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Posted: Thu Feb 21, 2019 12:05 am
Primary

Problem 1.
If a $1$ mile long train crosses a $2$ mile platform in $1$ minute,then what is the required time for $2016$ mile long train of same speed while crossing a $2016$ mile platform?

Problem 2
$(11+11) \times(12+12)\times (13+13)...\times(18+18)\times(19+19)=x$

What is the remainder ,if $x$ is divided by $10$?

Problem 3
The sum of the digits of some two digits number is $11$ and these number can be written as the product of only two prime numbers.What is the sum of these two digits numbers?

Problem 4
Jannat had some marbles and she gave half of these to Mahi.Then she divided rest of the marble in two groups so that one group had one more marble than the other group.She gave half of the marbles of the group that contains even number of marbles to Tasnina.Finally she had four marbles remaining.How many marbles she had at first?

Problem 5
There are $7$ violet balls, $6$ blue balls, $5$ ndigo balls, $4$ green balls, $3$ yellow balls, $2$ orange balls and $1$ red ball. They should be kept at minimum number of boxes in such a way that no box contains more than one ball of same color and each box contains same number of balls. What is the number of balls in each box?

Problem 6
A number is called Tamata Number if it stays the same when the number is written in reverse order.For example, $121$ is a Tamata Number because,we get $121$, if $121$ is written in reverse order.Now what is the minimum G.C.D. (Greatest Common Divisor) of two four digit Tamata Number?

Problem 7

The floor of a museum is shaped like a simple polygon where the sides of the polygon have no points of intrsection except the adjacent sides.The walls of the museum are along the sides of the polygon.Guards have to be employed to guard the valuable things in the museum.Every guard can cover up the infinity around him but if there is a wall,he cannot watch beyond the wall.If the number of sides of the polygon is $n$,then prove that it is possible to guard the museum completely with minimum $\lfloor x/3\rfloor$ .[ $\lfloor x\rfloor$ is the biggest integer which is not bigger then $x$]?

Junior

Problem 1 .

Screenshot_2019-02-20-22-44-47-2.png (5.83 KiB) Viewed 1539 times
Problem 2
$ABC$ is a triangular piece of paper with an area of $500$ square units and $AB = 20$ units. $DE$ is parallel to$AB$. The triangle is folded along$DE$. The part of the triangle below $AB$ has an area of $80$ square units. What is the area of $CDE$?
Screenshot_2019-02-20-22-44-47-1.png (7.05 KiB) Viewed 1539 times
Problem 3.
There is a simple polygon with $2016$ sides where there is intersection among the sides except the intersections of the adjacent sides. Maximum how many diagonals can be drawn inside the polygon such that if any two diagonals intersect , then their of intersection can't be any other point except the vertex of the polygon?

Problem 4 .
In the quadrilateral $ABCD$, $AB^2+CD^2=BC^2+AD^2$.Prove that the diagonals of the quadrilateral are perpendicular to each other.

Problem 5 .
$P$ is a four digit positive integer where $P=abcd$ and( $a\leq b\leq c\leq d$)Again, $Q = dcba$ and $Q-P =X$ . If the digits of $X$ are written in reverse then we get $Y$. Find all possible values of $X+Y$.

Problem 6
$p= 3^w,q=3^x, r = 3^y, s = 3^z$. where $w,x,y,z$ are positive integer. Find the minimum value of $(w+x+y+z)$ such that $( p^2 + q^3 + r^5 = s^7)$.

Problem 7
The floor of a museum is shaped like a simple polygon where the sides of the polygon have no points of intrsection except the adjacent sides.The walls of the museum are along the sides of the polygon.Guards have to be employed to guard the valuable things in the museum.Every guard can cover up the infinity around him but if there is a wall,he cannot watch beyond the wall.If the number of sides of the polygon is $n$,then prove that it is possible to guard the museum completely with minimum $\lfloor x/3\rfloor$ .[ $\lfloor x\rfloor$ is the biggest integer which is not bigger then $x$]

Problem 8
In $\bigtriangleup ABC$ , $\angle A = 20$, $\angle B = 80$, $\angle C = 80$, $BC = 12$ units. Perpendicular $BP$ is drawn on $AC$ from from $B$ which intersects $AC$ at the point $P$. $Q$ is a point on $AB$ in such a way that $QB = 6$ units. Find the value of $\angle CPQ$.

