A tourist office was investigating the numbers of sunny and rainy days in a year in each of six regions. The results are partly shown in the following table: Region , sunny or rainy , unclassified $A \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 336 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,29$ $B \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 321 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,44$ $C \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 335 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,30$ $D \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 343 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,22$ $E \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 329 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,36$ $F \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 330 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,35$ Looking at the detailed data, an officer observed that if one region is excluded, then the total number of rainy days in the other regions equals one third of the total number of sunny days in these regions. Determine which region is excluded.

A game is played with tokens according to the following rules. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a discard pile. The game ends when some player runs out of tokens. Players $ A$, $ B$, and $ C$ start with $ 15$, $ 14$, and $ 13$ tokens, respectively. How many rounds will there be in the game? $ \textbf{(A)}\ 36 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 38\qquad \textbf{(D)}\ 39\qquad \textbf{(E)}\ 40$

The circles $k_1$ and $k_2$ intersect at points $D$ and $P$. The common tangent of the two circles on the side of $D$ touches $k_1$ at $A$ and $k_2$ at $B$. The straight line $AD$ intersects $k_2$ for a second time at $C$. Let $M$ be the center of the segment $BC$. Show that $ \angle DPM = \angle BDC$ .

Suppose there exist constants $A$, $B$, $C$, and $D$ such that \[n^4=A\binom n4+B\binom n3+C\binom n2 + D\binom n1\] holds true for all positive integers $n\geq 4$. What is $A+B+C+D$? [i]Proposed by David Altizio[/i]

The numbers $1,2,\ldots,100$ are written in the cells of a $10\times 10$ table, each number is written once. In one move, Nazar may interchange numbers in any two cells. Prove that he may get a table where the sum of the numbers in every two adjacent (by side) cells is composite after at most $35$ such moves. [i]N. Agakhanov[/i]

Are there two such quadrangles that the sides of the first are less than the corresponding sides of the second, and the corresponding diagonals are larger? (Arseniy Akopyan)

Pinocchio claims that he can take some non-right-angled triangles , all of which are similar to one another and some of which may be congruent to one another, and put them together to form a rectangle. Is Pinocchio lying? (A Fedotov)

(A.Zaslavsky, 9--10) Given four points $ A$, $ B$, $ C$, $ D$. Any two circles such that one of them contains $ A$ and $ B$, and the other one contains $ C$ and $ D$, meet. Prove that common chords of all these pairs of circles pass through a fixed point.

Pentagon $ABCDE$ has $AB = BC = CD = DE$, $\angle ABC = \angle BCD = 108^o$, and $\angle CDE = 168^o$. Find the measure of angle $\angle BEA$ in degrees.

Find the smallest positive integer $n$ such that it is possible to paint each of the $64$ squares of an $8 \times 8$ board of one of $n$ colors so that any four squares that form an $L$ as in the following figure (or congruent figures obtained through rotations and/or reflections) have different colors. [img]https://cdn.artofproblemsolving.com/attachments/a/2/c8049b1be8f37657c058949e11faf041856da4.png[/img]

Suppose we have a convex quadrilateral $ABCD$ such that $\angle B = 100^\circ$ and the circumcircle of $\triangle ABC$ has a center at $D$. Find the measure, in degrees, of $\angle D$. [i]Note:[/i] The circumcircle of a $\triangle ABC$ is the unique circle containing $A$, $B$, and $C$.

Let $\{a_n\}$ be a bounded real sequence. (a) Prove that if X is a positive-measure subset of $\mathbb R$, then for almost all $x\in X$, there exist a subsequence $\{y_n\}$ of X such that $$\sum_{n=1}^\infty (n(y_n-x)-a_n)=1$$ (b) construct an unbounded sequence $\{a_n\}$ for which the above equation is also true.

