Let $a,b,x\in \mathbb{N}$ with $b>1$ and such that $b^{n}-1$ divides $a$. Show that in base $b$, the number $a$ has at least $n$ non-zero digits.

Let P be an interior point of a regular n-gon $ A_1 A_2 ...A_n$, the lines $ A_i P$ meet the regular n-gon at another point $ B_i$, where $ i\equal{}1,2,...,n$. Prove that sums of all $ PA_i\geq$ sum of all $ PB_i$.

[u]Version for Nordic Countries[/u] Six pine trees grow on the shore of a circular lake. It is known that a treasure is submerged at the mid-point $T$ between the intersection points of the altitudes of two triangles, the vertices of one being at three of the $6$ pines, and the vertices of the second one at the other three pines. At how many points $T$ must one dive to find the treasure? [u]Version for Tropical Countries[/u] A captain finds his way to Treasure Island, which is circular in shape. He knows that there is treasure buried at the midpoint of the segment joining the orthocentres of triangles $ABC$ and $DEF$, where $A$, $B$, $C$, $D$, $E$ and $F$ are six palm trees on the shore of the island, not necessarily in cyclic order. He finds the trees all right, but does not know which tree is denoted by which letter. What is the maximum number of points at which the captain has to dig in order to recover the treasure? (S Markelov)

Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$

Let $n$ be a positive integer. Alice writes $n$ real numbers $a_1, a_2,\dots, a_n$ in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors ($a_n$ is not a neighbor of $a_1$, nor vice versa). A number [i]changes sign[/i] if it changes from being nonnegative to negative or vice versa. In terms of $n$, determine the maximum number of times that $a_1$ can change sign, across all possible values of $a_1,a_2,\dots, a_n$ and all possible sequences of moves Alice may make.

Each of the spots in a $8\times 8$ chessboard is occupied by either a black or white “horse”. At most how many black horses can be on the chessboard so that none of the horses attack more than one black horse? [b]Remark:[/b] A black horse could attack another black horse.

Let $k$ be a positive integer and let $m=6k-1$. Let $$S(m)=\sum_{j=1}^{2k-1}(-1)^{j+1}\binom m{3j-1}.$$Prove that $S(m)$ is never zero.

On a tournament with $ n \ge 3$ participants, every two participants played exactly one match and there were no draws. A three-element set of participants is called a [i]draw-triple[/i] if they can be enumerated so that the first defeated the second, the second defeated the third, and the third defeated the first. Determine the largest possible number of draw-triples on such a tournament.

$\frac{2^1+2^0+2^{-1}}{2^{-2}+2^{-3}+2^{-4}}$ equals $\text{(A)} \ 6 \qquad \text{(B)} \ 8 \qquad \text{(C)} \ \frac{31}{2} \qquad \text{(D)} \ 24 \qquad \text{(E)} \ 512$

In the diagram of rectangles below, with lengths as labeled, let $A$ be the area of the rectangle labeled $A$, and so on. Find $36A+6B+C+6D$. [asy] size(3cm); real[] A = {0,8,13}; real[] B = {0,7,12}; for (int i = 0; i < 3; ++i) { draw((A[i],0)--(A[i],-B[2])); draw((0,-B[i])--(A[2],-B[i])); } label("8", (4,0), N); label("5", (10.5,0),N); label("7", (0,-3.5),W); label("5", (0,-9.5),W); label("$A$", (4,-3.5)); label("$B$", (10.5,-3.5)); label("$C$", (10.5,- 9.5)); label("$D$", (4, -9.5)); [/asy] [i]2018 CCA Math Bonanza Team Round #1[/i]

The numeral $47$ in base $a$ represents the same number as $74$ in base $b$. Assuming that both bases are positive integers, the least possible value of $a+b$ written as a Roman numeral, is $\textbf{(A) }\mathrm{XIII}\qquad\textbf{(B) }\mathrm{XV}\qquad\textbf{(C) }\mathrm{XXI}\qquad\textbf{(D) }\mathrm{XXIV}\qquad \textbf{(E) }\mathrm{XVI}$

Let $ABC$ be an acute triangle with $AB\le AC$ and let $c(O,R)$ be it's circumscribed circle (with center $O$ and radius $R$). The perpendicular from vertex $A$ on the tangent of the circle passing through point $C$, intersect it at point $D$. a) If the triangle $ABC$ is isosceles with $AB=AC$, prove that $CD=BC/2$. b) If $CD=BC/2$, prove that the triangle $ABC$ is isosceles.

The infinite periodic fractions $\frac{a} {b}$ and $\frac{c} {d}$ with $(a, b)=(c, d)=1$ are such that every finite block of digits in the first fraction after the decimal point appears in the second fraction as well (again after the decimal point). Show that $b=d$.

