Let $a$, $b$, $c$ and $d$ positive real numbers with $a > c$ and $b < d$. Assume that \[a + \sqrt{b} \ge c + \sqrt{d} \qquad \text{and} \qquad \sqrt{a} + b \le \sqrt{c} + d\] Prove that $a + b + c + d > 1$. [i]Proposed by Victor Domínguez[/i]

In each box of a $ 1 \times 2009$ grid, we place either a $ 0$ or a $ 1$, such that the sum of any $ 90$ consecutive boxes is $ 65$. Determine all possible values of the sum of the $ 2009$ boxes in the grid.

Problem 2. Let $P(x) = ax^2 + bx + c$ where $a, b, c$ are real numbers. If $$P(a) = bc, \hspace{0.5cm} P(b) = ac, \hspace{0.5cm} P(c) = ab$$ then prove that $$(a - b)(b - c)(c - a)(a + b + c) = 0.$$

The circle $ \Gamma $ is inscribed to the scalene triangle $ABC$. $ \Gamma $ is tangent to the sides $BC, CA$ and $AB$ at $D, E$ and $F$ respectively. The line $EF$ intersects the line $BC$ at $G$. The circle of diameter $GD$ intersects $ \Gamma $ in $R$ ($ R\neq D $). Let $P$, $Q$ ($ P\neq R , Q\neq R $) be the intersections of $ \Gamma $ with $BR$ and $CR$, respectively. The lines $BQ$ and $CP$ intersects at $X$. The circumcircle of $CDE$ meets $QR$ at $M$, and the circumcircle of $BDF$ meet $PR$ at $N$. Prove that $PM$, $QN$ and $RX$ are concurrent. [i]Author: Arnoldo Aguilar, El Salvador[/i]

An infinite sequence of distinct real numbers is given. Prove that it contains a subsequence of $1999$ terms which is either monotonically increasing or monotonically decreasing.

10. Prair picks a three-digit palindrome $n$ at random. If the probability that $2n$ is also a palindrome can be expressed as $\frac{p}{q}$ in simplest terms, find $p + q$. (A palindrome is a number that reads the same forwards as backwards; for example, $161$ and $2992$ are palindromes, but $342$ is not.) 11. If two distinct integers are picked randomly between $1$ and $50$ inclusive, the probability that their sum is divisible by $7$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. 12. Ali is playing a game involving rolling standard, fair six-sided dice. She calls two consecutive die rolls such that the first is less than the second a "rocket." If, however, she ever rolls two consecutive die rolls such that the second is less than the first, the game stops. If the probability that Ali gets five rockets is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, find $p+q$.

In a rectangular box $ABCDEFGH$ with edge lengths $AB = AD = 6$ and $AE = 49$, a plane slices through point $A$ and intersects edges $BF$, $FG$, $GH$, $HD$ at points $P$, $Q$, $R$, $S$ respectively. Given that $AP = AS$ and $PQ = QR = RS$, find the area of pentagon $APQRS$.

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 $b$.

Let $ v_1, v_2, \ldots, v_{1989}$ be a set of coplanar vectors with $ |v_r| \leq 1$ for $ 1 \leq r \leq 1989.$ Show that it is possible to find $ \epsilon_r$, $1 \leq r \leq 1989,$ each equal to $ \pm 1,$ such that \[ \left | \sum^{1989}_{r\equal{}1} \epsilon_r v_r \right | \leq \sqrt{3}.\]

For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p\plus{}q}{2} \right) \leq \frac{f(p) \plus{} f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always: \[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]

A strip is defined as the region between two parallel lines; the width of the strip is the distance between the two lines. Two strips of width $1$ intersect in a parallelogram whose area is $2.$ What is the angle between the strips? \[ \mathrm a. ~ 15^\circ\qquad \mathrm b.~30^\circ \qquad \mathrm c. ~45^\circ \qquad \mathrm d. ~60^\circ \qquad \mathrm e. ~90^\circ\]

Let $n \ge 6$ be an integer. We have at our disposal $n$ colors. We color each of the unit squares of an $n \times n$ board with one of the $n$ colors. a) Prove that, for any such coloring, there exists a path of a chess knight from the bottom-left to the upper-right corner, that does not use all the colors. b) Prove that, if we reduce the number of colors to $\lfloor 2n/3 \rfloor + 2$, then the statement from a) is true for infinitely many values of $n$ and it is false also for infinitely many values of $n$

Sum of $m$ pairwise different positive even numbers and $n$ pairwise different positive odd numbers is equal to $1987$. Find, with proof, the maximum value of $3m+4n$.

If $\cos 2^{\circ} - \sin 4^{\circ} -\cos 6^{\circ} + \sin 8^{\circ} \ldots + \sin 88^{\circ}=\sec \theta - \tan \theta$, compute $\theta$ in degrees. [i]2015 CCA Math Bonanza Team Round #10[/i]

There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\frac{4}{7}$, then the number common to both sets of four numbers is $\text{(A)}\ 5\frac{3}{7} \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 6\frac{4}{7} \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 7\frac{3}{7}$

Along a circle are $100$ white points. An integer $k$ is given, where $2 \le k \le 50$. In each move, we choose a block of $k$ adjacent points such that the first and the last are white, and we paint both of them black. For which values of $k$ is it possible for us to paint all $100$ points black after $50$ moves?

Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

Let $s(n)$ be the sum of all positive divisors of $n$, so $s(6) = 12$. We say $n$ is almost perfect if $s(n) = 2n - 1$. Let $\mod(n, k)$ denote the residue of $n$ modulo $k$ (in other words, the remainder of dividing $n$ by $k$). Put $t(n) = \mod(n, 1) + \mod(n, 2) + \cdots + \mod(n, n)$. Show that $n$ is almost perfect if and only if $t(n) = t(n-1)$.

Let $p,q,r \in \mathbb{R}^+$ and for every $n \in \mathbb{N}$ where $pqr=1$ , denote $$ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+ 1} \leq 1$$ [i](Art-Ninja)[/i]

Julius has a set of five positive integers whose mean is 100. If Julius removes the median of the set of five numbers, the mean of the set increases by 5, and the median of the set decreases by 5. Find the maximum possible value of the largest of the five numbers Julius has.

Two players play a game. They have $n > 2$ piles containing $n^{10}+1$ stones each. A move consists of removing all the piles but one and dividing the remaining pile into $n$ nonempty piles. The player that cannot move loses. Who has a winning strategy, the player that moves first or his adversary?

Let $ABC$ be a triangle whose angle at $A$ is right, and let $D$ be the foot of the altitude from $A$. A variable point $M$ traces the interior of the minor arc $AB$ of the circle $ABC$. The internal bisector of the angle $DAM$ crosses $CM$ at $N$. The line through $N$ and perpendicular to $CM$ crosses the line $AD$ at $P$. Determine the locus of the point where the line $BN$ crosses the line $CP$. [i]* * *[/i]

There are three sticks, each of which has an integer length which is at least $n$; the sum of their lengths is $n(n + 1)/2$. Prove that it is possible to break the sticks (possibly several times) so that the resulting sticks have length $1, 2,\dots, n$. [i]Note: a stick of length $a + b$ can be broken into sticks of lengths $a$ and $b$.[/i]

The pentagon $ABCDE$ has an inscribed circle and the diagonals $AD$ and $CE$ intersect in its centre $O$. Prove that the segment $BO$ and the side $DE$ are perpendicular. (Folklore)