Find whether the number of powers of 2, which have a digit sum smaller than $2019^{2019}$, is finite or infinite.

Points $A$ and $B$ lie on the graph of $y=\log_{2}x$. The midpoint of $\overline{AB}$ is $(6, 2)$. What is the positive difference between the $x$-coordinates of $A$ and $B$? $\textbf{(A)}~2\sqrt{11}\qquad\textbf{(B)}~4\sqrt{3}\qquad\textbf{(C)}~8\qquad\textbf{(D)}~4\sqrt{5}\qquad\textbf{(E)}~9$

[b]H[/b]orizontal parallel segments $AB=10$ and $CD=15$ are the bases of trapezoid $ABCD$. Circle $\gamma$ of radius $6$ has center within the trapezoid and is tangent to sides $AB$, $BC$, and $DA$. If side $CD$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $ABCD$.

Two cirlces $ C_1$ and $ C_2$, with center $ O_1$ and $ O_2$ respectively, intersect at $ A$ and $ B$. Let $ O_1$ lies on $ C_2$. A line $ l$ passes through $ O_1$ but does not pass through $ O_2$. Let $ P$ and $ Q$ be the projection of $ A$ and $ B$ respectively on the line $ l$ and let $ M$ be the midpoint of $ \overline{AB}$. Prove that $ MPQ$ is an isoceles triangle.

Find all integers $n$ for which $n^7-41$ is the square of an integer

Define the Fibonacci numbers such that $F_{1} = F_{2} = 1,$ $F_{k} = F_{k-1} + F_{k-2}$ for $k > 2.$ For large positive integers $n,$ the expression (containing $n$ nested square roots) $$\sqrt{2023 F^{2}_{2^{1}} + \sqrt{2023 F^{2}_{2^{2}} + \sqrt{2023 F_{2^{3}}^{2} \dots + \sqrt{2023 F^{2}_{2^{n}} }}}}$$ approaches $\frac{a + \sqrt{b}}{c}$ for positive integers $a,b,c$ where $\gcd(a,c) = 1.$ Find $a+b+c.$

A grasshopper is jumping about in a grid. From the point with coordinates $(a, b)$ it can jump to either $(a + 1, b),(a + 2, b),(a + 1, b + 1),(a, b + 2)$ or $(a, b + 1)$. In how many ways can it reach the line $x + y = 2014?$ Where the grasshopper starts in $(0, 0)$.

The positive integer $m$ is a multiple of 111, and the positive integer $n$ is a multiple of 31. Their sum is 2017. Find $n - m$.

For which real values of $x,y$ the expression$\frac{2-\left(\dfrac{x+y}{3}-1\right)^2}{\left(\dfrac{x-3}{2}+\dfrac{2y-x}{3}\right)^2+4}$ becomes maximum? Which is that maximum value?

Every side and diagonal of a regular $12$-gon is colored in one of $12$ given colors. Can this be done in such a way that, for every three colors, there exist three vertices which are connected to each other by segments of these three colors?

Let $f$ be a continuous real-valued function on $\mathbb R^3$.  Suppose that for every sphere $S$ of radius $1$, the integral of $f(x,y,z)$ over the surface of $S$ equals zero.  Must $f(x,y,z)$ be identically zero?

In a triangle $ABC$, $\angle B = 70^o$, $\angle C = 50^o$. A point $M$ is taken on the side $AB$ such that $\angle MCB = 40^o$ , and a point $N$ is taken on the side $AC$ such that $\angle NBC = 50^o$. Find $\angle NMC$.

[i]The Game.[/i] Eric and Greg are watching their new favorite TV show, [i]The Price is Right[/i]. Bob Barker recently raised the intellectual level of his program, and he begins the latest installment with bidding on following question: How many Carmichael numbers are there less than $100,000$? Each team is to list one nonnegative integer not greater than $100,000$. Let $X$ denote the answer to Bob’s question. The teams listing $N$, a maximal bid (of those submitted) not greater than $X$, will receive $N$ points, and all other teams will neither receive nor lose points. (A Carmichael number is an odd composite integer $n$ such that $n$ divides $a^{n-1}-1$ for all integers $a$ relatively prime to $n$ with $1<a<n$.)

