David Essner Mathematics Competition XXXI, 2011-2012
8:44 AM
Muhammad Yusuf
, Posted in
1. Tom, Alice and John took an exam. Alice scored 80.
Tom scored 10 more than the average of the three, while John scored 16 less
than the average of the three. The average of the three was then
(a) 72 (b) 74 (c) 75⅔ (d)
76⅓ (e) 78
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Soln: (b) Let N be the average. Then 3N = 80 + (N + 10) + (N − 16) or 3N = 2N + 74
2. Two sides of an isosceles triangle have length 2
and 5. What is the area of the triangle?
(a) 5 (b) 2√6 (c) √21 (d) 2√5 (e) There is more than one possible value.
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Soln: (b) The other
side is 5. If h is the altitude to the
side of length 2 then h2 = 52−12 = 24.
The area is then (1/2)(2)(2√6).
3.
4. What is the area of a rectangle if the diagonals
have length 1 and 60◦ is an angle
of their intersection?
(a) 1/2 (b) √2/2 (c) √3/2 (d) √3/4 (e) (√3 + 1)/2
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Soln: (d) The
rectangle is composed of two 30◦, 60◦ right
triangles each of which has hypotenuse 1 and sides of length 1/2 and √3/2. The area of each triangle is then √3/8.
5. A multiple choice test has 30 questions and 5
choices for each question. If a student answers all 30 questions and the score
is [number right - (number wrong/4)] then which of the following is a possible
score?
(a) −10 (b) 5.25 (c) 7.75 (d) 8.75 (e) 9.25
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Soln: (d) If R is the number right then the score is R − (30 − Rhave the same y–intercept b and the sum of the x–intercepts of L1 and L2 is 10, then b equals
(a) 5/6 (b) −6/5 (c) −2 (d) 3/2 (e) 2
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Soln: (c) The
equations of L1, L2 are y = x/2 + b and y b and −3b. From −5b = 10 it follows that b = −2.
7. What is the value of (log2 2 (d) 2 (e) 8/3
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Soln: (d) Let x
(a) 5/6 (b) −6/5 (c) −2 (d) 3/2 (e) 2
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Soln: (c) The
equations of L1, L2 are y = x/2 + b and y = x/3 + b. Setting y = 0 gives the x intercepts as −2b and −3b. From −5b = 10 it follows that b = −2.
7. What is the value of (log2 3)(log3 4)?
(a) 3/4 (b) 4/3 (c) 3/2 (d) 2 (e) 8/3
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Soln: (d) Let x = log2 3 and y = log3 4. Then 2x = 3 and 3y = 4. Hence 2xy = 3y = 4 and xy = 2.
8. If 0 < x <
π/2 and sin x = 2 cos x then (sin x)(cos x) equals
(a) 1/3 (b) 2/5 (c) 1/5 (d) 3/8 (e) √3/4
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Soln: (b) 1 − cos2 x = sin2 x = 4 cos2 x implies 5 cos2 x = 1 and hence cos2 x = 1/5 and sin2 x = 4/5. Thus (sin2 x)(cos2 x) = 4/25 and (sin x)(cos x) = 2/5.
9. How many positive integer pairs (m, n) satisfy the equation 2m + 7n = 835?
(a) 44 (b) 51 (c) 60 (d) 71 (e) 119
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Soln: (c) n can be any odd integer from 1 to 119 inclusive. There are (1 + 119)/2 = 60 such integers.
10. From a point P two tangent lines are drawn to a circle C. If A,B are the
tangent points, O is the center of C, and ∠APB = 300 then ∠AOB equals
(a) 600 (b) 900 (c) 1200 (d) 1500 (e) 1800
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Soln: (d) ∠AOB = 3600 − ∠APB − ∠PAO − ∠PBO = 3600 – 300 – 900 – 900 = 1500.
