Anya and Vanya’s houses are located on the straight road. The distance between their houses is divided by a shop and a school into three equal parts. If Anya and Vanya leave their houses at the same time and walk towards each other, they will meet near the shop. If Anya rides a scooter, then her speed will increase by $150\,\text{m/min}$, and they will meet near the school. Find Vanya’s speed of walking.

For $a_1,..., a_5 \in R$, $$\frac{a_1}{k^2 + 1}+ ... +\frac{a_5}{k^2 + 5}=\frac{1}{k^2}$$ for all $k \in \{2, 3, 4, 5, 6\}$. Calculate $$\frac{a_1}{2}+... +\frac{a_5}{6}.$$

$P_1,P_2,\ldots,P_{100}$ are $100$ points on the plane, no three of them are collinear. For each three points, call their triangle [b]clockwise[/b] if the increasing order of them is in clockwise order. Can the number of [b]clockwise[/b] triangles be exactly $2017$? [i]Proposed by Morteza Saghafian[/i]

Semicircles $POQ$ and $ROS$ pass through the center of circle $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? [asy] import graph; size(7.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-6.27,xmax=10.01,ymin=-5.65,ymax=10.98; draw(circle((0,0),2)); draw((-3,0)--(3,0),EndArrow(6)); draw((0,-3)--(0,3),EndArrow(6)); draw(shift((0.01,1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,179.76,359.76)); draw(shift((-0.01,-1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,-0.38,179.62)); draw((-1.4,1.43)--(1.41,1.41)); draw((-1.42,-1.41)--(1.4,-1.42)); label("$ P(-1,1) $",(-2.57,2.17),SE*lsf); label("$ Q(1,1) $",(1.55,2.21),SE*lsf); label("$ R(-1,-1) $",(-2.72,-1.45),SE*lsf); label("$S(1,-1)$",(1.59,-1.49),SE*lsf); dot((0,0),ds); label("$O$",(-0.24,-0.35),NE*lsf); dot((1.41,1.41),ds); dot((-1.4,1.43),ds); dot((1.4,-1.42),ds); dot((-1.42,-1.41),ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] $ \textbf{(A)}\ \frac{\sqrt 2}4 \qquad\textbf{(B)}\ \frac 12 \qquad\textbf{(C)}\ \frac{2}{\pi} \qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{\sqrt 2}{2} $

The number of distinct pairs of integers $(x,y)$ such that \[0 < x < y\quad \text{and}\quad \sqrt{1984} = \sqrt{x} + \sqrt{y}\] is $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }7$

Josh writes the numbers $1,2,3,\dots,99,100$. He marks out $1$, skips the next number $(2)$, marks out $3$, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number $(2)$, skips the next number $(4)$, marks out $6$, skips $8$, marks out $10$, and so on to the end. Josh continues in this manner until only one number remains. What is that number? $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 56 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 96$

The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $ 9$ trapezoids, let $ x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $ x$? [asy]unitsize(4mm); defaultpen(linewidth(.8pt)); int i; real r=5, R=6; path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0); for(i=0; i<9; ++i) { draw(rotate(20*i)*t); } draw((-r,0)--(R+1,0)); draw((-R,0)--(-R-1,0));[/asy]$ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 102 \qquad \textbf{(C)}\ 104 \qquad \textbf{(D)}\ 106 \qquad \textbf{(E)}\ 108$

Mazo performs the following operation on triplets of non-negative integers: If at least one of them is positive, it chooses one positive number, decreases it by one, and replaces the digits in the units place with the other two numbers. It starts with the triple $x$, $y$, $z$. Find a triple of positive integers $x$, $y$, $z$ such that $xy + yz + zx = 1000$ (*) and the number of operations that Mazo can subsequently perform with the triple $x, y, z$ is (a) maximal (i.e. there is no triple of positive integers satisfying (*) that would allow him to do more operations); (b) minimal (i.e. every triple of positive integers satisfying (*) allows him to perform at least so many operations).

