Let $I$ be the incenter of $\triangle ABC$ and $P$ be a point such that $PI$ is perpendicular to $BC$ and $PA$ is parallel to $BC$. Let the line parallel to $BC$, which is tangent to the incircle of $\triangle ABC$, intersect $AB$ and $AC$ at points $Q$ and $R$ respectively. Prove that $\angle BPQ = \angle CPR$.

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

Let $a$ and $b$ be positive real numbers with the sum equal to $1$. Prove that if $a^3$ and $b^3$ are rational, so are $a$ and $b$.

Prove that if $F(x)$ and $G(x)$ are polynomials with coefficients $0$ and $1$ such that $$F(x)G(x) = 1 +x + x^2 +...+ x^{n-1}$$ holds for some $n > 1$, then one of them can be represented in the form $$ (1 +x + x^2 +...+ x^{k-1}) T(x)$$ for some $k > 1$ where $T(x)$ is a polynomial with coefficients $0$ and $1$. (V Senderov, M Vialiy)

Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$ [i]Proposed by Carl Schildkraut[/i]

Let $(x_n)_{n\geq 1}$ be the sequence defined recursively as such: \[x_1=1, \ x_{n+1}=\frac{x_1}{n+1}+\frac{x_2}{n+2}+\cdots+\frac{x_n}{2n} \ \forall n\geq 1.\]Consider the sequence $(y_n)_{n\geq 1}$ such that $y_n=(x_1^2+x_2^2+\cdots x_n^2)/n$ for all $n\geq 1.$ Prove that [list=a] [*]$x_{n+1}^2<y_n/2$ and $y_{n+1}<(2n+1)/(2n+2)\cdot y_n$ for all $n\geq 1;$ [*]$\lim_{n\to\infty}x_n=0.$ [/list]

There are $30$ children, $a_1,a_2,...,a_{30}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.

Given is a triangle $ABC$. The points $P, Q$ lie on the extensions of $BC$ beyond $B, C$, respectively, such that $BP=BA$ and $CQ=CA$. Prove that the circumcenter of triangle $APQ$ lies on the angle bisector of $\angle BAC$.

Let $ \mathcal{H}$ be a family of finite subsets of an infinite set $ X$ such that every finite subset of $ X$ can be represented as the union of two disjoint sets from $ \mathcal{H}$. Prove that for every positive integer $ k$ there is a subset of $ X$ that can be represented in at least $ k$ different ways as the union of two disjoint sets from $ \mathcal{H}$. [i]P. Erdos[/i]

Terry drove along a scenic road using $9$ gallons of gasoline. Then Terry went onto the freeway and used $17$ gallons of gasoline. Assuming that Terry gets $6.5$ miles per gallon better gas mileage on the freeway than on the scenic road, and Terry’s average gas mileage for the entire trip was $30$ miles per gallon, find the number of miles Terry drove.

Minneapolis-St. Paul International Airport is $ 8$ miles southwest of downtown St. Paul and $ 10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis? $ \textbf{(A)}\ 13\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ 16\qquad \textbf{(E)}\ 17$

Find the value of $x+y$ for which the expression \[\frac{6x^2}{y^6} + \frac{6y^2}{x^6}+9x^2y^2+\frac{4}{x^6y^6}\] is minimized.

Let $Q$ be an $n$-by-$n$ real orthogonal matrix, and let $u\in \mathbb{R}^n$ be a unit column vector (that is, $u^Tu=1$). Let $P=I-2uu^T$, where $I$ is the $n$-by-$n$ identity matrix. Show that if $1$ is not an eigenvalue of $Q$, then $1$ is an eigenvalue of $PQ$.

Let $I$ denote the $2\times2$ identity matrix $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ and let $$M=\begin{pmatrix}I&A\\B&C\end{pmatrix},\enspace N=\begin{pmatrix}I&B\\A&C\end{pmatrix}$$where $A,B,C$ are arbitrary $2\times2$ matrices which entries in $\mathbb R$, the real numbers. Thus $M$ and $N$ are $4\times4$ matrices with entries in $\mathbb R$. Is it true that $M$ is invertible (i.e. there is a $4\times4$ matrix $X$ such that $MX=XM=I$) implies $N$ is invertible? Justify your answer.

The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor $ for $n \ge 1$. Find all numbers that appear more than twice at this sequence.

