Let $f(x)$ be a continuous function satisfying $f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)$ Find the value of $k$ for which $\int_0^{\pi} f(x)dx$ is maximized.

Find the number of positive integers $n < 2018$ such that $25^n + 9^n$ is divisible by $13$.

Given the true statement: If a quadrilateral is a square, then it is a rectangle. It follows that, of the converse and the inverse of this true statement is: $ \textbf{(A)}\ \text{only the converse is true} \qquad\textbf{(B)}\ \text{only the inverse is true }\qquad \textbf{(C)}\ \text{both are true} \qquad$ $\textbf{(D)}\ \text{neither is true} \qquad\textbf{(E)}\ \text{the inverse is true, but the converse is sometimes true} $

How many nonempty subsets $B$ of $\{0, 1, 2, 3, \dots, 12\}$ have the property that the number of elements in $B$ is equal to the least element of $B$? For example, $B = \{4, 6, 8, 11\}$ satisfies the condition. $\textbf{(A)}\ 256 \qquad\textbf{(B)}\ 136 \qquad\textbf{(C)}\ 108 \qquad\textbf{(D)}\ 144 \qquad\textbf{(E)}\ 156$

Fourteen white cubes are put together to form the figure on the right. The complete surface of the figure, including the bottom, is painted red. The figure is then separated into individual cubes. How many of the individual cubes have exactly four red faces? [asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(3/4,8/15,7/15); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white, thick(), nolight); draw(shift(2,0,0)*unitcube, white, thick(), nolight); draw(shift(0,0,1)*unitcube, white, thick(), nolight); draw(shift(2,0,1)*unitcube, white, thick(), nolight); draw(shift(0,1,0)*unitcube, white, thick(), nolight); draw(shift(2,1,0)*unitcube, white, thick(), nolight); draw(shift(0,2,0)*unitcube, white, thick(), nolight); draw(shift(2,2,0)*unitcube, white, thick(), nolight); draw(shift(0,3,0)*unitcube, white, thick(), nolight); draw(shift(0,3,1)*unitcube, white, thick(), nolight); draw(shift(1,3,0)*unitcube, white, thick(), nolight); draw(shift(2,3,0)*unitcube, white, thick(), nolight); draw(shift(2,3,1)*unitcube, white, thick(), nolight);[/asy] $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12$

A transposition of a vector is created by switching exactly two entries of the vector. For example, $(1,5,3,4,2,6,7)$ is a transposition of $(1,2,3,4,5,6,7).$ Find the vector $X$ if $S=(0,0,1,1,0,1,1)$, $T=(0,0,1,1,1,1,0),$ $U=(1,0,1,0,1,1,0),$ and $V=(1,1,0,1,0,1,0)$ are all transpositions of $X$. Describe your method for finding $X.$

$ABCD$ is a convex quadrilateral such that $\angle A = \angle C < 90^{\circ}$ and $\angle ABD = 90^{\circ}$. $M$ is the midpoint of $AC$. Prove that $MB$ is perpendicular to $CD$.

In $\triangle ABC\ T$ is a point lies on the internal angle bisector of $B$. Let $\omega$ be circle with diameter $BT$. $\omega$ intersects with $BA$ and $BC$ at $P$ and $Q$,respectively. A circle passes through $A$ and tangent to $\omega$ at $P$ intersects with $AC$ again at $X$ . A circle passes through $B$ and tangent to $\omega$ at $Q$ intersects with $AC$ again at $Y$ . Prove that $TX=TY$

Let $U$ be an open subset of the complex plane $\mathbb{C}$ including $\mathbb{D}=\{z \in \mathbb{C} : |z| \le 1\}$ and $f$ be analytic over $U$. Prove that if for every $z$ with a complex norm equal to $1$($|z|=1$) we have $0<Re(\bar{z}f(z))$, then $f$ has only one root in $\mathbb{D}$ and that's simple.

A sequence of real numbers $(a_k)_{k \ge 0}$ is called [i]log-concave[/i] if for every $k \ge 1$, the inequality $a_{k - 1}a_{k + 1} \le a_k^2$ holds. Let $n, l \in \mathbb{N}$. Prove that the sequence $(a_k)_{k \ge 0}$ with general term \[a_k = \sum_{i = k}^{k + l} {n \choose i}\] is log-concave. Proposed by [i]Svetlana Poznanovikj[/i]

A map $ F : P(X) \rightarrow P(X)$, where $ P(X)$ denotes the set of all subsets of $ X$, is called a $ \textit{closure operation}$ on $ X$ if for arbitrary $ A,B \subset X$, the following conditions hold: (i) $ A \subset F(A);$ (ii) $ A \subset B \Rightarrow F(A) \subset F(B);$ (iii) $ F(F(A))\equal{}F(A)$. The cardinal number $ \min \{ |A| : \;A \subset X\ ,\;F(A)\equal{}X\ \}$ is called the $ \textit{density}$ of $ F$ and is denoted by $ d(F)$. A set $ H \subset X$ is called $ \textit{discrete}$ with respect to $ F$ if $ u \not \in F(H\minus{}\{ u \})$ holds for all $ u \in H$. Prove that if the density of the closure operation $ F$ is a singular cardinal number, then for any nonnegative integer $ n$, there exists a set of size $ n$ that is discrete with respect to $ F$. Show that the statement is not true when the existence of an infinite discrete subset is required, even if $ F$ is the closure operation of a topological space satisfying the $ T_1$ separation axiom. [i]A. Hajnal[/i]

Seven cookies of radius $1$ inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie? [asy] draw(circle((0,0),3)); draw(circle((0,0),1)); draw(circle((1,sqrt(3)),1)); draw(circle((-1,sqrt(3)),1)); draw(circle((-1,-sqrt(3)),1)); draw(circle((1,-sqrt(3)),1)); draw(circle((2,0),1)); draw(circle((-2,0),1)); [/asy] $\textbf{(A) } \sqrt{2} \qquad \textbf{(B) } 1.5 \qquad \textbf{(C) } \sqrt{\pi} \qquad \textbf{(D) } \sqrt{2\pi} \qquad \textbf{(E) } \pi$

A $20 \times 20 \times 20$ cube is composed of $2000$ bricks of size $2 \times 2 \times 1$ . Prove that it is possible to pierce the cube with a needle so that the needle passes through the cube without passing through a brick . (A . Andjans , Riga)

Let $k > 1$ be a positive integer and $n \ge 2019$ be an odd positive integer. The non-zero rational numbers $x_1, x_2,..., x_n$ are not all equal, and satisfy the following chain of equalities: $$x_1 +\frac{k}{x_2}= x_2 +\frac{k}{x_3}= x_3 +\frac{k}{x_4}= ... = x_{n-1} +\frac{k}{x_n}= x_n +\frac{k}{x_1}.$$ What is the smallest possible value of $k$?

Evaluate $\int_0^{\frac{\pi}{2}} \sqrt{1-2\sin 2x+3\cos ^ 2 x}\ dx.$ [i]2011 University of Occupational and Environmental Health/Medicine　entrance exam[/i]

How many kites are there such that all of its four vertices are vertices of a given regular icosagon ($20$-gon)? $ \textbf{(A)}\ 105 \qquad\textbf{(B)}\ 100 \qquad\textbf{(C)}\ 95 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 85 $

Prove that there is no natural $n$ such that $8n + 7$ is the sum of three squares.