### Reg. Round 2015

```Cumann Oidí Matamaitice na hÉireann
Irish Mathematics Teachers Association
Foireann Mata 2015
Babhta Réigiúnach
Team Maths 2015
R
Regional Round
BABHTA 1
ROUND 1
1)
Write 2 n - 2 n-1 in form 2x
2)
120 people are having a meeting at which they are to be
separated into equal-sized groups having at least three but no
more than 12 to a group.
How many different group sizes are possible?
BABHTA 2
1)
ROUND 2
Find the sum of the distances from one vertex of a square of
side 2 cm to the midpoints of each of the sides of the square.
Answer in the form a +b c , where a, b and c ∈ N .
2)
A circle, which passes through the origin, cuts off intercepts
of lengths 4 and 6 units on the positive x- and y- axes
respectively. Find the equation of the circle.
Answer in form x2 + y2 + 2gx + 2fy + c = 0
Cumann Oidí Matamaitice na hÉireann
Irish Mathematics Teachers Association
Foireann Mata 2015
Team Maths 2015
Babhta Réigiúnach
R
Regional Round
BABHTA 3
ROUND 3
1) If a and b ∈ R find the numerical value of a + b when
4cos2 (θ)−3 = a+bsin(θ) , where Sin(θ)≠ 1
1−2sin(θ)
2
2)
For what positive value of k does the line x + y = k intersect
the circle x2 + y2 = 3 at one point only?
Answer in the form a , where a ∈ N
BABHTA 4
ROUND 4
1)
A rectangle with sides in the ratio 3 : 5 is inscribed in a circle.
The four vertices of the rectangle are on the circle.
Calculate the ratio of the area of the rectangle to the area of the
circle.
Answer in simplest form a , where a and b ∈ N
bπ
2)
The following array of integers is called a ladder; each
horizontal row is called a rung. Find the sum of the pair of
integers on the 7th row.
1
2
5
12
.
.
.
1
3
7
17
.
.
.
Cumann Oidí Matamaitice na hÉireann
Irish Mathematics Teachers Association
Foireann Mata 2015
Babhta Réigiúnach
Team Maths 2015
R
Regional Round
BABHTA 5
1)
ROUND 5
Find the sum of the roots of the equation
9 2x +1 - 28.3 2x + 3 = 0
Answer in simplest form a , where a and b ∈ Z
b
2)
A box contains three coins; one coin is fair, one coin is twoheaded, and one coin is weighted so that the probability of
heads appearing is 1 . A coin is selected at random and tossed.
3
Find the probability that a head will appear?
Answer in simplest form a , where a and b ∈ N
b
BABHTA 6
ROUND 6
1)
Two sides of a parallelogram have length 5 and the other two
sides have length 7.
The length of one diagonal is 11.
Calculate the length of the other diagonal.
Answer correct to one decimal place.
2)
Four semicircles are drawn in the interior of a
square using each side of the square as a
diameter.
The area of the square is 64 square units.
Find the area of the shaded region.
Answer in form a π - b, where a and b ∈ N
Cumann Oidí Matamaitice na hÉireann
Irish Mathematics Teachers Association
Foireann Mata 2015
Team Maths 2015
Babhta Réigiúnach
R
BABHTA 7
Regional Round
ROUND 7
1)
A and B are two points on a
straight road and B is 800 m
east of A.
D and C are two landmarks
which are due north of B. It is
known that BD = 600m and
DC =1000m.
Calculate the shortest distance,
DP , from D to a straight road
which joins AC.
Answer in simplest form a b ,
where a and b ∈ N
2)
If x is a positive real number and (x+ 1 )2 = 7 find the value
x
of
x3 + 13
x
Answer in simplest form a b where a and b ∈ N
3)
The complex number z satisfies z + z = 2 + 8i .
Find the value of z .
4)
Find the value of x + y + z if
1
1
1
+
+
=12
xy yz zx
and xyz =
Answer in simplest form a , where a and b ∈ N .
b
1
18
Cumann Oidí Matamaitice na hÉireann
Irish Mathematics Teachers Association
Foireann Mata 2015
Babhta Réigiúnach
BABHTA 8
1)
Team Maths 2015
R
Regional Round
ROUND 8
Find all the solutions to the equation Cos(A) + Sin(A) =
3
2
in
the domain O < A < π
Answers in terms of π .
