Prove that $n^3-n-3$ is not a perfect square for any integer $n$. [i]Calvin Deng.[/i]

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$$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$

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?

$n > 1$ is an odd number and $a_1, a_2, . . . , a_n$ are positive integers such that $gcd(a_1, a_2, . . . , a_n) = 1$. If $d = gcd (a_1^n + a_1.a_2. . . a_n, a_2^n + a_1.a_2. . . a_n, . . . , a_n^n + a_1.a_2. . . a_n) $ find all possible values of $d$.

A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$). Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime. [i]Bobby Shen.[/i]

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Compute $(2 + 0)(2 + 2)(2 + 0)(2 + 2)$. [b]p2.[/b] Given that $25\%$ of $x$ is $120\%$ of $30\%$ of $200$, find $x$. [b]p3.[/b] Jacob had taken a nap. Given that he fell asleep at $4:30$ PM and woke up at $6:23$ PM later that same day, for how many minutes was he asleep? [b]p4.[/b] Kevin is painting a cardboard cube with side length $12$ meters. Given that he needs exactly one can of paint to cover the surface of a rectangular prism that is $2$ meters long, $3$ meters wide, and $6$ meters tall, how many cans of paint does he need to paint the surface of his cube? [b]p5.[/b] How many nonzero digits does $200 \times 25 \times 8 \times 125 \times 3$ have? [b]p6.[/b] Given two real numbers $x$ and $y$, define $x \# y = xy + 7x - y$. Compute the absolute value of $0 \# (1 \# (2 \# (3 \# 4)))$. [b]p7.[/b] A $3$-by-$5$ rectangle is partitioned into several squares of integer side length. What is the fewest number of such squares? Squares in this partition must not overlap and must be contained within the rectangle. [b]p8.[/b] Points $A$ and $B$ lie in the plane so that $AB = 24$. Given that $C$ is the midpoint of $AB$, $D$ is the midpoint of $BC$, $E$ is the midpoint of $AD$, and $F$ is the midpoint of $BD$, find the length of segment $EF$. [b]p9.[/b] Vincent the Bug and Achyuta the Anteater are climbing an infinitely tall vertical bamboo stalk. Achyuta begins at the bottom of the stalk and climbs up at a rate of $5$ inches per second, while Vincent begins somewhere along the length of the stalk and climbs up at a rate of $3$ inches per second. After climbing for $t$ seconds, Achyuta is half as high above the ground as Vincent. Given that Achyuta catches up to Vincent after another $160$ seconds, compute $t$. [b]p10.[/b] What is the minimum possible value of $|x - 2022| + |x - 20|$ over all real numbers $x$? [b]p11.[/b] Let $ABCD$ be a rectangle. Lines $\ell_1$ and $\ell_2$ divide $ABCD$ into four regions such that $\ell_1$ is parallel to $AB$ and line $\ell_2$ is parallel to $AD$. Given that three of the regions have area $6$, $8$, and $12$, compute the sum of all possible areas of the fourth region. [b]p12.[/b] A diverse number is a positive integer that has two or more distinct prime factors. How many diverse numbers are less than $50$? [b]p13.[/b] Let $x$, $y$, and $z$ be real numbers so that $(x+y)(y +z) = 36$ and $(x+z)(x+y) = 4$. Compute $y^2 -x^2$. [b]p14.[/b] What is the remainder when $ 1^{10} + 3^{10} + 7^{10}$ is divided by $58$? [b]p15.[/b] Let $A = (0, 1)$, $B = (3, 5)$, $C = (1, 4)$, and $D = (3, 4)$ be four points in the plane. Find the minimum possible value of $AP + BP + CP + DP$ over all points $P$ in the plane. [b]p16.[/b] In trapezoid $ABCD$, points $E$ and $F$ lie on sides $BC$ and $AD$, respectively, such that $AB \parallel CD \parallel EF$. Given that $AB = 3$, $EF = 5$, and $CD = 6$, the ratio $\frac{[ABEF]}{[CDFE]}$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. (Note: $[F]$ denotes the area of $F$.) [b]p17.[/b] For sets $X$ and $Y$ , let $|X \cap Y |$ denote the number of elements in both $X$ and $Y$ and $|X \cup Y|$ denote the number of elements in at least one of $X$ or $Y$ . How many ordered pairs of subsets $(A,B)$ of $\{1, 2, 3,..., 8\}$ are there such that $|A \cap B| = 2$ and $|A \cup B| = 5$? [b]p18.[/b] A tetromino is a polygon composed of four unit squares connected orthogonally (that is, sharing a edge). A tri-tetromino is a polygon formed by three orthogonally connected tetrominoes. What is the maximum possible perimeter of a tri-tetromino? [b]p19.[/b] The numbers from $1$ through $2022$, inclusive, are written on a whiteboard. Every day, Hermione erases two numbers $a$ and $b$ and replaces them with $ab+a+b$. After some number of days, there is only one number $N$ remaining on the whiteboard. If $N$ has $k$ trailing nines in its decimal representation, what is the maximum possible value of $k$? [b]p20.[/b] Evaluate $5(2^2 + 3^2) + 7(3^2 + 4^2) + 9(4^2 + 5^2) + ... + 199(99^2 + 100^2)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n = (p^2 +2)^2 -9(p^2 -7)$ where $p$ is a prime number. Determine the smallest value of the sum of the digits of $n$ and for what prime number $p$ is obtained.

