David Essner Mathematics Competition XXXI, 2011-2012


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) 75    (d) 76            (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) 26 (c) 21 (d) 25 (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 = 5212 = 24.
The area is then (1/2)(2)(26).

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.
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2012 LSU Math Contest (2)



22 Water when freezing (and turning into ice) enlarges its volume by a factor of 1/10. By what factor will the volume of ice decrease when it is melted?

23 When Achilles started chasing a turtle they were 990 yards apart. Achilles covers 10 yards in each second. The turtle covers 1 yard in 10 seconds. How long will it take Achilles to catch the turtle?

24 For which natural number n is the distance between the numbers 20+21+22+ . . . +2n and 2012 the smallest?

25 A digital clock has a special button B. Pushing that button causes the clock to switch to the next exact hour, i.e. it either moves back 0 to 29 minutes or moves forward from 1 to 30 minutes. If at 10:15 the clock shows 8:20, what is the shortest time needed to set the clock to the right time by only pushing B, possibly a few times. The clock runs accurately - doesn’t slow or go fast.

26 Find the area of the shaded region.

27 The numbers, 2a - 2, 2a + 2, and a + 1 are lengths of three sides of a triangle. What can you say about the number a?

28 A polyhedron has n faces, one of which is a regular pentagon. What is the minimal possible value of n?

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2012 LSU Math Contest (1)


1.Two different numbers y and z are roots of the quadratic equation ax2 + c = 0. Which one of the following is not true in general
A y + z = 0
B y2 + z2 = 0
C y3 + z3 = 0
D y=z = -1
E y z < 0

2. All the vertices of a convex quadrilateral ABCD lie on two perpendicular lines. It follows that
A A circle can be inscribed in the quadrilateral ABCD
B ABCD can be inscribed in a circle
C ABCD is a square
D ABCD is a rhombus
E None of the above is true

3. The length of the earth’s equator is approximately 40000 km. What is the approximate length of the 600 parallel?
A 20000
B 23500
C 26200
D 30000
E 34000

4. If the greatest common divisor of two natural numbers a and b is 3 and a/b = 0.4, then ab is
A 10
B 18
C 30
D 36
E 90

5. How many different real roots does the polynomial x4 - x3 - x2 + E four
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Math Test 2012


1. A natural number is a palindrome if its digits read the same from left to right as they do from right to left. For example, 3521253 is a palindrome. How many 3­digit numbers are palindromes that are divisible by 11?
(A) none of them are divisible by 11
(B) 4
(C) 8
(D) 22
(E) 90

2. Triangle ABC has area 12. If side AB has length 6 and side BC has length 5, what must be the length of side AC?
(A) 5
(B) 37
(C) 97
(D) 109
(E) More than one answer is possible with the given information.

3. A club with 15 members must choose a delegation of 4 members to serve at a convention. Team members Jacob and Sarah refuse to serve on a delegation together. How many delegations are possible?
(A) 130
(B) 143
(C) 156
(D) 169
(E) None of the above

4. If
then
(A) 64
(B) 27
(C) 8
(D)127/64
(E) 1
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