The document consists of a math exam with multiple choice and open-ended questions covering a variety of topics, including geometry, algebra, probability, and basic arithmetic. Each question is followed by potential answers and solutions detailing how to arrive at the correct answer. The exam tests knowledge on divisors, area calculations, angles on a clock, and other mathematical concepts.

1. VTAMPS 15
Secondary 3 Set 1
Multiple Choice
1. Find the sum of all odd integers between 300 and 400 that are divisible by both 5 and 7.
A. 665 B. 700 C. 735 D. 770 E. 805
2. A rectangular land, 500 meters long and 600 meters wide, is to be divided between two kids as
inheritance in the ratio 1: 5. What is the area, in hectares, of the larger land?
A. 15 B. 20 C. 25 D. 30 E. 35
3. The slope of a line is 3 and it crosses the 𝑦-axis at (0, −5). At what point will this line cross the 𝑥-
axis?
A. (−
15
8
, 0)
B. (−
5
3
, 0)
C. (
5
3
, 0)
D. (
15
8
, 0)
E. (
21
10
, 0)
4. A square with an area of 8 sq. units is inscribed in a circle. Find the area of the circle.
A. 2𝜋 B. 4𝜋 C. 6𝜋 D. 8𝜋 E. 10𝜋
5. A fair six-sided die is rolled twice. What is the probability of getting composite numbers for both
rolls?
A.
1
25
B.
1
16
C.
1
9
D.
1
4
E.
1
2

2. 6. If 𝑎 + √𝑏 = √67 + 16√3, find the value of 𝑎2
+ 𝑏.
A. 3 B. 16 C. 16√3 D. 67 E. 67√3
7. What is the measure of the acute angle between the hour and minute hands of a correctly working
clock at 3: 16?
A. 1° B. 2° C. 3° D. 4° E. 5°
8. If 3−𝑥
= 5, what is the value of 125 × 27𝑥−1
?
A.
1
125
B.
1
27
C.
1
3
D.
1
5
E. 1
9. Find the area of a regular hexagon with side length of 6cm.
A. 6√3 𝑐𝑚2
B. 18√3 𝑐𝑚2
C. 30√3 𝑐𝑚2
D. 42√3 𝑐𝑚2
E. 54√3 𝑐𝑚2
10. A certain test is taken by children and adults. The average score of the children is 5 points while the
average score of the adults is 15 points. If the average score of all the children and adults together is
12 points, what percentage of the takers are children?
A. 30% B. 40% C. 50% D. 60% E. 70%
Open-ended Questions
11. How many positive divisors does 90 have?
12. A piece of cardboard is 3 ft by 5 ft in size. You are planning to cut out 90 triangles such that all of
them are of equal size and the cardboard is totally used up. What should be the area of each triangle
in square inches?
13. Given that 𝑎 > 𝑏, 𝑎 + 𝑏 = 7, and 𝑎2
+ 𝑏2
= 37, what is the value of 𝑎3
− 𝑏3
?
14. A polyhedron with 14 vertices and 20 edges. How many faces does it have?
15. In how many can you sit a group of 6 people around a circular table if two of them insist on sitting
next to each other?
16. If yesterday is Thursday, what day is 20232024
days from today?

3. 17. In a certain furniture shop, 4 workers can build 10 wooden tables in 4 days. At the same rate, how
many workers are needed to build 15 wooden tables in exactly 8 days?
18. What is the sum of the reciprocal of the roots of the equation 2𝑥2
− 10𝑥 + 14 = 0?
19. How many diagonals does a 14-sided convex polygon have?
20. In how many ways can you distribute 8 identical snacks to three different kids if a kid should get at
least one snack?
21. How many integers 𝑛 would make the expression
5𝑛+11
3𝑛+4
an integer?
22. In a toy store, the value of 1 red marble is the same as 2 blue marbles, the value of 3 blue marbles is
the same as 4 yellow marbles, and the value of 5 yellow marbles is the same as 6 white marbles. You
bought several red marbles yesterday and decided to have some of them replaced with blue and
white marbles today. If you need the same number of blue and white marbles, what is the least
number of red marbles you need to have replaced?
23. Given that 𝑎 $ 𝑏 =
𝑎3−𝑏3
𝑎2−𝑏2
, find the value of 2 (
𝑥2+𝑦2
𝑥+𝑦
) if 𝑥 $ 𝑦 =
89
10
and
1
𝑥
+
1
𝑦
=
10
19
.
24. A regular hexagon with an area of 6√3 is circumscribing a circle and at the same time is inscribed in
another circle. What is the sum of the areas of the two circles?
25. Suppose you randomly arrange three green cups, three orange cups, and three purple cups in a
single row. What is the probability that all the purple cups are right next to each other?

