## BdMO National Secondary:Problem Collection

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

### BdMO National Secondary:Problem Collection

Problem 1
Solve for ($x,y$) in real number where $x^x=y$ and $y^y=y.$
viewtopic.php?f=13&t=4186

Problem 2
Writing down all the integers from $1$ to $100$ we make a large integer $N$.
$N=123456789101112...9899100$.
What will be the remainder if we divide $N$ by $3$?
viewtopic.php?f=13&t=5616

Problem 3
A square has sides of length $2$. Let $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent side of the square. Say $L$ is the set of the midpoints of all segments in $S$. Find out the area enclosed by $L$.
viewtopic.php?f=13&t=592

Problem 4
Two parallel chords of a circle have length $10$ and $14$. The distance between them is $6$. The chord parallel to these chords and half way between them has length $\sqrt a$. Find $a$.
viewtopic.php?f=13&t=591

Problem 5
A ball is thrown upward vertically to a height of $650$ meters from ground.Each $2$ times it hits the ground it bounces $\dfrac 25$ of the height it fell in the previous stage.How much the ball will travel before it stops?
viewtopic.php?f=13&t=5617

Problem 6
What is the area bounded by the region $|x+y|+|x-y|=4$ ?Where $x$ and $y$ are real numbers.
viewtopic.php?f=13&t=5614

Problem 7
Find the smallest positive integer $n>1$, such that $\sqrt{1+2+3+...+n}$ is an integer.$(n<10)$.
viewtopic.php?f=13&t=5613

Problem 8
If $m +12 = p^a$ and $m -12 = p^b$ where $a,b,m$ are integers and $p$ is a prime number. Find all possible primes $p > 0$ . [Note: $p$ only takes three values]
viewtopic.php?f=13&t=597

Problem 9
If $x^2+3x-4$ is a factor of $x^3+bx^2+cx+11$,then find the values of $b$ and $c$.
viewtopic.php?f=13&t=5615

Problem 10
A drawer in a darkened room contains $100$ black socks, $80$ blue socks, $60$ red socks and $40$ purple socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least $10$ pairs? [A pair of socks = $2$ socks of the same color.]
viewtopic.php?f=13&t=5610

Problem 11
Find the area of the largest square inscribed in a triangle of sides $5,6$ and $7$.
viewtopic.php?f=13&t=5611

Problem 12
Find the remainder on dividing ($x^{100}-2x^{51}+1$) by ($x^2-1$)?
viewtopic.php?f=13&t=4187

Problem 13
Prove that if $a$ and $b$ are two integers,then $a\times b=LCM(a,b)\times GCD(a,b)$ .
viewtopic.php?f=13&t=5612

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO National Secondary:Problem Collection

Problem 1:
The Sum of the first $2008$ odd positive integer is subtracted from the sum of the first $2008$ even positive integers. Find the result.
viewtopic.php?f=13&t=599

Problem 2:
One coin is labeled with the number $1$, two different coins are labeled with the number $2$, three different coins are labeled with the number $3$,...,forty-nine different coins are labeled with the number $49$, and ffty different coins are labeled with the number $50$. All of these coins are then put into a black bag. The coins are then randomly drawn one by one. We need $10$ coins of any type. What is the minimum number of coins that must be drawn to make sure that we have at least $10$ coins of one type?
viewtopic.php?f=13&t=600

Problem 3:
Let $a$ be an integer. The number $m$ which has the form $m = 4a + 3$ is a multiple of $11$. If we divide $a^4$ By $11$, what is the remainder? Show with proof.
viewtopic.php?f=13&t=601

Problem 4:
The function $f(x)$ is a complicated nonlinear function. It satisfies, $f(x) + f(1-x) = 1$. Evaluate $\int_{0}^{1} f(x)dx$
viewtopic.php?f=13&t=602

Problem 5:
Asmaa, and her brother Ahmed are chess players. Asmaa's son Shamim and her daughter Sharmeen are also chess players. The worst player's twin (who is one of the $4$ chess players) and best player are of the opposite sex. The worst player and the best player are the same age. Who is the worst player?
viewtopic.php?f=13&t=603

Problem 6:
The three numbers $1,2,3$ are used to make a $5$ digit number. The five digit number must contain at least one $1$, at least one $2$, and at least one $3$. How many such five digit numbers can be made? (Hint: First count the number of words missing either a $1$ or a $2$ or a $3$.)
viewtopic.php?f=13&t=604

Problem 7:
We want to find all integer solutions $(m, n)$ to $1+ 5\cdot 2^m = n^2$ .
(A) Find an expression for $n^2 -1$ ;
(B) are $(n +1)$ and $(n -1)$ both even, or both odd, or is one even and the other odd?
(C) Let $a=\frac {n-1}{2}$, Find an expression for $a(a +1)$
(D) If $a$ is odd, is $a +1$ even or odd?
(E) From parts (C) and (D), is it possible for $a = 1$, or $a(a +1) = ?$
(F) Find the only possible values $a$ can take and then find what $m$ and $n$ should be.
viewtopic.php?f=13&t=605

Problem 8:
$ABCD$ is a cyclic quadrilateral. The diagonals $AC$ and $BD$ intersect at $E$.$AB = 39; AE = 45; AD = 60; BC = 56$. Find the length of $CD$.
viewtopic.php?f=13&t=606

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO National Secondary:Problem Collection

Problem 1:
$300$ politicians are sitting in a room. Each one is corrupted or honest. At least one is honest. Given any two politicians, at least one is corrupt. How many are corrupted and how many are honest?
viewtopic.php?f=13&t=609

Problem 2:
Find all integral solutions of the equation: $\frac{x^2}{2}+\frac{5}{y}=7$
viewtopic.php?f=13&t=610

