In a triangle $ABC$ with circumcircle $(O)$ suppose that $A$-altitude cut $(O)$ at $D$. Let altitude of $B,C$ cut $AC,AB$ at $E,F$. $H$ is orthocenter and $T$ is midpoint of $AH$. Parallel line with $EF$ passes through $T$ cut $AB,AC$ at $X,Y$. Prove that $\angle XDF = \angle YDE$.

A square with sides of length $6$ has the same area as a rectangle with a length of $9$. What is the width of the rectangle? $\textbf{(A) } 2\qquad\textbf{(B) } \frac73\qquad\textbf{(C) } 3\qquad\textbf{(D) } \frac{10}{3}\qquad\textbf{(E) } 4$

Find all naturals $k$ such that $3^k+5^k$ is the power of a natural number with the exponent $\ge 2$.

The vertices of a triangle are lattice points (they have integer coordinates). There are no other lattice points on the boundary of the triangle, but there is exactly one lattice point inside the triangle. Show that it must be the centroid.

Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$

An acute triangle $ ABC$ has three heights $ AD, BE$ and $ CF$ respectively. Prove that the perimeter of triangle $ DEF$ is not over half of the perimeter of triangle $ ABC.$

Let $S=\{1,2,3,...,12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product? [i]Proposed by Gabriel Wu[/i]

A boa of size $k$ is a graph with $k+1$ vertices $\{0,1,\dots,k-1,k\}$ and edges only between the vertices $i$ and $i+1$ for $0\leq i < k.$ The boa is place in a graph $G$ through a injection of graphs. (This is an injective function form the vertices of the boa to the vertices of the graph in such a way that if there is an edge between the vertices $x$ and $y$ in the boa then there must be an edge between $f(x)$ and $f(y)$ in $G$). The Boa can move in the graph $G$ using to type of movement each time. If the boa is initially on the vertices $f(0),f(1),\dots,f(k)$ then it moves in one of the following ways: (i) It choose $v$ a neighbor of $f(k)$ such that $v\not\in\{f(0),f(1),\dots,f(k-1)\}$ and the boa now moves to $f(0),f(1),\dots,f(k)$ with $f'(k)=v$ and $f'(i) = f(i+1)$ for $0 \leq i < k,$ or (ii) It choose $v$ a neighbor of $f(0)$ such that $v\not\in\{f(1),f(2),\dots,f(k)\}$ and the boa now moves to $f(0),f(1),\dots,f(k)$ with $f'(0)=v$ and $f'(i) = f'(i-1)$ for $0 < i \leq k.$ Prove that if $G$ is a connected graph with diameter $d$, then it is possible to put a size $\lceil d/2 \rceil$ boa in $G$ such that the boa can reach any vertex of $G$.

Maria Ivanovna drew on the blackboard a right triangle $ABC$ with a right angle $B$. Three students looked at her and said: $\bullet$ Yura said: "The hypotenuse of this triangle is $10$ cm." $\bullet$ Roma said: "The altitude drawn from the vertex $B$ on the side $AC$ is $6$ cm." $\bullet$ Seva said: "The area of the triangle $ABC$ is $25$ cm$^2$." Determine which of the students was mistaken if it is known that there is exactly one such person.

Let $ABC$ be a triangle and $D$ a point inside the triangle, located on the median of $A$. Prove that if $\angle BDC = 180^o - \angle BAC$, then $AB \cdot CD = AC \cdot BD$.

Points $ A,B,C,D,E$ and $ F$ lie, in that order, on $ \overline{AF}$, dividing it into five segments, each of length 1. Point $ G$ is not on line $ AF$. Point $ H$ lies on $ \overline{GD}$, and point $ J$ lies on $ \overline{GF}$. The line segments $ \overline{HC}, \overline{JE},$ and $ \overline{AG}$ are parallel. Find $ HC/JE$. $ \text{(A)}\ 5/4 \qquad \text{(B)}\ 4/3 \qquad \text{(C)}\ 3/2 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 2$

Several sets of prime numbers, such as $ \{ 7, 83, 421, 659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have? $ \textbf{(A)}\ 193\qquad\textbf{(B)}\ 207\qquad\textbf{(C)}\ 225\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 447$

Fathers childhood lasted for one sixth part of his life, and he married one $8$th after that and he immediately left to army. When one $12$th of his life passed, father returned from the army and $5$ years after he got a son. Son who lived for one half of fathers years, died $4$ years before his father. How many years lived his father, and how many years he had when his son was born?

