of the resulting triangle, then $\cos \phi \ge 1-\sin (a/2)$.

The expression $ \frac{2x^2-x}{(x+1)(x-2)}-\frac{4+x}{(x+1)(x-2)}$ cannot be evaluated for $ x=-1$ or $ x=2$, since division by zero is not allowed. For other values of $ x$: $\textbf{(A)}\ \text{The expression takes on many different values.} \\ \textbf{(B)}\ \text{The expression has only the value 2.} \\ \textbf{(C)}\ \text{The expression has only the value 1.} \\ \textbf{(D)}\ \text{The expression always has a value between } -1 \text{ and } +2. \\ \textbf{(E)}\ \text{The expression has a value greater than 2 or less than } -1.$

$\alpha,\beta$ are two different solutions to the equation $4x^2-4tx+1=0(t\in\mathbb{R})$, the domain of definition of the function $f(x)=\frac{2x-t}{x^2+1}$ is $[\alpha,\beta](\alpha<\beta)$. [b](a)[/b] Find $g(t)=\max f(x)-\min f(x)$. [b](b)[/b] Prove: for $u_i\in\left(0,\frac{\pi}{2}\right)(i=1,2,3)$, if $\sin u_1+\sin u_2+\sin u_3=1$, then $\frac{1}{g(\tan u_1)}+\frac{1}{g(\tan u_2)}+\frac{1}{g(\tan u_3)}<\frac{3}{4}\sqrt6$.

Fix positive integers $n$ and $k$, and $2n$ positive (not neccesarily distinct) real numbers $a_1,\cdots, a_n$, $b_1, \cdots, b_n$. An equation is written on a whiteboard: $$t=*\times*\times\cdots\times*$$ where $t$ is a fixed positive real number, with exactly $k$ asterisks. Ebi fills each asterisk with a number from $a_1, a_2,\cdots, a_n$, while Rubi fills each asterisk with a number from $b_1, b_2,\cdots, b_n$, so that the equation on the whiteboard is correct. Suppose for every positive real number $t$, the number of ways for Ebi and Rubi to do so are equal. Prove that the sequences $a_1,\cdots, a_n$ and $b_1, \cdots, b_n$ are permutations of each other. [i](Note: $t=a_1a_2a_3$ and $t=a_2a_3a_1$ are considered different ways to fill the asterisks, and the chosen terms need not be distinct, for example $t=a_1a_1a_2$.)[/i] [i]Proposed by Wong Jer Ren[/i]

Given a sequence $\{x_k\}$ such that $x_1 = 1$, $x_{n+1} = n \sin x_n+ 1$. Prove that the sequence is non-periodic.

Points on a spherical surface with radius $4$ are colored in $4$ different colors. Prove that there exist two points with the same color such that the distance between them is either $4\sqrt{3}$ or $2\sqrt{6}$. (Distance is Euclidean, that is, the length of the straight segment between the points)

The side length of the largest square below is $8\sqrt{2}$, as shown. Find the area of the shaded region. [asy] size(10cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-18.99425911800572,xmax=23.81538435842469,ymin=-15.51769962526155,ymax=6.464951807764648; pen zzttqq=rgb(0.6,0.2,0.); pair A=(0.,1.), B=(0.,0.), C=(1.,0.), D=(1.,1.), F=(1.,2.), G=(2.,3.), H=(0.,3.), I=(0.,5.), J=(-2.,3.), K=(-4.,5.), L=(-4.,1.), M=(-8.,1.), O=(-8.,-7.), P=(0.,-7.); draw(B--A--D--C--cycle); draw(A--C--(2.,1.)--F--cycle); draw(A--(2.,1.)--G--H--cycle); draw(A--G--I--J--cycle); draw(A--I--K--L--cycle); draw(A--K--M--(-4.,-3.)--cycle); draw(A--M--O--P--cycle); draw(A--O--(0.,-15.)--(8.,-7.)--cycle); filldraw(A--B--C--D--cycle,opacity(0.2)+black); filldraw(A--(2.,1.)--F--cycle,opacity(0.2)+black); filldraw(A--G--H--cycle,opacity(0.2)+black); filldraw(A--I--J--cycle,opacity(0.2)+black); filldraw(A--K--L--cycle,opacity(0.2)+black); filldraw(A--M--(-4.,-3.)--cycle, opacity(0.2)+black); filldraw(A--O--P--cycle,opacity(0.2)+black); draw(B--A); draw(A--D); draw(D--C); draw(C--B); draw(A--C); draw(C--(2.,1.)); draw((2.,1.)--F); draw(F--A); draw(A--(2.,1.)); draw((2.,1.)--G); draw(G--H); draw(H--A); draw(A--G); draw(G--I); draw(I--J); draw(J--A); draw(A--I); draw(I--K); draw(K--L); draw(L--A); draw(A--K); draw(K--M); draw(M--(-4.,-3.)); draw((-4.,-3.)--A); draw(A--M); draw(M--O); draw(O--P); draw(P--A); draw(A--O); draw(O--(0.,-15.)); draw((0.,-15.)--(8.,-7.)); draw((8.,-7.)--A); draw(A--B,black); draw(B--C,black); draw(C--D,black); draw(D--A,black); draw(A--(2.,1.),black); draw((2.,1.)--F,black); draw(F--A,black); draw(A--G,black); draw(G--H,black); draw(H--A,black); draw(A--I,black); draw(I--J,black); draw(J--A,black); draw(A--K,black); draw(K--L,black); draw(L--A,black); draw(A--M,black); draw(M--(-4.,-3.),black); draw((-4.,-3.)--A,black); draw(A--O,black); draw(O--P,black); draw(P--A,black); label("$8\sqrt{2}$",(-8,-7)--(0,-15)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] [i]Lightning 2.4[/i]

A square has vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(10,10)$ on the $x-y$ coordinate plane. A second quadrilateral is constructed with vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(15,15)$. Find the positive difference between the areas of the original square and the second quadrilateral. [i]Proposed byWilliam Hua[/i]

A quadrilateral with sides $a,b,c,d$ is inscribed in a circle of radius $R$. Prove that if $a^2+b^2+c^2+d^2=8R^2$, then either one of the angles of the quadrilateral is right or the diagonals of the quadrilateral are perpendicular.