Let $p$ be a prime number. Prove that it is possible to choose a permutation $a_1, a_2,...,a_p$ of $1,2,...,p$ such that the numbers $a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p$ all have different remainder upon division by $p$.

[b]Determine all four digit numbers [/b]$\bar{a}\bar{b}\bar{c}\bar{d}$[b] such that[/b] $$a(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^{6})=\bar{a}\bar{b}\bar{c}\bar{d}$$

Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

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 2020 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. The number $N$ is the smallest positive integer with the sum of its digits equal to $2020$. What is the first (leftmost) digit of $N$? p2. At a food stall, the price of $16$ banana fritters is $k$ RM , and the price of $k$ banana fritters is $ 1$ RM . What is the price of one banana fritter, in sen? Note: $1$ RM is equal to $100$ sen. p3. Given a trapezium $ABCD$ with $AD \parallel$ to $BC$, and $\angle A = \angle B = 90^o$. It is known that the area of the trapezium is 3 times the area of $\vartriangle ABD$. Find $$\frac{area \,\, of \,\, \vartriangle ABC}{area \,\, of \,\, \vartriangle ABD}.$$ p4. Each $\vartriangle$ symbol in the expression below can be substituted either with $+$ or $-$: $$\vartriangle 1 \vartriangle 2 \vartriangle 3 \vartriangle 4.$$ How many possible values are there for the resulting arithmetic expression? Note: One possible value is $-2$, which equals $-1 - 2 - 3 + 4$. p5. How many $3$-digit numbers have its sum of digits equal to $4$? [b]Part B[/b] (2 points each) p6. Find the value of $$+1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 +... - 2020$$ where the sign alternates between $+$ and $-$ after every three numbers. p7. If Natalie cuts a round pizza with $4$ straight cuts, what is the maximum number of pieces that she can get? Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting. p8. Given a square with area $ A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $ B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $A/B$. p9. This sequence lists the perfect squares in increasing order: $$0, 1, 4, 9, 16, ... ,a, 10^8, b, ...$$ Determine the value of $b - a$. p10. Determine the last digit of $5^5 + 6^6 + 7^7 + 8^8 + 9^9$. [b]Part C[/b] (3 points each) p11. Find the sum of all integers between $-\sqrt{1442}$ and $\sqrt{2020}$. p12. Three brothers own a painting company called Tiga Abdul Enterprise. They are hired to paint a building. Wahab says, "I can paint this building in $3$ months if I work alone". Wahib says, "I can paint this building in $2$ months if I work alone". Wahub says, "I can paint this building in $k$ months if I work alone". If they work together, they can finish painting the building in $1$ month only. What is $k$? p13. Given a rectangle $ABCD$ with a point $P$ inside it. It is known that $PA = 17$, $PB = 15$, and $PC = 6$. What is the length of $PD$? p14. What is the smallest positive multiple of $225$ that can be written using digits $0$ and $ 1$ only? p15. Given positive integers $a, b$, and $c$ with $a + b + c = 20$. Determine the number of possible integer values for $\frac{a + b}{c}$. [b]Part D[/b] (4 points each) p16. If we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest possible value of $p$. p17. A football is made by sewing together some black and white leather patches. The black patches are regular pentagons of the same size. The white patches are regular hexagons of the same size. Each pentagon is bordered by $5$ hexagons. Each hexagons is bordered by $3$ pentagons and $3$ hexagons. We need $12$ pentagons to make one football. How many hexagons are needed to make one football? p18. Given a right-angled triangle with perimeter $18$. The sum of the squares of the three side lengths is $128$. What is the area of the triangle? p19. A perfect square ends with the same two digits. How many possible values of this digit are there? p20. Find the sum of all integers $n$ that fulfill the equation $2^n(6 - n) = 8n$.

The sequence of positive integers $a_0, a_1, a_2, \ldots$ is recursively defined such that $a_0$ is not a power of $2$, and for all nonnegative integers $n$: (i) if $a_n$ is even, then $a_{n+1} $ is the largest odd factor of $a_n$ (ii) if $a_n$ is odd, then $a_{n+1} = a_n + p^2$ where $p$ is the smallest prime factor of $a_n$ Prove that there exists some positive integer $M$ such that $a_{m+2} = a_m $ for all $m \geq M$. [i]Proposed by Andrew Wen[/i]

A triplet of positive integers $(x, y, z)$ satisfying $x, y, z > 1$ and $x^3 - yz^3 = 2021$ is called [i]primary [/i] if at least two of the integers $x, y, z$ are prime numbers. a) Find at least one primary triplet. b) Show that there are infinitely many primary triplets.

The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation.

