Euclid 2010年真题

(a) If 3x = 27, what is the value of 3x+2?

(b) If 2531359x = 2731459 , what is the value of x?

y = x + 2, and the x-axis.

Determine the area of ∆ABC.

(a) Maria has a red package, a green package, and a blue package.

The sum of the masses of the three packages is 60kg.

The sum of the masses of the red and green packages is 25kg.

The sum of the masses of the green and blue packages is 50kg.

What is the mass of the green package, in kg?

(b) A palindrome is a positive integer that is the same when read forwards or backwards. For example, 151 is a palindrome. What is the largest palindrome less than 200 that is the sum of three consecutive integers?

(a) Bea the bee sets out from her hive, H, and

flies south for 1 hour to a field, F. She spends

30 minutes in the field, and then flies 45

minutes west to a garden, G. After spending

1 hour in the garden, she flies back to her

hive along a straight line route. Bea always

flies at the same constant speed. What is the

total length of time, in minutes, that she is

away from her hive?

(b) In the diagram, points P(p, 4), B(10, 0), and

O(0, 0) are shown. If ∆OPB is right-angled

at P, determine all possible values of p.

(a) Thurka bought some stuffed goats and some toy helicopters. She paid a total of $201. She did not buy partial goats or partial helicopters. Each stuffed goat cost $19 and each toy helicopter cost $17. How many of each did she buy?

(b) Determine all real values of x for which (x + 8)4 = (2x + 16)2.

(a) If f(x) = 2x + 1 and g(f(x)) = 4x2 + 1, determine an expression for g(x).

(b) A geometric sequence has 20 terms.

The sum of its first two terms is 40.

The sum of its first three terms is 76.

The sum of its first four terms is 130.

Determine how many of the terms in the sequence are integers.

(A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a constant. For example, 3, 6, 12 is a geometric sequence with three terms.)

(a) A snail’s shell is formed from six triangular

sections, as shown. Each triangle has interior

angles of 30° , 60°◦ and 90° . If AB has a length

of 1 cm, what is the length of AH, in cm?

(b) In rectangle ABCD, point E is on side DC.

Line segments AE and BD are perpendicular

and intersect at F. If AF = 4 and DF = 2,

determine the area of quadrilateral BCEF.

(a) Determine all real values of x for which .

(b) Determine all points (x, y) where the two curves y = log10(x4) and y = (log10 x)3 intersect.

(a) Oi-Lam tosses three fair coins and removes all of the coins that come up heads.

George then tosses the coins that remain, if any. Determine the probability that George tosses exactly one head.

(b) In the diagram, points B, P, Q, and C lie

on line segment AD. The semi-circle with

diameter AC has centre P and the semi-circle

with diameter BD has centre Q. The two

semi-circles intersect at R. If ∠P RQ = 40°,

determine the measure of ∠ARD.

(a) (i) If θ is an angle whose measure is not an integer multiple of 90° , prove that

(ii) Ross starts with an angle of measure 8° and doubles it 10 times until he obtains 8192° . He then adds up the reciprocals of the sines of these 11 angles.

That is, he calculates

Determine, without using a calculator, the measure of the acute angle α so that .

(b) In ∆ABC, BC = a, AC = b, AB = c, and a < (b + c).

Prove that ∠BAC < (∠ABC + ∠ACB).

10.

For each positive integer n, let T(n) be the number of triangles with integer side lengths, positive area, and perimeter n. For example, T(6) = 1 since the only such triangle with a perimeter of 6 has side lengths 2, 2 and 2.

(a) Determine the values of T(10), T(11) and T(12).

(b) If m is a positive integer with m ≥ 3, prove that T(2m) = T(2m - 3).