Problem 9
Area of $\bigtriangleup ABC$ is $2016$. $D,E,F$ are three points on the sides $BC,AB,AC$ respectively. Show that, the area of at least one triangle among $\bigtriangleup AEF$, $\bigtriangleup BDE$, $\bigtriangleup CDF$ is not larger than $504$ square units.

Problem 10

$a, b, c, d$ are four positive integers where $a < b < c < d$ and the sum of any three of them is divisible by the fourth. Find all possible values of $(a, b, c, d)$

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 $7$.
viewtopic.php?f=13&t=3878

Problem 2:
(a) How many positive integer factors does $6000$ have?

(b) How many positive integer factors of $6000$ are not perfect squares?
viewtopic.php?f=13&t=3878

Problem 3:
$\triangle ABC$ is isosceles $AB = AC$. $P$ is a point inside $\triangle ABC$ such that $\angle BCP = 30^{\circ}$ and $\angle APB = 150^{\circ}$ and $\angle CAP = 39^{\circ}$. Find $\angle BAP$.
viewtopic.php?f=13&t=3730

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,

$S = \left | x_2 - x_1 \right | + \left | x_3 - x_2 \right | + ................+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right |$.
viewtopic.php?f=13&t=3656

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?
viewtopic.php?f=13&t=3878

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 $IB$?
viewtopic.php?f=13&t=3878

Problem 7:
Aasma is a mathematician and devised an algorithm to find a husband. The strategy is:

$\circ$ 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 $r$.
$\circ$ Reject the first $k$ men and let $H$ be highest rank of these $k$ men.
$\circ$ 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 $1000^{th}$ person is selected.

Aasma 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) What is the probability that the highest ranking prospect among all $1000$ prospects is the $(m + 1)^{th}$ prospect?

(b) 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) What is the probability that the prospect with the highest rank is the $(m+1)^{th}$ person and that Aasma will choose the $(m+1)^{th}$ boy using this algorithm?

(d) The total probability that Aasma 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
ln $N$ $\approx \frac{1}{N-1} + \frac{1}{N-2} +........+ \frac{1}{2} +\frac{1}{1}$

(e) 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, $e = 2.718....$.
viewtopic.php?f=13&t=3878

Problem 8:
$\triangle ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, $AC = 3$. 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 at a second point $F$. Then $AF^{2} = \frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
viewtopic.php?f=13&t=3855

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 $7$.
viewtopic.php?f=13&t=3878

Problem 2:
(a) How many positive integer factors does $6000$ have?

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

viewtopic.php?f=13&t=3878

Problem 3:
$\triangle ABC$ is isosceles $AB = AC$. $P$ is a point inside $\triangle ABC$ such that $\angle BCP = 30^{\circ}$ and $\angle APB = 150^{\circ}$ and $\angle CAP = 39^{\circ}$. Find $\angle BAP$.
viewtopic.php?f=13&t=3730

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,

$S = \left | x_2 - x_1 \right | + \left | x_3 - x_2 \right | + ................+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right |$.
viewtopic.php?f=13&t=3656

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?
viewtopic.php?f=13&t=3878

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 $IB$?
viewtopic.php?f=13&t=3878

Problem 7:
Aasma is a mathematician and devised an algorithm to find a husband. The strategy is:

$\circ$ 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 $r$.
$\circ$ Reject the first $k$ men and let $H$ be highest rank of these $k$ men.
$\circ$ 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 $1000^{th}$ person is selected.

Aasma 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) What is the probability that the highest ranking prospect among all $1000$ prospects is the $(m + 1)^{th}$ prospect?

(b) 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) What is the probability that the prospect with the highest rank is the $(m+1)^{th}$ person and that Aasma will choose the $(m+1)^{th}$ boy using this algorithm?

(d) The total probability that Aasma 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
ln $N$ $\approx \frac{1}{N-1} + \frac{1}{N-2} +........+ \frac{1}{2} +\frac{1}{1}$

(e) 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, $e = 2.718....$.
viewtopic.php?f=13&t=3878

Problem 8:
$\triangle ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, $AC = 3$. 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 at a second point $F$. Then $AF^{2} = \frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
viewtopic.php?f=13&t=3855

Problem 9
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.
viewtopic.php?f=13&t=4140

### Re: BdMO National Olympiad 2016:Problemsets

Posted: Thu Feb 21, 2019 7:26 am
This topic is mainly for showcasing the problems. Please use individual topics on each problem for discussion (the link is just below the problem description or at the problem number).
Happy Problem Solving