[u]Set 6[/u] [b]B16.[/b] Let $\ell_r$ denote the line $x + ry + r^2 = 420$. Jeffrey draws the lines $\ell_a$ and $\ell_b$ and calculates their single intersection point. [b]B17.[/b] Let set $L$ consist of lines of the form $3x + 2ay = 60a + 48$ across all real constants a. For every line $\ell$ in $L$, the point on $\ell$ closest to the origin is in set $T$ . The area enclosed by the locus of all the points in $T$ can be expressed in the form nπ for some positive integer $n$. Compute $n$. [b]B18.[/b] What is remainder when the $2020$-digit number $202020 ... 20$ is divided by $275$? [u]Set 7[/u] [b]B19.[/b] Consider right triangle $\vartriangle ABC$ where $\angle ABC = 90^o$, $\angle ACB = 30^o$, and $AC = 10$. Suppose a beam of light is shot out from point $A$. It bounces off side $BC$ and then bounces off side $AC$, and then hits point $B$ and stops moving. If the beam of light travelled a distance of $d$, then compute $d^2$. [b]B20.[/b] Let $S$ be the set of all three digit numbers whose digits sum to $12$. What is the sum of all the elements in $S$? [b]B21.[/b] Consider all ordered pairs $(m, n)$ where $m$ is a positive integer and $n$ is an integer that satisfy $$m! = 3n^2 + 6n + 15,$$ where $m! = m \times (m - 1) \times ... \times 1$. Determine the product of all possible values of $n$. [u]Set 8[/u] [b]B22.[/b] Compute the number of ordered pairs of integers $(m, n)$ satisfying $1000 > m > n > 0$ and $6 \cdot lcm(m - n, m + n) = 5 \cdot lcm(m, n)$. [b]B23.[/b] Andrew is flipping a coin ten times. After every flip, he records the result (heads or tails). He notices that after every flip, the number of heads he had flipped was always at least the number of tails he had flipped. In how many ways could Andrew have flipped the coin? [b]B24.[/b] Consider a triangle $ABC$ with $AB = 7$, $BC = 8$, and $CA = 9$. Let $D$ lie on $\overline{AB}$ and $E$ lie on $\overline{AC}$ such that $BCED$ is a cyclic quadrilateral and $D, O, E$ are collinear, where $O$ is the circumcenter of $ABC$. The area of $\vartriangle ADE$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. What is $m + n + p$? [u]Set 9[/u] [i]This set consists of three estimation problems, with scoring schemes described.[/i] [b]B25.[/b] Submit one of the following ten numbers: $$3 \,\,\,\, 6\,\,\,\, 9\,\,\,\, 12\,\,\,\, 15\,\,\,\, 18\,\,\,\, 21\,\,\,\, 24\,\,\,\, 27\,\,\,\, 30.$$ The number of points you will receive for this question is equal to the number you selected divided by the total number of teams that selected that number, then rounded up to the nearest integer. For example, if you and four other teams select the number $27$, you would receive $\left\lceil \frac{27}{5}\right\rceil = 6$ points. [b]B26.[/b] Submit any integer from $1$ to $1,000,000$, inclusive. The standard deviation $\sigma$ of all responses $x_i$ to this question is computed by first taking the arithmetic mean $\mu$ of all responses, then taking the square root of average of $(x_i -\mu)^2$ over all $i$. More, precisely, if there are $N$ responses, then $$\sigma =\sqrt{\frac{1}{N} \sum^N_{i=1} (x_i -\mu)^2}.$$ For this problem, your goal is to estimate the standard deviation of all responses. An estimate of $e$ gives $\max \{ \left\lfloor 130 ( min \{ \frac{\sigma }{e},\frac{e}{\sigma }\}^{3}\right\rfloor -100,0 \}$ points. [b]B27.[/b] For a positive integer $n$, let $f(n)$ denote the number of distinct nonzero exponents in the prime factorization of $n$. For example, $f(36) = f(2^2 \times 3^2) = 1$ and $f(72) = f(2^3 \times 3^2) = 2$. Estimate $N = f(2) + f(3) +.. + f(10000)$. An estimate of $e$ gives $\max \{30 - \lfloor 7 log_{10}(|N - e|)\rfloor , 0\}$ points. PS. You had better use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c4h2777391p24371239]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve the equation in natural numbers $$ x+y+z+t=xyzt. $$

In trapezoid $ABCD$, $AD$ is parallel to $BC$. $\angle A = \angle D = 45^o$, while $\angle B = \angle C = 135^o$. If $AB = 6$ and the area of $ABCD$ is $30$, find $BC$. [img]https://cdn.artofproblemsolving.com/attachments/0/8/d667522259c773435bc53f5988831aceaef7b7.png[/img]

What is the last two digits of the number $(3 + 7 + 11 +
... + 2007)^2$?

Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.

Let $A$ be the subset of the natural numbers such that the sum of Its digits are multiples of$ 2009$. Find $x, y \in A$ such that $y - x > 0$ is minimum and $x$ is also minimum.

Suppose we have a regular $24$-gon labeled $A_1 \cdots A_{24}.$ We will draw $2$ similar $24$-gons within $A_1 \cdots A_{24}.$ For the sake of this problem, make $A_i=A_{i+24}.$ With our first configuration, we create $3$ stars by creating lines $\overline{A_iA_{i+9}}.$ A $24$-gon will be created in the center, which we denote as our first $24$-gon. With our second configuration, we create a start by creating lines $\overline{A_iA_{i+11}}.$ A $24$-gon will be created in the center, which we denote as our second $24$-gon. Find the ratio of the areas of the first $24$-gon to the second $24$-gon.

