There are $n$ light bulbs in a circle and one of them is marked. Let operation $A$: Take a positive divisor $d$ of the number $n,$ starting with the light bulb marked and clockwise, we count around the circumference from $1$ to $dn$, changing the state (on or off) to those light bulbs that correspond to the multiples of $d$. Let operation $B$ be: Apply operation$ A$ to all positive divisors of $n$ (to the first divider that is applied is with all the light bulbs off and the remaining divisors is with the state resulting from the previous divisor). Determine all the positive integers $n$, such that when applying the operation on $B$, all the light bulbs are on.

Let $\displaystyle a,b,c,d \in \mathbb R$ such that $\displaystyle a>c>d>b>1$ and $\displaystyle ab>cd$. Prove that $\displaystyle f : \left[ 0,\infty \right) \to \mathbb R$, defined through \[ \displaystyle f(x) = a^x+b^x-c^x-d^x, \, \forall x \geq 0 , \] is strictly increasing.

Some checkers placed on an $n \times n$ checkerboard satisfy the following conditions: (a) every square that does not contain a checker shares a side with one that does; (b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side. Prove that at least $(n^{2}-2)/3$ checkers have been placed on the board.

$x$, $y$ and $z$ are real numbers where the sum of any two among them is not $1$. Show that, \[ \dfrac{(x^2+y)(x+y^2)}{(x+y-1)^2}+\dfrac{(y^2+z)(y+z^2)}{(y+z-1)^2} + \dfrac{(z^2+x)(z+x^2)}{(z+x-1)^2} \ge 2(x+y+z) - \dfrac{3}{4}\]Find all triples $(x,y,z)$ of real numbers satisfying the equality case.

Byan is playing raven, raven, falcon with his three friends. His friends sit down in a circle, and Byan repeatedly walks clockwise around them, tapping each friend he passes on the head and then saying `raven' or `falcon', each with probability $\frac{1}{2}$. The game ends after Byan has said `falcon' twice. The probability one of his friends will be called a falcon twice can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m+n$. [i]Individual #6[/i]

On a semicircle with unit radius four consecutive chords $AB,BC, CD,DE$ with lengths $a, b, c, d$, respectively, are given. Prove that \[a^2 + b^2 + c^2 + d^2 + abc + bcd < 4.\]

$n$ is natural number and $p$ is prime number. If $1+np$ is square of natural number then prove that $n+1$ is equal to some sum of $p$ square of natural numbers.

$E,F$ are points on $AB,AC$ that satisfies $(B,E,F,C)$ cyclic. $D$ is the intersection of $BC$ and the perpendicular bisecter of $EF$, and $B',C'$ are the reflections of $B,C$ on $AD$. $X$ is a point on the circumcircle of $\triangle{BEC'}$ that $AB$ is perpendicular to $BX$,and $Y$ is a point on the circumcircle of $\triangle{CFB'}$ that $AC$ is perpendicular to $CY$. Show that $DX=DY$.

A cyclic quadrilateral $ABCD$ has diameter $AC$ with circumcircle $\omega$. Let $K$ be the foot of the perpendicular from $C$ to $BD$, and the tangent to $\omega$ at $A$ meets $BD$ at $T$. Let the line $AK$ meets $\omega$ at $X$ and choose a point $Y$ on line $AK$ such that $\angle TYA=90^{\circ}$. Prove that $AY=KX$. [i]Proposed by Anzo Teh Zhao Yang[/i]

We have squares $ABCD$ and $EFGH$. Square $ABCD$ has points with coordinates $A=(1,1,-1)$, $B=(1,-1,-1)$, $C=(-1,-1,-1)$ and $D=(-1,1,-1)$. Square $EFGH$ has points with coordinates $E=(\sqrt{2},0,1)$, $F=(0,-\sqrt{2},1)$, $G=(-\sqrt{2},0,1)$, and $H=(0,\sqrt{2},1)$. Consider the solid formed by joining point $A$ to $H$ and $E$, point $B$ to $E$ and $F$, point $C$ to $F$ and $G$, and point $D$ to $G$ and $H$. Compute the volume of this solid.

