Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 Azerbaijan National Olympiad
Find all  polynomials $P(x)$ with real coefficents such that $$P(P(x))=(x^2+x+1)\cdot P(x)$$where $x \in \mathbb{R}$
2015 Azerbaijan National Olympiad
Natural number $M$ has $6$ divisors, such that sum of them are equal to $3500$.Find the all values of $M$.
2015 Azerbaijan National Olympiad
In the convex quadrilateral $ABCD$ angle $\angle{BAD}=90$,$\angle{BAC}=2\cdot\angle{BDC}$ and $\angle{DBA}+\angle{DCB}=180$. Then find the angle $\angle{DBA}$
2020 Azerbaijan National Olympiad
$13$ fractions are corrected by using each of the numbers $1,2,...,26$ once.Example:$\frac{12}{5},\frac{18}{26}.... $

What is the maximum number of fractions which are integers?
2020 Azerbaijan National Olympiad
$a,b,c$ are positive integer.

Solve the equation:

$ 2^{a!}+2^{b!}=c^3 $
2020 Azerbaijan National Olympiad
$a,b,c$ are positive numbers.$a+b+c=3$

Prove that:

$\sum \frac{a^2+6}{2a^2+2b^2+2c^2+2a-1}\leq 3 $
2020 Azerbaijan National Olympiad
There is a non-equilateral triangle $ABC$.Let $ABC$'s Incentri $I$.Point $D$ is on the $BC$ side.The circle drawn outside the triangle $IBD$ and $ICD$ intersects the sides $AB$ and $AC$ at points $E$ and $F.$The circle drawn outside the triangle $DEF$ intersects the sides $AB$ and $AC$ at $N$ and $M$.Prove that $EM\parallel FN $.
2020 Azerbaijan National Olympiad
$a,b,c$ are non-negative integers.

Solve: $a!+5^b=7^c$



Proposed by Serbia
2019 Bangladesh Mathematical Olympiad
Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers.
2019 Bangladesh Mathematical Olympiad
Prove that,if $a,b,c$ are positive real numbers,

$$  \dfrac{a}{bc}+ \dfrac{b}{ca}+\dfrac{c}{ab}\geq \dfrac{2}{a}+\dfrac{2}{b}-\dfrac{2}{c}$$
2019 Bangladesh Mathematical Olympiad
Let $\alpha$ and $\omega$ be two circles such that $\omega$ goes through the center of $\alpha$.$\omega$ intersects $\alpha$ at $A$ and $B$.Let $P$ any point on the circumference $\omega$.The lines $PA$ and $PB$ intersects $\alpha$ again at $E$ and $F$ respectively.Prove that $AB=EF$.
2019 Bangladesh Mathematical Olympiad
$A$ is a positive real number.$n$ is positive integer number.Find the set of possible values of the infinite sum $x_0^n+x_1^n+x_2^n+...$ where $x_0,x_1,x_2...$ are all positive real numbers so that the infinite series $x_0+x_1+x_2+...$ has sum $A$.
2019 Bangladesh Mathematical Olympiad
Prove that for all positive integers $n$ we can find a permutation of {$1,2,...,n$} such that the average of two numbers doesn't appear in-between them.For example {$1,3,2,4$}works,but {$1,4,2,3$} doesn't because $2$ is between $1$ and $3$.
2019 Bangladesh Mathematical Olympiad
When a function $f(x)$ is differentiated $n$ times ,the function we get id denoted $f^n(x)$.If $f(x)=\dfrac {e^x}{x}$.Find the value of

$$\lim_{n \to \infty} \dfrac {f^ {2n}(1)}{(2n)!}$$
2019 Bangladesh Mathematical Olympiad
Given three cocentric circles $\omega_1$,$\omega_2$,$\omega_3$ with radius $r_1,r_2,r_3$ such that $r_1+r_3\geq {2r_2}$.Constrat a line that intersects $\omega_1$,$\omega_2$,$\omega_3$ at $A,B,C$ respectively such that $AB=BC$.
2019 Bangladesh Mathematical Olympiad
The set of natural numbers $\mathbb{N}$ are partitioned into a finite number of subsets.Prove that there exists a subset of $S$ so that for any natural numbers $n$,there are infinitely many multiples of $n$ in $S$.
2019 Bangladesh Mathematical Olympiad
Let $ABCD$ is a convex quadrilateral.The internal angle bisectors of $\angle {BAC}$ and $\angle {BDC}$ meets at $P$.$\angle {APB}=\angle {CPD}$.Prove that $AB+BD=AC+CD$.
2019 Bangladesh Mathematical Olympiad
Given $2020*2020$ chessboard, what is the maximum number of warriors you can put on its cells such that no two warriors attack each other.