In a cube, let $M$ be the midpoint of one of the segments. Choose two vertices of the cube, $A$ and $B$. What is the number of distinct possible triangles $\triangle AMB$ up to congruency? [i]Proposed by Harry Kim[/i]

Call a non-constant polynomial [i]real[/i] if all its coecients are real. Let $P$ and $Q$ be polynomials with complex coefficients such that the composition $P \circ Q$ is real. Show that if the leading coefficient of $Q$ and its constant term are both real, then $P$ and $Q$ are real.

Tracy has been baking a rectangular cake whose surface is dissected by grid lines in square fields. The number of rows is $ 2^n$ and the number of columns is $ 2^{n \plus{} 1}$ where $ n \geq 1, n \in \mathbb{N}.$ Now she covers the fields with strawberries such that each row has at least $ 2n \plus{} 2$ of them. Show that there four pairwise distinct strawberries $ A,B,C$ and $ D$ which satisfy those three conditions: (a) Strawberries $ A$ and $ B$ lie in the same row and $ A$ further left than $ B.$ Similarly $ D$ lies in the same row as $ C$ but further left. (b) Strawberries $ B$ and $ C$ lie in the same column. (c) Strawberries $ A$ lies further up and further left than $ D.$

Let the incircle of a nonisosceles triangle $ABC$ touch $AB$, $AC$ and $BC$ at points $D$, $E$ and $F$ respectively. The corresponding excircle touches the side $BC$ at point $N$. Let $T$ be the common point of $AN$ and the incircle, closest to $N$, and $K$ be the common point of $DE$ and $FT$. Prove that $AK||BC$.

A light car and a heavy truck have the same momentum. The truck weighs ten times as much as the car. How do their kinetic energies compare? $\textbf{(A)}$ The truck's kinetic energy is larger by a factor of $100$ $\textbf{(B)}$ They truck's kinetic energy is larger by a factor of $10$ $\textbf{(C)}$ They have the same kinetic energy $\textbf{(D)}$ The car's kinetic energy is larger by a factor of $10$ $\textbf{(E)}$ The car's kinetic energy is larger by a factor of $100$

Find all the positive integers $a{}$ and $b{}$ such that $(7^a-5^b)/8$ is a prime number. [i]Cosmin Manea and Dragoș Petrică[/i]

Given an equilateral triangle we select an arbitrary point on its interior. We draw theperpendiculars from that point to the three sides of the triangle. Show that the sum of the lengths of these perpendiculars is equal to the height of the triangle.

Find all integer pairs $(x,y)$ such that $x^3+y^3=(x+y)^2$.

A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $ 2: 1$. The ratio of the rectangle's length to its width is $ 2: 1$. What percent of the rectangle's area is inside the square? $ \textbf{(A)}\ 12.5 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 75 \qquad \textbf{(E)}\ 87.5$

Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.

Consider a function $ f(x)\equal{}xe^{\minus{}x^3}$ defined on any real numbers. (1) Examine the variation and convexity of $ f(x)$ to draw the garph of $ f(x)$. (2) For a positive number $ C$, let $ D_1$ be the region bounded by $ y\equal{}f(x)$, the $ x$-axis and $ x\equal{}C$. Denote $ V_1(C)$ the volume obtained by rotation of $ D_1$ about the $ x$-axis. Find $ \lim_{C\rightarrow \infty} V_1(C)$. (3) Let $ M$ be the maximum value of $ y\equal{}f(x)$ for $ x\geq 0$. Denote $ D_2$ the region bounded by $ y\equal{}f(x)$, the $ y$-axis and $ y\equal{}M$. Find the volume $ V_2$ obtained by rotation of $ D_2$ about the $ y$-axis.

Let $n$ be an integer with $n\geq 3$ and assume that $2n$ vertices of a regular $(4n + 1)-$gon are coloured. Show that there must exist three of the coloured vertices forming an isosceles triangle.

Let $P(x)$ and $Q(x)$ be arbitrary polynomials with real coefficients, and let $d$ be the degree of $P(x)$. Assume that $P(x)$ is not the zero polynomial. Prove that there exist polynomials $A(x)$ and $B(x)$ such that: (i) both $A$ and $B$ have degree at most $d/2$ (ii) at most one of $A$ and $B$ is the zero polynomial. (iii) $\frac{A(x)+Q(x)B(x)}{P(x)}$ is a polynomial with real coefficients. That is, there is some polynomial $C(x)$ with real coefficients such that $A(x)+Q(x)B(x)=P(x)C(x)$.

In a square board of size 1001 x 1001, we color some $m$ cells in such a way that: i. Of any two cells that share an edge, at least one is colored. ii. Of any 6 consecutive cells in a column or a row, at least 2 consecutive ones are colored. Determine the smallest possible value of $m$.

Solve in the natural numbers the equation $ \log_{6n-19} (n!+1) =2. $ [i]Dragoș Crișan[/i]

In a 100-dimensional hypercube, each edge has length $ 1$. The box contains $2^{100} + 1$ hyperspheres with the same radius $ r$. The center of one hypersphere is the center of the hypercube, and it touches all the other spheres. Each of the other hyperspheres is tangent to $100$ faces of the hypercube. Thus, the hyperspheres are tightly packed in the hypercube. Find $ r$.

Let \(ABC\) be an acute triangle with circumradius \(R\). Let \(D\) be the midpoint of \(BC\) and \(F\) the midpoint of \(AB\). The perpendicular to \(AC\) through \(F\) and the perpendicular to \(BC\) through \(B\) intersect at \(N\). Prove that \(ND = R\).

Determine all pairs $(x, y)$ of positive integers with $y \vert x^2 +1$ and $x^2 \vert y^3 +1$.