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.
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- Sat Feb 12, 2011 4:50 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2011/9
- Replies: 12
- Views: 8712
- Sat Feb 12, 2011 4:49 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2011/10
- Replies: 3
- Views: 3631
BdMO National Higher Secondary 2011/10
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 diagon...
- Mon Feb 07, 2011 12:13 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary: Problem Collection
- Replies: 3
- Views: 15665
Re: BdMO National Higher Secondary: Problem Collection
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 $2$? http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=620 Problem 2: Isosceles triangle $ABC$ is right angled at $B$ and $AB = 3$. A ...
- Mon Feb 07, 2011 12:11 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2010/10
- Replies: 2
- Views: 3627
BdMO National Higher Secondary 2010/10
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, \...
- Mon Feb 07, 2011 12:10 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2010/8
- Replies: 1
- Views: 2790
BdMO National Higher Secondary 2010/8
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.
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.
- Mon Feb 07, 2011 12:09 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2010/9
- Replies: 1
- Views: 2927
BdMO National Higher Secondary 2010/9
Problem 9:
Find the number of odd coefficients in expansion of $(x + y)^{2010}$.
Find the number of odd coefficients in expansion of $(x + y)^{2010}$.
- Mon Feb 07, 2011 12:09 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2010/7
- Replies: 1
- Views: 3052
BdMO National Higher Secondary 2010/7
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 poi...
- Mon Feb 07, 2011 12:08 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2010/5
- Replies: 3
- Views: 3823
BdMO National Higher Secondary 2010/5
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)
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)
- Mon Feb 07, 2011 12:08 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2010/6
- Replies: 1
- Views: 2805
BdMO National Higher Secondary 2010/6
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.
$a$ 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.
- Mon Feb 07, 2011 12:08 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Higher Secondary 2010/4
- Replies: 2
- Views: 9201
BdMO National Higher Secondary 2010/4
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 $x$.
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 $x$.