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Fiona

Questions (317)

Suppose we have a bag with $10$ slips of paper in it. Eight slips have a $7$ on them and the other two have a $9$ on them.What
1. 1 answer
2. 36 views
Suppose we have a bag with $10$ slips of paper in it. Eight slips have a $7$ on them and the other two have a $9$ on them.How
1. 1 answer
2. 46 views
Fill in the blanks with integers to get an identity.(n - k + 2)^2 = __ n^2 + ___ k^2 + ___ n + ___k + ___ nk + ___
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2. 19 views
A school orders 99 textbooks, all for the same price. When thebill for the total order comes, the first and last digits are
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2. 28 views
In the diagram, AOB is a sector of a circle with aob = 60 degrees and radius 12. OY is drawn perpendicular to AB and intersects
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2. 35 views
A standard six-sided die is rolled $8$ times. You are told that among the rolls, there was two $1$'s, two $2$'s, two $3$'s, and
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2. 32 views
We call a number cozy if every digit in the number is either a $3$ or next to a $3.$ For example, the numbers $333,$ $83,$
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2. 28 views
Joanna has several beads that she wants to assemble into a bracelet. There are seven beads: five of the beads have the same
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2. 35 views
In a series of coin flips, a run is a series of one or more consecutive coin flips that all have the same result. For example,
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2. 28 views
Let IJKLMN be a hexagon with side lengths IJ = LM = 3, JK = LM = 4, and KL = IN = 6. Also, all the interior angles of the
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2. 37 views
In how many ways can 4 balls be placed in 4 boxes if 3 balls are indistinguishably white, 1 ball is black, and the boxes are
1. 1 answer
2. 28 views
In how many ways can three pairs of siblings from different families be seated in two rows of three chairs, if siblings may not
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2. 37 views
Sophie's favorite (positive) number is a two-digit number. If she reverses the digits, the result is 72 less than her favorite
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2. 27 views
At a store, you are interested in purchasing a candy bar and a piece of bubble gum. Together, they cost $2.00. The candy bar
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2. 28 views
When expanded as a decimal, the fraction $\frac{1}{7}$ has a repetend (the repeating part of the decimal) of $142857$. The last
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2. 18 views
Let x and y be integers. If x and y satisfy 37x + 82y = -172 and x + y = -1, then find x.
1. 1 answer
2. 20 views
Find the number of bases b \ge 2 such that 144_b is prime.
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2. 20 views
ABCD is a square. How many squares have two or more vertices in the set {A, B, C, D}?
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2. 17 views
Find the smallest positive solution to tan(2x) + tan(3x) = sec(3x) in radians.
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2. 13 views
A circular table is pushed into a corner of the room, where two walls meet at a right angle. A point P on the edge of the table
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2. 40 views
Find the number of ways of choosing three circles below, so that no two circles are next to each other. The diagram shows 12
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2. 43 views
Define $x\star y=\frac{\sqrt{x^2+3xy+y^2-2x-2y+4}}{xy+4}$. Compute \[((\cdots ((2007\star 2006)\star 2005)\star\cdots )\star
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2. 26 views
Find all possible values of b, where (a,b,c) satisfiesfloor(a)*b*c = 3 a*floor(b)*c = 4 a*b*floor(c) = 5 and a, b, c are
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2. 44 views
My Skibidi Team called The Sigma's won 2/9 of its games this seasons. If we lost 15 more games than we won, how many games did
1. 1 answer
2. 12 views
I drove to the beach at a rate of 60 miles per hour. If I had driven at a rate of 40 miles per hour instead, then I would have
1. 1 answer
2. 26 views
When the product (3N* + 8N - 3)(pN - 1) is divided by (N + 1), the remainder is 24. What is the remainder when pao is divided by
1. 1 answer
2. 15 views
avatar+8For distinct positive integers a, b, and c, N = 5a + 3b + 5c = 4a + 5b + 4c. If 131 < N < 150, then what is the maximum
1. 1 answer
2. 31 views
Simplify 4/(1 - 3^(1/3) + 9^(1/3))
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2. 9 views
On a True/False test, Amy answered the first three questions wrong but answered the rest of the questions correctly. On the same
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2. 60 views
Six teams play a single round-robin tournament, in which each team plays every other team exactly once, there are no ties, and
1. 1 answer
2. 84 views
In quadrilateral BCED, sides BD and CE are extended past B and C, respectively, to meet at point A. If BD = 18, BC = 8, CE = 2,
1. 1 answer
2. 19 views
Let n be a positive integer, and let S = {1, 2, 3, ..., n}. Three subsets A, B, C of S are chosen at random.(a) Find the
1. 1 answer
2. 59 views
The product of all the prime numbers between 1 and 100 is equal to P. What is the remainder when P is divided by 16?