Warrior is a special chess piece which can move either $3$ steps  forward and one step sideward and $2$ step forward and $2$ step sideward in any direction.
2018 Bangladesh Mathematical Olympiad
Solve:



$x^2(2-x)^2=1+2(1-x)^2$



Where $x$ is real number.
2018 Bangladesh Mathematical Olympiad
BdMO National 2018 Higher Secondary P2



$AB$ is a diameter of a circle and $AD$ & $BC$ are two tangents of that circle.$AC$ & $BD$ intersect on a point of the circle.$AD=a$ & $BC=b$.If $a\neq b$ then $AB=?$
2018 Bangladesh Mathematical Olympiad
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 $S$ ?
2018 Bangladesh Mathematical Olympiad
Yukihira is counting the minimum number of lines $m$, that can be drawn on the plane so that they intersect in exactly $200$ distinct points.What is $m$?
2018 Bangladesh Mathematical Olympiad
Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $E$ . The two circles passing through $B$ meet again at $F$ . The two circles passing through $C$ meet again at  $G$. The two circles passing through $D$ meet again at $H$. Suppose, $ E, F, G,H $ are all distinct. Is the quadrilateral $EFGH$ similar to $ABCD$ ? Show with proof.
2018 Bangladesh Mathematical Olympiad
Find all the pairs of integers $(m,n)$ satisfying the equality

$3(m^2+n^2)-7(m+n)=-4$
2018 Bangladesh Mathematical Olympiad
Evaluate



$\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\ sinx\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} dx$
2018 Bangladesh Mathematical Olympiad
a tournament is playing between n persons. Everybody plays with everybody one time. There is no draw here. A number $k$ is called $n$ good if there is any tournament such that in that tournament they have any player in the tournament that has lost all of $k$'s.

prove that

1. $n$ is greater than or equal to $2^{k+1}-1$

2.Find all $n$ such that $2$ is a n-good
2016 Bangladesh Mathematical Olympiad
BdMO National $2016$ Higher Secondary



Problem 1:

(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 $7$.
2016 Bangladesh Mathematical Olympiad
Bangladesh National Mathematical Olympiad 2016 Higher Secondary



Problem 2:

(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
$\triangle ABC$ is isosceles $AB = AC$. $P$ is a point inside $\triangle ABC$ such that

$\angle BCP = 30$ and $\angle APB = 150$ and $\angle CAP = 39$. Find $\angle BAP$.
2016 Bangladesh Mathematical Olympiad
BdMO National 2016 Higher Secondary



Problem 4:

Consider the set of integers $  \left \{ 1, 2, ......... , 100 \right \} $. Let $   \left \{ x_1, x_2, ......... , x_{100} \right \}$ be some arbitrary arrangement of the integers $ \left \{ 1, 2, ......... , 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 | + ................+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | $.
2016 Bangladesh Mathematical Olympiad
Bangladesh National Mathematical Olympiad 2016 Higher Secondary



Problem 5:

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?
2016 Bangladesh Mathematical Olympiad
BdMO National 2016 Higher Secondary Problem 6





$\triangle ABC$ is an isosceles triangle with $AC = BC$ and $\angle ACB < 60^{\circ}$. $I$ and $O$ are the incenter and circumcenter of $\triangle ABC$. The circumcircle of $\triangle BIO$ intersects $BC$ at $D \neq B$.



(a) Do the lines $AC$ and $DI$ intersect? Give a proof.



(b) What is the angle of intersection between the lines $OD$ and $IB$?
2016 Bangladesh Mathematical Olympiad
Juli is a mathematician and devised an algorithm to find a husband. The strategy is:

• Start interviewing a maximum of $1000$ prospective husbands. Assign a ranking $r$ to each person that is a positive integer. No two prospects will have same the rank $r$.

• Reject the first $k$ men and let $H$ be highest rank of these $k$ men.

• After rejecting the first $k$ men, select the next prospect with a rank greater than $H$ and then stop the search immediately. If no candidate is selected after $999$ interviews, the $1000th$ person is selected.



Juli wants to find the value of $k$ for which she has the highest probability of choosing the highest ranking prospect among all $1000$ candidates without having to interview all $1000$ prospects.

(a) (6 points:) What is the probability that the highest ranking prospect among all $1000$ prospects is the $(m + 1)th$ prospect?

(b) (6 points:) Assume the highest ranking prospect is the $(m + 1)th$ person to be interviewed. What is the probability that the highest rank candidate among the first $m$ candidates is one of the first $k$ candidates who were rejected?

(c) (6 points:) What is the probability that the prospect with the highest rank is the $(m+1)th$ person and that Juli will choose the $(m+1)th$ man using this algorithm?

(d) (16 points:) The total probability that Juli will choose the highest ranking prospect among the $1000$ prospects is the sum of the probability for each possible value of $m+1$ with $m+1$ ranging between $k+1$ and $1000$.

Find the sum.  To simplify your answer use the formula

$In N \approx \frac{1}{N-1}+\frac{1}{N-2}+...+\frac{1}{2}+1$

(e) (6 points:) Find that value of $k$ that maximizes the probability of choosing the highest ranking prospect without interviewing all $1000$ candidates. You may need to know that the maximum of the function $x ln \frac{A}{x-1}$ is approximately $\frac{A + 1}{e}$, where $A$ is a constant and $e$ is Euler’s number, $e = 2.718....$
2016 Bangladesh Mathematical Olympiad
Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where m and n are relatively prime positive integers. Find $m + n$.
2016 Bangladesh Mathematical Olympiad
The integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$



(a)(3 POINTS:)Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$

Where $j$ is not a function of $x$,is $Z(j)=e^{j^{2}/4a} Z(0)$



(b)(10 POINTS):Show that,

$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n}$

Where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times...\times3\times 1$



(c)(7 POINTS):What is the number of ways to form $n$ pairs from $2n$ distinct objects?Interept the previous part of the problem in term of this answer.
2013 Bangladesh Mathematical Olympiad
Higher Secondary P1



A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon $ABCDE$, $AB=AE$, $BC=DE$, $P$ and $Q$ are midpoints of $AE$ and $AB$ respectively. $PQ||CD$, $BD$ is perpendicular to both $AB$ and $DE$. Prove that $ABCDE$ is a degenerate pentagon.
2013 Bangladesh Mathematical Olympiad
Higher Secondary P2



Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$.
2013 Bangladesh Mathematical Olympiad
Higher Secondary P3



Let $ABCDEF$ be a regular hexagon with $AB=7$. $M$ is the midpoint of $DE$. $AC$ and $BF$ intersect at $P$, $AC$ and $BM$ intersect at $Q$, $AM$ and $BF$ intersect at $R$. Find the value of $[APB]+[BQC]+[ARF]-[PQMR]$. Here $[X]$ denotes the area of polygon $X$.
2013 Bangladesh Mathematical Olympiad
Higher Secondary P4



If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$, find $\dfrac{a}{b}$.
2013 Bangladesh Mathematical Olympiad
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 $x$.
2013 Bangladesh Mathematical Olympiad
There are $n$ cities in a country. Between any two cities there is at most one road. Suppose that the total number of roads is $n.$ Prove that there is a city such that starting from there it is possible to come back to it without ever travelling the same road twice.
2013 Bangladesh Mathematical Olympiad
Higher Secondary P7



If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.
2013 Bangladesh Mathematical Olympiad
$\triangle ABC$ is an acute angled triangle. Perpendiculars drawn from its vertices on the opposite sides are $AD$, $BE$ and $CF$. The line parallel to $ DF$ through $E$ meets $BC$ at $Y$ and $BA$ at $X$. $DF$ and $CA$ meet at $Z$. Circumcircle of $XYZ$ meets $AC$ at $S$. Given, $\angle B=33 ^\circ.$ find the angle $\angle FSD $ with proof.
2013 Bangladesh Mathematical Olympiad
Six points $A, B, C, D, E, F$ are chosen on a circle anticlockwise. None of $AB, CD, EF$ is a diameter. Extended

$AB$ and $DC$ meet at $Z, CD$ and $FE$ at $X, EF$ and $BA$ at $Y.  AC$ and $BF$ meets at $P, CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R.$ If $O$ is the point of intersection of $YQ$ and $ZR,$ find the $\angle XOP.$
2013 Bangladesh Mathematical Olympiad
Higher Secondary P10



$X$ is a set of $n$ elements. $P_m(X)$ is the set of all $m$ element subsets (i.e. subsets that contain exactly $m$ elements) of $X$. Suppose $P_m(X)$ has $k$ elements. Prove that the elements of $P_m(X)$ can be ordered in a sequence $A_1, A_2,...A_i,...A_k$ such that it satisfies the two conditions:

(A) each element of $P_m(X)$ occurs exactly once in the sequence,

(B) for any $i$ such that $0<i<k$, the size of the set $A_i \cap A_{i+1}$ is $m-1$.
2011 Bangladesh Mathematical Olympiad
Higher Secondary: 2011



Time: 4 Hours



Problem 1:

Prove that for any non-negative integer $n$ the numbers $1, 2, 3, ..., 4n$ can be divided in tow mutually exclusive classes with equal number of members so that the sum of numbers of each class is equal.





Problem 2:

In the first round of a chess tournament, each player plays against every other player exactly once. A player gets $3, 1$ or $-1$ points respectively for winning, drawing or losing a match. After the end of the first round, it is found that the sum of the scores of all the players is $90$. How many players were there in the tournament?





Problem 3:

$E$ is the midpoint of side $BC$ of rectangle $ABCD$. $A$ point $X$ is chosen on $BE$. $DX$ meets extended $AB$ at $P$. Find the position of $X$ so that the sum of the areas of  $\triangle BPX$ and  $\triangle DXC$ is maximum with proof.





Problem 4:

Which one is larger 2011! or, $(1006)^{2011}$? Justify your answer.





Problem 5:

In a scalene triangle $ABC$ with $\angle A = 90^{\circ}$, the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R$. The lines $RS$ and $BC$ intersect at $N$ while the lines $AM$ and $SR$ intersect at $U$. Prove that the triangle $UMN$ is isosceles.





Problem 6:

$p$ is a prime and sum of the numbers from $1$ to $p$ is divisible by all primes less or equal to $p$. Find the value of $p$ with proof.





Problem 7:

Consider a group of $n > 1$ people. Any two people of this group are related by mutual friendship or mutual enmity. Any friend of a friend and any enemy of an enemy is a friend. If $A$ and $B$ are friends/enemies then we count it as $1$ friendship/enmity. It is observed that the  number of friendships and number of enmities are equal in the group. Find all possible values of $n$.





Problem 8:

$ABC$ is a right angled triangle with $\angle A = 90^{\circ}$ and $D$ be the midpoint of $BC$. A point $F$ is chosen on $AB$. $CA$ and $DF$ meet at $G$ and $GB \parallel AD$. $CF$ and $AD$ meet at $O$ and $AF = FO$. $GO$ meets $BC$ at $R$. Find the sides of  $ABC$ if the area of $GDR$ is $\dfrac{2}{\sqrt{15}}$





Problem 9:

The repeat of a natural number is obtained by writing it twice in a row (for example, the repeat of $123$ is $123123$). Find a positive integer (if any) whose repeat is a perfect square.





Problem 10:

Consider a square grid with $n$ rows and $n$ columns, where $n$ is odd (similar to a chessboard). Among the $n^2$ squares of the grid, $p$ are black and the others are white. The number of black squares is maximized while their arrangement is such that horizontally, vertically or diagonally neighboring black squares are separated by at least one white square between them. Show that there are infinitely many triplets of integers $(p, q, n)$ so that the number of white squares is $q^2$.





The problems of the Junior categories are available in BdMO Online forum:
2015 Bangladesh Mathematical Olympiad
BdMO National 2015 Secondary Problem 1.



A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement:

Witness One    said  exactly one  of the four witnesses is a liar.



Witness  Two  said  exactly  two  of the four witnesses is a liar.



Witness Three    said  exactly three of the four witnesses is a liar.



Witness  Four  said  exactly four  of the four witnesses is a liar.



