Jan rolls a fair six-sided die and calls the result $r$. Then, he picks real numbers $a$ and $b$ between 0 and 1 uniformly at random and independently. If the probability that the polynomial $\tfrac{x^2}{r} - x\sqrt{a} + b$ has a real root can be expressed as simplified fraction $\frac{p}{q}$, find $p$. Let $T = TNYWR$. Compute the number of ordered triples $(a,b,c)$ such that $a$, $b$, and $c$ are distinct positive integers and $a + b + c = T$.

Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$

Let $\vartriangle ABC$ be a triangle. Determine all points $P$ in the plane such that the triangles $\vartriangle ABP$, $\vartriangle ACP$ and $\vartriangle BCP$ all have the same area.

Let \( x \), \( y \), \( p \) be positive integers that satisfy the equation \( x^4 = p + 9y^4 \), where \( p \) is a prime number. Show that \( \frac{p^2 - 1}{3} \) is a perfect square and a multiple of 16.

Find all prime $p$ numbers such that $p^3-4p+9$ is perfect square.

Let $n$ be a positive integer. In an $n\times n$ garden, a fountain is to be built with $1\times 1$ platforms covering the entire garden. Ana places all the platforms at a different height. Afterwards, Beto places water sources in some of the platforms. The water in each platform can flow to other platforms sharing a side only if they have a lower height. Beto wins if he fills all platforms with water. Find the least number of water sources that Beto needs to win no matter how Ana places the platforms.

Let $d \geq 3$ be a positive integer. The binary strings of length $d$ are splitted into $2^{d-1}$ pairs, such that the strings in each pair differ in exactly one position. Show that there exists an $\textit{alternating cycle}$ of length at most $2d-2$, i.e. at most $2d-2$ binary strings that can be arranged on a circle so that any pair of adjacent strings differ in exactly one position and exactly half of the pairs of adjacent strings are pairs in the split.

Let $a$ and $b$ be real numbers bigger than $1$. Find maximal value of $c \in \mathbb{R}$ such that $$\frac{1}{3+\log _{a} b}+\frac{1}{3+\log _{b} a} \geq c$$

The limit of the sum of an infinite number of terms in a geometric progression is $ \frac {a}{1 \minus{} r}$ where $ a$ denotes the first term and $ \minus{} 1 < r < 1$ denotes the common ratio. The limit of the sum of their squares is: $ \textbf{(A)}\ \frac {a^2}{(1 \minus{} r)^2} \qquad\textbf{(B)}\ \frac {a^2}{1 \plus{} r^2} \qquad\textbf{(C)}\ \frac {a^2}{1 \minus{} r^2} \qquad\textbf{(D)}\ \frac {4a^2}{1 \plus{} r^2} \qquad\textbf{(E)}\ \text{none of these}$

Consider a positive integer $a > 1$. If $a$ is not a perfect square then at the next move we add $3$ to it and if it is a perfect square we take the square root of it. Define the trajectory of a number $a$ as the set obtained by performing this operation on $a$. For example the cardinality of $3$ is $\{3, 6, 9\}$. Find all $n$ such that the cardinality of $n$ is finite. The following part problems may attract partial credit. $\textbf{(a)}$Show that the cardinality of the trajectory of a number cannot be $1$ or $2$. $\textbf{(b)}$Show that $\{3, 6, 9\}$ is the only trajectory with cardinality $3$. $\textbf{(c)}$ Show that there for all $k \geq 3$, there exists a number such that the cardinality of its trajectory is $k$. $\textbf{(d)}$ Give an example of a number with cardinality of trajectory as infinity.

The International Mathematical Olympiad is being organized in Japan, where a folklore belief is that the number $4$ brings bad luck. The opening ceremony takes place at the Grand Theatre where each row has the capacity of $55$ seats. What is the maximum number of contestants that can be seated in a single row with the restriction that no two of them are $4$ seats apart (so that bad luck during the competition is avoided)?

The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002\text{ in}^3$. Find the minimum possible sum in inches of the three dimensions. $\textbf{(A) }36\qquad\textbf{(B) }38\qquad\textbf{(C) }42\qquad\textbf{(D) }44\qquad\textbf{(E) }92$

Let $a_n = 2^{3n-1} + 3^{6n-2} + 5^{6n-3}$. Compute gcd$(a_1, a_2, ... , a_{25})$

Four circles of radius $1$ are each tangent to two sides (line segments) of a square and externally tangent to a circle of radius $3$. What is the area of the space that is inside the square but not contained in any of the circles?

Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.

An arbitrary positive number $a$ is given. A sequence ${a_n}$ is defined by equalities $a_1=\frac{a}{a+1}$ and $a_{n+1}=\frac{aa_n}{a^2+a_n-aa_n}$ for all $n \geq 1$ Find the minimal constant $C$ such that inequality $$a_1+a_1a_2+\ldots+a_1\ldots a_m<C$$ holds for all positive integers $m$ regardless of $a$

in $ABC$ let $E$ and $F$ be points on line $AC$ and $AB$ respectively such that $BE$ is parallel to $CF$. suppose that the circumcircle of $BCE$ meet $AB$ again at $F'$ and the circumcircle of $BCF$ meets $AC$ again at $E'$. show that $BE'$ Is parallel to $CF'$.

Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)\] for all $x,y\in\mathbb{R}$.

If the operation $x * y$ is defined by $x * y = (x+1)(y+1) - 1$, then which one of the following is FALSE? $\textbf{(A)} \ x * y = y *x$ for all real $x$ and $y$. $\textbf{(B)} \ x * (y + z) = ( x * y ) + (x * z)$ for all real $x,y,$ and $z$ $\textbf{(C)} \ (x-1) * (x+1) = (x * x) - 1$ for all real $x$. $\textbf{(D)} \ x * 0 = x$ for all real $x$. $\textbf{(E)} \ x * (y * z) = (x * y) * z$for all real $x,y,$ and $z$.

Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17}$ .Find the maximum possible value of $n$ .

Two triangles are drawn on a plane in such a way that the area covered by their union is an n-gon (not necessarily convex). Find all possible values of the number of vertices n.

On the parabola $y = x^2$ lie three different points $P, Q$ and $R$. Their projections $P', Q'$ and $R'$ on the $x$-axis are equidistant and equal to $s$ , i.e. $| P'Q'| = | Q'R'| = s$. Determine the area of $\vartriangle PQR$ in terms of $s$

An archery target can be represented as three concentric circles with radii $3$, $2$, and $1$ which split the target into $3$ regions, as shown in the figure below. What is the area of Region $1$ plus the area of Region $3$? [i]2015 CCA Math Bonanza Team Round #1[/i]

How many pairs of positive integers $ (a,b)$ are there such that $ \gcd(a,b) \equal{} 1$ and \[ \frac {a}{b} \plus{} \frac {14b}{9a} \]is an integer? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ \text{infinitely many}$

The base three representation of $x$ is \[ 12112211122211112222. \]The first digit (on the left) of the base nine representation of $x$ is $\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 5$