Compute the value of $$\sin^2 20^{\circ} + \cos^2 50^{\circ} + \sin 20^{\circ} \cos 50^{\circ}.$$ [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 2)[/i]

Find all two digit numbers $ \overline{AB}$ such that $ \overline{AB}$ divides $ \overline{A0B}$.

In a circular tower with an internal diameter of $ 2$ m, there is a spiral staircase with a height of $ 6$ m. The height of each stair step is $ 0.15$ m. In the horizontal projection, the steps form adjacent circular sections with an angle of $ 18^\circ $. The narrower ends of the steps are mounted in a round pillar with a diameter of $ 0.64$ m, the axis of which coincides with the axis of the tower. Calculate the greatest length of a straight rod that can be moved up these stairs from the bottom to the top (do not take into account the thickness of the rod or the thickness of the boards from which the stairs are made).

There are $n$ girls $G_1,\ldots, G_n$ and $n$ boys $B_1,\ldots,B_n$. A pair $(G_i,B_j)$ is called $\textit{suitable}$ if and only if girl $G_i$ is willing to marry boy $B_j$. Given that there is exactly one way to pair each girl with a distinct boy that she is willing to marry, what is the maximal possible number of suitable pairs?

Three prime numbers $p,q,r$ and a positive integer $n$ are given such that the numbers \[ \frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq} \] are integers. Prove that $p=q=r $. [i]Nazar Agakhanov[/i]

$p,q$ satisfies $px+q\geq \ln x$ at $a\leq x\leq b\ (0<a<b)$. Find the value of $p,q$ for which the following definite integral is minimized and then the minimum value. \[\int_a^b (px+q-\ln x)dx\]

Reduced to lowest terms, $ \frac {a^2\minus{}b^2}{ab}\minus{} \frac {ab\minus{}b^2}{ab\minus{}a^2}$ is equal to: $\textbf{(A)}\ \dfrac{a}{b} \qquad \textbf{(B)}\ \dfrac{a^2-2b^2}{ab} \qquad \textbf{(C)}\ a^2 \qquad \textbf{(D)}\ a-2b \qquad \textbf{(E)}\ \text{None of these}$

Let $ABCD$ be a trapezoid with the measure of base $AB$ twice that of base $DC$, and let $E$ be the point of intersection of the diagonals. If the measure of diagonal $AC$ is $11$, then that of segment $EC$ is equal to $\textbf{(A) }3\textstyle\frac{2}{3}\qquad\textbf{(B) }3\frac{3}{4}\qquad\textbf{(C) }4\qquad\textbf{(D) }3\frac{1}{2}\qquad \textbf{(E) }3$

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

Let $k$ and $n$ be two given distinct positive integers greater than $1$. There are finitely many (not necessarily distinct) integers written on the blackboard. Kázmér is allowed to erase $k$ consecutive elements of an arithmetic sequence with a difference not divisible by $k$. Similarly, Nándor is allowed to erase $n$ consecutive elements of an arithmetic sequence with a difference that is not divisible by $n$. The initial numbers on the blackboard have the property that both Kázmér and Nándor can erase all of them (independently from each other) in a finite number of steps. Prove that the difference of biggest and the smallest number on the blackboard is at least $\varphi(n)+\varphi(k)$. [i]Proposed by Boldizsár Varga, Budapest[/i]

