Find all positive integers $a, b,c$ greater than $1$, such that $ab + 1$ is divisible by $c, bc + 1$ is divisible by $a$ and $ca + 1$ is divisible by $b$.

Find the remainder after division of the polynomial $x+x^3 +x^9 +x^{27} +x^{81} +x^{243}$ by $x-1$.

In the grid to the right, the shortest path through unit squares between between the pair of 2's has length 2. Fill in some of the unit squares in the grid so that (i) exactly half of the squares in each row and column contain a number, (ii) each of the number 1 through 12 appears exactly twice, and (iii) for $n=1,2,\cdot\cdot\cdot,12$, the shortest path between the pair of $n$'s has length exactly $n$.

Let $c_1<c_2<c_3$ be the three smallest positive integer values of $c$ such that the distance between the parabola $y=x^2+2020$ and the line $y=cx$ is a rational multiple of $\sqrt{2}$. Compute $c_1+c_2+c_3$.

Find all triples $(a,b,c)$ of real numbers such that $|2a + b| \ge 4$ and $|ax^2 + bx + c| \le 1$ $ \forall x \in [-1, 1]$.

The evil sorceress Morgana lives in a square-shaped palace divided into a \(101 \times 101\) grid of rooms, each initially at a temperature of \(20^\circ\)C. Merlin attempts to freeze Morgana by casting a spell that permanently sets the central cell's temperature to \(0^\circ\)C. At each subsequent nanosecond, the following steps occur in order: 1. For every cell except the central one, the new temperature is computed as the arithmetic mean of the temperatures of its adjacent cells (those sharing a side). 2. All new temperatures (except the central cell) are updated simultaneously to the calculated values. Morgana can freely move between rooms but will freeze if all room temperatures drop below \(10^{-2025}\) degrees. The ice spell lasts for \(10^{75}\) nanoseconds, after which temperatures revert to their initial values. Will Merlin succeed in freezing Morgana?

Let $D$ be a point inside an acute triangle $ABC$, such that $\angle BAD = \angle DBC$ and $\angle DAC = \angle BCD$. Let $P$ be a point on the circumcircle of the triangle $ADB$. Suppose $P$ are itself outside the triangle $ABC$. A line through $P$ intersects the ray $BA$ in $X$ and ray $CA$ in $Y$, so that $\angle XPB = \angle PDB$. Show that $BY$ and $CX$ intersect on $AD$.

$n$ real numbers are written in increasing order in a line. The same numbers are written in the second line below in unknown order. The third line contains the sums of the pairs of numbers above from two previous lines. It comes out, that the third line is arranged in increasing order. Prove that the second line coincides with the first one.

The value of $x$ in the interval $[0, 2\pi]$ that minimizes the value of $x + 2\cos x$ can be written in the form $a\pi/b,$ where $a$ and $b$ are relatively prime positive integers. Compute $a + b.$

Find the distance of the centre of gravity of a uniform $100$-dimensional solid hemisphere of radius $1$ from the centre of the sphere with $10\%$ relative error.

Asheville, Bakersfield, Charter, and Darlington are four small towns along a straight road in that order. The distance from Bakersfield to Charter is one-third the distance from Asheville to Charter and one-quarter the distance from Bakersfield to Darlington. If it is $12$ miles from Bakersfield to Charter, how many miles is it from Asheville to Darlington?

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $(f(x + y))^2 = f(x^2) + f(y^2)$ for all $x, y \in \mathbb{Z}$.

