Yusuf Sila

Math Problems from Rocket City

noreply@blogger.com (Muhammad Yusuf) — Sun, 6 May 2012 22:25:00 -0700

1. Tux is camping on Plotu, and his campsite is in the shape of a circle. The base of his tent is in the shape of a square, which happens to be inscribed in the circle. If the side length of the square is the positive solution for x in the equation 3x2=13x+30, then what is the area of the region outside his tent, but inside his campsite (the shaded region between the circle and the square)?

(3 points)

2. How many solutions of the form (x,y), with x and y both being positive integers, are there to the equation 2x + 10 = 2010?

(3 points)

3. Dis, a galactic mail carrier, wants to get from his house (Point A) to the post office (Point B) located on the 3 x 3 grid. Evil Ed, an especially bothersome alien, is located on Point C. If Dis can only move right and down along grid lines, how many different paths can he take from his house to the post office while avoiding Evil Ed?

(4 points)

David Essner Mathematics Competition XXXI, 2011-2012

noreply@blogger.com (Muhammad Yusuf) — Mon, 16 Apr 2012 08:44:00 -0700

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 2^x = 3 and 3^y = 4. Hence 2^xy = 3^y = 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.

2012 LSU Math Contest (2)

noreply@blogger.com (Muhammad Yusuf) — Mon, 16 Apr 2012 08:29:00 -0700

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?

2012 LSU Math Contest (1)

noreply@blogger.com (Muhammad Yusuf) — Mon, 16 Apr 2012 08:23:00 -0700

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

Math Test 2012

noreply@blogger.com (Muhammad Yusuf) — Mon, 16 Apr 2012 08:17:00 -0700

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

(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

(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

(D) 169

(E) None of the above

4. If

then

(A) 64

(B) 27

(D)127/64

(E) 1

C of C Math Marathon 2012

noreply@blogger.com (Muhammad Yusuf) — Mon, 16 Apr 2012 08:08:00 -0700

1. Express 2012 as a difference of two integer squares.

2. Each account on a social networking website has a Personal Identification Number, which is required to be a string of eight decimal digits not containing two consecutive digits that are both odd. So for example 31415926 is not an acceptable PIN but 02252012 is an acceptable PIN. Find the number of acceptable PINs.

3. What is the smallest number of integer squares whose sum is 2012?

4. At a small party there are fifteen door prizes, each of which is randomly given to one of the five guests. What is the probability that each of the five guests receives at least one door prize?

Nassau County Interscholastic Mathematics League : Team

noreply@blogger.com (Muhammad Yusuf) — Tue, 3 Apr 2012 16:24:00 -0700

1. If x % of x is 2.25, and x > 0, find x.

2. Find the sum of the three smallest composite numbers that have no prime factor less than 10.

3. The product of n distinct integers is 18. Find the maximum value of n.

4. Points A and D are on opposite sides of BC, AB ⊥ BC, and CD ⊥ BC, AB = 9, BC = 21, and CD = 11. Find AD.

5. One of the roots of x² + Bx + C = 0, where B and C are integers, is √2 + 1. Find|B + C|.

6. In triangle ABC, AB = 5, BC = 6, and AC = 7. Point O is outside the triangle and in the interior of angle A. A circle with center O is tangent to side BC at Q, and to sides AB and AC extended at P and R, respectively. Find BQ.

7. The polynomial P(x) has degree 3 and leading coefficient 1. If P(1) = 1, P(2) = 2, and P(3) = 3, find P(4).

Math Contest : Nassau County Interscholastic Mathematics League (3)

noreply@blogger.com (Muhammad Yusuf) — Tue, 3 Apr 2012 16:16:00 -0700

5. Chauncey goes up a flight of 8 stairs. With each step, he goes up either one or two stairs. Find the number of different sequences of steps he can take to the top.

Note that leaping one stair and then two stairs is different from leaping two stairs and then one stair.

6. In convex quadrilateral ABCD, diagonal AC is perpendicular to diagonal BD, AB = 10, BC = 5, and CD = 11. Find AD.

Math Contest : Nassau County Interscholastic Mathematics League

noreply@blogger.com (Muhammad Yusuf) — Tue, 3 Apr 2012 16:14:00 -0700

3. Find the sum of the coefficients in the expansion of (3x – 1)⁴

4. Point A is on a circle whose center is O, AB is tangent to the circle, AB = 6, D is inside the circle, OD = 2, DB intersects the circle at C, and BC = DC = 3. If r is the length of the radius of the circle, find r².