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2. 20 views
what is the smallest integer n that n^2-17 is a prime number squared?
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2. 14 views
In how many ways can you choose 2 people out of a group of 10 people?
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2. 14 views
Let n be a positive integer such that n= ABC_7 and n=CBA_11. Find the largest possible value of n in base 10.
1. 1 answer
2. 25 views
a large community college has professors and lecturers.The total number of faculty members is 228 the school reported that they
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2. 43 views
In triangle ABE, C and D are point on side BE. If BD = 8, CE = 12, [ABC] = 10, and [ADE] = 24, then find [ACD].
1. 1 answer
2. 19 views
A square is inscribed in a right triangle. Find the area of the square.legs are length 1 and 3.
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2. 23 views
Prove that given any set of $17$ integers, there exist nine of them whose sum is divisible by $2.$
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2. 37 views
Consider a house with a finite number of rooms. Some of the rooms have an even number of doors, and some of the rooms have an
1. 1 answer
2. 50 views
Twelve points are given in the plane, so that no three points lie on the same line.Five distinct segments joining pairs of these
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2. 46 views
A standard 8 times 8 chessboard has 64 unit squares.Eight rooks are randomly placed on different squares of a chessboard. A rook
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2. 82 views
Ava’s father George picks her up at the train station when she comes home from school, then George drives Ava home. They
1. 1 answer
2. 27 views
Let S = {1, 2, 3, ..., n}. Three subsets A, B, C of S are chosen at random.Find the probability that A is a subset of B, and B
1. 1 answer
2. 31 views
Let S = {1, 2, 3, ..., n}. Three subsets A, B, C of S are chosen at random.Find the probability that the union of A, B, and C is
1. 1 answer
2. 18 views
among 3 digit numbers, how many numbers are there with exactly 9 factors?
1. 1 answer
2. 17 views
On the xy-plane, the origin is labeled with an M. The points (1,0), (-1,0), (0,1), and (0,-1) are labeled with A's. The points
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2. 34 views
A fair coin is flipped until either three heads in a row or four tails in a row occur. What is the expected number of times the
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2. 62 views
The two circles below are externally tangent. A common external tangent intersects line PQ at R. Find QRThe circles have
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2. 74 views
Consider a 4 colored cube of 343 small cubes each of dimension 1x1x1having one particular color. No two consective cubes on the
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2. 36 views
Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, then turns 3 random lamps on.
1. 1 answer
2. 43 views
A fair coin is tossed repeatedly until either heads comes up three times in a row or tails comes up three times in a row. What
1. 1 answer
2. 93 views
Consider a 4 colored cube of 343 small cubes each of dimension 1x1x1 having one particular color. No two consective cubes on the
1. 1 answer
2. 33 views
Consider a 4 colored cube of 343 small cubes each of dimension 1x1x1 having one particular color. No two consective cubes on the
1. 1 answer
2. 17 views
Two circles are drawn, with the same center. A chord of the large circle is drawn, so that it is tangent to the small circle. If
1. 1 answer
2. 43 views
A rectangular room measures 12-feet by 6-feet. How many square yards of carpet are needed to cover the floor of the room?
1. 1 answer
2. 51 views
Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages 40 miles per hour, he arrives at
1. 1 answer
2. 53 views
Kwan must read a book that is 400 pages long. He reads at a rate of one page per minute. If he begins reading on a Saturday and
1. 1 answer
2. 43 views
How many integers n with 70 <= n <= 90 can be written as n = ab + 2a + 3b for at least one ordered pair of positive integers
1. 1 answer
2. 25 views
Dr. Jones lives in a country with a progressive tax system. That is, he does not pay any taxes on the first $20,000 in income he
1. 1 answer
2. 42 views
The primes 3, 5, and 17 have the property such that their quadratic nonresidues are also primitive roots. Find the next two
1. 1 answer
2. 33 views
Let f(x) = k(x) if x > 3, f(x) = x^2 - 6x + 12 if x <= 3. Find the function k(x) such that f is its own inverse.