The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner? [asy] draw((0,0)--(0,1)--(1,1)--(1,2)--(2,2)--(2,1)--(4,1)--(4,0)--(2,0)--(2,-1)--(1,-1)--(1,0)--cycle); draw((1,0)--(1,1)--(2,1)--(2,0)--cycle); draw((3,1)--(3,0)); label("$1$",(1.5,1.25),N); label("$2$",(1.5,.25),N); label("$3$",(1.5,-.75),N); label("$4$",(2.5,.25),N); label("$5$",(3.5,.25),N); label("$6$",(.5,.25),N); [/asy] $\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15$

Let $S$ be the set of all ordered triples $\left(a,b,c\right)$ of positive integers such that $\left(b-c\right)^2+\left(c-a\right)^2+\left(a-b\right)^2=2018$ and $a+b+c\leq M$ for some positive integer $M$. Given that $\displaystyle\sum_{\left(a,b,c\right)\in S}a=k$, what is \[\displaystyle\sum_{\left(a,b,c\right)\in S}a\left(a^2-bc\right)\] in terms of $k$? [i]2018 CCA Math Bonanza Lightning Round #4.1[/i]

Let $E$ be the point of intersection of the diagonals of the cyclic quadrilateral $ABCD$, and let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$, respectively. Prove that the radii of the circles circumscribed around the triangles $KLE$ and $MNE$ are equal.

We want to choose telephone numbers for a city. The numbers have $10$ digits and $0$ isn’t used in the numbers. Our aim is: We don’t choose some numbers such that every $2$ telephone numbers are different in more than one digit OR every $2$ telephone numbers are different in a digit which is more than $1$. What is the maximum number of telephone numbers which can be chosen? In how many ways, can we choose the numbers in this maximum situation?

Calculate the following indefinite integrals. [1] $\int \sin x\cos ^ 3 x dx$ [2] $\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx$ [3] $\int x^2 \sqrt{x^3+1}dx$ [4] $\int \frac{e^{2x}-3e^{x}}{e^x}dx$ [5] $\int (1-x^2)e^x dx$

Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that: [LIST] [*] in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and [/*] [*]in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$, one third are $M$ and one third are $O$.[/*] [/LIST] [b]Note.[/b] The rows and columns of an $n \times n$ table are each labelled $1$ to $n$ in a natural order. Thus each cell corresponds to a pair of positive integer $(i,j)$ with $1 \le i,j \le n$. For $n>1$, the table has $4n-2$ diagonals of two types. A diagonal of first type consists all cells $(i,j)$ for which $i+j$ is a constant, and the diagonal of this second type consists all cells $(i,j)$ for which $i-j$ is constant.

The difference between two prime numbers is $11$. Find their sum.

Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that \[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\] [i]Proposed by Alexey Gladkich, Israel[/i]

Let $A$ be a subset of the edges of an $n\times n $ table. Let $V(A)$ be the set of vertices from the table which are connected to at least on edge from $A$ and $j(A)$ be the number of the connected components of graph $G$ which it's vertices are the set $V(A)$ and it's edges are the set $A$. Prove that for every natural number $l$: $$\frac{l}{2}\leq min_{|A|\geq l}(|V(A)|-j(A)) \leq \frac{l}{2}+\sqrt{\frac{l}{2}}+1$$

We consider $S$ a set of odd positive interger numbers with $n\geq 3$ elements such that no element divides another element. We say that a set $S$ is $beautiful$ if for any 3 elements from $S$, there is one the divides the sum of the other 2. We call a beautiful set $S$ $maximal$ if we can't add another number to the set such that $S$ will still be beautiful. Find the values of $n$ for which there exists a $maximal$ set.

Side $AB$ of triangle $ABC$ has length $8$ inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is: $\textbf{(A)}\ \dfrac{51}{4} \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ \dfrac{53}{4} \qquad \textbf{(D)}\ \dfrac{40}{3} \qquad \textbf{(E)}\ \dfrac{27}{2} $

Let $A$ be an non-invertible of order $n$, $n>1$, with the elements in the set of complex numbers, with all the elements having the module equal with 1 a)Prove that, for $n=3$, two rows or two columns of the $A$ matrix are proportional b)Does the conclusion from the previous exercise remains true for $n=4$?

In a half-ball of radius $3$ is inscribed a cylinder with base lying on the base plane of the half-ball, and another such cylinder with equal volume. If the base-radius of the first cylinder is $\sqrt3$, what is the base-radius of the other one?