A light has been placed on every lattice point (point with integer coordinates) on the (infinite) 2$D$ plane. Dene the Chebyshev distance between points $(x_1,y_1)$ and $(x_2, y_2)$ to be $\ max (|x_1 - x_2|, |y_1 -y_2|)$. Each light is turned on with probability $\frac{1}{2^{d/2}}$ , where $d$ is the Chebyshev distance from that point to the origin. What is expected number of lights that have all their directly adjacent lights turned on? (Adjacent points being points such that $|x_1-x_2|+|y_1- y_2| =1$.)

Find all functions $f : \mathbb{N} \rightarrow \mathbb{R}$ such that for all triples $a,b,c$ of positive integers the following holds : $$f(ac)+f(bc)-f(c)f(ab) \ge 1$$ Proposed by [i]Mojtaba Zare[/i]

Suppose $x$ is a positive real number such that $\{x\}, [x]$ and $x$ are in a geometric progression. Find the least positive integer $n$ such that $x^n > 100$. (Here $[x]$ denotes the integer part of $x$ and $\{x\} = x - [x]$.)

The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$.

There are$ 1$ cm long bars with a number$ 1$, $2$ or $3$ written on each one from them. There is an unlimited supply of bars with each number. Two triangles formed by three bars are considered different if none of them can be built with the bars of the other triangle. (a) How many different triangles formed by three bars are possible? (b) An equilateral triangle of side length $3$ cm is formed using $18$ bars, , divided into $9$ equilateral triangles, different by pairs, $1$ cm long on each side. Find the largest sum possible from the numbers written on the $9$ bars of the border of the big triangle. [center][img]https://cdn.artofproblemsolving.com/attachments/1/1/c2f7edeea3c70ba4689d7b12fe8ac8be72f115.png[/img][/center]

There are $n$ ordered tuples of positive integers $(a,b,c,d)$ that satisfy $$a^2+ b^2+ c^2+ d^2=13 \cdot 2^{13}.$$ Let these ordered tuples be $(a_1,b_1,c_1,d_1), (a_2,b_2,c_2,d_2), \dots, (a_n,b_n,c_n,d_n)$. Compute $\sum_{i=1}^{n}(a_i+b_i+c_i+d_i)$. [i]Proposed by Kaylee Ji[/i]

Any natural number $n, n\ge 3$ can be presented in different ways as a sum several summands (not necessarily different). Find the greatest possible value of these summands. Folklore

For how many positive integers $n$ is \[\left( 1999+\frac{1}{2}\right)^{n}+\left(2000+\frac{1}{2}\right)^{n}\] an integer?

Find the number of pairs of integers $(x, y)$ such that $x^2 + 2y^2 < 25$.

If $ N > 1$, then ${ \sqrt [3] {N \sqrt [3] {N \sqrt [3] {N}}}} =$ $ \textbf{(A)}\ N^{\frac {1}{27}}\qquad \textbf{(B)}\ N^{\frac {1}{9}}\qquad \textbf{(C)}\ N^{\frac {1}{3}}\qquad \textbf{(D)}\ N^{\frac {13}{27}}\qquad \textbf{(E)}\ N$

Find all the $4$-digit natural numbers, written in base $10$, that are equal to the cube of the sum of its digits.

Given a prime $p \geq 5$ , show that there exist at least two distinct primes $q$ and $r$ in the range $2, 3, \ldots p-2$ such that $q^{p-1} \not\equiv 1 \pmod{p^2}$ and $r^{p-1} \not\equiv 1 \pmod{p^2}$.

Define a sequence of polynomials $P_0\left(x\right)=x$ and $P_k\left(x\right)=P_{k-1}\left(x\right)^2-\left(-1\right)^kk$ for each $k\geq1$. Also define $Q_0\left(x\right)=x$ and $Q_k\left(x\right)=Q_{k-1}\left(x\right)^2+\left(-1\right)^kk$ for each $k\geq1$. Compute the product of the distinct real roots of \[P_1\left(x\right)Q_1\left(x\right)P_2\left(x\right)Q_2\left(x\right)\cdots P_{2018}\left(x\right)Q_{2018}\left(x\right).\] [i]2018 CCA Math Bonanza Tiebreaker Round #2[/i]