Misha solved the equation $x^2 + ax + b = 0$ and told Dima the set of four numbers - two roots and two coefficients of this equation (but not said which of them are roots and which are coefficients). Will he be able to Dima, find out what equation Misha solved if all the numbers in the set turned out to be different?

Marine biologists are studying a new species of shellfish whose first generation consists of $100$ shellfish, and their colony reproduces as follows: if a given generation consists of $N$ shellfish (where $5\mid N$ always holds), they divide themselves into $N/5$ groups of $5$ shellfish each. Each group collectively produces $15$ offspring, who form the next generation. Some of the shellfish contain a pearl, but a shellfish can only contain a pearl if none of its direct ancestors contained a pearl. The value of a pearl is determined by the generation of the shellfish containing it: in the $n^{\mathrm{th}}$ generation, its value is $1/3^n$. Find the maximum possible total value of the pearls in the colony. [i]Proposed by: Csongor Beke, Cambridge[/i]

Prove that if for any positive $p$ all roots of the equation $ax^2 + bx + c + p = 0$ are real and positive then $a = 0$.

A sequence $\{a_n\}_{n\geq 1}$ of positive reals is defined by the rule $a_{n+1}a_{n-1}^5=a_n^4a_{n-2}^2$ for integers $n>2$ together with the initial values $a_1=8$ and $a_2=64$ and $a_3=1024$. Compute \[\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}\]

A rectangle has an area of $16$ and a perimeter of $18$; determine the length of the diagonal of the rectangle. [i]2015 CCA Math Bonanza Individual Round #8[/i]

On a circle $4$ points are chosen and for each point we wrote the multiple of its distances to the rest. Could the written numbers be $1,2,3, 4$ in some order?

Two players $A$ and $B$ play the following game: $A$ chooses a point, with integer coordinates, on the plane and colors it green, then $B$ chooses $10$ points of integer coordinates, not yet colored, and colors them yellow. The game always continues with the same rules; $A$ and $B$ choose one and ten uncolored points and color them green and yellow, respectively. a. The objective of $A$ is to achieve $111^2$ green points that are the intersections of $111$ horizontal lines and $111$ vertical lines (parallel to the coordinate axes). $B$'s goal is to stop him. Determine which of the two players has a strategy that ensures you achieve your goal. b. The objective of $A$ is to achieve $4$ green points that are the vertices of a square with sides parallel to the coordinate axes. $B$'s goal is to stop him. Determine which of the two players has a strategy that will ensure that they achieve their goal.

Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying $$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$ for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$. [i]Ankan Bhattacharya[/i]

A rhombus has vertices at $(0,0)$, $(6, 8)$, $(16, 8)$, and $(10, 0)$. A line with slope $m$ passes through the point $(3, 1)$ and splits the rhombus into $2$ regions of equal area. Find $m$.

In $\triangle{ABC}$, $AX=XY=YB=BC$, and $m\angle{ABC}=120^{\circ}$. What is $m\angle{BAC}$? [asy] pair A, B, C, X, Y; A = origin; X = dir(30); Y = X + dir(0); B = Y + dir(60); C = B + dir(330); draw(A--B--C--cycle); draw(X--Y--B); label("$A$",A,W); label("$B$",B,N); label("$C$",C,E); label("$X$",X,NW); label("$Y$",Y,SE); [/asy] $\text{(A) }15\qquad\text{(B) }20\qquad\text{(C) }25\qquad\text{(D) }30\qquad\text{(E) }35$

For covering the floor of a rectangular room rectangular tiles of sizes $2 \times 2$ and $4 \times 1$ were used. Show that it's not possible to cover the floor if there is one plate less of one type and one more of the other type.

How many four-digit positive integers have no adjacent equal even digits? For example, count numbers such as $1164$ and $2035$ but not $6447$ or $5866$.

A natural number $n$ is given. For all integer triplets $(a,b,c)$ such that $0 < |a| , |b|, |c| < 2023$ and satisfying below, show that the product of all possible integer $a$ is a perfect square. (The value of $a$ allows duplication) $$(a+nb)(a-nc) + abc = 0$$

Find all positive integers $n$ such that there exists integers $n_1,\ldots,n_k\ge 3$, for some integer $k$, satisfying \[n=n_1n_2\cdots n_k=2^{\frac{1}{2^k}(n_1-1)\cdots (n_k-1)}-1.\]

A circle centered at $ A$ with a radius of 1 and a circle centered at $ B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is [asy] size(220); real r1 = 1; real r2 = 3; real r = (r1*r2)/((sqrt(r1)+sqrt(r2))**2); pair A=(0,r1), B=(2*sqrt(r1*r2),r2); dot(A); dot(B); draw( circle(A,r1) ); draw( circle(B,r2) ); draw( (-1.5,0)--(7.5,0) ); draw( A -- (A+dir(210)*r1) ); label("$1$", A -- (A+dir(210)*r1), N ); draw( B -- (B+r2*dir(330)) ); label("$4$", B -- (B+r2*dir(330)), N ); label("$A$",A,dir(330)); label("$B$",B, dir(140)); draw( circle( (2*sqrt(r1*r),r), r )); [/asy] $ \displaystyle \textbf{(A)} \ \frac {1}{3} \qquad \textbf{(B)} \ \frac {2}{5} \qquad \textbf{(C)} \ \frac {5}{12} \qquad \textbf{(D)} \ \frac {4}{9} \qquad \textbf{(E)} \ \frac {1}{2}$

In rectangle $ ABCD$, $ AD \equal{} 1$, $ P$ is on $ \overline{AB}$, and $ \overline{DB}$ and $ \overline{DP}$ trisect $ \angle ADC$. What is the perimeter of $ \triangle BDP$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)); dotfactor=4; pair D=(0,0), C=(sqrt(3),0), B=(sqrt(3),1), A=(0,1), P=(sqrt(3)/3,1); pair[] dotted={A,B,C,D,P}; draw(A--B--C--D--cycle); draw(B--D--P); dot(dotted); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$P$",P,N);[/asy]$ \textbf{(A)}\ 3 \plus{} \frac {\sqrt3}{3} \qquad\textbf{(B)}\ 2 \plus{} \frac {4\sqrt3}{3}\qquad\textbf{(C)}\ 2 \plus{} 2\sqrt2\qquad\textbf{(D)}\ \frac {3 \plus{} 3\sqrt5}{2} \qquad\textbf{(E)}\ 2 \plus{} \frac {5\sqrt3}{3}$

Let $x_{1}$ be a positive real number and for every integer $n \geq 1$ let $x_{n+1} = 1 + x_{1}x_{2}\ldots x_{n-1}x_{n}$. If $x_{5} = 43$, what is the sum of digits of the largest prime factors of $x_{6}$?