Let the number of ways for a rook to return to its original square on a $4\times 4$ chessboard in 8 moves if it starts on a corner be $k$. Find the number of positive integers that are divisors of $k$. A "move" counts as shifting the rook by a positive number of squares on the board along a row or column. Note that the rook may return back to its original square during an intermediate step within its 8-move path. [i]Proposed by Bradley Guo[/i]

Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).

If $f(x) = ax^4-bx^2+x+5$ and $f(-3) = 2$, then $f(3) =$ $\textbf{(A)}\ -5 \qquad \textbf{(B)}\ -2 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 8$

Solve for real values of parameter $a$, the inequality : $$\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}$$

Prove that among ten consecutive natural numbers there is always at least one, and at most four, numbers that are not divisible by any of the numbers $ 2 $, $ 3 $, $ 5 $, $ 7 $.

Let $P$ be a convex $2006$-gon. The $1003$ diagonals connecting opposite vertices and the $1003$ lines connecting the midpoints of opposite sides are concurrent, that is, all $2006$ lines have a common point. Prove that the opposite sides of $P$ are parallel and congruent.

[b]p1.[/b] There are $16$ students in a class. Each month the teacher divides the class into two groups. What is the minimum number of months that must pass for any two students to be in different groups in at least one of the months? [b]p2.[/b] Find all functions $f(x)$ defined for all real $x$ that satisfy the equation $2f(x) + f(1 - x) = x^2$. [b]p3.[/b] Arrange the digits from $1$ to $9$ in a row (each digit only once) so that every two consecutive digits form a two-digit number that is divisible by $7$ or $13$. [b]p4.[/b] Prove that $\cos 1^o$ is irrational. [b]p5.[/b] Consider $2n$ distinct positive Integers $a_1,a_2,...,a_{2n}$ not exceeding $n^2$ ($n>2$). Prove that some three of the differences $a_i- a_j$ are equal . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider two overlapping regular tetrahedrons of side length $2$ in space. They are centered at the same point, and the second one is oriented so that the lines from its center to its vertices are perpendicular to the faces of the first tetrahedron. What is the volume encompassed by the combined solid?

Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$

Samuel Tsui and Jason Yang each chose a different integer between $1$ and $60$, inclusive. They don’t know each others’ numbers, but they both know that the other person’s number is between $1$ and $60$ and distinct from their own. They have the following conversation: Samuel Tsui: Do our numbers have any common factors greater than $1$? Jason Yang: Definitely not. However their least common multiple must be less than$ 2023$. Samuel Tsui: Ok, thismeans that the sumof the factors of our two numbers are equal. What is the sumof Samuel Tsui’s and Jason Yang’s numbers? [i]Proposed by Samuel Tsui[/i]

In an acute triangle $ABC$, let a line $\ell$ pass through the orthocenter and not through point $A$. The line $\ell$ intersects line $BC$ at $P(\neq B, C)$. A line passing through $A$ and perpendicular to $\ell$ meets the circumcircle of triangle $ABC$ at $R(\neq A)$. Let the feet of the perpendiculars from $A, B$ to $\ell$ be $A', B'$, respectively. Define line $\ell_1$ as the line passing through $A'$ and perpendicular to $BC$, and line $\ell_2$ as the line passing through $B'$ and perpendicular to $CA$. Prove that if $Q$ is the reflection of the intersection of $\ell_1$ and $\ell_2$ across $\ell$, then $\angle PQR = 90^{\circ}$.

In $R^n$ space is given a finite set of points $X$. It is known that for any subset $Y \subseteq X$ of at most $n+1$ points, there is a unit ball $B_Y$ containing $Y$ and not containing the origin. Prove that there is a unit a ball $B_X$ containing $X$ and not containing the origin.

An arithmetic sequence of exactly $10$ positive integers has the property that any two elements are relatively prime. Compute the smallest possible sum of the $10$ numbers. [i]Proposed by Kyle Lee[/i]

The point $ O$ is the center of the circle circumscribed about $ \triangle ABC$, with $ \angle BOC \equal{} 120^\circ$ and $ \angle AOB \equal{} 140^\circ$, as shown. What is the degree measure of $ \angle ABC$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair B=dir(80), A=dir(220), C=dir(320), O=(0,0); draw(unitcircle); draw(A--B--C--O--A--C); draw(O--B); draw(anglemark(C,O,A,2)); label("$A$",A,SW); label("$B$",B,NNE); label("$C$",C,SE); label("$O$",O,S); label("$140^{\circ}$",O,NW,fontsize(8pt)); label("$120^{\circ}$",O,ENE,fontsize(8pt));[/asy]$ \textbf{(A)}\ 35 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 60$

Express the sum of $99$ terms$$\frac{1\cdot 4}{2\cdot 5}+\frac{2\cdot 7}{5\cdot 8}+\ldots +\frac{k(3k+1 )}{(3k-1)(3k+2)}+\ldots +\frac{99\cdot 298}{296\cdot 299}$$ as an irreducible fraction.

We call an $m$-dimensional smooth manifold [i]parallelizable[/i] if it admits $m$ smooth tangent vector fields that are linearly independent at all points. Show that if $M$ is a closed orientable $2n$-dimensional smooth manifold of Euler characteristic $0$ that has an immersion into a parallelizable smooth $(2n+1)$-dimensional manifold $N$, then $M$ is itself parallelizable.

How many four digit numbers $\overline{abcd}$ simultaneously satisfy the equalities $a+b=c+d$ and $a^2+b^2=c^2+d^2$?