BdMO National Higher Secondary: Problem Collection

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BdMO National Higher Secondary: Problem Collection

Unread post by BdMO » Sun Feb 06, 2011 10:40 pm

Bangladesh National Mathematical Olympiad 2007 : Higher Secondary

Problem 1:
In the figure $AB=8,\ BC=7$ and $CA=6.\ \Delta PAB$ is similar to $\Delta PCA$. What is $PC$?
viewtopic.php?f=13&t=584

Problem 2:
$WZ$ is the diameter of circle with center $O$. $OY=5$, arc $XY$ creates angle $60^{\circ}$ at the center. If $\angle ZYO=60^{\circ}$, then $XY=?$.
viewtopic.php?f=13&t=585

Problem 3:
In $\Delta ABC,\ \angle PAC=\angle PBC$. The perpendicular from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$ respectively. $D$ is the midpoint of $AB$. What is the value of $\frac {DL}{DM}$?
viewtopic.php?f=13&t=586

Problem 4:
$(m,n)$ represents the largest common divisor of integers $m$ and $n$. For example $(2,3)=1$ and $(10,15)=5$. Suppose $n(n+1)(n+2)$ is a square, where $n$=integer.
a) What is $(n,n+1)$?
b) What is $(n+1,n+2)$?
c) What is $(n+1,n(n+2))$?

From your answers $a,b,c$ is it possible for $n(n+1)(n+2)$ to be a square?
viewtopic.php?f=13&t=587

Problem 5:
If $x_1, x_2$ are the zeros of the polynomial $x^2-6x+1$, then prove that for every nonnegative integer $n$, $x_1^n+x_2^n$ is an integer and not divisible by $5$.
viewtopic.php?f=13&t=588

Problem 6:
Writing down all the integers from $19$ to $92$ we make a large integer $N$.\[N=192021\cdots 909192\]If $N$ is divisible by $3^k$ then what is the maximum value of $k$?
viewtopic.php?f=13&t=589

Problem 7:
$f(x)=x^6+x^5+\cdots +x+1$Find the remainder when dividing $f(x^7)$ by $f(x)$.
viewtopic.php?f=13&t=590

Problem 8:
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 9:
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 10:
Find the area bounded by the curves $y=|x-1|$ and $x^2+y^2=2x$ (where $y\ge 0$)
viewtopic.php?f=13&t=593

Problem 11:
Solve the inequality\[2\cos x \le |\sqrt {1+\sin 2x} -\sqrt {1- \sin 2x}|\]
viewtopic.php?f=13&t=594

Problem 12:
Find the minima and maxima of $\left(\frac{\sin 10x}{\sin x}\right )^2$ in the interval $[0,\pi]$.
viewtopic.php?f=13&t=595

Problem 13:
Sohag and Pias had some coconuts. They sold each coconut at the price which is equal to the number of the total coconuts. Sohag and Pias began to take $20$ Taka each alternately from the obtained money.
Sohag started the process. After a while he found that there was not enough money to take like before. Then he took the remaining money and to make the sharing fair he gave his pen to Pias. If they took $25$ Taka each alternately the situation would be almost same but in that case Sohag Had to give his pencil to Pias. If the pen costs $5$ Taka more than the pencil. What is the price of the pencil? [The costs of pen and pencil are integers]
viewtopic.php?f=13&t=596

Problem 14:
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

LaTeXed by: Zzzz
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Re: BdMO National Higher Secondary: Problem Collection

Unread post by Moon » Sun Feb 06, 2011 11:21 pm

Bangladesh National Mathematical Olympiad 2008: Higher Secondary

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

Problem 9:
Let $ABCD$ be a convex quadrilateral with $AB = BC = CD$. Note, $AC \not = BD$. Let $E$ be the intersection point of the diagonals of $ABCD$. $AE = DE$. If $\angle BAD+\angle ADC=\theta$ , find $\theta$.
viewtopic.php?f=13&t=607

Problem 10:
A quadrilateral $ABCD$ with $\angle BAD + \angle ADC > 180\circ$ circumscribes a circle of center $I$. A line through $I$ meets $AB$ and $CD$ at points $X$ and $Y$ respectively. If $IX = IY$ then what is
\[ \frac{AX\cdot DY}{BX \cdot CY}\]
viewtopic.php?f=13&t=608

LaTeXed by: Zzzz

(The official pdf of the problem set contains some errors. I have the hard copy of the corrected version. So here you can find the correct problems.)
Attachments
BdMO_2008_HS.pdf
(49.22 KiB) Downloaded 251 times
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Re: BdMO National Higher Secondary: Problem Collection

Unread post by Moon » Sun Feb 06, 2011 11:36 pm

Bangladesh National Mathematical Olympiad 2009: Higher Secondary

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 \
viewtopic.php?f=13&t=619

LaTeXed by: Zzzz
Attachments
2009_national_higher_secondary.pdf
(61.92 KiB) Downloaded 238 times
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.

BdMO
Posts: 134
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Re: BdMO National Higher Secondary: Problem Collection

Unread post by BdMO » Mon Feb 07, 2011 12:13 am

Bangladesh National Mathematical Olympiad 2010: Higher Secondary

Problem 1:
Let $S=1^1+2^2+3^3+ ... +2010^{2010}$ . What is the remainder when $S$ is divided by $2$?
viewtopic.php?f=13&t=620

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:
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 4:
Given a point $P$ inside a circle $\Gamma$, two perpendicular chords through $P$ divide $\Gamma$ into distinct regions $a,\ b,\ c,\ d$ clockwise such that $a$ contains the centre of $\Gamma$.
Prove that \[ [a] + [c] \ge [ b ] + [d] \] Where $[x]$ = area of $x$.
viewtopic.php?f=13&t=623

Problem 5:
How many regular polygons can be constructed from the vertices of a regular polygon with $2010$ sides? (Assume that the vertices of the $2010$-gon are indistinguishable)
viewtopic.php?f=13&t=625

Problem 6:
$a$ and $b$ are two positive integers both less than $2010$; $a\ne b$. Find the number of ordered pairs $(a, b)$ such that $a^2 + b^2$ is divisible by $5$. Find $a + b$ so that $a^2 + b^2$ is maximum.
viewtopic.php?f=13&t=624

Problem 7:
Let $ABC$ be a triangle with $AC > AB$: Let $P$ be the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle CAB$: Let $X$ and $Y$ be the feet of the perpendiculars from $P$ to lines $AB$ and $AC$ respectively. Let $Z$ be the intersection point of lines $XY$ and $BC$: Determine the value of \[\frac{BZ}{ZC}\]
viewtopic.php?f=13&t=626

Problem 8:
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 9:
Find the number of odd coefficients in expansion of $(x + y)^{2010}$.
viewtopic.php?f=13&t=627

Problem 10:
$a_1, a_2,\cdots , a_k, \cdots , a_n$ is a sequence of distinct positive real numbers such that $a_1 < a_2 < \cdots <a_k$ and $a_k > a_{k+1} > \cdots > a_n$. A Grasshopper is to jump along the real axis, starting at the point $O$ and making $n$ jumps to the right of lengths $a_1, a_2, \cdots , a_n$ respectively. Prove that, once he reaches the rightmost point, he can come back to point $O$ by making $n$ jumps to the left of lengths $a_1, a_2, \cdots , a_n$ in some order such that he never lands on a point which he already visited while jumping to the right. (The only exceptions are point $O$ and the rightmost point)

LaTeXed by: Zzzz
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higher secondary_2010.pdf
(94.33 KiB) Downloaded 285 times

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