Let $a,b,c$ be real positive numbers. Prove that the following inequality holds: \[{ \sum_{\rm cyc}\sqrt{5a^2+5c^2+8b^2\over 4ac}\ge 3\cdot \root 9 \of{8(a+b)^2(b+c)^2(c+a)^2\over (abc)^2} }\]

[b]a)[/b] Give an example of function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ that admits a primitive $ F:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ having the property that $ F^e $ is a primitive of $ f^e. $ [b]b)[/b] Prove that there is no derivable function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that has a primitive $ G:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ e^G $ is a primitive of $ e^g. $

Find the largest positive real $ k$, such that for any positive reals $ a,b,c,d$, there is always: \[ (a\plus{}b\plus{}c) \left[ 3^4(a\plus{}b\plus{}c\plus{}d)^5 \plus{} 2^4(a\plus{}b\plus{}c\plus{}2d)^5 \right] \geq kabcd^3\]

Let $\Delta ABC$ be an acute-angled triangle with $AB < AC$, $H$ be a point on $BC$ such that $AH\ \bot BC$ and $G$ be the centroid of $\Delta ABC$. Let $P,Q$ be the point of tangency of the inscribed circle of $\Delta ABC$ with $AB,AC$, correspondingly. Define $M,N$ to be the midpoint of $PB,QC$, correspondingly. Let $D,E$ be points on the inscribed circle of $\Delta ABC$ such that $\angle BDH + \angle ABC = 180^{\circ}$, $\angle CEH + \angle ACB = 180^{\circ}$. Prove that lines $MD,NE,HG$ share a common point.

In $\triangle ABC$, $X, Y$, are points on BC such that $BX=XY=YC$, $M , N$ are points on $AC$ such that $AM=MN=NC$. $BM$ and $BN$ intersect $AY$ at $S$ and $R$ and respectively. If the area of $\triangle ABC$ is $1$, find the area of $SMNR$.

$ABCD$ is a quadrilateral. The bisectors of $\angle A$ and $\angle B$ meet at $E$. The line through $E$ parallel to $CD$ meets $AD$ at $L$ and $BC$ at $M$. Show that $LM = AL + BM$.

Let circles $C_1$ and $C_2$ be internally tangent at point $P$, with $C_1$ being the smaller circle. Consider a line passing through $P$ which intersects $C_1$ at $Q$ and $C_2$ at $R$. Let the line tangent to $C_2$ at $R$ and the line perpendicular to $\overline{PR}$ passing through $Q$ intersect at a point $S$ outside both circles. Given that $SR = 5$, $RQ = 3$, and $QP = 2$, compute the radius of $C_2$.

Let $P$ be an arbitrary point on the side $BC$ of triangle $ABC$ and let $D$ be the tangency point between the incircle of the triangle $ABC$ and the side $BC$. If $Q$ and $R$ are respectively the incenters in the triangles $ABP$ and $ACP$, prove that $\angle QDR$ is a right angle. Prove that the triangle $QDR$ is isosceles if and only if $P$ is the foot of the altitude from $A$ in the triangle $ABC$.

Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that \[{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}\] is divisible by $p^{2}$.

Let $PAB$ and $PBC$ be two similar right-angled triangles (in the same plane) with $\angle PAB = \angle PBC = 90^\circ$ such that $A$ and $C$ lie on opposite sides of the line $PB$. If $PC = AC$, calculate the ratio $\frac{PA}{AB}$.

Let the tangent line passing through a point $A$ outside the circle with center $O$ touches the circle at $B$ and $C$. Let $[BD]$ be the diameter of the circle. Let the lines $CD$ and $AB$ meet at $E$. If the lines $AD$ and $OE$ meet at $F$, find $|AF|/|FD|$.

How many digits are in the base $10$ representation of $3^{30}$ given $\log 3 = 0.47712$? [i]2015 CCA Math Bonanza Lightning Round #1.4[/i]