Sixteen real numbers are arranged in a magic square of side $4$ so that the sum of numbers in each row, column or main diagonal equals $k$. Prove that the sum of the numbers in the four corners of the square is also $k$.

Solve equation in real numbers $\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{…+\sqrt{4^nx+3}}}}}-\sqrt{x}=1$

Let $a_1,a_2,\ldots,a_n$ ($n\geq 2$) be nonnegative real numbers whose sum is $\frac{n}{2}$. For every $i=1,\ldots,n$ define $$b_i=a_i+a_ia_{i+1}+a_ia_{i+1}a_{i+2}+\cdots+ a_ia_{i+1}\cdots a_{i+n-2}+2a_ia_{i+1}\cdots a_{i+n-1}$$ where $a_{j+n}=a_j$ for every $j$. Prove that $b_i\geq 1$ holds for at least one index $i$.

Let $ABC$ be a triangle with $\angle{BAC} = 40^{\circ}$ and $\angle{ABC}=60^{\circ}$. Let $D$ and $E$ be the points lying on the sides $AC$ and $AB$, respectively, such that $\angle{CBD} = 40^{\circ}$ and $\angle{BCE} = 70^{\circ}$. Let $F$ be the point of intersection of the lines $BD$ and $CE$. Show that the line $AF$ is perpendicular to the line $BC$.

Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$. Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$ for all positive integers $n$.

Let $O$ be the circumcenter of an acute-angled triangle $ABC$. Let $AH$ be the altitude of this triangle, $M,N,P,Q$ be the midpoints of the segments $AB, AC, BH, CH$, respectively. Let $\omega_1$ and $\omega_2$ be the circumferences of the triangles $AMN$ and $POQ$. Prove that one of the intersection points of $\omega_1$ and $\omega_2$ belongs to the altitude $AH$. (A. Voidelevich)

Acute triangle $ABC$ with $AB>BC$ is inscribed in circle $\Omega$. Points $D$ and $E$, that lie on $(BC)$ and $(AB)$ are the feet of altitudes from $A$ and $C$ in triangle $ABC$, and $M$ is the midpoint of the segment $DE$. Half-line $(AM$ intersects the circle $\Omega$ for the second time in $N$. Show that the circumcenter of triangle $MDN$ lies on the line $BC$.

Find all ordered pairs $(a, b)$ of positive integers such that $a^2 + b^2 + 25 = 15ab$ and $a^2 + ab + b^2$ is prime.

[asy]draw(Circle((0,0), 1)); dot((0,0)); label("$O$", (0,0), S); label("$A$", (-1,0), W); label("$B$", (1,0), E); label("$a$", (-0.5,0), S); draw((-1,-1.25)--(-1,1.25)); draw((1,-1.25)--(1,1.25)); draw((-1,0)--(1,0)); draw((-1,0)--(-1,0)+2.3*dir(30)); label("$C$", (-1,0)+2.3*dir(30), E); label("$D$", (-1,0)+1.8*dir(30), N); dot((-1,0)+.4*dir(30)); label("$E$", (-1,0)+.4*dir(30), N); [/asy] In this figure $AB$ is a diameter of a circle, centered at $O$, with radius $a$. A chord $AD$ is drawn and extended to meet the tangent to the circle at $B$ in point $C$. Point $E$ is taken on $AC$ so that $AE=DC$. Denoting the distances of $E$ from the tangent through $A$ and from the diameter $AB$ by $x$ and $y$, respectively, we can deduce the relation: $\text{(A)}\ y^2=\dfrac{x^3}{2a-x} \qquad \text{(B)}\ y^2=\frac{x^3}{2a+x}\qquad \text{(C)}\ y^4=\frac{x^2}{2-x}\qquad\\ \text{(D)}\ x^2=\dfrac{y^2}{2a-x}\qquad \text{(E)}\ x^2=\frac{y^2}{2a+x}$

Let $ABC$ be an acute triangle. Let $D, E,$ and $F$ be the feet of altitudes from $A, B,$ and $C$ to sides $BC, CA,$ and $AB$, respectively, and let $Q$ be the foot of altitude from A to line $EF$ . Given that $AQ = 20, BC = 15,$ and $AD = 24$, compute the perimeter of triangle $DEF.$