Two different $3 \times 3$ grids are chosen within a $5 \times 5$ grid. What is the least number of unit grids contained in the overlap of the two $3 \times 3$ grids? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }6$

In isosceles trapezoid $ABCD$, parallel bases $\overline{AB}$ and $\overline{CD}$ have lengths $500$ and $650$, respectively, and $AD=BC=333$. The angle bisectors of $\angle{A}$ and $\angle{D}$ meet at $P$, and the angle bisectors of $\angle{B}$ and $\angle{C}$ meet at $Q$. Find $PQ$.

In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that \[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]

Let \[ f(x)\equal{}a_0\plus{}a_1x\plus{}a_2x^2\plus{}a_{10}x^{10}\plus{}a_{11}x^{11}\plus{}a_{12}x^{12}\plus{}a_{13}x^{13} \; (a_{13} \not\equal{}0) \] and \[ g(x)\equal{}b_0\plus{}b_1x\plus{}b_2x^2\plus{}b_{3}x^{3}\plus{}b_{11}x^{11}\plus{}b_{12}x^{12}\plus{}b_{13}x^{13} \; (b_{3} \not\equal{}0) \] be polynomials over the same field. Prove that the degree of their greatest common divisor is at least $ 6$. [i]L. Redei[/i]

Let $ABC$ be a right triangle with $\angle ACB = 90^o$ and M the center of $AB$. Let $G$ br any point on the line $MC$ and $P$ a point on the line $AG$, such that $\angle CPA = \angle BAC$ . Further let $Q$ be a point on the straight line $BG$, such that $\angle BQC = \angle CBA$ . Show that the circles of the triangles $AQG$ and $BPG$ intersect on the segment $AB$.

Prove that there is a multiple of $2^{2022}$ that has $2022$ digits, and can be written using digits $1$ and $2$ only.

Let $\Omega_1$ be a circle with centre $O$ and let $AB$ be diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$ different from $O$. Suppose another circle $\Omega_2$ with centre $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$ intersecting $\Omega_1$ again at $A_1$ and B1 respectively such that $A_1$ and $B_1$ are on the opposite sides of $AB$. Given that $A_1 B = 5, AB_1 = 15$ and $OP = 10$, find the radius of $\Omega_1$.

Chris has a bag with 4 black socks and 6 red socks (so there are $10$ socks in total). Timothy reaches into the bag and grabs two socks [i]without replacement[/i]. Find the probability that he will not grab two red socks. [i]Proposed by Chris Tong[/i]

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $N \ge 3$ be the answer to Problem 21. A regular $N$-gon is inscribed in a circle of radius $1$. Let $D$ be the set of diagonals, where we include all sides as diagonals. Then, let $D'$ be the set of all unordered pairs of distinct diagonals in $D$. Compute the sum $$\sum_{\{d,d'\}\in D'} \ell (d)^2 \ell (d')^2,$$where $\ell (d)$ denotes the length of diagonal $d$. [b]p20.[/b] Let $N$ be the answer to Problem $19$, and let $M$ be the last digit of $N$. Let $\omega$ be a primitive $M$th root of unity, and define $P(x)$ such that$$P(x) = \prod^M_{k=1} (x - \omega^{i_k}),$$where the $i_k$ are chosen independently and uniformly at random from the range $\{0, 1, . . . ,M-1\}$. Compute $E \left[P\left(\sqrt{\rfloor \frac{1250}{N} \rfloor } \right)\right].$ [b]p21.[/b] Let $N$ be the answer to Problem $20$. Define the polynomial $f(x) = x^{34} +x^{33} +x^{32} +...+x+1$. Compute the number of primes $p < N$ such that there exists an integer $k$ with $f(k)$ divisible by $p$.

Let $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$.

Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$ Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases. (Edit: Proposed by sir Leonard Giugiuc, Romania)

Consider the equation $x^4-24x^3+210x^2+mx+n=0$. Given that the roots of this equation are nonnegative reals, find the maximum possible value of a root of this equation across all values of $m$ and $n$. [i]Proposed by Andrew Zhao[/i]

Two identical blue blocks, two identical red blocks, two identical green blocks, and two identical purple blocks are placed next to each other in a row. Find the number of distinct arrangements of these blocks where no blue block is placed next to a red block, and no green block is placed next to a purple block.