International Mathematical Olympiad National Selection Test Malaysia 2021 Round 1 Primary 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. Faris has six cubes on his table. The cubes have a total volume of $2021$ cm$^3$. Five of the cubes have side lengths $5$ cm, $5$ cm, $6$ cm, $6$ cm, and $11$ cm. What is the side length of the sixth cube (in cm)? p2. What is the sum of the first $200$ even positive integers? p3. Anushri writes down five positive integers on a paper. The numbers are all different, and are all smaller than $10$. If we add any two of the numbers on the paper, then the result is never $10$. What is the number that Anushri writes down for certain? p4. If the time now is $10.00$ AM, what is the time $1,000$ hours from now? Note: Enter the answer in a $12$-hour system, without minutes and AM/PM. For example, if the answer is $9.00$ PM, just enter $9$. p5. Aminah owns a car worth $10,000$ RM. She sells it to Neesha at a $10\%$ profit. Neesha sells the car back to Aminah at a $10\%$ loss. How much money did Aminah make from the two transactions, in RM? [b]Part B[/b] (2 points each) p6. Alvin takes 250 small cubes of side length $1$ cm and glues them together to make a cuboid of size $5$ cm  $\times 5$ cm  $\times 10$ cm. He paints all the faces of the large cuboid with the color green. How many of the small cubes are painted by Alvin? p7. Cikgu Emma and Cikgu Tan select one integer each (the integers do not have to be positive). The product of the two integers they selected is $2021$. How many possible integers could have been selected by Cikgu Emma? p8. A three-digit number is called [i]superb[/i] if the first digit is equal to the sum of the other two digits. For example, $431$ and $909$ are superb numbers. How many superb numbers are there? p9. Given positive integers $a, b, c$, and $d$ that satisfy the equation $4a = 5b =6c = 7d$. What is the smallest possible value of $ b$? p10. Find the smallest positive integer n such that the digit sum of n is divisible by $5$, and the digit sum of $n + 1$ is also divisible by $5$. Note: The digit sum of $1440$ is $1 + 4 + 4 + 0 = 9$. [b]Part C[/b] (3 points each) p11. 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? p12. 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? p13. Clarissa opens a pet shop that sells three types of pets: goldshes, hamsters, and parrots. The pets inside the shop together have a total of $14$ wings, $24$ heads, and $62$ legs. How many goldshes are there inside Clarissa's shop? p14. A positive integer $n$ is called [i]special [/i] 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? p15. 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 D[/b] (4 points each) p16. 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? p17. 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. p18. Haz 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 $k$? p19. What is the smallest positive multiple of $24$ that can be written using digits $4$ and $5$ only? p20. 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? PS. Problems 11-20 were also used in [url=https://artofproblemsolving.com/community/c4h2676837p23203256]Juniors [/url]as 1-10.

Two circles touch internally at point $A$. A chord $ BC $ of the larger circle is drawn tangent to the smaller one at point $ D $. Prove that $ AD $ is the bisector of angle $ BAC $.

In the figure below $\triangle ABC$, $\triangle DEF$, and $\triangle GHI$ are overlapping equilateral triangles, $C$ and $F$ lie on $\overline{BD}$, $F$ and $I$ lie on $\overline{EG}$, and $C$ and $I$ lie on $\overline{AH}$. Length $AB = 2FC$, $DE = 3FC$, and $GH = 4FC$. Given that the area of $\triangle FCI$ is $3$, find the area of the hexagon $ABGHDE$. [asy] size(5cm); pen dps = fontsize(10); defaultpen(dps); pair A,B,C,D,E,F,G,H,I; G=origin; H=(4,0); I=(2,2*sqrt(3)); F=(1.5,3*sqrt(3)/2); C=F+(1,0); B=F-(1,0); D=C+(2,0); A=F+(0,sqrt(3)); E=C+(0.5,3*sqrt(3)/2); draw(A--H--G--E--D--B--cycle); label("$A$",A,N*.5); label("$B$",B,S*.5); label("$C$",C,SW*.5); label("$D$",D,S*.5); label("$E$",E,N*.5); label("$F$",F,SE*.5); label("$G$",G,S*.5); label("$H$",H,S*.5); label("$I$",I,N*2); [/asy]

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$, such that $f(x+y) = f(y) f(x f(y))$ for every two real numbers $x$ and $y$.

