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Question 5: There are 1950 orange trees, tangerine trees and litchi trees in a garden where two thirds of the orange trees are equal to three fifths of the tangerine trees and [r]
(1)1 PEOPLE’S COMMITTEE OF THANH OAI DISTRICT
COMMITTEE DIVISION OF EDUCATION AND TRAINING 2019 IMSO Mathematics Competition
Part A: Mathematics Essay Problems
Question 1: Now, a father’s age is three and a half times his son’s age Six years ago, the father was six times as old as his son What are their present ages?
Solution
Let the father’s age be 7k, then his son’s age be 2k (k ∈ N*)
Six years ago, the father’s age was (7k – 6), and the son’s age was (2k – 6) Because six years ago, the father was six times as old as his son, we have:
7k – = (2k – 6) 7k – = 12k – 36 5k = 30 or k =
Hence, the father is 42 years old and his son is 12 years old now
Question 2:
Compare the following two fractions: A = 10
2005 + 1
102006 + 1 and B =
102004 + 102005 + 1
Solution
10A = 10
2006 + 10
102006 + = +
102006 + (1)
10B = 10
2005 + 10
102005 + = +
102005 + (2)
From (1) and (2), we obtain 10A < 10B or A < B
Question 3:What is the value of each of the expressions below? a, P = ˗
7 + 72 ˗
1 73 +
1 74 ˗
1
75 + ˗
1 799 +
1 7100
b, N = + 53 + 56 + 59 + + 599
(2)2
a, P = ˗ +
1 72 ˗
1 73 +
1 74 ˗
1
75 + ˗ 799 +
1 7100
7P = ˗ + ˗
1 72 +
1 73 ˗
1
74 + ˗
1 798 +
1 799
7P + P = + 7100
8P =
101 + 1
7100 or P =
7101 + 1
8 7100
b, N = + 53 + 56 + 59 + + 599
53 N = 53 + 56 + 59 + + 599 + 5102
125 N – N = 5102 -
124 N = 5102 - 1 or N = 5 102 - 1
124
Question 4:
a, Given P = 1 23 +
5 22 32 +
7
32 42 + +
19 92 102
Prove that P <
b, What is the number of terms in the sum S = + + + such that S is a three-digit number where its three three-digits are the same?
Solution
a, P =
2 - 12
1 22 +
32 - 22 22 32 +
42 - 32
32 42 + +
102 - 92 92 102
P = 112 -
1 22 +
1 22 -
1 32 +
1 32 -
1
42 + +
1 92 -
1 102
P = - 1001 Hence, P <
b, Let the number that needs finding be aaa = 111 a (with a ≠ 0) Let the number of terms in the sum S be n
(3)3
It follows that n (n + 1) is divisible by 37 Indeed, 37 is a prime number, and (n + 1) must be smaller than 74 Therefore, n = 37 or n + = 37
If n = 37, then n + = 38 It follows that n (n + 1): = 37 x 38 : = 703 (this is not a satisfying answer)
If n + = 37, then n = 36 Hence, n (n + 1): = 37 x 36 : = 666 (this is a satisfying answer)
Thus, the number of terms in the sum is 36
Question 5:
Let MNP be a triangle with angle N = 50 and MN = 3cm On the opposite ray of ray MN is point Q such that MQ = 7cm Ray Nq bisects angle PNM On the left half-plane with boundary line NQ, not containing point P, draw ray Np’ such that angle QNp’ is half the size angle PNM
a, Let I and J be the mid-points of MN and MQ respectively Determine the measure of IJ?
b, Determine the measure of angle QNp
c, Prove that Np and Np’ are two opposite rays
Solution
a, Because I is the midpoint of MN, we have MI = MN2 = 32 = 1.5 (cm) (1) Because J is the midpoint of MQ, similarly we have MJ = MQ
2 =
2 = 3.5 (cm) (2) From (1) and (2), we induce that MI < MJ or I is between M and J
N
M
P
Q
p’ J
(4)4
It follows that IJ = MJ - MI = 3.5 - 1.5 = (cm)
b, Np is the bisector of angle PNM, so MNp = PNM : = 25
Because NM and NQ are two opposite rays, MNp and pNQ are adjacent supplementary angles
It follows that pNQ = 180 – 25 = 155 c, From the assumption, we have QNp’ =
2 PNM = 25 Moreover, pNQ = 155 (as proved in section b)
Hence, QNp’ + pNQ = 180
Additionally, ray Np and ray Np’ lie on the two opposite half-planes bounded by NQ Thus, pNp’ = QNp’ + QNP = 180
Or Np and Np’ are two opposite rays Part B: Short answer problems
Question 1: Find the last digit of expression P = 19871987 + 20182019
Answer: 5
Question 2: Reduce expression Q =
4 315 + 46 97
47 275 - (23)4 814
Answer: 4 5
Question 3: Given x + (x + 1) + (x + 2) + + (x + 100) = 55 101 Solve for x
Answer:5
Question 4: Let xoy and yoz be two adjacent supplementary angles such that yoz = 45 xoy Determine the measures of these two angles
Answer: 100 and 80
Question 5: There are 1950 orange trees, tangerine trees and litchi trees in a garden where two thirds of the orange trees are equal to three fifths of the tangerine trees and equal to six sevenths of the litchi trees How many fruit trees of each kind are there in
(5)