2015 JUNIOR DIVISION FIRST ROUND PAPER

7/25/2020 8:04:00 AM

What is the value of ?

  • 0
  • 32
  • 33

Someone set the alarm clock for 1:30 pm and fell asleep at 12:35 pm. When he was awaken by the alarm clock, for how long had this person been sleeping?

  • 1 hour and 5 minutes
  • 55 minutes
  • 95 minutes
  • 105 minutes
  • 11 hours and 5 minutes

In a quadrilateral ABCD, AB // DC, BC // ED, ∠C = 110°. What is the measure of ∠A? 

  • 20°
  • 35°
  • 40°
  • 55°
  • 70°

In a sale, each dress is reduced to 49% of its price and if two dresses are purchased at the same time, both are reduced to 45% of their prices. Lily buys two dresses together and pays 90 dollars for both. By doing this instead of buying them separately, how many dollars has she saved?

  • 10
  • 8
  • 6
  • 4
  • 3.6

Sixteen points are arranged in a 3 cm by 3 cm formation. Four of them are removed, leaving behind twelve points as shown in the diagram. If we choose three of these twelve points as vertices of a triangle, what is the largest possible area of this triangle, in ?

Class A has 17 students more than Class B, which has 15 students less than Class C. Of the following five numbers, which can be the total number of students in these three classes? 

  • 150
  • 151
  • 152
  • 153
  • 154

From 0, 1, 2, 3, 4 and 5, we choose two different numbers x and y. What is the largest possible value of ?

  • 75
  • 163
  • 175
  • 187
  • 200

Divide the rectangle ABCD into four isosceles right triangles and one square, as in the diagram left. If the area of square EFGH is 100 , what is the area of rectangle ABCD, in ?

  • 750
  • 1000
  • 1100
  • 1200
  • 1600

A group of students are staying in a hotel. If five of them share a room, then there is no room for six of them. If six of them share a room, there are just enough rooms, one of which has less than six students. Of the following five numbers, which cannot be the number of students? 

  • 46
  • 51
  • 56
  • 61
  • 66

In a pentagon, one angle is 48°. The second angle is three times as large as the first. The third angle is 30° less than the second. The fourth angle is 10° less than the fifth. What is the measure of the fourth angle, in degrees?

  • 112
  • 122
  • 132
  • 142
  • 152

There are three shirts, three pairs of trousers and three pairs of shoes. Of each type, one is red, one is black and one is white. In how many different ways can we choose one of each type so that something white is chosen? 

  • 8
  • 9
  • 18
  • 19
  • 27

 In triangle ABC, AB is perpendicular to BC. D and E are points on BC such that ∠BAD = ∠DAE = ∠EAC and ∠ADC - ∠C = 56°. What is the measure of ∠BAC?

  • 42°
  • 45°
  • 51°
  • 60°
  • 84°

If and , what is the value of  

  • 1
  • 2
  • 3
  • 4
  • 5

C and D are points on AB such that AC : CD : DB = 1 : 2 : 3. Semicircles are drawn on the same side of AB with respective diameters AB, AC, CD and DB. What fraction of the area of the largest semicircle is the total area of the other three semicircles?

Each coin is worth either 1 dollar, 5 dollars or 10 dollars. Their total worth is 60 dollars. They may be divided into three, four or five piles of equal worth. What is the minimum number of coins? 

  • 6
  • 11
  • 15
  • 16
  • 20

From a cube of side length 10 cm, a cylinder with diameter 6 cm and depth 8 cm is hollowed out. What is the volume, in , of the remaining part of the cube? Take .

  • 426.08
  • 517.46
  • 573.94
  • 717.46
  • 773.92

If a, b and c are all positive integers, which of the following numbers can be the value of (a + b + c)(a + b - c)(a - b + c)(-a + b + c)?

  • 24
  • 54
  • 48
  • 60
  • 100

A three-layer structure consists of 14 unit cubes. The bottom layer consists of 9 cubes in a 3 by 3 configuration. The middle layer consists of 4 cubes in a 2 by 2 configuration. The top layer consists of a single cube. The exposed surface area of this structure is painted, including the bottom. What is the total area of the unpainted surface of the individual cubes?

  • 20
  • 31
  • 42
  • 53
  • 64

In an election between four candidates, they are supported respectively by 11, 12, 13 and 14 of the first 50 voters. Six more votes are to be cast, each for one of the four candidates. In how many ways can the candidate currently with 13 supporters become the uncontested winner? 

  • 16
  • 17
  • 18
  • 19
  • 20

Let x, y and z be distinct positive prime numbers such that x + y + z and are also prime numbers. What is the minimum value of x + y + z?

  • 17
  • 19
  • 23
  • 29
  • 31

ABCDEF is a regular hexagon. G is the midpoint of AB and H is the point on AF such that FH=2AH. If the area of triangle AHG is 1. What is the area, in , of ABCDEF?

Answer:

What is the value of abc where a, b and c are positive real numbers such that a(b + c) = 48, b(c + a) = 70 and c(a + b) = 88?

Answer:

What is the value of where a and b are real numbers such that ?

Answer:

What is the maximum value of a if

Answer:

For any permutation of 1, 2, 3, 4, 5, 6, 7 and 8, add the second number to the first, multiply the sum by the third number, add the fourth number to the product, multiply the sum by the fifth number, and so on. What is the minimum value of the final sum?

Answer: