The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches? [asy] usepackage("mathptmx"); defaultpen(linewidth(0.5)); size(5cm); defaultpen(fontsize(14pt)); label("$\textbf{Math}$", (2.1,3.7)--(3.9,3.7)); label("$\textbf{Team}$", (2.1,3)--(3.9,3)); filldraw((1,2)--(2,1)--(3,2)--(4,1)--(5,2)--(4,3)--(5,4)--(4,5)--(3,4)--(2,5)--(1,4)--(2,3)--(1,2)--cycle, mediumgray*0.5 + lightgray*0.5); draw((0,0)--(6,0), gray); draw((0,1)--(6,1), gray); draw((0,2)--(6,2), gray); draw((0,3)--(6,3), gray); draw((0,4)--(6,4), gray); draw((0,5)--(6,5), gray); draw((0,6)--(6,6), gray); draw((0,0)--(0,6), gray); draw((1,0)--(1,6), gray); draw((2,0)--(2,6), gray); draw((3,0)--(3,6), gray); draw((4,0)--(4,6), gray); draw((5,0)--(5,6), gray); draw((6,0)--(6,6), gray); [/asy] $\textbf{(A) } 10 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15$

Let $OX, OY, OZ$ be mutually orthogonal lines in space. Let $C$ be a fixed point on $OZ$ and $U,V$ variable points on $OX, OY,$ respectively. Find the locus of a point $P$ such that $PU, PV, PC$ are mutually orthogonal.

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

In two circles intersecting at points $A$ and $B$, parallel chords $A_1B_1$ and $A_2B_2$ are drawn. The lines $AA_1$ and $BB_2$ intersect at the point $X, AA_2$ and $BB_1$ intersect at the point $Y$. Prove that $XY // A_1B_1$.

Let $f_1,f_2,\cdots ,f_{10}$ be bijections on $\mathbb{Z}$ such that for each integer $n$, there is some composition $f_{\ell_1}\circ f_{\ell_2}\circ \cdots \circ f_{\ell_m}$ (allowing repetitions) which maps $0$ to $n$. Consider the set of $1024$ functions \[ \mathcal{F}=\{f_1^{\epsilon_1}\circ f_2^{\epsilon_2}\circ \cdots \circ f_{10}^{\epsilon_{10}}\} \] where $\epsilon _i=0$ or $1$ for $1\le i\le 10.\; (f_i^{0}$ is the identity function and $f_i^1=f_i)$. Show that if $A$ is a finite set of integers then at most $512$ of the functions in $\mathcal{F}$ map $A$ into itself.

Given a sequence of $19$ positive (not necessarily distinct) integers not greater than $93$, and a set of $93$ positive (not necessarily distinct) integers not greater than $19$. Show that we can find non-empty subsequences of the two sequences with equal sum.

Find all continuous functions $f : (0,+\infty) \to (0,+\infty)$ such that if $a, b, c$ are the lengths of the sides of any triangle then it is satisfied that $$\frac{f(a+b-c)+f(b+c-a)+f(c+a-b)}{3}=f\left(\sqrt{\frac{ab+bc+ca}{3}}\right)$$

Prove that $(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n}$ is an even integer and that $(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=w\sqrt{3}$ for some positive integer $w$, for all integers $n \geq 1$.

Let be nonnegative real numbers $ a,b,c, $ holding the inequality: $ \sum_{\text{cyc}} \frac{a}{b+c+1} \le 1. $ Prove that $ \sum_{\text{cyc}} \frac{1}{b+c+1} \ge 1. $

You have 5000 distinct finite sets. Their intersection is empty. However, the intersection of any two is nonempty. What is the smallest possible number of elements contained in their union?

Does there exist an infinite sequence of primes $p_1, p_2, p_3, \dots $ for which, \[p_{n+1}=2p_n+1\] for each $n$?

Determine the smallest value of $M$ for which for any choice of positive integer $n$ and positive real numbers $x_1<x_2<\ldots<x_n \le 2023$ the inequality $$\sum_{1\le i < j \le n , x_j-x_i \ge 1} 2^{i-j}\le M$$ holds.

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.

$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.

Find all solutions of the system $\sqrt[3]{\frac{yz^4}{x^2}}+2wx=0 $ $\sqrt[3]{\frac{xz^4}{y}}+5wy=0 $ $\sqrt[3]{\frac{xy}{x}}+7wz^{-1/3}=0$ $x^{12}+\frac{125}{4}y^5+\frac{343}{2}z^4=16$ where $x, y, z \ge 0$ and $w \in R$ [hide=PS] I attached the system, in case I have any typos[/hide]

Let $x,y,z$ be positive reals such that $xyz=1$. Show that $$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$ When does equality happen?

The diagonals of a convex quadrilateral divide it into four triangles. Prove that the nine point centers of these four triangles either lie on one straight line, or are the vertices of a parallelogram.

Let $\Omega$ be a circle with radius $18$ and let $\mathcal{S}$ be the region inside $\Omega$ that the centroid of $\triangle XYZ$ sweeps through as $X$ varies along all possible points lying outside of $\Omega$, $Y$ varies along all possible points lying on $\Omega$ and $XZ$ is tangent to the circle. Compute the greatest integer less than or equal to the area of $\mathcal{S}$. [I]Proposed by [b]FedeX333X[/b][/I]

Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

Every edge of a tetrahedron is tangent to a given sphere. Prove that the three line segments joining the points of tangency of the three pairs of opposite edges of the tetrahedron are concurrent. [i](7 points)[/i]

Find all real polynomials $P(x)$ that satisfy $$P(x^3-2)=P(x)^3-2$$

How many ordered triples $(x,y,z)$ of integers satisfy the system of equations below? \[ \begin{array}{l} x^2-3xy+2yz-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array} \] $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{a finite number greater than 2} \qquad \text{(E)} \ \text{infinately many}$

Given $n$ positive real numbers $x_1,x_2,x_3,...,x_n$ such that $$\left (1+\frac{1}{x_1}\right )\left(1+\frac{1}{x_2}\right)...\left(1+\frac{1}{x_n}\right)=(n+1)^n$$ Determine the minimum value of $x_1+x_2+x_3+...+x_n$. [i]Proposed by Loh Kwong Weng[/i]

Find all polynomials $p$ with integer coefficients such that the number $p(a) - p(b)$ is divisible by $a + b$ for all integers $a, b$, provided that $a + b \neq 0$.

Find all triples of positive integers $(a,b,c)$ such that: $a^2bc-2ab^2c-2abc^2+b^3c+bc^3+2b^2c^2=11$