BdMO National Olympiad 2014:Problemsets

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BdMO National Olympiad 2014:Problemsets

Unread post by samiul_samin » Thu Feb 21, 2019 8:55 pm

Bangladesh National Mathematical Olympiad 2014:
Primary


Problem 1
If a number is multiplied by itself then the obtained product is a square number.For example,$2\times2=4$ is a square number.The sum of three consequative positive integers is a square number.Which is the smallest such square number?

Problem 2
When, $x$ is divided by $10$, the quotient is $y$,with a remainder of $4$. If $x$ and $y$ are both positive integers,what is the remainder when $x$ is divided by $5?$

Problem 3
Rubai and Bidushi have some marbles.Bidushi told Rubai ,"if you give me some marbles,I will return one more marble tuan as many as you gave me. "Rubai said,"Alright I will first give you $6 $ marbles.Then Rubai gave Bidushi $6$ marbles and Bidushi returned $7$ marbles to Rubai.Thus after they have exchanged marble $5$ times,Bidushi had no marbles left.How many marbles Bidushi have in the beginning?


Problem 4
Subrata has invented a new type of clock,according to which, there are $15$ hours in each day and $80$ minutes in each hour. For example Subrata's clock shows $10:00$ when the actual time is $16:00$ in a traditional clock. If the time is $20:36$ in a traditional clock, then what will be the time in Subrata's clock ?


Problem 5
How many four digits numbers are there for which ,the number formed by its last two digits in the same order when multiplied by three gives us the number formed by its first two digits in the same order?For example, $3612$ is such a number where the number formed by the last two digits in the same order is $12$ and when multiplied by $3$ gives $36$.


Problem 6
.A work has to be done in $18$days.A contructor assigned $20$ men to do the tusk.But,after $10$ days it was found that only half of the work was done.So,how many men should he add so that the work will be finshed in time?

Problem 7
A new series is to be formed by removing some terms from the series $1,2,3,4,.........,30$
such that no terms of the new series is obtained if the new series is doubled.Maximum how many terms can be in the new series?

Problem 8
Is it possible to completely cover a $14*14$ grid by "T" shaped blocks from the diagram such that no block overlaps any other bolcks? Explain your answer with logic.
Screenshot_2019-02-21-20-51-36-1.png
Screenshot_2019-02-21-20-51-36-1.png (1.09 KiB) Viewed 647 times
Last edited by samiul_samin on Thu Feb 21, 2019 10:25 pm, edited 1 time in total.

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Re: BdMO National Olympiad 2014:Problemsets

Unread post by samiul_samin » Thu Feb 21, 2019 9:12 pm

Bangladesh National Mathematical Olympiad 2014:
Junior


Problem 1
How many triangles can be made in total by choosing three out of four sticks of $2m,4m,6m$ and $8m $ length?

Problem 2
How many four digits numbers are there for which ,the number formed by its last two digits in the same order when multiplied by four gives us the number formed by its first two digits in the same order?For example, $4812$ is such a number where the number formed by the last two digits in the same order is $12$ and when multiplied by $4$ gives $48$.

Problem 3
Subrata has invented a new type of clock,according to which, there are $15$ hours in each day and $80$ minutes in each hour. For example Subrata's clock shows $10:00$ when the actual time is $16:00$ in a traditional clock. If the time is $20:36$ in a traditional clock, then what will be the time in Subrata's clock ?

Problem 4
The unit digit of a six digit number is $1$ and it is removed, leaving a five digit number. The removed unit digit is $1$ is then placed at the far left of the five digit number,making a new six digit number. If the new number is $\dfrac 31$ of the original number,what is the original number?

Problem 5
A new series is to be formed by removing some terms from the series $1,2,3,4,.........,30$
such that no terms of the new series is obtained if the new series is doubled.Maximum how many terms can be in the new series?