2)
Find the value of x2 + x 4 + x6 +........+ x98if x1, x2, x3....... is an
arithmetic progression with common difference 1, given that
x1 + x2 + x3.......+ x98 =137 .
3)
The lengths of the sides of a triangle are 10, 17 and 21.
What is the length of the shortest altitude of the triangle?
4)
Find all the real positive values of p and r which satisfy the
following equations:
p + pr + pr 2 = 26
p2r + p2r 2 + p2r 3 = 156
Answers in the form (p , r ).
Cumann Oidí Matamaitice na hÉireann
Irish Mathematics Teachers Association
Foireann Mata 2015
Babhta Réigiúnach
SCOILT
Team Maths 2015
R
Regional Round
TIEBREAK
1)
The number 13 is prime. If you reverse the digits you also get
a prime number, 31. Find the largest prime number that
satisfies this condition if the sum of the two primes is 110.
2)
If Alex stands on a table and Brian stands on the floor then
Alex is 80 cm taller than Brian.
If Brian stands on the same table and Alex stands on the floor
then Brian is 1 m taller than Alex.
How high is the table?
3)
An equilateral triangle is cut into four equilateral triangles,
each with a perimeter of 12 cm.
What is the perimeter, in cm, of the original equilateral
triangle?
4)
If the product 156x 285x 557 was evaluated, it would end with
a string of consecutive zeros.
How many zeros are in this string?
5)
The numbers 1 to 10 are placed around a circle. Sue crosses
out 1, then 4, and then 7. Continuing in a clockwise direction
she crosses out every third number of those remaining, until
only two numbers are left.
What is the sum of the two remaining numbers?
6)
If P = 3x + 3-x and Q = 3x - 3-x what is the numerical value of
P 2 - Q2 ?
7)
The graph of 5x – 3y – 7 = 0 is translated 3 units up and 2
units to the right.
What is the equation of the new graph in the form
ax + by + c =0, where a , b and c ∈ Z.
Cumann Oidí Matamaitice na hÉireann
Irish Mathematics Teachers Association
Foireann Mata 2015
Babhta Réigiúnach
8)
9)
Team Maths 2015
R
Regional Round
7
is written as a decimal?
13
a = log8 (225) and b = log2 (15) . Write a in terms of b.
What is the 2015th digit when
10) a , b and c are real numbers which satisfy the following
equations:
a–b+c= 2
b–c+a=-3
c–a+b= 5
Find the numerical value of a + b + c.
11) Find the altitude of the equilateral triangle whose area and
perimeter have the same numerical value.
12) If 16x+1= 3, what is the value of 2 4x + 2 ?
13)
If 2 x = 3, 3y =5 and 5z =8 what is the numerical value of the
product xyz?
14) In the triangle ABC, AB = 11, AC = 9 and the length of the
altitude from A to [ BC ] = 7. Calculate the length of the side
[ BC ] .
Answer in simplest form a b , where a and b ∈ N.
Cumann Oidí Matamaitice na hÉireann
Irish Mathematics Teachers Association
Foireann Mata 2015
Team Maths 2015
Babhta Réigiúnach
R
Regional Round
Answers Team Maths Regional Round 2015
Round 1
Q1
Round 2 Q1
2n - 1
Q2
7
2+2 5
Q2
x 2 + y 2 - 4x - 6y = 0
Round 3 Q1 3
Q2
Round 4 Q1
30
17π
Round 5 Q1
−
6
Q2 408
1
2
Q2
11
18
Round 6 Q1 5.2
Q2
32π - 64
Round 7 Q1 200 5
Q3 17
Q2
Q4
4 7
Round 8 Q1
π 5π
,
12 12
Q3 8
2
3
Q2 93
Q4 (18 , 1 ) and (2 , 3)
3
Cumann Oidí Matamaitice na hÉireann
Irish Mathematics Teachers Association
Foireann Mata 2015
Team Maths 2015
Babhta Réigiúnach
R
Regional Round
Tiebreak
Q1 73
Q2 90
Q3 24
Q6 4
Q7 5x – 3y -8 =0
Q10 4
Q11 6
Q12
3
4
Q4 10
Q5 10
Q8 6
Q13 3
Q9 a =
Q14 10 2
2
b
3
```