There are 2023 natural written in a row. The first number is 12, and each number starting from the third is equal to the product of the previous two numbers, or to the previous number increased by 4. What is the largest number of perfect squares that can be among the 2023 numbers? [i]Based on the British Mathematical Olympiad[/i]

Points on a square with side length $ c$ are either painted blue or red. Find the smallest possible value of $ c$ such that how the points are painted, there exist two points with same color having a distance not less than $ \sqrt {5}$. $\textbf{(A)}\ \frac {\sqrt {10} }{2} \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \sqrt {5} \qquad\textbf{(D)}\ 2\sqrt {2} \qquad\textbf{(E)}\ \text{None}$

On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$, it is possible to state that there is at least one rook in each $k\times k$ square ?

Find all the pairs positive integers $(x, y)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2}$ , where $(x, y)$ is the greatest common divisor of $x, y$ and $[x, y]$ is the least common multiple of $x, y$.

A recipe that makes $ 5$ servings of hot chocolate requires $ 2$ squares of chocolate, $ \frac{1}{4}$ cup sugar, $ 1$ cup water and $ 4$ cups milk. Jordan has $ 5$ squares of chocolate, $ 2$ cups of sugar, lots of water and $ 7$ cups of milk. If she maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate she can make? $ \textbf{(A)}\ 5 \frac18 \qquad \textbf{(B)}\ 6\frac14 \qquad \textbf{(C)}\ 7\frac12 \qquad \textbf{(D)}\ 8 \frac34 \qquad \textbf{(E)}\ 9\frac78$

Positive integers are colored in black and white. We know that the sum of two numbers of different colors is always black, and that there are infinitely many numbers that are white. Prove that the sum and product of two white numbers are also white numbers.

Let $a$ and $b$ be relatively prime positive integers such that \[\dfrac ab=\dfrac1{2^1}+\dfrac2{3^2}+\dfrac3{2^3}+\dfrac4{3^4}+\dfrac5{2^5}+\dfrac6{3^6}+\cdots,\] where the numerators always increase by $1$, and the denominators alternate between powers of $2$ and $3$, with exponents also increasing by $1$ for each subsequent term. Compute $a+b$.

In $\triangle ABC$, $c-a$ is equal to height on side $AC$. Then, the value of $\sin\frac{C-A}{2}+\cos\frac{C+A}{2}$ is $\text{(A)}1\qquad\text{(B)}\frac{1}{2}\qquad\text{(C)}\frac{1}{3}\qquad\text{(D)}-1$

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?

Prove that for all integers $ a > 1$ and $ b > 1$ there exists a function $ f$ from the positive integers to the positive integers such that $ f(a\cdot f(n)) \equal{} b\cdot n$ for all $ n$ positive integer.

In a triangle $ABC$, let $H$ be the point of intersection of altitudes, $I$ the center of incircle, $O$ the center of circumcircle, $K$ the point where the incircle touches $BC$. Given that $IO$ is parallel to $BC$, prove that $AO$ is parallel to $HK$.

The increasing sequence $2,3,5,6,7,10,11,\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence.

Let circles $ \Gamma $ and $ \omega $ are circumcircle and incircle of the triangle $ABC$, the incircle touches sides $BC,CA,AB$ at the points $A_1,B_1,C_1$. Let $A_2$ and $B_2$ lies the lines $A_1I$ and $B_1I$ ($A_1$ and $A_2$ lies different sides from $I$, $B_1$ and $B_2$ lies different sides from $I$) such that $IA_2=IB_2=R$. Prove that : (a) $AA_2=BB_2=IO$; (b) The lines $AA_2$ and $BB_2$ intersect on the circle $ \Gamma ;$

Opposite sides of a regular hexagon are $12$ inches apart. The length of each side, in inches, is $\textbf{(A) }7.5\qquad\textbf{(B) }6\sqrt{2}\qquad\textbf{(C) }5\sqrt{2}\qquad\textbf{(D) }\frac{9}{2}\sqrt{3}\qquad \textbf{(E) }4\sqrt{3}$

If two points are picked randomly on the perimeter of a square, what is the probability that the distance between those points is less than the side length of the square?

We consider the following triangular array \[ \begin{array}{cccccccc} 0 & 1 & 1 & 2 & 3 & 5 & 8 & \ldots \\ \ & 0 & 1 & 1 & 2 & 3 & 5 & \ldots \\ \ & \ & 2 & 3 & 5 & 8 & 13 & \ldots \\ \ & \ & \ & 4 & 7 & 11 & 18 & \ldots \\ \ & \ & \ & \ & 12 & 19 & 31 & \ldots \\ \end{array} \] which is defined by the conditions i) on the first two lines, each element, starting with the third one, is the sum of the preceding two elements; ii) on the other lines each element is the sum of the two numbers found on the same column above it. a) Prove that all the lines satisfy the first condition i); b) Let $a,b,c,d$ be the first elements of 4 consecutive lines in the array. Find $d$ as a function of $a,b,c$.

Assume $A,B,C$ are three collinear points that $B \in [AC]$. Suppose $AA'$ and $BB'$ are to parrallel lines that $A'$, $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$, $A'$ and $C$. Also $O_2$ is circumcenter of circle passing through $B$, $B'$ and $C$. If area of $A'CB'$ is equal to area of $O_1CO_2$, then find all possible values for $\angle CAA'$

A simple polygon in the plane is a figure bounded by a closed nonself-intersecting broken line. (a) Do there exist two congruent simple $7$-gons in the plane such that all the seven vertices of one of the $7$-gons are the vertices of the other one and yet these two $7$-gons have no common sides? (b) Do there exist three such $7$-gons? (V Proizvolov)

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 $2^n(6 - n) = 8n$.

Given three parallel straight lines. Construct a square three of whose vertices belong to these lines.

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

Some cells of a checkered plane are marked so that figure $A$ formed by marked cells satisfies the following condition:$1)$ any cell of the figure $A$ has exactly two adjacent cells of $A$; and $2)$ the figure $A$ can be divided into isosceles trapezoids of area $2$ with vertices at the grid nodes (and acute angles of trapezoids are equal to $45$) . Prove that the number of marked cells is divisible by $8$.

Four whole numbers, when added three at a time, give the sums $180$, $197$, $208$, and $222$. What is the largest of the four numbers? $\text{(A)} \ 77 \qquad \text{(B)} \ 83 \qquad \text{(C)} \ 89 \qquad \text{(D)} \ 95 \qquad \text{(E)} \ \text{cannot be determined}$

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]