CrashOveride
Active Member
Yeah type it up sedated
And I guess we're tipping Archman for medal this yr?
And I guess we're tipping Archman for medal this yr?
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UNSW school maths comp (1 Viewer)
- Thread starter Grey Council
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Grey Council
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I asked him
he said he could easily get gold on a good day.
*faints*
he said he could easily get gold on a good day.
*faints*
turtle_2468
Member
hehe... always the confident one he is.. 
Grey Council
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HAHAH!turtle_2468 said:hehe... always the confident one he is..
someone didn't get gold...
*ducks*
lol, he quite pointedly mentioned that you went there three times, and managed silver.
but to be fair, his complete saying was:
"If its a good day, and the questions go my way, I can easily get gold"
So I guess he wasn't THAT much overconfident.
hrm
SeDaTeD
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Ivan Guo I'm guessing?
haha, I figured them all out except for Q6, too bad that was after the test finished.
Anyway, time to waste my time:
Q1.
Solve for x, y and z
x + 1/y = 1
y + 1/z = 2
z + 1/x = 5
Q2a)
Two sequences are defined by
p_1 = 1, q_1 = 1
and for n >= 1,
p_n+1 = p_n + 2q_n,
q_n+1 = p_n + q_n (underscores are subscripts)
Show that for all n,
(p_n)^2 - 2(q_n)^2 = (-1)^n
Q2b)
Mike lives in a long street and his house has a 3-digit number. The odd-numbered houses are on one side of the street, the even-numbered on the other. Mike notices that the sum of the house numbers on his side of the street up to and including his house is equal to the sum of the house numbers on his side of the street from the other end of the street down to and including his house. What is Mike's house number?
Q3.
A triangle with sides 13. 14 and 15 sits around the top half of a sphere of radius 5 (that is, it touches the sphere at three points, as shown on the diagram)
How far is the plane of the triangle from the centre of the sphere?
(sorry no diagram)
Q4
A triangle is either isosceles (which for the purpose of this question we shall take to include equilateral) or scalene (all sides different).
a) How many triangles are there with integer length sides and perimeter 24?
b) How many triangles are there with integer length sides and perimeter 36?
c) How many isosceles triangles are there with integer length sides and perimeter 12n? (where n is an integer)
d) How many triangles are there altogether with integer length sides and perimeter 12n? (where n is an integer)
Q5
The numbers a and b satisfy
a^3 - 3ab^2 = 52,
b^3 - 3ba^2 = 47
a) Find a^2 + b^2
b) Hence or otherwise find a and b
Q6
Five different four-digit integers all have the same initial digit, and their sum is divisible by four of them. Find all possible such sets of integers.
That is all.
haha, I figured them all out except for Q6, too bad that was after the test finished.
Anyway, time to waste my time:
Q1.
Solve for x, y and z
x + 1/y = 1
y + 1/z = 2
z + 1/x = 5
Q2a)
Two sequences are defined by
p_1 = 1, q_1 = 1
and for n >= 1,
p_n+1 = p_n + 2q_n,
q_n+1 = p_n + q_n (underscores are subscripts)
Show that for all n,
(p_n)^2 - 2(q_n)^2 = (-1)^n
Q2b)
Mike lives in a long street and his house has a 3-digit number. The odd-numbered houses are on one side of the street, the even-numbered on the other. Mike notices that the sum of the house numbers on his side of the street up to and including his house is equal to the sum of the house numbers on his side of the street from the other end of the street down to and including his house. What is Mike's house number?
Q3.
A triangle with sides 13. 14 and 15 sits around the top half of a sphere of radius 5 (that is, it touches the sphere at three points, as shown on the diagram)
How far is the plane of the triangle from the centre of the sphere?
(sorry no diagram)
Q4
A triangle is either isosceles (which for the purpose of this question we shall take to include equilateral) or scalene (all sides different).
a) How many triangles are there with integer length sides and perimeter 24?
b) How many triangles are there with integer length sides and perimeter 36?
c) How many isosceles triangles are there with integer length sides and perimeter 12n? (where n is an integer)
d) How many triangles are there altogether with integer length sides and perimeter 12n? (where n is an integer)
Q5
The numbers a and b satisfy
a^3 - 3ab^2 = 52,
b^3 - 3ba^2 = 47
a) Find a^2 + b^2
b) Hence or otherwise find a and b
Q6
Five different four-digit integers all have the same initial digit, and their sum is divisible by four of them. Find all possible such sets of integers.
That is all.
Grey Council
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I'm guessing your from Ruse.
Only school Ivan Guo is famous in is Ruse.
lol, and you could do a fair bit of those questions, so I'm quite sure your from Ruse.
Only school Ivan Guo is famous in is Ruse.
lol, and you could do a fair bit of those questions, so I'm quite sure your from Ruse.
Grey Council
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heehee, Ivan came first last year.
Regardless, I think you'll be interested in knowing that Ivan Guo has most likely just got a gold medal in the International Mathematical Olympiad.
*faints with shock at his genius'ity*
Regardless, I think you'll be interested in knowing that Ivan Guo has most likely just got a gold medal in the International Mathematical Olympiad.
*faints with shock at his genius'ity*