For $i=0,1,\dots,5$ let $l_i$ be the ray on the Cartesian plane starting at the origin, an angle $\theta=i\frac{\pi}{3}$ counterclockwise from the positive $x$-axis. For each $i$, point $P_i$ is chosen uniformly at random from the intersection of $l_i$ with the unit disk. Consider the convex hull of the points $P_i$, which will (with probability 1) be a convex polygon with $n$ vertices for some $n$. What is the expected value of $n$?

Let $a,b,c,d,e,f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of $\color{white}\ .\quad \ \color{black}\ \quad abc+bcd+cde+def+efa+fab$ and determine all $6$-tuples $(a,b,c,d,e,f)$ for which this maximal value is achieved.

Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.

IMONST = [b]I[/b]nternational [b]M[/b]athematical [b]O[/b]lympiad [b]N[/b]ational [b]S[/b]election [b]T[/b]est Malaysia 2021 Round 1 Juniors Time: 2.5 hours [hide=Rules] $\bullet$ For each problem you have to submit the answer only. The answer to each problem is a non-negative integer. $\bullet$ No mark is deducted for a wrong answer. $\bullet$ The maximum number of points is (1 + 2 + 3 + 4) x 5 = 50 points.[/hide] [b]Part A[/b] (1 point each) p1. Adam draws $7$ circles on a paper, with radii $ 1$ cm, $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm, and $7$ cm. The circles do not intersect each other. He colors some circles completely red, and the rest of the circles completely blue. What is the minimum possible difference (in cm$^2$) between the total area of the red circles and the total area of the blue circles? p2. The number $2021$ has a special property that the sum of any two neighboring digits in the number is a prime number ($2 + 0 = 2$, $0 + 2 = 2$, and $2 + 1 = 3$ are all prime numbers). Among numbers from $2021$ to $2041$, how many of them have this property? p3. Clarissa opens a pet shop that sells three types of pets: goldshes, hamsters, and parrots. The pets inside the shop together have a total of $14$ wings, $24$ heads, and $62$ legs. How many goldshes are there inside Clarissa's shop? p4. A positive integer $n$ is called special if $n$ is divisible by $4$, $n+1$ is divisible by $5$, and $n + 2$ is divisible by $6$. How many special integers smaller than $1000$ are there? p5. Suppose that this decade begins on $ 1$ January $2020$ (which is a Wednesday) and the next decade begins on $ 1$ January $2030$. How many Wednesdays are there in this decade? [b]Part B[/b] (2 points each) p6. Given an isosceles triangle $ABC$ with $AB = AC$. Let D be a point on $AB$ such that $CD$ is the bisector of $\angle ACB$. If $CB = CD$, what is $\angle ADC$, in degrees? p7. Determine the number of isosceles triangles with the following properties: all the sides have integer lengths (in cm), and the longest side has length $21$ cm. p8. Haz marks $k$ points on the circumference of a circle. He connects every point to every other point with straight lines. If there are $210$ lines formed, what is $k$? p9. What is the smallest positive multiple of $24$ that can be written using digits $4$ and $5$ only? p10. In a mathematical competition, there are $2021$ participants. Gold, silver, and bronze medals are awarded to the winners as follows: (i) the number of silver medals is at least twice the number of gold medals, (ii) the number of bronze medals is at least twice the number of silver medals, (iii) the number of all medals is not more than $40\%$ of the number of participants. The competition director wants to maximize the number of gold medals to be awarded based on the given conditions. In this case, what is the maximum number of bronze medals that can be awarded? [b]Part C[/b] (3 points each) p11. Dinesh has several squares and regular pentagons, all with side length $ 1$. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so? [img]https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg [/img] p12. If $x +\frac{1}{x} = 5$, what is the value of $x^3 +\frac{1}{x^3} $ ? p13. There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue? p14. The two diagonals of a rhombus have lengths with ratio $3 : 4$ and sum $56$. What is the perimeter of the rhombus? p15. How many integers $n$ (with $1 \le n \le 2021$) have the property that $8n + 1$ is a perfect square? [b]Part D[/b] (4 points each) p16. Given a segment of a circle, consisting of a straight edge and an arc. The length of the straight edge is $24$. The length between the midpoint of the straight edge and the midpoint of the arc is $6$. Find the radius of the circle. p17. Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$. How many passcodes satisfy these conditions? p18. A tree grows in the following manner. On the first day, one branch grows out of the ground. On the second day, a leaf grows on the branch and the branch tip splits up into two new branches. On each subsequent day, a new leaf grows on every existing branch, and each branch tip splits up into two new branches. How many leaves does the tree have at the end of the tenth day? p19. Find the sum of (decimal) digits of the number $(10^{2021} + 2021)^2$? p20. Determine the number of integer solutions $(x, y, z)$, with $0 \le x, y, z \le 100$, for the equation$$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$

There are five dots arranged in a line from left to right. Each of the dots is colored from one of five colors so that no $3$ consecutive dots are all the same color. How many ways are there to color the dots?

In a fixed plane, consider a convex quadrilateral $ABCD$. Choose a point $O$ in the plane and let $K, L, M$, and $N$ be the circumcentres of triangles $AOB, BOC, COD$, and $DOA$, respectively. Prove that there exists exactly one point $O$ in the plane such that $KLMN$ is a parallelogram.

Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with $x^3 -x$. Let $$ \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 -x } \right) =\frac{f(x)}{g(x)}$$ for polynomials $f(x)$ and $g(x).$ Find the smallest possible degree of $f(x)$.

Let $K, L,M,N$ designate the centers of the squares erected on the four sides (outside) of a rhombus. Prove that the polygon $KLMN$ is a square.

The numbers $1,2,\ldots,9$ are arranged so that the $1$st term is not $1$ and the $9$th term is not $9$. What is the probability that the third term is $3$? $\text{(A) }\frac{17}{75}\qquad\text{(B) }\frac{43}{399}\qquad\text{(C) }\frac{127}{401}\qquad\text{(D) }\frac{16}{19}\qquad\text{(E) }\frac{6}{7}$

a) Prove that for all real numbers $k,l,m$ holds : $$(k+l+m)^2 \ge 3 (kl+lm+mk)$$ When does equality holds? b) If $x,y,z$ are positive real numbers and $a,b$ real numbers such that $$a(x+y+z)=b(xy+yz+zx)=xyz,$$ prove that $a \ge 3b^2$. When does equality holds?

Let $ A$, $ B$ be the number of digits of $ 2^{1998}$ and $ 5^{1998}$ in decimal system. $ A \plus{} B \equal{} ?$ $\textbf{(A)}\ 1998 \qquad\textbf{(B)}\ 1999 \qquad\textbf{(C)}\ 2000 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ 3998$

Find all functions $f:R\rightarrow R$ such that \[ f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy)) \] for all $x,y\in R$.

Find the sum of all integers $n$ with $2 \le n \le 999$ and the following property: if $x$ and $y$ are randomly selected without replacement from the set $\left\{ 1,2,\dots,n \right\}$, then $x+y$ is even with probability $p$, where $p$ is the square of a rational number. [i]Proposed by Ivan Koswara[/i]

Let $f(x)$ be a quadratic polynomial such that $f(f(1)) = f(-f(-1)) = 0$ and $f(1) \neq -f(-1)$. Suppose furthermore that the quadratic $2f(x)$ has coefficients that are nonzero integers. Find $f(0)$. [i]Proposed by Andrew Wu[/i]

Let $D$ be a point on the side $BC$ of $\triangle ABC$. If $AB=8$, $AC=7$, $BD=2$, and $CD=1$, find $AD$.

Let be given a tetrahedron whose any two opposite edges are equal. A plane varies so that its intersection with the tetrahedron is a quadrilateral. Find the positions of the plane for which the perimeter of this quadrilateral is minimum, and find the locus of the centroid for those quadrilaterals with the minimum perimeter.

A number of spheres with radius $ 1$ are being placed in the form of a square pyramid. First, there is a layer in the form of a square with $ n^2$ spheres. On top of that layer comes the next layer with $ (n\minus{}1)^2$ spheres, and so on. The top layer consists of only one sphere. Compute the height of the pyramid.

Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$ [i]Andrew Carratu[/i]

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Let $ ABC$ be a triangle with $ \angle A = 60^{\circ}$.Prove that if $ T$ is point of contact of Incircle And Nine-Point Circle, Then $ AT = r$, $ r$ being inradius.

How many 3-digit positive integers have the property that the sum of their digits is greater than the product of their digits? [i]2016 CCA Math Bonanza Lightning #3.1[/i]

Let $m$ be a positive integer. We say that a positive integer $x$ is $m$-good if $a^m$ divides $x$ for some integer $a > 1$. We say a positive integer $x$ is $m$-bad if it is not $m$-good. (a) Is it true that for every positive integer $n$ there exist $n$ consecutive $m$-bad positive integers? (b) Is it true that for every positive integer $n$ there exist $n$ consecutive $m$-good positive integers?

Let $FIG$ be a triangle and let $D$ be a point on $\overline{FG}$. The line perpendicular to $\overline{FI}$ passing through the midpoint of $\overline{FD}$ and the line perpendicular to $\overline{IG}$ passing through the midpoint of $\overline{DG}$ intersect at $T$. Prove that $FT = GT$ if and only if $\overline{ID}$ is perpendicular to $\overline{FG}$.

(a) Prove that for any positive numbers $x_1,x_2,...,x_k$ ($k > 3$), $$\frac{x_1}{x_k+x_2}+ \frac{x_2}{x_1+x_3}+...+\frac{x_k}{x_{k-1}+x_1}\ge 2$$ (b) Prove that for every $k$ this inequality cannot be sharpened, i.e. prove that for every given $k$ it is not possible to change the number $2$ in the right hand side to a greater number in such a way that the inequality remains true for every choice of positive numbers $x_1,x_2,...,x_k$. (A Prokopiev)

Suppose that on a parabola with vertex $V$ and a focus $F$ there exists a point $A$ such that $AF=20$ and $AV=21$. What is the sum of all possible values of the length $FV?$ $\textbf{(A) }13 \qquad \textbf{(B) }\frac{40}3 \qquad \textbf{(C) }\frac{41}3 \qquad \textbf{(D) }14\qquad \textbf{(E) }\frac{43}3$ Proposed by [b]djmathman[/b]