If $ a\equal{}\minus{}2$, the largest number in the set $ \left \{ \minus{}3a,4a,\frac{24}{a},a^2,1 \right \}$ is \[ \textbf{(A)}\ \minus{}3a \qquad \textbf{(B)}\ 4a \qquad \textbf{(C)}\ \frac{24}{a} \qquad \textbf{(D)}\ a^2 \qquad \textbf{(E)}\ 1 \]

Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that (a) $f(1)=1$ (b) $f(n+2)+(n^2+4n+3)f(n)=(2n+5)f(n+1)$ for all $n \in \mathbb{N}$. (c) $f(n)$ divides $f(m)$ if $m>n$.

The graph of the degree $2021$ polynomial $P(x)$, which has real coefficients and leading coefficient 1, meets the x-axis at the points $(1, 0)$, $(2, 0)$, $(3, 0)$ , $...$ , $(2020, 0)$ and nowhere else. The mean of all possible values of $P(2021)$ can be written in the form $a!/b$, where $a$ and $b$ are positive integers and $a$ is as small as possible. Compute $a + b$.

There are $n!$ empty baskets in a row, labelled $1, 2, . . . , n!$. Caesar first puts a stone in every basket. Caesar then puts 2 stones in every second basket. Caesar continues similarly until he has put $n$ stones into every nth basket. In other words, for each $i = 1, 2, . . . , n,$ Caesar puts $i$ stones into the baskets labelled $i, 2i, 3i, . . . , n!.$ Let $x_i$ be the number of stones in basket $i$ after all these steps. Show that $n! \cdot n^2 \leq \sum_{i=1}^{n!} x_i^2 \leq n! \cdot n^2 \cdot \sum_{i=1}^{n} \frac{1}{i} $

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

Let the incenter and the $A$-excenter of $\triangle ABC$ be $I$ and $I_A$, respectively. Let $BI$ intersect $AC$ at $E$ and $CI$ intersect $AB$ at $F$. Suppose that the reflections of $I$ with respect to $EF$, $FI_A$, $EI_A$ are $X$, $Y$, $Z$, respectively. Show that $\odot(XYZ)$ and $\odot(ABC)$ are tangent to each other.

On the side $BC$ of a triangle $ABC$ the midpoint $M$ and arbitrary point $K$ is marked. Lines that pass through $K$ parallel to the sides of the triangle intersect the line $AM$ at $L$ and $N$. Prove that $ML=MN$.

In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points. A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on. Find the minimum value of a [i]good[/i] number $m$.

Is it possible to put the numbers $0,1$ or $2$ in the unit squares of the cross-lined paper $100\times 100$ in such a way, that every rectangle $3\times 4$ (and $4\times 3$) would contain three zeros, four ones and five twos?

[b](a)[/b] Prove that for all positive integers $m,n$ we have \[\sum_{k=1}^n k(k+1)(k+2)\cdots (k+m-1)=\frac{n(n+1)(n+2) \cdots (n+m)}{m+1}\] [b](b)[/b] Let $P(x)$ be a polynomial with rational coefficients and degree $m.$ If $n$ tends to infinity, then prove that \[\frac{\sum_{k=1}^n P(k)}{n^{m+1}}\] Has a limit.

Prove that if $c > \dfrac{8}{3}$, then there exists a real number $\theta$ such that $\lfloor\theta^{c^n}\rfloor$ is prime for every positive integer $n$.