What is the value of $(2^{-1})^{-2}$? [i]2015 CCA Math Bonanza Lightning Round #1.1[/i]

For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where \[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]

Let $n\geq 3$ and $A=\{1,2,\dots ,n\}$. For any function $f:A\rightarrow A$ we define $$A_f=\{|f(1)-f(2)|,|f(2)-f(3)|,\dots ,|f(n-1)-f(n)|,|f(n)-f(1)|\}.$$ Determine the smallest and greatest value of the cardinal of $A_f$ as $f$ goes through all bijective functions from $A$ to $A$. [i]Silviu Cristea[/i]

Determine all positive integers $n$ such that $5^n - 1$ can be written as a product of an even number of consecutive integers.

Let $ABC$ be triangle such that $|AB| = 5$, $|BC| = 9$ and $|AC| = 8$. The angle bisector of $\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\widehat{CAB}$ meets $BC$ at $Y$. Let $Z$ be the intersection of lines $XY$ and $AC$. What is $|AZ|$? $ \textbf{a)}\ \sqrt{104} \qquad\textbf{b)}\ \sqrt{145} \qquad\textbf{c)}\ \sqrt{89} \qquad\textbf{d)}\ 9 \qquad\textbf{e)}\ 10 $

Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that \[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

[b]p13[/b] Let $ABCD$ be a square. Points $E$ and $F$ lie outside of $ABCD$ such that $ABE$ and $CBF$ are equilateral triangles. If $G$ is the centroid of triangle $DEF$, then find $\angle AGC$, in degrees. [b]p14 [/b]The silent reading $s(n)$ of a positive integer $n$ is the number obtained by dropping the zeros not at the end of the number. For example, $s(1070030) = 1730$. Find the largest $n < 10000$ such that $s(n)$ divides $n$ and $n\ne s(n)$. [b]p15[/b] Let $ABCDEFGH$ be a regular octagon with side length $12$. There exists a region $R$ inside the octagon such that for each point $X$ in $R$, exactly three of the rays $AX$, $BX$, $CX$, $DX$, $GX$, and $HX$ intersect segment $EF$. If the area of region $R$ can be expressed as $a -b\sqrt{c}$ for positive integers $a, b, c$ with $c$ squarefree, find $a + b + c$.

Square $ABCD$ has center $O$, $AB=900$, $E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F$, $m\angle EOF =45^\circ$, and $EF=400$. Given that $BF=p+q\sqrt{r}$, wherer $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.

Find all positive integers n with the following property: There exists a positive integer $k$ and mutually distinct integers $x_1,x_2,\ldots,x_n$ such that the set $\{x_i+x_j\mid1\le i<j\le n\}$ is a set of distinct powers of $k$.

After the final exam, Mr. Liang asked each of his 17 students to guess the average final exam score. David, a very smart student, received a 100 and guessed the average would be 97. Each of the other 16 students guessed $30+\frac{n}{2}$ where $n$ was that student’s score. If the average of the final exam scores was the same as the average of the guesses, what was the average score on the final exam? [i]Proposed by Eric Wang[/i]

$101$ people, sitting at a round table in any order, had $1,2,... , 101$ cards, respectively. A transfer is someone give one card to one of the two people adjacent to him. Find the smallest positive integer $k$ such that there always can through no more than $ k $ times transfer, each person hold cards of the same number, regardless of the sitting order.

Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half. [i]Proposed by Gerhard Wöginger, Austria[/i]

Find all prime numbers $p$ such that $(x + y)^{19} - x^{19} - y^{19}$ is a multiple of $p$ for any positive integers $x$, $y$.

Let $x$, $y$, and $z$ be real numbers such that $$12x - 9y^2 = 7$$ $$6y - 9z^2 = -2$$ $$12z - 9x^2 = 4$$ Find $6x^2 + 9y^2 + 12z^2$.

Find all pairs of positive integers $(m,n)$ such that one can partition a $m\times n$ board with $1\times 2$ or $2\times 1$ dominoes and draw one of the diagonals on each of the dominos so that none of the diagonals share endpoints.

The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules: [list] [*] $D(1) = 0$; [*] $D(p)=1$ for all primes $p$; [*] $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$. [/list] Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$.

Let $ABCD$ be a cyclic quadrilateral, with circumcircle $\omega$ and circumcenter $O$. Let $AB$ intersect $CD$ at $E$, $AD$ intersect $BC$ at $F$, and $AC$ intersect $BD$ at $G$. The points $A_1, B_1, C_1, D_1$ are chosen on rays $GA$, $GB$, $GC$, $GD$ such that: $\bullet$ $\displaystyle \frac{GA_1}{GA} = \frac{GB_1}{GB} = \frac{GC_1}{GC} = \frac{GD_1}{GD}$ $\bullet$ The points $A_1, B_1, C_1, D_1, O$ lie on a circle. Let $A_1B_1$ intersect $C_1D_1$ at $K$, and $A_1D_1$ intersect $B_1C_1$ at $L$. Prove that the image of the circle $(A_1B_1C_1D_1)$ under inversion about $\omega$ is a line passing through the midpoints of $KE$ and $LF$. [i]Proposed by Anzo Teh Zhao Yang & Ivan Chan Kai Chin[/i]

Halfway through a $ 100$-shot archery tournament, Chelsea leads by $ 50$ points. For each shot a bullseye scores $ 10$ points, with other possible scores being $ 8, 4, 2, 0$ points. Chelsea always scores at least $ 4$ points on each shot. If Chelsea's next $ n$ shots are bulleyes she will be guaranteed victory. What is the minimum value for n? $ \textbf{(A)}\ 38\qquad \textbf{(B)}\ 40\qquad \textbf{(C)}\ 42\qquad \textbf{(D)}\ 44\qquad \textbf{(E)}\ 46$

Let $P$ and $Q$ be isogonal conjugates inside triangle $ABC$. Let $\omega$ be the circumcircle of $ABC$. Let $A_1$ be a point on arc $BC$ of $\omega$ satisfying $\angle BA_1P = \angle CA_1Q$. Points $B_1$ and $C_1$ are defined similarly. Prove that $AA_1$, $BB_1$, $CC_1$ are concurrent.

If $p = (1- \cos x)(1+ \sin x)$ and $q = (1+ \cos x)(1- \sin x)$, write the expression $$\cos^2 x - \cos^4 x - \sin2x + 2$$ in terms of $p$ and $q$.

Heather compares the price of a new computer at two different stores. Store A offers $ 15\%$ off the sticker price followed by a $ \$90$ rebate, and store B offers $ 25\%$ off the same sticker price with no rebate. Heather saves $ \$15$ by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars? $ \textbf{(A)}\ 750 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1050 \qquad \textbf{(E)}\ 1500$

At Ignus School there are $425$ students. Of these students $351$ study mathematics, $71$ study Latin, and $203$ study chemistry. There are $199$ students who study more than one of these subjects, and $8$ students who do not study any of these subjects. Find the number of students who study all three of these subjects.

Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AB = 3CD$. Let $E$ be then midpoint of the diagonal $BD$. If $[ABCD] = n \times [CDE]$, what is the value of $n$? (Here $[t]$ denotes the area of the geometrical figure$ t$.)

Find the index of the singular point $0$ of the vector field \[(xy+yz+xz)\]

Prove that for all positive reals $a,b,c,d$, we have $\frac{a+b+c+d}{abcd}\leq \frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}+\frac{1}{d^{3}}$