Nassau County Interscholastic Mathematics League : Contest 1 - 2010/2011

noreply@blogger.com (Muhammad Yusuf) — Tue, 3 Apr 2012 16:11:00 -0700

1. Compute 19.27² + 11.73² + (38.54)(11.73).

2. Point A is outside a circle and AB and AC are tangent to the circle at B and C, respectively. Points P and R are on AB and BC, respectively, PR is tangent to the circle at Q, and AB = 20. Find the perimeter of triangle APR.

ABACUS - JANUARY 2012 : LEVEL 7 - 8

noreply@blogger.com (Muhammad Yusuf) — Tue, 3 Apr 2012 09:11:00 -0700

PROBLEM 1. The ages of a father and his two different-aged sons are the powers of the same prime number. Last year everybody's age was a prime number. How old are they now?

PROBLEM 2. Could the sum of seven consecutive integers be a prime number?

PROBLEM 3. A 5-digit number is divisible by 7, 8, and 9. The number created from the first two digits is a prime number, 1 greater than a square number, and the sum of these two digits is a two-digit number. Find this 5-digit number.

PROBLEM 4. Every digit of a 5-digit number is either 1 or a prime number. Not only that, but any number created by any 2, 3, or 4 consecutive digits of this number are prime numbers also. Find this number, and check if it is a prime number or not.

PROBLEM 5. Can you put the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 into two groups so that the sums of the numbers in each group is the same? Can you do this to get the same product in each group?

PROBLEM 6. Are there any five consecutive whole numbers, which can be put into two groups so that the product of the numbers in each group is the same?

PROBLEM 7. In some cases, when 22022 and 20222 are divided by the same 3-digit number, they give the same remainder. Which one of these divisors can be determined by the remainder?

PROBLEM 8. Find a, b, c, and d, such that a^bb = bccbdc

NUMBER CARDS

noreply@blogger.com (Muhammad Yusuf) — Tue, 3 Apr 2012 09:05:00 -0700

Take 10 number cards with the following numbers on them (one number on each card): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Make a pile of them by putting one on top of another, and hold the pile in your hands. Now put the top card on the table, put the next top card on the bottom of the pile in your hands, put the next top card on the table, the next top card on the bottom of the pile, and so on, until you run out of cards. In what order do you have to stack the cards at the beginning if you want the cards to be on the table at the end in the following order : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10?

HONEST AND LIAR

noreply@blogger.com (Muhammad Yusuf) — Tue, 3 Apr 2012 09:04:00 -0700

Two kinds of people live on an island: honest and liar. Honest people always say the truth, liars always lie. One day we asked everybody in a group of 5 people from this island who know each other: "How many of you are honest?" We received the following answers: 0, 1, 2, 3, 4.

How many of them are honest?

TEAM POINTS

noreply@blogger.com (Muhammad Yusuf) — Tue, 3 Apr 2012 09:02:00 -0700

In a league the teams received a total of 420 points. You got 2 points for a victory, 1 point for a tie, and 0 points for a loss. How many teams were there in the league if every team played every team twice?

Hot Dog Package

noreply@blogger.com (Muhammad Yusuf) — Mon, 26 Mar 2012 02:36:00 -0700

A store sells hot dog buns 8 per package, sells hot dogs 10 per package, sells plastic plates 25 per package, and sells bags of potato chips 14 per package. What is the smallest positive total number of these packages one can buy at this store so that the purchase includes equally many hot dog buns as hot dogs as plates as bags of potato chips, without buying any partial packages?

Invest and Earning

noreply@blogger.com (Muhammad Yusuf) — Mon, 26 Mar 2012 02:34:00 -0700

Bob invests $100,000, earning 5% simple annual interest. Upon adding the interest to the account after one year, he withdraws 2% of what is now in the account, investing the rest for another year at 5% simple interest. After the account has this new year of interest added to it, he again withdraws 2% of what is now in the account, investing the rest for another year at 5% simple interest, and so on. Rounded to the nearest dollar, how much is in the account after his 7^th withdrawal?

Perpendicular medians

noreply@blogger.com (Muhammad Yusuf) — Fri, 9 Mar 2012 21:25:00 -0800

Suppose the medians AA' and BB' of triangle ABC intersect at right angles. If BC = 3 and AC = 4, what is the length of side AB?