Find all pairs $(m,n)$ of positive integers such that $m+n$ and $mn+1$ are both powers of $2$.

A “number triangle” $(t_{n, k}) (0 \le k \le n)$ is defined by $t_{n,0} = t_{n,n} = 1 (n \ge 0),$ \[t_{n+1,m} =(2 -\sqrt{3})^mt_{n,m} +(2 +\sqrt{3})^{n-m+1}t_{n,m-1} \quad (1 \le m \le n)\] Prove that all $t_{n,m}$ are integers.

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

Find all triplets $(x, y, p)$ of positive integers such that $p$ is a prime number and $\frac{xy^3}{x+y}=p.$

Find, with proof, all triples of integers $ (a,b,c)$ such that $ a, b$ and $ c$ are the lengths of the sides of a right angled triangle whose area is $ a \plus{} b \plus{} c$

Let $u_0, u_1, u_2, \ldots$ be integers such that $u_0 = 100$; $u_{k+2} \geqslant 2 + u_k$ for all $k \geqslant 0$; and $u_{\ell+5} \leqslant 5 + u_\ell$ for all $\ell \geqslant 0$. Find all possible values for the integer $u_{2023}$.

Determine the prime numbers $p, q, r$ with the property that: $p(p-7) + q (q-7) = r (r-7)$.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

Determine for what $n\ge 3$ integer numbers, it is possible to find positive integer numbers $a_1 < a_2 < ...< a_n$ such $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1$ and $a_1 a_2\cdot\cdot\cdot a_n$ is a perfect square.

How many positive integers $N$ in the segment $\left[10, 10^{20} \right]$ are such that if all their digits are increased by $1$ and then multiplied, the result is $N+1$? [i](F. Bakharev)[/i]

$n>1$ is odd number. There are numbers $n,n+1,n+2,...,2n-1$ on the blackboard. Prove that we can erase one number, such that the sum of all numbers will be not divided any number on the blackboard.

Prove that sum of $12$ consecutive integers cannot be a square. Give an example of $11$ consecutive integers whose sum is a perfect square.

The prime numbers $p$ and $q$ and the integer $a$ are chosen such that $p> 2$ and $a \not\equiv 1$ (mod $q$), but $a^p \equiv 1$ (mod $q$). Prove that $(1 + a^1)(1 + a^2)...(1 + a^{p - 1})\equiv 1$ (mod $q$) .

Square $ABCD$ has side length $4$. Points $E$ and $F$ are the midpoints of sides $AB$ and $CD$, respectively. Eight $1$ by $2$ rectangles are placed inside the square so that no two of the eight rectangles overlap (see diagram). If the arrangement of eight rectangles is chosen randomly, then there are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that none of the rectangles crosses the line segment $EF$ (as in the arrangement on the right). Find $m + n$. [asy] size(200); defaultpen(linewidth(0.8)+fontsize(10pt)); real r = 7; path square=origin--(4,0)--(4,4)--(0,4)--cycle; draw(square^^shift((r,0))*square,linewidth(1)); draw((1,4)--(1,0)^^(3,4)--(3,0)^^(0,2)--(1,2)^^(1,3)--(3,3)^^(1,1)--(3,1)^^(2,3)--(2,1)^^(3,2)--(4,2)); draw(shift((r,0))*((2,4)--(2,0)^^(0,2)--(4,2)^^(0,1)--(4,1)^^(0,3)--(2,3)^^(3,4)--(3,2))); label("A",(4,4),NE); label("A",(4+r,4),NE); label("B",(0,4),NW); label("B",(r,4),NW); label("C",(0,0),SW); label("C",(r,0),SW); label("D",(4,0),SE); label("D",(4+r,0),SE); label("E",(2,4),N); label("E",(2+r,4),N); label("F",(2,0),S); label("F",(2+r,0),S); [/asy]

Prove that for every integer $n \geq 2$ there exist $n$ different positive integers such that for any two of these integers $a$ and $b$ their sum $a+b$ is divisible by their difference $a - b.$

For each positive integer $k$, let $S(k)$ be the sum of the digits of $k$ in the decimal system. Prove that there exists a positive integer $k$, which does not have the digit $9$ in its decimal representation, such that: $$S(2^{24^{2017}}k)=S(k)$$

Assume $a_1,a_2,...,a_{14}$ are positive integers such that $\sum_{i=1}^{14}3^{a_i} = 6558$. Prove that the numbers $a_1,a_2,...,a_{14}$ consist of the numbers $1,...,7$, each taken twice.

Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.