Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his starting position? (I forgot answer choices)

[asy] real t=pi/12;real u=8*t; real cu=cos(u);real su=sin(u); draw(unitcircle); draw((cos(-t),sin(-t))--(cos(13*t),sin(13*t))); draw((cu,su)--(cu,-su)); label("A",(cos(13*t),sin(13*t)),W); label("B",(cos(-t),sin(-t)),E); label("C",(cu,su),N); label("D",(cu,-su),S); label("E",(cu,sin(-t)),NE); label("2",((cu-1)/2,sin(-t)),N); label("6",((cu+1)/2,sin(-t)),N); label("3",(cu,(sin(-t)-su)/2),E); //Credit to Zimbalono for the diagram[/asy] Chords $AB$ and $CD$ in the circle above intersect at $E$ and are perpendicular to each other. If segments $AE$, $EB$, and $ED$ have measures $2$, $3$, and $6$ respectively, then the length of the diameter of the circle is $\textbf{(A) }4\sqrt{5}\qquad\textbf{(B) }\sqrt{65}\qquad\textbf{(C) }2\sqrt{17}\qquad\textbf{(D) }3\sqrt{7}\qquad \textbf{(E) }6\sqrt{2}$

For how many integers $n$ does the expression \[\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}} \] represent a real number, where log denotes the base $10$ logarithm? $ \textbf{(A) }900 \qquad \textbf{(B) }2\qquad \textbf{(C) }902 \qquad \textbf{(D) } 2 \qquad \textbf{(E) }901$

Jeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every 1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is $\frac{m}{n}$, where m and n are relatively prime positive integers. Find $m + n$.

An ellipse, whose semi-axes have length $a$ and $b$, rolls without slipping on the curve $y=c\sin{\left(\frac{x}{a}\right)}$. How are $a,b,c$ related, given that the ellipse completes one revolution when it traverses one period of the curve?

The sequence $a_1,a_2,\ldots$ is defined by $a_1=2019$, $a_2=2020$, $a_3=2021$, $a_{n+3}=a_n(a_{n+1}a_{n+2}+1)$ for $n\ge 1$. Determine the value of the infinite sum \[\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots \]

Given a segment $AB$ of the length 1, define the set $M$ of points in the following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$

A flagpole is originally $ 5$ meters tall. A hurricane snaps the flagpole at a point $ x$ meters above the ground so that the upper part, still attached to the stump, touches the ground $ 1$ meter away from the base. What is $ x$? $ \textbf{(A)}\ 2.0 \qquad \textbf{(B)}\ 2.1 \qquad \textbf{(C)}\ 2.2 \qquad \textbf{(D)}\ 2.3 \qquad \textbf{(E)}\ 2.4$

In a football season, even number $n$ of teams plays a simple series, i.e. each team plays once against each other team. Show that ona can group the series into $n-1$ rounds such that in every round every team plays exactly one match.

The silicon-oxygen bonds in $\ce{SiO2}$ are best described as ${ \textbf{(A)}\ \text{coordinate covalent}\qquad\textbf{(B)}\ \text{ionic}\qquad\textbf{(C)}\ \text{nonpolar covalent}\qquad\textbf{(D)}}\ \text{polar covalent}\qquad $

Find the number of functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ having the property that $ (f\circ f\circ f)(n)=n+3, $ for any natural numbers $ n. $

In the unit squre For the given natural number $n \geq 2$ find the smallest number $k$ that from each set of $k$ unit squares of the $n$x$n$ chessboard one can achoose a subset such that the number of the unit squares contained in this subset an lying in a row or column of the chessboard is even

Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, and let $M$ be the midpoint of $EF$. Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ at $S$ and $T$, respectively. If the quadrilateral $KSAT$ is cycle, prove that $\angle{KEF}=\angle{KFE}=\angle{A}$.

Let $\triangle ABC$ with $AB=AC$ and $BC=14$ be inscribed in a circle $\omega$. Let $D$ be the point on ray $BC$ such that $CD=6$. Let the intersection of $AD$ and $\omega$ be $E$. Given that $AE=7$, find $AC^2$. [i]Proposed by Ephram Chun and Euhan Kim[/i]

In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won?