Problem 6
Is it possible to completely cover a $14*14$ grid by "T" shaped blocks from the diagram such that no block overlaps any other bolcks? Explain your answer with logic.
Screenshot_2019-02-21-20-51-36-1.png
Screenshot_2019-02-21-20-51-36-1.png (1.09 KiB) Viewed 643 times
Problem 7
In the figure, if $AB= 10$, what is the length of the side $CD$?
Screenshot_2019-02-21-20-56-07-1.png
Screenshot_2019-02-21-20-56-07-1.png (7.88 KiB) Viewed 643 times

Problem 8
$AVIK$ is a square. The point $E$ is taken on $VK$ in such a way that $3VE=EK$. $F$ is the midpoint of $AK$. What is the value of $\angle {FEI}$?

Problem 9
If $N$ is an even integer , prove that $48$ divides $N(N^2+20)$.



Problem 10
Oindri has $100$ chocolates. She finished eating all her chocolates in $58$ days by eating at least one chocolate each day. Prove that, in how many consecutive days did she eat exactly $15$ chocolates?
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Re: BdMO National Olympiad 2014:Problemsets

Unread post by samiul_samin » Thu Feb 21, 2019 9:36 pm

Bangladesh National Mathematical Olympiad 2014:
Secondary


Problem 1
If $x$ is divided by $10$ then the quotient is $y$ and remainder is $3$.If $x$ and $y$ are both positive integers then what will be the remainder if $x$ is divided by $5$?

Problem 2
Subrata has invented a new type of clock,according to which, there are $15$ hours in each day and $80$ minutes in each hour. For example Subrata's clock shows $10:00$ when the actual time is $16:00$ in a traditional clock. If the time is $18:42$ in a traditional clock, then what will be the time in Subrata's clock ?

Problem 3
Closing his eyes Towsif begins to place knights on a Chess board of $ {19} \times {21}$. After placing how many knights
Towsif will be sure that on the next move at least one knight will attack another one. (In one move knight goes
straight for ${2}$ steps and the 3rd step should be at right angle to the previous path.)

Problem 4
In $$ΔABC$$, $$∠B = 90$$. A circle is drawn taking $$AB$$ as a chord. $$O$$ is the center of the circle. $$O$$ and $$C$$ isn't on the same side of $$AB$$. $$BD$$ is perpendicular to $$AC$$. Prove that, $$BD$$ will be a tangent to the circle if and only if $$∠BAO = ∠BAC$$.

Problem 5
$97+98+99+.........+114+115=2014$.Here sum of $19$ consequative numbers is $2014$.Find the largest number of consequative positive integers whose sum is exactly $2014$ and justify why you think this must be the largest number.

Problem 6
Let $\triangle ABC $ be an acute angled triangle with $\angle C=60^{\circ}$.Perpendicular $AA_1$ & $BB_1$ are drawn from point $A$ and $B$ to the sides $BC$ & $AC$ respectively.Let $M$ be the midpoint of $AB$.What is the value of $\dfrac {\angle A_1MB_1}{\angle A_1CB_1}$?

Problem 7
In $ \triangle ABC$ $E, F$ are two points on $BC$ and $AB$ such that $ EF || AC . Q$ is a point on $AB$ such that $ \frac{AQ}{PQ} = \frac{30}{13} $ . $PQ$ is parallel to $EF$ where $P$ lies on $CB$ . $X$ is taken on extended $EQ$ such that $CX = 20.4$ . Given $ \frac{CY}{EY} = \frac{XY}{CY} , PX = 15.6$ ; if $\angle YCE = 22.5 , \angle PXQ = ?$

Problem 8
If the lengths of two altitudes drawn from two vertices of a triangle on their opposite sides are $2014$ and $1$ unit, then what will be the length of the altitude drawn from the third vertex of the triangle on its opposite side?

Problem 9
There are $n$ players in a chess tournament.Every player plays every other player exactly once and there are no draws.Prove that ,the players can be lebeled $1,2,... ... ...,n$ so that $i$ beats $i+1$ for each $i \in$ {$1,2,3,.........,n-1$}


Problem 10
Suppose that,you are on the left most bottom point of a $n\times n$ grid.You have to reach the rightmost and topmost point.But the rule is you can move just only toward the upper or right direction.Can't move down or to the left.Ans as there are mines at the squares which are along diagonal you can't go these places too.Determine,how many ways are there to reach the destination.
Screenshot_2019-02-21-17-01-22-1.png
Screenshot_2019-02-21-17-01-22-1.png (3.24 KiB) Viewed 640 times
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Re: BdMO National Olympiad 2014:Problemsets

Unread post by samiul_samin » Thu Feb 21, 2019 10:18 pm

Bangladesh National Mathematical Olympiad 2014:
Higher Secondary


Problem 1
If $x$ is divided by $10$ then the quotient is $y$ and remainder is $3$.If $x$ and $y$ are both positive integers then what will be the remainder if $x$ is divided by $5$?