Pascal's triangle

noreply@blogger.com (Muhammad Yusuf) — Fri, 9 Mar 2012 21:16:00 -0800

Show that any two elements (both greater than one) drawn from the same row of Pascal's triangle have greatest common divisor greater than one. For example, the greatest common divisor of 28 and 70 is 14.

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

Nonagon diagonals

noreply@blogger.com (Muhammad Yusuf) — Fri, 9 Mar 2012 21:11:00 -0800

In regular nonagon ABCDEFGHI, show that AF = AB + AC.

Area of a trapezoid

noreply@blogger.com (Muhammad Yusuf) — Fri, 9 Mar 2012 21:09:00 -0800

A trapezoid¹ is divided into four triangles by its diagonals. Let the triangles adjacent to the parallel sides have areas A and B. Find the area of the trapezoid in terms of A and B.

(1) A trapezoid is a quadrilateral with at least one pair of parallel sides. In some countries, such a quadrilateral is known as a trapezium.

set

noreply@blogger.com (Muhammad Yusuf) — Fri, 9 Mar 2012 21:07:00 -0800

Suppose that for every (including the empty set and the whole set) subset X of a finite set S there is a subset X* of S and suppose that if X is a subset of Y then X* is a subset of Y* . Show that there is a subset A of S satisfying A * = A.

Three children

noreply@blogger.com (Muhammad Yusuf) — Fri, 9 Mar 2012 21:04:00 -0800

On the first day of a new job, a colleague invites you around for a barbecue. As the two of you arrive at his home, a young boy throws open the door to welcome his father. “My other two kids will be home soon!” remarks your colleague.

Waiting in the kitchen while your colleague gets some drinks from the basement, you notice a letter from the principal of the local school tacked to the noticeboard. “Dear Parents,” it begins, “This is the time of year when I write to all parents, such as yourselves, who have a girl or girls in the school, asking you to volunteer your time to help the girls' soccer team.” “Hmmm,” you think to yourself, “clearly they have at least one of each!”

This, of course, leaves two possibilities: two boys and a girl, or two girls and a boy. Are these two possibilities equally likely, or is one more likely than the other?

Note: This is not a trick puzzle. You should assume all things that it seems you're meant to assume, and not assume things that you aren't told to assume. If things can easily be imagined in either of two ways, you should assume that they are equally likely. For example, you may be able to imagine a reason that a colleague with two boys and a girl would be more likely to have invited you to dinner than one with two girls and a boy. If so, this would affect the probabilities of the two possibilities. But if your imagination is that good, you can probably imagine the opposite as well. You should assume that any such extra information not mentioned in the story is not available.

Two ladders in an alley

noreply@blogger.com (Muhammad Yusuf) — Fri, 9 Mar 2012 21:03:00 -0800

Two ladders are placed cross-wise in an alley to form a lopsided X-shape. The walls of the alley are not quite vertical, but are parallel to each other. The ground is flat and horizontal. The bottom of each ladder is placed against the opposite wall. The top of the longer ladder touches the alley wall 5 feet vertically higher than the top of the shorter ladder touches the opposite wall, which in turn is 4 feet vertically higher than the intersection of the two ladders. How high vertically above the ground is that intersection?

Five men, a monkey, and some coconuts

noreply@blogger.com (Muhammad Yusuf) — Fri, 9 Mar 2012 21:02:00 -0800

Five men crash-land their airplane on a deserted island in the South Pacific. On their first day they gather as many coconuts as they can find into one big pile. They decide that, since it is getting dark, they will wait until the next day to divide the coconuts.

That night each man took a turn watching for rescue searchers while the others slept. The first watcher got bored so he decided to divide the coconuts into five equal piles. When he did this, he found he had one remaining coconut. He gave this coconut to a monkey, took one of the piles, and hid it for himself. Then he jumbled up the four other piles into one big pile again.

To cut a long story short, each of the five men ended up doing exactly the same thing. They each divided the coconuts into five equal piles and had one extra coconut left over, which they gave to the monkey. They each took one of the five piles and hid those coconuts. They each came back and jumbled up the remaining four piles into one big pile.

What is the smallest number of coconuts there could have been in the original pile?

PROBLEM OF THE WEEK

noreply@blogger.com (Muhammad Yusuf) — Fri, 9 Mar 2012 21:00:00 -0800

Two discs of radius one and a disc of radius one half are drawn on a plane so that each of them is touching the other two at one point–think of two quarters and a penny all flat on a table and all touching at their edges. Find the radius of the largest circle which is tangent to all three of the circles which are the edges of the discs.