5l <l:l <l:< - ?
<- l l+7
5<- ;oou
o0 <l:l0 l+7 ll4l5
4l*- : <l:l0 l+7 ll4l5 <l-l7
Category Higher Secondary -4 -4 -4 -4 : : : : 8 88 8 7l 7l 7l 7l Time -4 Hours
Suggestions: Earlier problems are intended to be easier than later problems; try
the beginning problems First; all problems have equal weight
1. ~ 2008|7 <l\* *l l *l+ *** ~ 2008|7 **l l *l+
|*l *l < |*l+ **7lz The sum of the first 2008 odd positive integer
is subtracted from the sum of the first 2008 even positive integers. Find the
result.
2. =*|7 ~ 5l |*1, <|7 |n ~ 5l *7*|7 |* 2, |7|7 |n ~ 5l *7*|7 |* 3,
, l"|7 |n ~ 5l *7*|7 |* 49 = l"|7 |n ~ 5l
*7*|7 |* 50 |l N* <*l ~ 5l** =*|7 *l*l l* ** *l ***
9>* =*|7 =*|7 ** ~ 5l *"l < *~* *|7 ~ 5l |* |17 <"l l* *,
* **l =* *l* N7 10|7 ~ 5l "l*l <**z 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?
3. <l l* a =*|7 l Nl m = 4a + 3 " m, 11-= <|7* a
4
-** 11 ll
l *l <* **7l N|"8 l**z |3l|7 ~l * 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.
4. ) (x f =*|7 |7 -||l +l" 1 ) 1 ( ) ( = + x f x f <*
1
0
) ( dx x f -= ~l
| * The function ) (x f is a complicated nonlinear function. It satisfies,
1 ) 1 ( ) ( = + x f x f . Evaluate
1
0
) ( dx x f
5. N~l " 7l l< N<*~9 9ll **ll N~l ** "l~|~ " *~* "l~|" 9ll
** *>* ll **ll* ~ (= >l* =*, = *l **ll *
5l <l:l <l:< - ?<- l l+7
5<- ;oou
o0 <l:l0 l+7 ll4l5
4l*- : <l:l0 l+7 ll4l5 <l-l7
||7 |** ll = *l **ll* =*< *>* ll ** **z
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?
6. 1,2 " 3 =< |7|7 NF |9* =*|7 5 NF ||"8 l 7| *l < l|7*7 *~*
1,2 " 3 =*l ** N*< =*~ **7l|7 5 NF ||"8 l 7| *l l*z (l<l
- * l ~*< 1,2 " 3 *< *<*l *~ l *l **7 l* 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.)
7.
2
2 . 5 1 n
m
= + ~|** l ~l<l ) , ( n m * **7 <* (A) 1
2
n ?
(B) ) 1 ( + n " ) 1 ( n = < *l l **l, l|* =*|7 *l N N|7 **l
7l * * (C) |9
2
1
=
n
a < 7* ? ) 1 ( = + a a (D) |9 a **l < 7* 1 + a
*l l **lz (E) (C) " (D) *** 1 = a l 1 1 = + a <"l *| Yz (F) a -=
"*~l= Yl ~l|7 * **l = m " n *| <"l |> 7l * **l We want to
find all integer solutions (m; n) to
2
2 . 5 1 n
m
= + . First: (A) Find an expression
for 1
2
n ; (B) are ) 1 ( + n and ) 1 ( n both even, or both odd, or is one even
and the other odd? (C) Let
2
1
=
n
a , Find an expression for ) 1 ( + a a ; (D) If a
is odd, is 1 + a even or odd? (E) From parts (C) and (D), is it possible for 1 = a ,
or ? ) 1 ( = + a a (F) Find the only possible values a can take and then find what
m and n should be.
8. ABCD >7 * * AC " BD, E |_*7 *9 *** AB = 39; AE = 45; AD =
60; BC = 56 <* CD=? ABCD is a convex quadrilateral. The diagonals AC
and BD intersect at E. AB = 39; AE = 45; AD = 60; BC = 56. Find the length
of CD.
9: ABCD >7 * AB = BC = CD 7* AC<>BD. >7 * ** *9|_ E. |9
AE = DE = <BAD+<ADC= <, 7* =z Let ABCD be a convex quadrilateral
5l <l:l <l:< - ?<- l l+7
5<- ;oou
o0 <l:l0 l+7 ll4l5
4l*- : <l:l0 l+7 ll4l5 <l-l7
with AB = BC = CD. Note, AC <> BD. Let E be the intersection point of the
diagonals of ABCD. AE = DE if <BAD+<ADC= , Find
Problem 10: A quadrilateral ABCD with \BAD + \ADC > 180 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 (AX DY )=(BX CY )?
