A circle and a square have the same perimeter. Then: $ \textbf{(A)}\ \text{their areas are equal} \qquad\textbf{(B)}\ \text{the area of the circle is the greater}$ $ \textbf{(C)}\ \text{the area of the square is the greater}$ $ \textbf{(D)}\ \text{the area of the circle is } \pi \text{ times the area of the square} \\ \qquad\textbf{(E)}\ \text{none of these}$

Determine whether there exists a prime $q$ so that for any prime $p$ the number $$\sqrt[3]{p^2+q}$$ is never an integer.

A function $f:\mathbb{R} \to \mathbb{R}$ has the following property: $$\text{For every } x,y \in \mathbb{R} \text{ such that }(f(x)+y)(f(y)+x) > 0, \text{ we have } f(x)+y = f(y)+x.$$ Prove that $f(x)+y \leq f(y)+x$ whenever $x>y$.

Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.

Let $p$ be an odd prime. Prove that \[1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.\]

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

Let $P_i(x_i,y_i)\ (i=1,2,\cdots,2023)$ be $2023$ distinct points on a plane equipped with rectangular coordinate system. For $i\neq j$, define $d(P_i,P_j) = |x_i - x_j| + |y_i - y_j|$. Define $$\lambda = \frac{\max_{i\neq j}d(P_i,P_j)}{\min_{i\neq j}d(P_i,P_j)}$$. Prove that $\lambda \geq 44$ and provide an example in which the equality holds.

[b]p19.[/b] Sequences $a_n$ and $b_n$ of positive integers satisfy the following properties: (1) $a_1 = b_1 = 1$ (2) $a_5 = 6, b_5 \ge 7$ (3) Both sequences are strictly increasing (4) In each sequence, the difference between consecutive terms is either $1$ or $2$ (5) $\sum^5_{n=1}na_n =\sum^5_{n=1}nb_n = S$ Compute $S$. [b]p20.[/b] Let $A$, $B$, and $C$ be points lying on a line in that order such that $AB = 4$ and $BC = 2$. Let $I$ be the circle centered at B passing through $C$, and let $D$ and $E$ be distinct points on $I$ such that $AD$ and $AE$ are tangent to $I$. Let $J$ be the circle centered at $C$ passing through $D$, and let $F$ and $G$ be distinct points on $J$ such that $AF$ and $AG$ are tangent to $J$ and $DG < DF$. Compute the area of quadrilateral $DEFG$. [b]p21.[/b] Twain is walking randomly on a number line. They start at $0$, and flip a fair coin $10$ times. Every time the coin lands heads, they increase their position by 1, and every time the coin lands tails, they decrease their position by $1$. What is the probability that at some point the absolute value of their position is at least $3$? PS. You should use hide for answers.

Find all the natural numbers $N$ which satisfy the following properties: (i) $N$ has exactly $6$ distinct factors $1, d_1, d_2, d_3, d_4, N$ and (ii) $1 + N = 5(d_1 + d_2+d_3 + d_4)$. Justify your answers.

(a) Is it true that, for arbitrary integer $n{}$ greater than $1$ and distinct positive integers $i{}$ and $j$ not greater than $n{}$, the set of any $n{}$ consecutive integers contains distinct numbers $i^{'}$ and $j^{'}$ whose product $i^{'}j^{'}$ is divisible by the product $ij$? (b) Is it true that, for arbitrary integer $n{}$ greater than $2$ and distinct positive integers $i, j, k$ not greater than $n{}$, the set of any $n{}$ consecutive integers contains distinct numbers $i^{'},j^{'},k^{'}$ whose product $i^{'}j^{'}k^{'}$ is divisible by the product $ijk$?

Let K be the field formed by the addition of a root of the polynomial $x^4 - 2x^2 - 1$ to the rational field. Prove that there are no exceptional units in the ring of integers of K. (A unit $\varepsilon$ is called exceptional if $1-\varepsilon$ is also a unit.)

Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold: (a) the union of any two white subsets is white; (b) the union of any two black subsets is black; (c) there are exactly $N$ white subsets.