Let $\, a_1, a_2, a_3, \ldots \,$ be a sequence of positive real numbers satisfying $\, \sum_{j=1}^n a_j \geq \sqrt{n} \,$ for all $\, n \geq 1$. Prove that, for all $\, n \geq 1, \,$ \[ \sum_{j=1}^n a_j^2 > \frac{1}{4} \left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right). \]

Given an isosceles triangle $ABC$ (with $AB=BC$). A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through the circumcentre of triangle $ABC$. [b]proposed by Sergej Berlov[/b]

Does there exist a functional equation[sup]1[/sup] that has a solution and the range of any of its solutions is the set of integers? [sup]1[/sup][size=75]A [i]functional equation[/i] has the form $\mbox{\footnotesize \(E = 0\)}$, where $\mbox{\footnotesize \(E\)}$ is a function form. The set of function forms is the smallest set $\mbox{\footnotesize \(\mathcal{F}\)}$ which contains the variables $\mbox{\footnotesize \(x_1, x_2, \dots\)}$, the real numbers $\mbox{\footnotesize \(r \in \mathbb{R}\)}$, and for which $\mbox{\footnotesize \(E, E_1, E_2 \in \mathcal{F}\)}$ implies $\mbox{\footnotesize \(E_1+E_2 \in \mathcal{F}\)}$, $\mbox{\footnotesize \(E_1 \cdot E_2 \in \mathcal{F}\)}$, and $\mbox{\footnotesize \(f(E) \in \mathcal{F}\)}$, where $\mbox{\footnotesize \(f\)}$ is a fixed function symbol. The solution of the functional equation $\mbox{\footnotesize \(E = 0\)}$ is a function $\mbox{\footnotesize \(f: \mathbb{R} \to \mathbb{R}\)}$ such that $\mbox{\footnotesize \(E = 0\)}$ holds for all values of the variables. E.g. $\mbox{\footnotesize \(f\big(x_1 + f(\sqrt{2} \cdot x_2 \cdot x_2)\big) + (-\pi) + (-1) \cdot x_1 \cdot x_1 \cdot x_2 = 0\)}$ is a functional equation.[/size]

Point $C_{1}$ of hypothenuse $AC$ of a right-angled triangle $ABC$ is such that $BC = CC_{1}$. Point $C_{2}$ on cathetus $AB$ is such that $AC_{2} = AC_{1}$; point $A_{2}$ is defined similarly. Find angle $AMC$, where $M$ is the midpoint of $A_{2}C_{2}$.

Find the derivative of the solution of the equation $\ddot{x} = \dot{x}^2 + x^3$ with initial condition $x(0) = 0$, $\dot{x}(0) = A$ with respect to $A$ for $A = 0$.

The sequence $(a_n)$ of complex numbers is considered in the complex plane, in which is: $$a_0 = 1, \,\,\, a_n = a_{n-1} +\frac{1}{n}(\cos 45^o + i \sin 45^o )^n.$$ Prove that the sequence of the real parts of the terms of $(a_n)$ is convergent and its limit is a number between $0.85$ and $1.15$.

Given a collection of sets $X = \{A_1, A_2, ..., A_n\}$. A set $\{a_1, a_2, ..., a_n\}$ is called a single representation of $X$ if $a_i \in A_i$ for all i. Let $|S| = mn$, $S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n$ with $|A_i| = |B_i| = m$ for all $i$. Prove that $S = C_1 \cup C_2 \cup ... \cup C_n$ where for every $i, C_i $ is a single represenation for $\{A_j\}_{j=1}^n $and $\{B_j\}_{j=1}^n$.

$D$ is midpoint of $AC$ for $\triangle ABC$. Bisectors of $\angle ACB,\angle ABD$ are perpendicular. Find max value for $\angle BAC$ [i](S. Berlov)[/i]

$AB$ is a chord of $O$ and $AB$ is not a diameter of $O$. The tangent lines to $O$ at $A$ and $B$ meet at $C$. Let $M$ and $N$ be the midpoint of the segments $AC$ and $BC$, respectively. A circle passing through $C$ and tangent to $O$ meets line $MN$ at $P$ and $Q$. Prove that $\angle PCQ = \angle CAB$.

There are $n$ students $ A_1 , A_2 , \cdots , A_n $ and some of them shaked hands with each other. ($ A_i $ and $ A_j$ can shake hands more than one time.) Let the student $ A_i $ shaked hands $ d_i $ times. Suppose $ d_1 + d_2 + \cdots + d_n > 0 $. Prove that there exist $ 1 \le i < j \le n $ satisfying the following conditions: (a) Two students $ A_i $ and $ A_j $ shaked hands each other. (b) $ \frac{(d_1 + d_2 + \cdots + d_n ) ^2 }{n^2 } \le d_i d_j $

Logan is constructing a scaled model of his town. The city's water tower stands $ 40$ meters high, and the top portion is a sphere that holds $ 100,000$ liters of water. Logan's miniature water tower holds $ 0.1$ liters. How tall, in meters, should Logan make his tower? $ \textbf{(A)}\ 0.04\qquad \textbf{(B)}\ \frac{0.4}{\pi}\qquad \textbf{(C)}\ 0.4\qquad \textbf{(D)}\ \frac{4}{\pi}\qquad \textbf{(E)}\ 4$

Find the largest positive integer $n$, such that there exists a finite set $A$ of $n$ reals, such that for any two distinct elements of $A$, there exists another element from $A$, so that the arithmetic mean of two of these three elements equals the third one.