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

The graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. What is $\tan(\angle ABC)$? $\textbf{(A)}\frac{1}{7}~\textbf{(B)}\frac{1}{4}~\textbf{(C)}\frac{3}{7}~\textbf{(D)}\frac{1}{2}~\textbf{(E)}\frac{4}{7}$

In a row of $2009$ weights, the weight of each weight is an integer grams and does not exceed $1$ kg. The weights of any two adjacent weights differ by exactly $1$ g, and the total weight of all weights in grams is an even number. Prove that weights can be separated into two piles, the sums of the weights in which are equal.

Prove that for each positive integer $n$, there exists a positive integer with the following properties: [list] [*] it has exactly $n$ digits, [*] none of the digits is 0, [*] it is divisible by the sum of its digits.[/list]

Let $ABCD$ be a quadtrilateral with no parallel sides. The diagonals intersect at $E$, and $P, Q$ are points on sides $AB, CD$ respectively such that $\frac{AP}{PB} = \frac{CQ}{QD}$. $PQ$ meet $AC$ and $BD$ at $R,S$. Prove that $(EAB),(ECD),(ERS)$ all meet a point other than $E$.

Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.

Let $ABCD$ be a square with side length $1$. Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.

In a cafeteria, a glass of lemonade, three sandwiches and seven biscuits have cost $1$ shilling and $2$ pence, and a glass of lemonade, four sandwiches and $10$ biscuits they are worth $1$ shilling and $5$ pence. Find the price of: a) a glass of lemonade, a sandwich and a cake; b) two glasses of lemonade, three sandwiches and five biscuits. ($1$ shilling = $12$ pence).

Let's say a language $L \subseteq \{0,1\}^*$ is in $\textbf{P}_{angel}$ if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$, a sequence of strings $\{\alpha_n\}_{n \in \mathbb{N}}$ with $\alpha_n \in \{0,1\}^{p(n)}$, and a deterministic polynomial time Turing Machine $M$ such that for every $x \in \{0,1\}^n$ $$x \in L \Leftrightarrow M(x, \alpha_n) = 1$$ Let us call $\alpha_n$ to be the [i]angel string [/i]for all $x$ of the length $n$. Note that the [i]angel string[/i] is $\textbf{not}$ similar to a [i]witness[/i] or [i]certificate [/i]as used in the definition of $\textbf{NP}$ For example, all unary languages, even $UHALT$ which is undecidable, are in $\textbf{P}_{angel}$ because the \textit{angel string} can simply be a single bit that tells us if the given unary string is in $UHALT$ or not. \\\\ A set $S \subseteq \Sigma^*$ is said to be [b]sparse[/b] if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$ such that for each $n \in \mathbb{N}$, the number of strings of length $n$ in $S$ is bounded by $p(n)$. In other words, $|S^{=n}| \leq p(n)$, where $S^{=n} \subseteq S$ contains all the strings in $S$ that are of length $n$. [list=1] [*] Given $k \in \mathbb{N}$ sparse sets $S_1, S_2 \ldots S_k$, show that there exists a sparse set $S$ and a deterministic polynomial time TM $M$ with oracle access to $S$ such that given an input $\langle x,i \rangle$ the TM $M$ will accept it if and only if $x \in S_i$. \\Define the set $S$ (note that it need not be computable), and give the description of $M$ with oracle $S$. \\Note that a TM $M$ with oracle access to $S$ can query whether $s \in S$ and get the correct answer in return in constant time. [/*] [*] Let us define a variant of $\textbf{P}_{angel}$ called $\textbf{P}_{bad-angel}$ with a constraint that there should exists a polynomial time algorithm that can [b]compute[/b] the angel string for any length $n \in \mathbb{N}$. In other words, there is a poly-time algorithm $A$ such that $\alpha_n = A(n)$. \\Is $\textbf{P} =\textbf{P}_{bad-angel}$? Is $\textbf{NP}=\textbf{P}_{bad-angel}$? Justify. [/*] [*] Let the language $L \in$ $\textbf{P}_{angel}$. Show that there exists a sparse set $S_L$ and a deterministic polynomial time TM $M$ with oracle access to $S_L$ that can decide the language $L$. [/*]

Let $S$ be the set of positive perfect squares that are of the form $\overline{AA}$, i.e. the concatenation of two equal integers $A$. (Integers are not allowed to start with zero.) (a) Prove that $S$ is infinite. (b) Does there exist a function $f:S\times S \rightarrow S$ such that if $a,b,c \in S$ and $a,b | c$, then $f(a,b) | c$? (If such a function $f$ exists, we call $f$ an LCM function)

$A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP<DP$.

Let $S(n)$ be the sum of digits of natural number $n{}$. Is there a natural number $n{}$ for which $n+S(n)+S(S(n))=2011?$