4. VTAMPS 15
Secondary 3 Set 1
Multiple Choice
1. Find the sum of all odd integers between 300 and 400 that are divisible by both 5 and 7.
A. 665 B. 700 C. 735 D. 770 E. 805
Answer: B
Solution:
Note that a number is divisible by both 5 and 7 when it is a multiple of 35.
Hence, we only consider the odd multiples of 35 between 300 and 400, and those are
35 × 9 = 315
and
35 × 11 = 385.
Then their sum is
315 + 385 = 700.
2. A rectangular land, 500 meters long and 600 meters wide, is to be divided between two kids as
inheritance in the ratio 1: 5. What is the area, in hectares, of the larger land?
A. 15 B. 20 C. 25 D. 30 E. 35
Answer: C.
Solution:
𝐴𝑟𝑒𝑎𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 = 𝐿𝑒𝑛𝑔𝑡ℎ × 𝑊𝑖𝑑𝑡ℎ
= 500 𝑚 × 600 𝑚
= 300000 𝑚2
Since the land is going to be divided as 1: 5,
Then
1𝑥 + 5𝑥 = 6𝑥
6𝑥 = 300000 𝑚2
𝑥 = 50000 𝑚2
5𝑥 = 250000 𝑚2
Converting to hectares,
250000 𝑚2
×
1 ℎ𝑎
10000 𝑚2
= 25 ℎ𝑎.

5. 3. The slope of a line is 3 and it crosses the 𝑦-axis at (0, −5). At what point will this line cross the 𝑥-
axis?
A. (−
15
8
, 0)
B. (−
5
3
, 0)
C. (
5
3
, 0)
D. (
15
8
, 0)
E. (
21
10
, 0)
Answer: C. (
5
3
, 0)
Solution:
Using the point-slope form,
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1),
we suppose 𝑦 = 0, to get the 𝑥-intercept.
Then
0 − (−5) = 3(𝑥 − 0)
5 = 3𝑥
𝑥 =
5
3
.
Hence, the line will cross at point (
5
3
, 0).
OR
Using the slope-intercept form,
𝑦 = 𝑚𝑥 + 𝑏,
we suppose 𝑦 = 0, to get the 𝑥-intercept.
Then
0 = 3(𝑥) + (−5)
3𝑥 = 5
𝑥 =
5
3
.
Once again, the line will cross at point (
5
3
, 0).
4. A square with an area of 8 sq. units is inscribed in a circle. Find the area of the circle.
A. 2𝜋 B. 4𝜋 C. 6𝜋 D. 8𝜋 E. 10𝜋
Answer: B.
Solution:
Note that the side length of the square is √8.
Since the square is inscribed in the circle, then the diagonal of the square is the diameter of the
circle. Solving for the diagonal, we have
𝐷𝑖𝑎𝑔𝑜𝑛𝑎𝑙 = √𝑠2 + 𝑠2
= √2𝑠2
= √2(8)

6. = √16
= 4.
Since the diagonal is the diameter, then half of it would be the radius, that is
𝑅𝑎𝑑𝑖𝑢𝑠 =
𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟
2
=
4
2
= 2.
Solving for the area,
𝜋𝑟2
= 𝜋(2)2
= 𝜋(4)
= 4𝜋.
5. A fair six-sided die is rolled twice. What is the probability of getting composite numbers for both
rolls?
A.
1
25
B.
1
16
C.
1
9
D.
1
4
E.
1
2
Answer: C.
Solution:
Note that there are only two composite numbers on a six-sided die, which are 4 and 6.
Hence, the probability of getting composite numbers for both rolls would be
2
6
×
2
6
=
4
36
=
1
9
.
6. If 𝑎 + √𝑏 = √67 + 16√3, find the value of 𝑎2
+ 𝑏.
A. 3 B. 16 C. 16√3 D. 67 E. 67√3
Answer: D.
Solution:
𝑎 + √𝑏 = √67 + 16√3
𝑎2
+ 2𝑎√𝑏 + 𝑏 = 67 + 16√3
𝑎2
+ 𝑏 + 2𝑎√𝑏 = 67 + 16√3
By observation, we can see that
2𝑎√𝑏 = 16√3
and
𝑎2
+ 𝑏 = 67.
7. What is the measure of the acute angle between the hour and minute hands of a correctly working
clock at 3: 16?
A. 1° B. 2° C. 3° D. 4° E. 5°
Answer: B.
Solution:
Using the formula
𝜃 = |30°𝐻 −
11
2
°𝑀|,

7. we have,
𝜃 = |30°(3) −
11
2
°(16)|
= |90° − 88°|
= 2°
8. If 3−𝑥
= 5, what is the value of 125 × 27𝑥−1
?
A.
1
125
B.
1
27
C.
1
3
D.
1
5
E. 1
Answer: A.
Solution:
125 × 27𝑥−1
= 53
× 33(𝑥−1)
= (3−𝑥)3
× 33𝑥−3
= 3−3𝑥
× 33𝑥−3
= 3(−3𝑥)+(3𝑥−3)
= 3−3
=
1
33
=
1
27
9. Find the area of a regular hexagon with side length of 6cm.
A. 6√3 𝑐𝑚2
B. 18√3 𝑐𝑚2
C. 30√3 𝑐𝑚2
D. 42√3 𝑐𝑚2
E. 54√3 𝑐𝑚2
Answer: E.
Solution:
𝐴𝑟𝑒𝑎𝐻𝑒𝑥𝑎𝑔𝑜𝑛 =
3√3
2
𝑠2
=
3√3
2
(6 𝑐𝑚)2
=
3√3
2
(36 𝑐𝑚2)
= 54√3 𝑐𝑚2
10. A certain test is taken by children and adults. The average score of the children is 5 points while the
average score of the adults is 15 points. If the average score of all the children and adults together is
12 points, what percentage of the takers are children?
A. 30% B. 40% C. 50% D. 60% E. 70%
Answer: A.
Solution:
Let 𝑥 be the number of adults and let 𝑦 be the number of children.

8. Then
15𝑥 + 5𝑦
𝑥 + 𝑦
= 12
15𝑥 + 5𝑦 = 12𝑥 + 12𝑦
3𝑥 = 7𝑦
𝑥
𝑦
=
7
3
That is, 70% of the takers are adults, and 30% of the takers are children.
Open-ended Questions
11. How many positive divisors does 90 have?
Answer: 12
Solution:
90 = 2 × 32
× 5
Now, let’s add 1 to each of the exponents of each prime factor, and multiply them altogether, that is
(1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12.
12. A piece of cardboard is 3 ft by 5 ft in size. You are planning to cut out 90 triangles such that all of
them are of equal size and the cardboard is totally used up. What should be the area of each triangle
in square inches?
Answer: 24
Solution:
𝐴𝑟𝑒𝑎𝑐𝑎𝑟𝑑𝑏𝑜𝑎𝑟𝑑 = 𝑙𝑒𝑛𝑔𝑡ℎ × 𝑤𝑖𝑑𝑡ℎ
= 3 𝑓𝑡 × 5 𝑓𝑡
= 15 𝑓𝑡2
𝐴𝑟𝑒𝑎𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 =
𝐴𝑟𝑒𝑎𝑐𝑎𝑟𝑑𝑏𝑜𝑎𝑟𝑑
90
=
15𝑓𝑡2
90
=
1
6
𝑓𝑡2
Converting to square inches,
1
6
𝑓𝑡2
×
12𝑖𝑛
1𝑓𝑡
×
12𝑖𝑛
1𝑓𝑡
=
12
6
𝑖𝑛 × 12𝑖𝑛
= 24𝑖𝑛2
.
13. Given that 𝑎 > 𝑏, 𝑎 + 𝑏 = 7, and 𝑎2
+ 𝑏2
= 37, what is the value of 𝑎3
− 𝑏3
?
Answer: 215
Solution:
Observe that
𝑎 + 𝑏 = 7

9. (𝑎 + 𝑏)2
= 49
𝑎2
+ 2𝑎𝑏 + 𝑏2
= 49
𝑎2
+ 𝑏2
+ 2𝑎𝑏 = 37 + 12
2𝑎𝑏 = 12
𝑎𝑏 = 6
Then
𝑎 − 𝑏 = √(𝑎 − 𝑏)2
= √(𝑎2 + 𝑏2 − 2𝑎𝑏)
= √37 − 2(6)
= √37 − 12
= √25
= 5
Now
(𝑎3
− 𝑏3) = (𝑎 − 𝑏)(𝑎2
+ 𝑎𝑏 + 𝑏2)
= 5(37 + 6)
= 5(43)
= 215.
14. A polyhedron with 14 vertices and 20 edges. How many faces does it have?
Answer: 8
Solution:
Using the Euler’s Polyhedron Formula
𝐹 + 𝑉 = 𝐸 + 2,
we get
𝐹 = 𝐸 + 2 − 𝑉
𝐹 = 20 + 2 − 14 = 8.
15. In how many can you sit a group of 6 people around a circular table if two of them insist on sitting
next to each other?
Answer: 48
Solution:
In this scenario, we are to consider these two as one, and so instead of having 6 people, now we
have 5.
Following directly with the formula, we have
(𝑛 − 1)! = (5 − 1)! = 4! = 24.
Now, we also have to take into consideration the arrangement of the two people who insisted to seat
with each other, that is 2.
Therefore,
24 × 2 = 48.
16. If yesterday is Thursday, what day is 20232024
days from today?
Answer: Friday
Solution:

10. Observe that 2023 is divisible by 7.
Hence, to determine what day is 20232024
days from today, we have
20232024
≡ 0 𝑚𝑜𝑑 7.
Thus, 0 days from today, Friday, is Friday.
17. In a certain furniture shop, 4 workers can build 10 wooden tables in 4 days. At the same rate, how
many workers are needed to build 15 wooden tables in exactly 8 days?
Answer: 3
Solution:
Using the formula
𝑊𝑜𝑟𝑘
𝑊𝑜𝑟𝑘𝑒𝑟𝑠 × 𝑇𝑖𝑚𝑒
,
we have
10
4 × 4
=
15
𝑊 × 8
𝑊 =
15(16)
10 × 8
𝑊 =
3(2)
2
= 3.
18. What is the sum of the reciprocal of the roots of the equation 2𝑥2
− 10𝑥 + 14 = 0?
Answer:
5
7
Solution:
Let 𝑎, 𝑏 be the roots of the equation.
Then
𝑎 + 𝑏 = −
−10
2
= 5,
and
𝑎𝑏 =
14
2
= 7.
Solving for the sum of the reciprocal of the roots, we have
1
𝑎
+
1
𝑏
=
𝑎 + 𝑏
𝑎𝑏
=
5
7
.
19. How many diagonals does a 14-sided convex polygon have?
Answer: 77
Solution:
Using the formula
𝑑 =
𝑛(𝑛 − 3)
2
,
We have
𝑑 =
14(14 − 3)
2
= 7(11) = 77.
20. In how many ways can you distribute 8 identical snacks to three different kids if a kid should get at
least one snack?

11. Answer: 21
Solution:
(
𝑛 − 1
𝑘 − 1
) = (
8 − 1
3 − 1
) = 7𝐶2
=
7!
5! 2!
=
7 × 6
2
= 21
21. How many integers 𝑛 would make the expression
5𝑛+11
3𝑛+4
an integer?
Answer: 2
Solution:
Rewriting,
5𝑛 + 11
3𝑛 + 4
5
3
3𝑛 + 4 5𝑛 + 11
5𝑛 +
20
3
13
3
Hence,
5𝑛 + 11
3𝑛 + 4
=
5
3
+
13
3(3𝑛 + 4)
.
Now, for the expression to be an integer, 3𝑛 + 4 must be 13 or 1, that is
3𝑛 + 4 = 13
3𝑛 = 9
𝑛 = 3
and
3𝑛 + 4 = 1
3𝑛 = −3
𝑛 = −1.
Therefore, there are only 2 integers for 𝑛 that would make the expression an integer.
22. In a toy store, the value of 1 red marble is the same as 2 blue marbles, the value of 3 blue marbles is
the same as 4 yellow marbles, and the value of 5 yellow marbles is the same as 6 white marbles. You
bought several red marbles yesterday and decided to have some of them replaced with blue and
white marbles today. If you need the same number of blue and white marbles, what is the least
number of red marbles you need to have replaced?
Answer: 13
Solution:
𝑅: 𝐵 = 1: 2
𝐵: 𝑌 = 3: 4
⟹ 𝑅: 𝐵: 𝑌 = 1(3): (2)(3): 4(2) = 3: 6: 8
𝑌: 𝑊 = 5: 6
⟹ 𝑅: 𝐵: 𝑌: 𝑊 = 3(5): 6(5): (8)(5): 6(8)