The base-$7$ representation of number $n$ is $\overline{abc}_{(7)}$, and the base-$9$ representation of number $n$ is $\overline{cba}_{(9)}$. What is the decimal (base-$10$) representation of $n$?

Positive integers $a_1, a_2, \dots, a_{2023}$ satisfy the following conditions. [list] [*] $a_1 = 5, a_2 = 25$ [*] $a_{n + 2} = 7a_{n+1}-a_n-6$ for each $n = 1, 2, \dots, 2021$ [/list] Prove that there exist integers $x, y$ such that $a_{2023} = x^2 + y^2.$

If $x,y,K,m \in N$, let us define: $a_m= \underset{k \, twos}{2^{2^{,,,{^{2}}}}}$, $A_{km} (x)= \underset{k \, twos}{ 2^{2^{,,,^{x^{a_m}}}}}$, $B_k(y)= \underset{m \, fours}{4^{4^{4^{,,,^{4^y}}}}}$, Determine all pairs $(x,y)$ of non-negative integers, dependent on $k>0$, such that $A_{km} (x)=B_k(y)$

Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.

Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence \[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \] forms an arithmetic progression. Prove that the terms of the sequence are equal.

Let $a$ be any integer. Define the sequence $x_0,x_1,\ldots$ by $x_0=a$, $x_1=3$, and for all $n>1$ \[x_n=2x_{n-1}-4x_{n-2}+3.\] Determine the largest integer $k_a$ for which there exists a prime $p$ such that $p^{k_a}$ divides $x_{2011}-1$.

The following sequence lists all the positive rational numbers that do not exceed $\frac12$ by first listing the fraction with denominator 2, followed by the one with denominator 3, followed by the two fractions with denominator 4 in increasing order, and so forth so that the sequence is \[ \frac12,\frac13,\frac14,\frac24,\frac15,\frac25,\frac16,\frac26,\frac36,\frac17,\frac27,\frac37,\cdots. \] Let $m$ and $n$ be relatively prime positive integers so that the $2012^{\text{th}}$ fraction in the list is equal to $\frac{m}{n}$. Find $m+n$.

Which fractions $ \dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\sqrt{2}$ (without using any table).

Find the sum of first two integers $n > 1$ such that $3^n$ is divisible by $n$ and $3^n - 1$ is divisible by $n - 1$.

Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.

Find the smallest natural number which is a multiple of $2009$ and whose sum of (decimal) digits equals $2009$ [i]Proposed by Milos Milosavljevic[/i]

Let $a_0=29$, $b_0=1$ and $$a_{n+1} = a_n+a_{n-1}\cdot b_n^{2019}, \qquad b_{n+1}=b_nb_{n-1}$$ for $n\geq 1$. Determine the smallest positive integer $k$ for which $29$ divides $\gcd(a_k, b_k-1)$ whenever $a_1,b_1$ are positive integers and $29$ does not divide $b_1$.

Determine all integers $n\ge 2$ with the following property: the number $2^k\cdot n-1$ is prime for all $k\in\{2,3,\ldots,n\}$.

Determine all $6$-tuples $(x,y,z,u,v,w)$ of integers satisfying the equation \[x^3+7y^3+49z^3=2u^3+14v^3+98w^3.\]

Let $n$ be a natural number. We define sequences $\langle a_i\rangle$ and $\langle b_i\rangle$ of integers as follows. We let $a_0=1$ and $b_0=n$. For $i>0$, we let $$\left( a_i,b_i\right)=\begin{cases} \left(2a_{i-1}+1,b_{i-1}-a_{i-1}-1\right) & \text{if } a_{i-1}<b_{i-1},\\ \left( a_{i-1}-b_{i-1}-1,2b_{i-1}+1\right) & \text{if } a_{i-1}>b_{i-1},\\ \left(a_{i-1},b_{i-1}\right) & \text{if } a_{i-1}=b_{i-1}.\end{cases}$$ Given that $a_k=b_k$ for some natural number $k$, prove that $n+3$ is a power of two.

Is there an infinite periodic sequence of digits for which the following condition condition is fulfilled: for any natural number $n{}$ a natural number divisible by $2^n{}$ can be cut from this sequence of digits (as a word)? [i]Proposed by P. Kozhevnikov[/i]

$a, b$ are odd numbers that satisfy $(a-b)^2 \le 8\sqrt {ab}$. For $n=ab$, show that the equation $$x^2-2([\sqrt n]+1)x+n=0$$ has two integral solutions. $[r]$ denotes the biggest integer, not strictly bigger than $r$.

$p$ is a prime and $m$ is a non-negative integer $< p-1$. Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.