Assume that each of the statements is either true or false.How many of the winesses are liars?
2015 Bangladesh Mathematical Olympiad
BdMO National Higher Secondary Problem 3



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
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:$100$
2015 Bangladesh Mathematical Olympiad
There are $36$ participants at a BdMO event. Some of the participants shook hands with each other. But no two participants shook hands with each other more than once. Each participant recorded the number of handshakes they made. It was found that no two participants with the same number of handshakes made, had shaken hands with each other. Find the maximum possible number of handshakes at the party with proof. (When two participants shake hands with each other, this will be counted as one handshake.)
2015 Bangladesh Mathematical Olympiad
A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.
2015 Bangladesh Mathematical Olympiad
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 $m+n$?
2015 Bangladesh Mathematical Olympiad
In triangle $\triangle ABC$, the points $A', B', C'$ are on sides $BC, AC, AB$ respectively. Also, $AA', BB', CC'$ intersect at the point $O$(they are concurrent at $O$). Also, $\frac {AO}{OA'}+\frac {BO}{OB'}+\frac {CO}{OC'} = 92$. Find the value of $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$.
Bangladesh Mathematical Olympiad 2020 Final
A pair of positive integers $(m,n)$ is called 'steakmaker' if they maintain the equation 1 + 2$^m$ = n$^2$. For which values of m and n, the pair $(m,n)$ are steakmaker, find the sum of $mn$
Bangladesh Mathematical Olympiad 2020 Final
Consider rectangle $ABCD$.$ E$ is the mid-point of $AD$ and $F$ is the mid-point of $ED$. $CE$ cuts $AB$ in $G$ and $BF$ cuts $CD$ in $H$ point. We can write ratio of areas of $BCG$ and $BCH$ triangles as $\frac{m}{n}$. Find the value of $10m + 10n + mn$.
Bangladesh Mathematical Olympiad 2020 Final
Prottasha has a 10 sided dice. She throws the dice two times and sum the numbers she gets. Which number has the most probability to come out?
Bangladesh Mathematical Olympiad 2020 Final
Once in a restaurant Dr. Strange found out that there were 12 types of food items from 1 to 12 on the menu. He decided to visit the restaurant 12 days in a row and try a different food everyday. 1st day, he tries one of the items from the first two. On the 2nd day, he eats either item 3 or the item he didn’t tried on the 1st day. Similarly, on the 3rd day, he eats either item 4 or the item he didn’t tried on the 2nd day. If someday he's not able to choose items that way, he eats the item that remained uneaten from the menu. In how many ways can he eat the items for 12 days?
Bangladesh Mathematical Olympiad 2020 Final
For a positive real number $  [x] $ be its integer part. For example,  $[2.711] = 2, [7] = 7, [6.9] = 6$. $z$ is the maximum real number such that [$\frac{5}{z}$] + [$\frac{6}{z}$] = 7. Find the value of$ 20z$.
Bangladesh Mathematical Olympiad 2020 Final
Point $P$ is taken inside the square $ABCD$ such that $BP + DP=25$, $CP - AP = 15$ and $\angle$ABP = $\angle$ADP. What is the radius of the circumcircle of $ABCD$?
Bangladesh Mathematical Olympiad 2020 Final
Tiham is trying to find 6 digit positive integers$ PQRSTU$ (where $PQRSTU $are not necessarily distinct). But he only wants the numbers where the sum of the 3 digit number$ PQR$, and the 3 digit number $STU$ is divisible by 37. How many such numbers Tiham can find?
Bangladesh Mathematical Olympiad 2020 Final
Let $ABC$ be a triangle where$\angle$B=55 and $\angle$ C = 65. D is the mid-point of BC. Circumcircle of ACD and ABD cuts AB and AC at point F and E respectively. Center of circumcircle of AEF is O. $\angle$FDO = ?
Bangladesh Mathematical Olympiad 2020 Final
You have 2020 piles of coins in front Of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins and so on. So, the 2020th  pile contains 2020 coins. Guess a positive integer k, in which piles contain at least k coins, take away exact k coins from these piles. Find the minimum number of turns you need to take way all of these coins?
Bangladesh Mathematical Olympiad 2020 Final
Sokal da tries to find out the largest positive integer n such that if n transforms to base-7, then it looks like twice of base-10. $156$ is such a number because $(156)_{10}$ = $(312)_7$ and 312 = 2$\times$156. Find out Sokal da's number.
Bangladesh Mathematical Olympiad 2020 Final
A prime number$ q $is called 'Kowai' number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one 'Kowai' number can be found. Find the summation of all 'Kowai' numbers.
Bangladesh Mathematical Olympiad 2020 Final
$2^{2921}$ has $581$ digits and starts with a $4$. How many $2^n$'s starts with a $4$, where $0$ is the last digit?
1995 Belarus National Olympiad
Mark six points in a plane so that any three of them are vertices of a nondegenerate isosceles triangle.
1995 Belarus National Olympiad
Find all positive integers $n$ so that both $n$ and $n + 100$ have odd numbers of divisors.