Let $ n$ be the largest integer that is the product of exactly $ 3$ distinct prime numbers, $ d$, $ e$, and $ 10d\plus{}e$, where $ d$ and $ e$ are single digits. What is the sum of the digits of $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$

Suppose two convex quadrangles in the plane $P$ and $P'$, share a point $O$ such that, for every line $l$ trough $O$, the segment along which $l$ and $P$ meet is longer then the segment along which $l$ and $P'$ meet. Is it possible that the ratio of the area of $P'$ to the area of $P$ is greater then $1.9$?

We are given a circle $k$ and a point $A$ outside of $k$. Next we draw three lines through $A$: one secant intersecting the circle $k$ at points $B$ and $C$, and two tangents touching the circle$k$ at points $D$ and $E$. Let $F$ be the midpoint of $DE$. Show that the line $DE$ bisects the angle $\angle BFC$.

Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle). Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.

The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as \[P_0(x)=x,\quad P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1),\quad n\ge 1.\] Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$.

Five unit squares are arranged in a plus shape as shown below: [asy] size(3cm); real s=0.1; draw(s*(0,1)--s*(0,2)); draw(s*(1,0)--s*(1,3)); draw(s*(2,0)--s*(2,3)); draw(s*(3,1)--s*(3,2)); draw(s*(1,0)--s*(2,0)); draw(s*(0,1)--s*(3,1)); draw(s*(0,2)--s*(3,2)); draw(s*(1,3)--s*(2,3)); [/asy] What is the area of the smallest circle containing the interior and boundary of the plus shape? [i]2020 CCA Math Bonanza Team Round #3[/i]

A $ 20\times20\times20$ block is cut up into 8000 non-overlapping unit cubes and a number is assigned to each. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $ 1\times20\times20$ slices parallel to the faces of the block contain this unit cube. Find the sume of all numbers of the cubes outside these slices.

Suppose that $a,b,c$ are real numbers in $(0,\frac{\pi}{2})$ such that $a+b+c=\frac{\pi}{4}$ and $\tan{a}=\frac{1}{x},\tan{b}=\frac{1}{y},\tan{c}=\frac{1}{z}$ , where $x,y,z$ are positive integer numbers. Find $x,y,z$.

Let $P_c(x)=x^4+ax^3+bx^2+cx+1$ and $Q_c(x)=x^4+cx^3+bx^2+ax+1$ with $a,b$ real numbers, $c \in \{1,2, \dots, 2017\}$ an integer and $a \ne c$. Define $A_c=\{\alpha | P_c(\alpha)=0\}$ and $B_c=\{\beta | P(\beta)=0\}$. (a) Find the number of unordered pairs of polynomials $P_c(x), Q_c(x)$ with exactly two common roots. (b) For any $1 \le c \le 2017$, find the sum of the elements of $A_c \Delta B_c$.

Let $ABC$ be a triangle. Suppose that $D$, $E$, and $F$ are points on segments $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively such that triangles $AEF$, $BFD$, and $CDE$ have equal inradii. Prove that the sum of the inradii of $\triangle AEF$ and $\triangle DEF$ is equal to the inradius of $\triangle ABC$. [i]Aprameya Tripathy[/i]

In an acute-angled triangle $ABC$, $CF$ is an altitude, with $F$ on $AB$, and $BM$ is a median, with $M$ on $CA$. Given that $BM=CF$ and $\angle MBC=\angle FCA$, prove that triangle $ABC$ is equilateral.

A certain island has some ports along the coast and some towns inland. All roads on this island are one-way, and they do not meet except at a port or a town. Moreover, once you leave a certain port or town by road, there is no way you can return there by road. For any two ports $i$ and $j$, let $f_{ij}$ denote the number of different routes along the roads between $i$ and $j$. (a) Suppose there are four ports on the island: $1, 2, 3$ and $4$, in clockwise order. Show that $$f_{14}f_{23} \ge f_{13}f_{24}$$ (b) Suppose there were six ports on the island: $1, 2, 3, 4, 5$ and $6$, in clockwise order. Show that $$f_{16}f_{25}f_{34} + f_{15}f_{24}f_{36} + f_{14}f_{26}f_{35}\ge f_{16}f_{24}f_{35}+ f_{15}f_{26}f_{34} + f_{14}f_{25}f_{36}$$ (S Fomin}

Prove that for all natural $n$ there exists a polynomial $P(x)$ divisible by $(x-1)^n$ such that its degree is not greater than $2^n$ and each of its coefficients is equal to $1$, $0$ or $-1$. (D. Fomin, Leningrad)