Suppose $f_1(x), f_2(x), \dots, f_n(x)$ are functions of $n$ real variables $x = (x_1, \dots, x_n)$ with continuous second-order partial derivatives everywhere on $\mathbb{R}^n$. Suppose further that there are constants $c_{ij}$ such that \[ \frac{\partial f_i}{\partial x_j} - \frac{\partial f_j}{\partial x_i} = c_{ij} \] for all $i$ and $j$, $1\leq i \leq n$, $1 \leq j \leq n$. Prove that there is a function $g(x)$ on $\mathbb{R}^n$ such that $f_i + \partial g/\partial x_i$ is linear for all $i$, $1 \leq i \leq n$. (A linear function is one of the form \( a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n.) \)

If $ \frac {x^2 \minus{} bx}{ax \minus{} c} \equal{} \frac {m \minus{} 1}{m \plus{} 1}$ has roots which are numerical equal but of opposite signs, the value of $ m$ must be: $ \textbf{(A)}\ \frac {a \minus{} b}{a \plus{} b} \qquad\textbf{(B)}\ \frac {a \plus{} b}{a \minus{} b} \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ \frac {1}{c} \qquad\textbf{(E)}\ 1$

Alice chooses a prime number $p > 2$ and then Bob chooses a positive integer $n_0$. Alice, in the first move, chooses an integer $n_1 > n_0$ and calculates the expression $s_1 = n_0^{n_1} + n_1^{n_0}$; then Bob, in the second move, chooses an integer $n_2 > n_1$ and calculates the expression $s_2 = n_1^{n_2} + n_2^{n_1}$; etc. one by one. (Each player knows the numbers chosen by the other in the previous moves.) The winner is the one who first chooses the number $n_k$ such that $p$ divides $s_k(s_1 + 2s_2 + · · · + ks_k)$. Who has a winning strategy? Proposed by [i]Borche Joshevski, Macedonia[/i]

Two polynomials of the same degree $A(x)=a_nx^n+ \cdots + a_1x+a_0$ and $B(x)=b_nx^n+\cdots+b_1x+b_0$ are called [i]friends[/i] is the coefficients $b_0,b_1, \ldots, b_n$ are a permutation of the coefficients $a_0,a_1, \ldots, a_n$. $P(x)$ and $Q(x)$ be two friendly polynomials with integer coefficients. If $P(16)=3^{2020}$, the smallest possible value of $|Q(3^{2020})|$.

As shown on the figure, square $PQRS$ is inscribed in right triangle $ABC$, whose right angle is at $C$, so that $S$ and $P$ are on sides $BC$ and $CA$, respectively, while $Q$ and $R$ are on side $AB$. Prove that $A B\geq3QR$ and determine when equality occurs. [asy] defaultpen(linewidth(0.7)+fontsize(10)); size(150); real a=8, b=6; real y=a/((a^2+b^2)/(a*b)+1), r=degrees((a,b))+180; pair A=b*dir(-r)*dir(90), B=a*dir(180)*dir(-r), C=origin, S=y*dir(-r)*dir(180), P=(y*b/a)*dir(90-r), Q=foot(P, A, B), R=foot(S, A, B); draw(A--B--C--cycle^^R--S--P--Q); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$S$", S, dir(point--S)); label("$R$", R, dir(270)); label("$Q$", Q, dir(270)); label("$P$", P, dir(point--P));[/asy]

Let $n$ be a positive integer number. The decimal representation of $2^n$ contains $m$ same numbers from the right. What is the largest value of $m$? $\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None}$

If $f(x) = ax^2 + bx + c$ satisfies the condition $|f(x)| < 1; \forall x \in [-1, 1]$, prove that the equation $f(x) = 2x^2 - 1$ has two real roots.

There exists a unique line tangent to the graph of $y=x^4-20x^3+24x^2-20x+25$ at two distinct points. Compute the product of the $x$-coordinates of the two tangency points.

A given convex pentagon $ ABCDE$ has the property that the area of each of five triangles $ ABC, BCD, CDE, DEA$, and $ EAB$ is unity [i](equal to 1)[/i]. Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.

William writes the number $1$ on a blackboard. Every turn, he erases the number $N$ currently on the blackboard and replaces it with either $4N + 1$ or $8N + 1$ until it exceeds $1000$, after which no more moves are made. If the minimum possible value of the final number on the blackboard is $M$, find the remainder when $M$ is divided by $1000$.

A regular $2017$-gon is partitioned into triangles by a set of non-intersecting diagonals. Prove that among those triangles only one is acute-angled.

Let $a, b, c$ be the side-lengths of a triangle, $r$ be the inradius and $r_a, r_b, r_c$ be the corresponding exradius. Show that \[ \frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \leq 2 \cdot \frac{\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}}{r_a+r_b+r_c-3r} \]