Problem 2
Closing his eyes Tawsif begins to plca knights on a Chess board of $21\times 23$.After plcing how many knights Tawsif will be sure that in next move at least one knight will attack another one?

Problem 3
In $$ΔABC$$, $$∠B = 90$$. A circle is drawn taking $$AB$$ as a chord. $$O$$ is the center of the circle. $$O$$ and $$C$$ isn't on the same side of $$AB$$. $$BD$$ is perpendicular to $$AC$$. Prove that, $$BD$$ will be a tangent to the circle if and only if $$∠BAO = ∠BAC$$.

Problem 4
$97+98+99+.........+114+115=2014$.Here sum of $19$ consequative numbers is $2014$.Find the largest number of consequative positive integers whose sum is exactly $2014$ and justify why you think this must be the largest number.

Problem 5
Let $\triangle ABC $ be an acute angled triangle with $\angle C=60^{\circ}$.Perpendicular $AA_1$ & $BB_1$ are drawn from point $A$ and $B$ to the sides $BC$ & $AC$ respectively.Let $M$ be the midpoint of $AB$.What is the value of $\dfrac {\angle A_1MB_1}{\angle A_1CB_1}$?

Problem 6
There are $n$ players in a chess tournament.Every player plays every other player exactly once and there are no draws.Prove that ,the players can be lebeled $1,2,... ... ...,n$ so that $i$ beats $i+1$ for each $i \in$ {$1,2,3,.........,n-1$}

Problem 7
In $ \triangle ABC$ $E, F$ are two points on $BC$ and $AB$ such that $ EF || AC . Q$ is a point on $AB$ such that $ \frac{AQ}{PQ} = \frac{30}{13} $ . $PQ$ is parallel to $EF$ where $P$ lies on $CB$ . $X$ is taken on extended $EQ$ such that $CX = 20.4$ . Given $ \frac{CY}{EY} = \frac{XY}{CY} , PX = 15.6$ ; if $\angle YCE = 22.5 , \angle PXQ = ?$

Problem 8
You are stuck in a $2d$ plane and your movement is limited to the two dimension grid shown above.You start at point point ($0,0$) and have to reach the end point of ($10,0$),if your current position is ($x,y$) ,you may move to ($x+1,y+1$),($x+1,y$) and ($x+1,y-1$) if the dimension point is inside the grid.For example you may move from ($0,0$) to ($1,1$),($1,0$) & ($1,-1$);you may move from ($3,1$) to ($4,1$) and ($4,0$).So,how many different paths exist from ($0,0$) to ($10,0$)?
Screenshot_2019-02-21-17-01-22-2.png
Screenshot_2019-02-21-17-01-22-2.png (4.06 KiB) Viewed 636 times
Problem 9
Suppose that,you are on the left most bottom point of a $n\times n$ grid.You have to reach the rightmost and topmost point.But the rule is you can move just only toward the upper or right direction.Can't move down or to the left.Ans as there are mines at the squares which are along diagonal you can't go these places too.Determine,how many ways are there to reach the destination.
Screenshot_2019-02-21-17-01-22-1.png
Screenshot_2019-02-21-17-01-22-1.png (3.24 KiB) Viewed 636 times

Problem 10
In a space, there are $N$ points. Prove that, any one can divide these $N$ points with $N-1$ planes where these $N-1$ planes are parallel to each other.
Last edited by samiul_samin on Thu Feb 21, 2019 10:28 pm, edited 1 time in total.

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Re: BdMO National Olympiad 2014:Problemsets

Unread post by samiul_samin » Thu Feb 21, 2019 10:21 pm

This topic is mainly for showcasing the problems. Please use individual topics on each problem for discussion (the problem numbers are the links).
Happy Problem Solving :D :D :D

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