Math Olympiad Reviewer Number Theory Logical Reasoning

1. 1. Adrian is thinking of a number. After adding 23 to the number, the sum is then multiplied by 14. The final answer
is 364. What number is he thinking of?
a) 26 b) 49 c) 3 d) 5119
364 /14 = 26
26 -23 =3

2. 2. Express 40 seconds as fraction of 2 minutes.
a) 1/5 b) 1/3 c) ½ d) 2/3
40/120 =1/3

3. 3.
1
4
more than the product of
2
3
and
7
8
is
(2/3)(7/8) = 14/24
7
12
+
1
4
=
7+3
12
=10/12 or 5/6

4. 4. John spent
5
8
of his salary and gave
1
6
of the remainder to his mother. If he had Php 300 left in the end, how much did
he spend and give to his mother altogether?
spent
remainder
1
/
6
300
60 + 600 = 660

5. 30 cm
24 cm
a) 216 cm2 b) 360 cm2 c) 270 cm2 d) 432cm2
5. The perimeter of the triangle below is 72 cm. Find its area.
hypotenuse
3,4,5 -6,8,10-----12,16,20,
18,24,30
5,12,13-----10,24,26
7,24,25
A = (18)(24) divided by 2
(18)(24)
2

6. A1 =
(10)(10)
2
= 50
A2 =
(10)(18)
2
= 90
90-50 = 40𝑐𝑚2

7. 7. A piece of wood 28 m long is cut into 2 pieces in the ratio 3:4. What is the length of the shorter
piece of wood? a) 6m b) 12m c) 16m d) 4
3
7
28 = 12

8. 35
35

9. 9. The sum of 4 tens, 50 tenths,3 hundredths and 8 thousandths is
a) 4.538 b) 45.038 c) 40.538 d) 45.308
40
0.50
0.03
0.008

10. 10. There are 63 ducks, 27 geese and some chickens at a farm. If 40% of the total number of poultry is chickens,
what is the difference between the number of chickens and the number of geese at the farm?
a) 27 b) 33 c) 36 d) 60
40: c = 60 : 90
60c = 3600
C= 60

11. What is the value of x2 – x +
1
𝑥
if x =
1
2
?
(
1
2
)2
-
1
2
+
1
1
2
=
1
4
-
1
2
+ 2 = 7/4 or 1
3/4

12. 2. The measures of the angles of a triangle are in the ratio 2:3:4. What is the measure of the largest angle?
4
9
180 = 80

13. 3. Solve for x:
2(𝑥−3 )
3
−
𝑥+4
4
= 7
8(𝑥 − 3 )
−
3 𝑥 + 4
= 84
8x -24 -3x -12 = 84
5x -36 = 84
5x = 84 + 36
5x = 120
X = 24

14. 4. Solve the equation
11
𝑥+1
−
3
𝑥−1
=
10
𝑥2−1
11(x-1) – 3(x+1) = 10
11x – 11 – 3x – 3 = 10
8x = 10 + 11 + 3
8x = 24
X = 3
10(𝑥2
−1)
𝑥2 − 1
11
3 + 1
−
3
3 − 1
=
10
32 − 1
11
4
−
3
2
=
10
8

15. 5. Find the largest integral solution to 5 – 3 (4 – 5x ) < 23.
5 – 3 (4 – 5x ) < 23.
5 – 12 +15x < 23
15x < 23+7
15x < 30
x < 2