12. = 15: 30: 40: 48
From this, we now have the value of Red, Blue, and White marbles altogether.
Since there is no need for us to use the value of the Yellow marbles now, we can just have the ratio
for Red:White as
15: 48
Or
5: 16.
Now, note that the ratio of Red:Blue is 1:2.
Also, the LCM of 2 and 16 is 16.
Hence, when we have Blue = White, the earliest that would happen is when
𝑅𝑒𝑑: 𝐵𝑙𝑢𝑒
1: 2
8: 16
And
𝑅𝑒𝑑: 𝑊ℎ𝑖𝑡𝑒
5: 16
Adding the red marbles,
8 + 5 = 13.
Therefore, if we need to have the same number of blue and white marbles, the least number of red
marbles to replace is 13.
23. Given that 𝑎 $ 𝑏 =
𝑎3−𝑏3
𝑎2−𝑏2
, find the value of 2 (
𝑥2+𝑦2
𝑥+𝑦
) if 𝑥 $ 𝑦 =
89
10
and
1
𝑥
+
1
𝑦
=
10
19
.
Answer: 14
Solution:
Observe that
1
𝑥
+
1
𝑦
=
𝑥 + 𝑦
𝑥𝑦
=
10
19
.
Hence, 𝑥 + 𝑦 = 10𝑛 and 𝑥𝑦 = 19𝑛.
Now,
𝑥 $ 𝑦 =
𝑥3
− 𝑦3
𝑥2 − 𝑦2
𝑥 $ 𝑦 =
(𝑥 − 𝑦)(𝑥2
+ 𝑥𝑦 + 𝑦2)
(𝑥 + 𝑦)(𝑥 − 𝑦)
𝑥 $ 𝑦 =
𝑥2
+ 𝑦2
+ 𝑥𝑦
𝑥 + 𝑦
89
10
=
𝑥2
+ 𝑦2
+ 19𝑛
10𝑛
(89)(10𝑛) = 10(𝑥2
+ 𝑦2
+ 19𝑛)
890𝑛 − 190𝑛 = 10(𝑥2
+ 𝑦2)
700𝑛 = 10(𝑥2
+ 𝑦2
)
𝑥2
+ 𝑦2
= 70𝑛
Then

13. 2 (
𝑥2
+ 𝑦2
𝑥 + 𝑦
) = 2 (
70𝑛
10𝑛
) = 2(7) = 14.
24. A regular hexagon with an area of 6√3 is circumscribing a circle and at the same time is inscribed in
another circle. What is the sum of the areas of the two circles?
Answer: 7𝜋
Solution:
Note that the regular hexagon is made up of 6 equilateral triangles, such that each of these triangles
has an area of
6√3
6
= √3.
Also, 𝐴𝑟𝑒𝑎𝑒𝑞𝑢𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 =
√3
4
𝑠2
.
Then the side lengths of such triangles are
𝑠 = √
4𝐴𝑟𝑒𝑎𝑒𝑞𝑢𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
√3
= √
4√3
√3
= √4 = 2.
Furthermore, the heights of such triangles are
ℎ =
2𝐴𝑟𝑒𝑎𝑒𝑞𝑢𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
𝑠
=
2√3
2
= √3
since 𝐴𝑟𝑒𝑎𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 =
1
2
𝑠ℎ.
Observe that the radius of the bigger circle is the side length of the triangle and the radius of the
smaller circle is the height of the triangle, that is
𝑠 = 𝑅 & ℎ = 𝑟.
Finally, the sum of the areas of the circles would be
𝜋𝑅2
+ 𝜋𝑟2
= 𝜋(2)2
+ 𝜋(√3)
2
= 4𝜋 + 3𝜋
= 7𝜋.
25. Suppose you randomly arrange three green cups, three orange cups, and three purple cups in a
single row. What is the probability that all the purple cups are right next to each other?
Answer:
1
12
Solution:
Let us group the purple cups as one, such that we now have 7 cups, with 3 green, and 3 orange cups.
Then
𝑠𝑢𝑐𝑐𝑒𝑠𝑠
𝑡𝑜𝑡𝑎𝑙
=
7!
3! 3!
9!
3! 3! 3!
=
7!
3! 3!
×
3! 3! 3!
9!
=
7! 3! 3! 3!
3! 3! 9!
=
3 × 2
9 × 8
=
1
12
.