Problem 3:
Triangle $ABC$ is acute with the property that the bisector of $\angle BAC$ and the altitude from $B$ to side $AC$ and the perpendicular bisector of $AB$ intersect at one point. Determine the angle $\angle BAC$.
viewtopic.php?f=13&t=611

Problem 4:
Triangle $ABC$ is acute and $M$ is its circumcenter. Determine what point $P$ inside the triangle satisfy $1\le \frac {\angle APB}{\angle ACB} \le 2,\ 1\le \frac{\angle BPC}{\angle BAC}\le 2,\ 1\le \frac {\angle CPA}{\angle CBA} \le 2$
viewtopic.php?f=13&t=612

Problem 5:
In triangle $ABC,\ \angle A = 90\circ$. $M$ is the midpoint of $BC$. Choose $D$ on $AC$ such that $AD=AM$. The circumcircles of triangles $AMC$ and $BDC$ intersect at $C$ and at a point $P$. What is the ratio: $\frac {\angle ACB}{\angle PCB}=?$
viewtopic.php?f=13&t=613

Problem 6:
Forty MOVers (Mathematical Olympiad Volunteers) are sitting in a circle. Munir Hasan randomly chooses $3$ volunteers to help in the awards ceremony. In how many ways can the volunteers be chosen such that at least $2$ of the volunteers were sitting next to each before being chosen?
viewtopic.php?f=13&t=614

Problem 7:
How many positive prime numbers can be written as an alternating sequence of $1$'s and $0$'s where the first and last digit is $1$?
An alternating sequence of $1$'s and $0$'s is for example: $N = 1010101$ and has the property that $99N = 99999999$.
viewtopic.php?f=13&t=615

Problem 8:
The region $A$ is bounded by the $x$-axis, the line $y=\frac {x}{2}$ and the ellipse $\frac {x^2}{9}+y^2=1$. The region $B$ is bounded by the $y$-axis, the line $y = mx$ and the ellipse $y=\frac {x}{2}$ and the ellipse $\frac {x^2}{9}+y^2=1$. Find $m$ such that area of region $A$ is the equal to the area of region $B$.
viewtopic.php?f=13&t=616

Problem 9:
Each square of an $n×n$ chessboard is either red or green. The board is colored such that in any $2×2$ block of adjacent squares there are exactly $2$ green squares and $2$ red squares. How many ways can the chessboard be colored in this way? Note the number of ways for a $2×2$ chessboard is $6$ and the number of ways for a $3×3$ chessboard is $14$ which is bigger than $2^3$.
viewtopic.php?f=13&t=617

Problem 10:
$H$ is the orthocenter of acute triangle $ABC$. The triangle is inscribed in a circle with center $K$ with radius $R = 1$. Let $D$ is the intersection of the lines passing through $HK$ and $BC$. Also, $DK\cdot (DK - DH) = 1$. Find the area of the region $ABHC$.
viewtopic.php?f=13&t=618

Problem 11:
Find $S$ where
$= \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\dfrac{m^2n}{{3^m({n{3^m}}+{m3^n})}}$

viewtopic.php?f=13&t=619

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO National Secondary:Problem Collection

Problem 1:
There are $20$ people in a party excluding you.It is known that you know the same number of people as you don't know.How many of them do you know?
viewtopic.php?f=13&t=5604

Problem 2:
Isosceles triangle $ABC$ is right angled at $B$ and $AB = 3$. A circle of unit radius is drawn with its centre on any of the vertices of this triangle. Find the maximum value of the area of that part of the triangle that is not shared by the circle.
viewtopic.php?f=13&t=622

Problem 3:
Solve for real $x$:$\dfrac {|x^2-1|}{x-2}=x$
viewtopic.php?f=13&t=5605

Problem 4:
A series is formed in the following manner:
$A(1)=1$
$A(n)=f(m)$ numbers of $f(m)$ followed by $f(m)$ numbers of $0$.
$m$ is the number of digits in $A(n-1)$.
Find $A(30)$. Here $f(m)$ is the remainder when $m$ is divided by $9$.
viewtopic.php?f=13&t=621

Problem 5:
Triangle $ABC$ is right angled at $B$.The bisector of $\angle {BAC}$ meets $BC$ at $D$.Let $G$ denote the centroid (common point of medians) of the triangle $ABC$.Suppose $GD$ is parallel to $AB$ .Find $\angle C$.
viewtopic.php?f=13&t=5609

Problem 6:
Find ,with proof,all the perfect squares each of which is the product of four consequetive odd natural numbers.
viewtopic.php?f=13&t=5607

Problem 7:
Replace each asterick with proof in six digit number $13**45*$ by different digits such that the resulting number is divisible by $792$.
viewtopic.php?f=13&t=5606

Problem 8:
Triangle $ABC$ is right angled at $A$.Let $D$ be a point on $BC$.
$E$ and $F$ are reflections of $D$ on $AC$ and $AB$ respectively.
Prove that $[ABC]\geq [DEF]$.Find all position of $D$ for equality
(Here $[x]$ denotes the area of $x$).
viewtopic.php?f=13&t=5608

Problem 9:
Find all prime numbers $p$ and integers $a$ and $b$ (not necessarily positive) such that $p^a + p^b$ is the square of a rational number.
viewtopic.php?f=13&t=628

Problem 10:
In a set of $131$ natural numbers,no numbers has a prime factor greater than $42$.Prove that it is possible to choose four numbers from this set such that their product is a perfect square.
viewtopic.php?f=13&t=138

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO National Secondary:Problem Collection

This topic is for showcasing the problems. Please use individual topics on each problem for discussion (the link is just below the problem description). I have copied many problems from here.
!!!Happy Problem Solving!!!

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO National Secondary:Problem Collection

Many problems are just same as Higher Secondary category.