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Ten Points Each. Total 150 Points.: C B A A, B, and C Are Real Numbers. B A X X y y

The largest exterior angle of a pentagon with interior angles in proportion of 5:3:4:2:6 is 120 degrees. The value of (a + b) - c, where (a - 2)2 + b + 3 + |c + 5| = 0, is 0. The value of (1 - y)2/y - 1, where y < x-2 + 2-x + 1/2, after simplifying is 1. If ab = 1, the value of 1/1+a + 1/1+b is 2. A polygon with perimeter 80 cm and sides forming an arithmetic sequence with common difference 3 cm and longest side 22 cm

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Anwar Sanusi
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160 views2 pages

Ten Points Each. Total 150 Points.: C B A A, B, and C Are Real Numbers. B A X X y y

The largest exterior angle of a pentagon with interior angles in proportion of 5:3:4:2:6 is 120 degrees. The value of (a + b) - c, where (a - 2)2 + b + 3 + |c + 5| = 0, is 0. The value of (1 - y)2/y - 1, where y < x-2 + 2-x + 1/2, after simplifying is 1. If ab = 1, the value of 1/1+a + 1/1+b is 2. A polygon with perimeter 80 cm and sides forming an arithmetic sequence with common difference 3 cm and longest side 22 cm

Uploaded by

Anwar Sanusi
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Ten Points Each. Total 150 Points.

1. If the five interior angles of a pentagon (5–sided figure) are in proportion of


5 : 3 : 4 : 2 : 6, what is the degree measurement of its largest exterior angle?
2. Suppose ( a − 2) 2 + b + 3 + | c + 5 |= 0 , where a, b, and c are real numbers.
Find ( a + b) − c .
1
3. lf y < x−2 + 2− x + where x and y are real numbers, find and simplify
2
(1 − y ) 2
.
y −1
1 1
4. If ab = 1, what is the value for + ?
1 + a 1 + b2
2

5. Consider a polygon having a perimeter of 80 centimeters and side lengths


forming an arithmetic sequence with a common difference of 3 centimeters. If
the longest side has a length of 22 centimeters, how many sides does this
polygon have?
6. As shown in the figure on the right, the center of a quarter
circle is at the intersection of the two diagonals of the square
ABCD. Suppose the radius of this quarter circle is longer
than OC and the length of the side of this square is 24.
What is the area of the overlapped (shaded) region?
A 1 1 2 1 3 1 4 1 5
7. Suppose = (1 － x＋ x－ x＋ x－ x＋
1 + x + x2 + x3 + x4 + x5 + x6 2 4 8 16 32
1 1 1
− x 6 − x 5 − x 4 + 100
1 6 q
x )＋ 64 32 16 and A is a polynomial with as the
64 1+ x + x + x + x + x + x
2 3 4 5 6
p
coefficient of its x5 term with p and q being positive relatively prime integers.
Find the value of p + 2q.
8. Suppose two of a triangle's interior angles are 45° and 75° and the longest side
has a length of 3 + 3 . If the sum of the lengths of the other two sides is
x + y , find the value for x + y.
2015 Final 1
9. Among the natural numbers from 1 to 3005, how many are multiples of 5 whose
digits also sum to a multiple of 5 ?
10. Suppose a, b, and c are all real numbers and a2＋2b＝7, b2＋4c＝–7 , and
c2＋6a＝–14. Find the value for a2＋b2＋c2.
F (m)
11. Suppose F(1) = 5 and F(m–n) = when m and n are integers. If
F (n)
F(10) can be written in form ab where a is prime, what is the value for
2b – a?
 x ( y + z − x ) = 68 − 2 x
2


12. If x, y, and z satisfy the system of equations  y ( z + x − y ) = 102 − 2 y 2 , find
 z ( x + y − z ) = 119 − 2 z 2

|x – y + z|.
13. If p and q are natural numbers and both roots of equation px2 – qx + 2995 = 0
are prime numbers, what is the value for 21p2 + q?
32017 − 32016 − 32015 + 32014 + 3n+( −1)
n
14. If n is a positive integer and is a perfect
square, find the value for n.
15. Suppose real numbers a, b, and c satisfy the equations a + b + c = 15 and
ab + bc + ca = 27. Let the maximum value for c be M and minimum value be m,
find the value for M – m.

2 2015 Final

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