Olympiad problems

2016 Bangladesh Mathematical Olympiad, 4

Consider the set of integers $ \left \{ 1, 2, \dots , 100 \right \} $. Let $ \left \{ x_1, x_2, \dots , x_{100} \right \}$ be some arbitrary arrangement of the integers $ \left \{ 1, 2, \dots , 100 \right \}$, where all of the $x_i$ are different. Find the smallest possible value of the sum $$S = \left | x_2 - x_1 \right | + \left | x_3 - x_2 \right | + \cdots+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | .$$

2018 Bangladesh Mathematical Olympiad, 3

Tags: number theory , counting , Bdmo , Dice

BdMO National 2018 Higher Secondary P3 Nazia rolls four fair six-sided dice. She doesn’t see the results. Her friend Faria tells her that the product of the numbers is $144$. Faria also says the sum of the dice, $S$ satisfies $14\leq S\leq 18$ . Nazia tells Faria that $S$ cannot be one of the numbers in the set {$14,15,16,17,18$} if the product is $144$. Which number in the range {$14,15,16,17,18$} is an impossible value for $S$ ?

2013 Bangladesh Mathematical Olympiad, 5

Tags: number theory , Bdmo

Higher Secondary P5 Let $x>1$ be an integer such that for any two positive integers $a$ and $b$, if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$. Find with proof the number of positive integers that divide $x$.

2016 Bangladesh Mathematical Olympiad, 2

Tags: number theory , modular arithmetic , Divisibility , Bdmo , contests

(a) How many positive integer factors does $6000$ have? (b) How many positive integer factors of $6000$ are not perfect squares?

2016 Bangladesh Mathematical Olympiad, 1

Tags: Bdmo , contests , contest problem , number theory , modular arithmetic

(a) Show that $n(n + 1)(n + 2)$ is divisible by $6$. (b) Show that $1^{2015} + 2^{2015} + 3^{2015} + 4^{2015} + 5^{2015} + 6^{2015}$ is divisible by $7$.

2015 Bangladesh Mathematical Olympiad, 2

Tags: number theory , Bdmo , contest problem , Contest Preparation , Integer , solutions

[b][u]BdMO National Higher Secondary Problem 3[/u][/b] Let $N$ be the number if pairs of integers $(m,n)$ that satisfies the equation $m^2+n^2=m^3$ Is $N$ finite or infinite?If $N$ is finite,what is its value?

2015 Bangladesh Mathematical Olympiad, 3

Tags: number theory , algebra , Bdmo , contests , 2015 Contests , modular arithmetic

Let $n$ be a positive integer.Consider the polynomial $p(x)=x^2+x+1$. What is the remainder of $ x^3$ when divided by $x^2+x+1$.For what positive integers values of $n$ is $ x^{2n}+x^n+1$ divisible by $p(x)$? Post no:[size=300]$100$[/size]

2015 Bangladesh Mathematical Olympiad, 6

Tags: geometry , contests , Bdmo , trapezium , angles

Trapezoid $ABCD$ has sides $AB=92,BC=50,CD=19,AD=70$ $AB$ is parallel to $CD$ A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$.Given that $AP=\dfrac mn$ (Where $m,n$ are relatively prime).What is $m+n$?

2016 Bangladesh Mathematical Olympiad, 5

Tags: Bdmo , combinatorics , permutations , circle , contests

Suppose there are $m$ Martians and $n$ Earthlings at an intergalactic peace conference. To ensure the Martians stay peaceful at the conference, we must make sure that no two Martians sit together, such that between any two Martians there is always at least one Earthling. (a) Suppose all $m + n$ Martians and Earthlings are seated in a line. How many ways can the Earthlings and Martians be seated in a line? (b) Suppose now that the $m+n$ Martians and Earthlings are seated around a circular round-table. How many ways can the Earthlings and Martians be seated around the round-table?