Shikaku and his son Shikamaru must climb a staircase that has $2022$ steps; the steps are listed $1$, $2$, $...$ , $2022$ and the floor is considered step $0$. This bores them both a lot, so so they decide to organize a game. They begin by tying a rope between them, so that At most they can be separated from each other by a distance of $7$ steps, that is, if they are in the steps $m$ and$ n$, then it must always be true that $|m-n| \le 7$. For the game they establish the following rules: a) They move alternately in turns. b) In his corresponding turn, the player must move to a higher step than in the one that (the same) was previously. c) If a player has just moved to the $n$-th step, then on the next turn the other player cannot be moved to any of the steps $n-1$, $n$ or $n + 1$, except when it is for reach the last step. d) Whoever reaches the last step (listed with $2022$) wins. Shikamaru is bored to start, so his father starts. Determine which of the two players has a winning strategy and describe it.

For a certain complex number $c$, the polynomial \[ P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\] has exactly 4 distinct roots. What is $|c|$? $\textbf{(A) } 2 \qquad \textbf{(B) } \sqrt{6} \qquad \textbf{(C) } 2\sqrt{2} \qquad \textbf{(D) } 3 \qquad \textbf{(E) } \sqrt{10}$

Let $F$ be a finite field having an odd number $m$ of elements. Let $p(x)$ be an irreducible (i.e. nonfactorable) polynomial over $F$ of the form $$x^2+bx+c, ~~~~~~ b,c \in F.$$ For how many elements $k$ in $F$ is $p(x)+k$ irreducible over $F$?

Let us consider a triangle $\Delta{PQR}$ in the co-ordinate plane. Show for every function $f: \mathbb{R}^2\to \mathbb{R}\;,f(X)=ax+by+c$ where $X\equiv (x,y) \text{ and } a,b,c\in\mathbb{R}$ and every point $A$ on $\Delta PQR$ or inside the triangle we have the inequality: \begin{align*} & f(A)\le \text{max}\{f(P),f(Q),f(R)\} \end{align*}

Compute the product of the three smallest prime factors of \[21!\cdot 14!+21!\cdot 21+14!\cdot 14+21\cdot 14.\] [i]Proposed by Daniel Liu

Let $c, s$ be real functions defined on $\mathbb{R}\setminus\{0\}$ that are nonconstant on any interval and satisfy \[c\left(\frac{x}{y}\right)= c(x)c(y) - s(x)s(y)\text{ for any }x \neq 0, y \neq 0\] Prove that: $(a) c\left(\frac{1}{x}\right) = c(x), s\left(\frac{1}{x}\right) = -s(x)$ for any $x = 0$, and also $c(1) = 1, s(1) = s(-1) = 0$; $(b) c$ and $s$ are either both even or both odd functions (a function $f$ is even if $f(x) = f(-x)$ for all $x$, and odd if $f(x) = -f(-x)$ for all $x$). Find functions $c, s$ that also satisfy $c(x) + s(x) = x^n$ for all $x$, where $n$ is a given positive integer.

How many six-digit perfect squares can be formed using all the numbers $1,2,3,4,5,6$ as digits? $\textbf{(A)}~5$ $\textbf{(B)}~19$ $\textbf{(C)}~7$ $\textbf{(D)}~\text{None of the above}$

A $6n$-digit number is divisible by $7$. Prove that if its last digit is moved to the beginning of the number then the new number is also divisible by $7$.

The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L,$ without traveling along any portion of a road more than once. (Paula [i]is[/i] allowed to visit a city more than once.) How many different routes can Paula take? [asy] import olympiad; unitsize(50); for (int i = 0; i < 3; ++i) { for (int j = 0; j < 4; ++j) { pair A = (j,i); dot(A); } } for (int i = 0; i < 3; ++i) { for (int j = 0; j < 4; ++j) { if (j != 3) { draw((j,i)--(j+1,i)); } if (i != 2) { draw((j,i)--(j,i+1)); } } } label("$A$", (0,2), W); label("$L$", (3,0), E); [/asy] $\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$

Suppose that every two opposite edges of a tetrahedron are orthogonal. Show that the midpoints of the six edges lie on a sphere.