Euclid 2011年真题
1.
(a) If (x + 1) + (x + 2) + (x + 3) = 8 + 9 + 10, what is the value of x?
(b) If 
, what is the value of x?
(c) The point (a, 2) is the point of intersection of the lines with equations y = 2x - 4 and y = x + k. Determine the value of k.
2.
(a) An equilateral triangle of side length 1 is
cut out of the middle of each side of a square
of side length 3, as shown. What is the
perimeter of the resulting figure?
(b) In the diagram, DC = DB, ∠DCB = 15◦,
and ∠ADB = 130◦ . What is the measure of
∠ADC?
(c) In the diagram, ∠EAD = 90°, ∠ACD = 90°,
and ∠ABC = 90° . Also, ED = 13, EA = 12,
DC = 4, and CB = 2. Determine the length
of AB.
3.
(a) If 2 ≤ x ≤ 5 and 10 ≤ y ≤ 20, what is the maximum value of
?
(b) The functions f and g satisfy
f(x) + g(x) = 3x + 5
for all values of x. Determine the value of 2f(2)g(2).
4.
(a) Three different numbers are chosen at random from the set {1, 2, 3, 4, 5}. The numbers are arranged in increasing order.
What is the probability that the resulting sequence is an arithmetic sequence?
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3,5,7,9 is an arithmetic sequence with four terms.)
(b) In the diagram, ABCD is a quadrilateral
with AB = BC = CD = 6, ∠ABC = 90°,
and ∠BCD = 60°. Determine the length
of AD.
5.
(a) What is the largest two-digit number that becomes 75% greater when its digits are reversed?
(b) A triangle has vertices A(0, 3), B(4, 0),
C(k, 5), where 0 < k < 4. If the area of
the triangle is 8, determine the value of k.
6.
(a) Serge likes to paddle his raft down the Speed River from point A to point B. The speed of the current in the river is always the same. When Serge paddles, he always paddles at the same constant speed. On days when he paddles with the
current, it takes him 18 minutes to get from A to B. When he does not paddle, the current carries him from A to B in 30 minutes. If there were no current, how long would it take him to paddle from A to B?
(b) Square OP QR has vertices O(0, 0), P(0, 8), Q(8, 8), and R(8, 0). The parabola with equation y = a(x - 2)(x - 6) intersects the sides of the square OPQR at points K, L, M, and N. Determine all the values of a for which the area of the
trapezoid KLMN is 36.
7.
(a) A 75 year old person has a 50% chance of living at least another 10 years.
A 75 year old person has a 20% chance of living at least another 15 years.
An 80 year old person has a 25% chance of living at least another 10 years.
What is the probability that an 80 year old person will live at least another 5 years?
(b) Determine all values of x for which 
.
8.
The Sieve of Sundaram uses the following infinite table of positive integers:
The numbers in each row in the table form an arithmetic sequence. The numbers in each column in the table form an arithmetic sequence. The first four entries in each of the first four rows and columns are shown.
(a) Determine the number in the 50th row and 40th column.
(b) Determine a formula for the number in the Rth row and Cth column.
(c) Prove that if N is an entry in the table, then 2N + 1 is composite.
9.
Let 
denote the greatest integer less than or equal to x. For example, 
and 
.
Suppose that 
and 
for each positive integer n.
(a) Determine the value of g(2011).
(b) Determine a value of n for which f(n) = 100.
(c) Suppose that A = {f(1), f(2), f(3), . . .} and B = {g(1), g(2), g(3), . . .}; that is, A is the range of f and B is the range of
g. Prove that every positive integer m is an element of exactly one of A or B.
10.
In the diagram, 2∠BAC = 3∠ABC and K lies on BC such that ∠KAC = 2∠KAB. Suppose that BC = a, AC = b, AB = c, AK = d, and BK = x.
(a) Prove that 
and 
.
(b) Prove that (a2 - b2 )(a2 - b2 + ac) = b2c2.
(c) Determine a triangle with positive integer side lengths a, b, c and positive area that satisfies the condition in part (b).
