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TIMO - 2018 - Grade 9 - Preliminary Round - Paper 4

Question 1

Given A, B and C are three even non-zero digits, we have: 1. CBA is divisible by 9; 2. CAB is not divisible by 4; 3. ABC is not divisible by 8. Find the 3-digit number BCB given that A, B and C are different digits.
  1. 464
  2. 686
  3. 646
  4. 868

Question 2

There are n lines that are not parallel with each other on a plane. There are no 3 lines intersecting at a point. If they intersect 66 times, find n.
  1. 10
  2. 11
  3. 12
  4. 13

Question 3

Andy goes southeast for 22km, then goes southwest for 18km, goes northwest for 34km and goes northeast for 23km. How far is he now from the original position?
  1. 13km
  2. 97km
  3. 17km
  4. 95km

Question 4

In front of Graham’s Bicycle Bazaar there are some unicycles, some bicycles and some tricycles. Lucy sees that there are seven saddles in total, thirteen wheels in total and more bicycles than tricycles. How many unicycles are there?
  1. 1
  2. 2
  3. 3
  4. 4

Question 5

Amy is reading a novel. She reads 1 page on the first day, 2 pages on the second day, 3 pages on the third day, etc, until the total number of pages she reads is a multiple of 50. At least how many days does it take her?
  1. 99
  2. 100
  3. 25
  4. 24

Question 7

Factorize $x^2-2y^2-xy-x+5y-2$.
  1. $(x-2y-1)(x+y+2)$
  2. $(x+2y+1)(x-y-2)$
  3. $(x-2y+1)(x+y-2)$
  4. $(x-2y+1)(x-y-2)$

Question 8

Simplify $\sqrt{28-16\sqrt3}$.
  1. $4\sqrt3-2$
  2. $4-2\sqrt3$
  3. $2\sqrt3-4$
  4. $2-4\sqrt3$

Question 9

If $x^4+mx^3+nx^2-10x+12$ is divisible by $x^2-1$, find $m$.
  1. 13
  2. 10
  3. -13
  4. -10

Question 10

Find the greatest value of $-x^2+14x-25$.
  1. -24
  2. 171
  3. -25
  4. 24

Question 11

The 5-digit number $\overline{1A0B9}$ is divisible by 11. Find its remainder when divided by 9.
  1. 2
  2. 3
  3. 4
  4. 5

Question 12

Find the remainder when $2^{543}$ is divided by 11.
  1. 2
  2. 4
  3. 6
  4. 8

Question 13

Given $2x\equiv322\pmod{19}$, find the minimum value of $\lvert x\rvert$.
  1. 10
  2. 18
  3. 9
  4. 1

Question 14

It is February now. Which month will it be after $13^{17}$ months?
  1. January
  2. April
  3. March
  4. February

Question 15

Let $x,y$ be real numbers with $x+y=7$ and $x^2+y^2=29$. Find $x^3-2x^2y-2xy^2+y^3$.
  1. 7
  2. -7
  3. 29
  4. -29

Question 16

Find the area enclosed by the x-axis, y-axis and the line $y=\frac43x+8$.
  1. 24
  2. 48
  3. 16
  4. 36

Question 17

Find the maximum value of $\sin^2x+\cos x$.
  1. $\frac32$
  2. $\frac12$
  3. $\frac34$
  4. $\frac54$

Question 18

An interior angle of a regular polygon with $n^2-n$ sides is 12 degrees more than thirteen times an exterior angle. Given $n>0$, find $n$.
  1. 5
  2. 6
  3. 11
  4. 12

Question 19

A triangle has sides with lengths 14, 48 and 50. Find the value of the radius of circumscribed circle of that triangle.
  1. 25
  2. 37
  3. 32
  4. 30

Question 20

Find the shortest distance from the origin $O$ to the line $3y=4x-15$.
  1. 3
  2. 4
  3. 5
  4. 6