Shade in some of the regions in the grid to the right so that the shaded area is equal for each of the 11 rows and columns. Regions must be fully shaded or fully unshaded, at least one region must be shaded, and the area of shaded regions must be at most half of the whole grid. [asy] size(200); defaultpen(linewidth(0.45)); real[][] arr = { {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0},}; for (int i=0; i<11; ++i){ for (int j=0; j<11; ++j){ if(arr[10-j][i] == 1){ fill((i,j)--(i+1,j)--(i+1, j+1)--(i,j+1)--cycle, grey); } } } draw((0,0)--(4,0)--(4,3)--(3,3)--(3,1)--(1,1)--(1,2)--(2,2)--(2,1)--(2,3)--(3,3)--(1,3)--(1,2)--(1,3)--(0,3)--(0,0)--(0,5)--(1,5)--(1,4)--(4,4)--(4,3)--(6,3)--(6,2)--(5,2)--(5,1)--(4,1)--(4,0)--(6,0)--(6,3)--(9,3)--(9,0)--(8,0)--(8,2)--(7,2)--(7,1)--(8,1)--(7,1)--(7,0)--(6,0)--(11,0)--(11,2)--(10,2)--(10,1)--(9,1)--(9,3)--(11,3)--(11,2)--(11,4)--(8,4)--(8,3)--(8,4)--(6,4)--(6,3)--(6,4)--(3,4)--(3,5)--(2,5)--(2,6)--(1,6)--(1,5)--(1,7)--(0,7)--(0,5)--(0,8)--(2,8)--(2,6)--(3,6)--(3,7)--(4,7)--(4,6)--(3,6)--(4,6)--(4,5)--(3,5)--(5,5)--(5,4)--(5,5)--(7,5)--(7,4)--(7,5)--(9,5)--(9,4)--(9,5)--(11,5)--(11,4)--(11,6)--(10,6)--(10,7)--(9,7)--(9,5)--(9,7)--(8,7)--(8,5)--(8,6)--(7,6)--(7,7)--(8,7)--(6,7)--(6,6)--(7,6)--(6,6)--(6,5)--(5,5)--(5,6)--(4,6)--(4,8)--(2,8)--(2,9)--(0,9)--(0,8)--(0,10)--(1,10)--(1,9)--(1,10)--(2,10)--(2,9)--(3,9)--(3,8)--(3,11)--(0,11)--(0,10)--(0,11)--(5,11)--(5,10)--(3,10)--(4,10)--(4,8)--(5,8)--(5,6)--(5,7)--(6,7)--(6,8)--(5,8)--(5,9)--(4,9)--(6,9)--(6,10)--(5,10)--(5,11)--(8,11)--(8,10)--(6,10)--(7,10)--(7,9)--(6,9)--(7,9)--(7,8)--(6,8)--(7,8)--(7,7)--(8,7)--(8,8)--(7,8)--(8,8)--(8,10)--(9,10)--(9,9)--(8,9)--(10,9)--(10,8)--(9,8)--(9,7)--(10,7)--(10,6)--(11,6)--(11,8)--(10,8)--(11,8)--(11,10)--(9,10)--(11,10)--(11,11)--(8,11)); [/asy] You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

A student did not notice multiplication sign between two three-digit numbers and wrote it as a six-digit number. Result was 7 times more that it should be. Find these numbers. [i](2 points)[/i]

Let $ \{A_n\}_{n \in \mathbb{N}}$ be a sequence of measurable subsets of the real line which covers almost every point infinitely often. Prove, that there exists a set $ B \subset \mathbb{N}$ of zero density, such that $ \{A_n\}_{n \in B}$ also covers almost every point infinitely often. (The set $ B \subset \mathbb{N}$ is of zero density if $ \lim_{n \to \infty} \frac {\#\{B \cap \{0, \dots, n \minus{} 1\}\}}{n} \equal{} 0$.)

Three primes $p,q,$ and $r$ satisfy $p+q = r$ and $1 < p < q$. Then $p$ equals $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 17 $

For a triangle $ \triangle ABC (\angle B > \angle C) $, $ D $ is a point on $ AC $ satisfying $ \angle ABD = \angle C $. Let $ I $ be the incenter of $ \triangle ABC $, and circumcircle of $ \triangle CDI $ meets $ AI $ at $ E ( \ne I )$. The line passing $ E $ and parallel to $ AB $ meets the line $ BD $ at $ P $. Let $ J $ be the incenter of $ \triangle ABD $, and $ A' $ be the point such that $ AI = IA' $. Let $ Q $ be the intersection point of $ JP $ and $ A'C $. Prove that $ QJ = QA' $.

Let $C$ is at a circle. $[AB]$ is a diameter this circle. $D$ is a point at $[AB]$. Perpendicular from $C$ to $[AB]$'s foot on the $[AB]$ is $E$, perpendicular from $A$ to $[CD]$'s foot on the $[CD]$ is $F$. Prove that, $$[DC][FC]=[BD][EA]$$

Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$.