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TIMO - 2020 - Grade 7 - Preliminary Round - Paper 4

Question 1

Given $A,D$ are two nonzero digits, the two-digit numbers formed by them satisfy:
1. $\overline{DA}$ is prime.
2. $\overline{AD}$ is the product of two primes greater than 5.
Find $\overline{DA}$.
  1. 91
  2. 19
  3. 53
  4. 35

Question 2

Andy goes north for 20km, then goes east for 15km. How far is he now from the original position?
  1. 15km
  2. 20km
  3. 25km
  4. 35km

Question 3

How many times does hour hand and minute hand of a clock form a right angle in a day?
  1. 42
  2. 44
  3. 46
  4. 48

Question 4

Amy has been reading a novel. She read 1 page in the first day, 2 pages in the second day, 3 pages in the third day, etc, until the total pages she read is a multiple of 56. How many pages has she read until then?
  1. 1176
  2. 840
  3. 1288
  4. 1400

Question 5

There are 4 pairs of white chopsticks, 6 pairs of yellow chopsticks and 3 pairs of brown chopsticks mixed together. Close your eyes. If you want to get 1 pair of chopsticks of each colour, at least how many pieces of chopsticks are needed to be taken?
  1. 6
  2. 11
  3. 22
  4. 24

Question 7

Positive integers $x,y$ satisfy $2x+4y=48$. Find the largest value of $x+y$.
  1. 16
  2. 19
  3. 25
  4. 23

Question 8

How many integer solutions $x$ satisfy $777>9x+25>-29$?
  1. 91
  2. 89
  3. 87
  4. 85

Question 9

Positive real numbers $a,b$ satisfy $a^2-b^2=102$ and $a=b+6$. Find $a+b$.
  1. 15
  2. 17
  3. 19
  4. 21

Question 10

Factorize $x^2+11x+30$.
  1. $(x+5)(x+6)$
  2. $(x-5)(x+6)$
  3. $(x-5)(x-6)$
  4. $(x+5)(x-6)$

Question 11

Find the sum of all positive factors of the 7777.
  1. 119
  2. 7897
  3. 9792
  4. 9987

Question 12

From 1 to 2018, how many integers are divisible by 3 and 4 but not by 8?
  1. 84
  2. 82
  3. 168
  4. 160

Question 13

$B$ is an integer such that $B^{17}$ is a 21-digit number with units digit 6. Find $B$.
  1. 12
  2. 24
  3. 16
  4. 26

Question 14

Given $x>0$ and $x+\frac1x=3$, find $x^2+\frac1{x^2}$.
  1. 7
  2. 9
  3. 5
  4. 6

Question 15

The difference between two square numbers is 348. What is minimum value of the larger square?
  1. 1024
  2. 7744
  3. 1936
  4. 676

Question 16

A cube with side length 12 is cut into cubes with side length 1. Find the increase in total surface area of the cubes.
  1. 9504
  2. 864
  3. 10368
  4. 5184

Question 17

An iron wire is bent into a circle of radius 14. If it is now bent into a rectangle, what is the maximum area of the rectangle? Take $\pi=\frac{22}{7}$.
  1. 44
  2. 88
  3. 1936
  4. 484

Question 18

Small cubes with side length 1 are combined according to the pattern shown below. If there are 31 layers, find the volume.

Question illustration
  1. 1890
  2. 1891
  3. 2015
  4. 2020

Question 19

The hypotenuse of a right triangle is 10. What is the maximum area of the triangle?
  1. 25
  2. 24
  3. 30
  4. 50

Question 20

An interior angle of a regular n-sided polygon is 5 times of an exterior angle. Find n.
  1. 10
  2. 12
  3. 14
  4. 15