1. 1 answer
2. 22 views
Each face of a cube is painted randomly one of the colors red, orange, yellow, green, blue or purple. What is the probability
1. 1 answer
2. 47 views
The focus of the parabola x^2=4y is F=(0,1). A line passing through F intersects the parabola at M and N. The line passing
1. 1 answer
2. 39 views
Write sqrt(2)/(sqrt(2) + sqrt(3) - sqrt(5)) - 1/2 with a rational denominator.
1. 1 answer
2. 17 views
For a certain value of , the system:x+y+3z=10 -4x+2y+5z=7 kx+z=3 has no solutions. What is this value of k?
1. 1 answer
2. 17 views
Let p(x) be defined on 2 <= x <= 10 such that p(x) = x + 1 if floor(x) is prime, p(x) = p(y) + (x + 1 - floor(x)) otherwise,
1. 1 answer
2. 22 views
The inverse of a modulo 44 is b. What is the inverse of 9a modulo 44 in terms of b?
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2. 35 views
Find all integers n, 0 <= n < 163, such that n is its own inverse modulo 163.
1. 1 answer
2. 34 views
All the sides of a triangle are integers. If the perimeter of the triangle is 16, then how many different possible triangles are
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2. 28 views
Find the exponential form of the complex numbere^17πi/60+e^27πi/60+e^37πi/60+e^47πi/60+e^57πi/60 with proof.
1. 1 answer
2. 25 views
Triangle ABC has altitudes AD, BE, and CF. If AD=14, BE=16, and CF is a positive integer, then find the largest possible value
1. 1 answer
2. 27 views
Marian throws a dart that lands randomly on a dartboard shaped like an isosceles trapezoid with side lengths 12 inches, 12
1. 1 answer
2. 42 views
Points M, N, and O, are the midpoints of sides KL, LJ, and JK, respectively, of triangle JKL. Points P, Q, and R are the
1. 1 answer
2. 62 views
Triangle ABC has altitudes AD, BE, and CF. and If AD= 12, BE=16, and CF is a positive integer, then find the largest possible
1. 1 answer
2. 27 views
In triangle STU, let M be the midpoint of ST and let N be on TU such that SN is an altitude of triangle STU. If ST and SU are
1. 1 answer
2. 23 views
What is the product of the two smallest prime factors of 2^1024 - 1?
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2. 36 views
Gracie starts at her house and walks one block east. After each block she walks, she flips a coin. She walks one block east if
1. 1 answer
2. 36 views
How many integers $n$ with $70 \leq n \leq 90$ can be written as $n = ab + 2a + 3b$ for at least one ordered pair of positive
1. 1 answer
2. 47 views
What is the largest possible value not in the domain of log (x-2)/(x^2-5)
1. 1 answer
2. 19 views
Find all x such that neither 2x nor -20x is in the interval (-infty,-1).Show drafts
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2. 34 views
Points T and U lie on a circle centered at O, and point P is outside the circle such that PT and PU are tangent to the circle.
1. 1 answer
2. 48 views
Points A, B, and C are given in the coordinate plane. There exists a point Q and a constant k such that for any point P,PA² +
1. 1 answer
2. 38 views
Let u and w be complex numbers such that|u|=5, |w|=3, and |u+w|=6. Calculate |u+2w| with proof.
1. 1 answer
2. 19 views
A stick is broken at two points, chosen at random. If the length of thestick is 6 units, then what is the probability that all
1. 1 answer
2. 40 views
Five workers have been hired to complete a job. If one additional worker is hired, they could complete the job 12 days earlier.
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2. 32 views
In the figure, if the measure of arc FG is 118 degrees , the measure of arc FQ is 25 degrees, and FR=FG then what is angle RPG,
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2. 35 views
Find the complex number z that satisfies:(1+i)z-2z*=-11+25i The notation z* stands for the conjugate of z.
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2. 43 views
For all complex numbers z, let f(z) = z^2 if z is not real, and f(z) = -z^2 if z is real. Find f(f(f(f(1+i)))).
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2. 16 views
Let $m$ and $n$ be positive integers. If $m$ has exactly $7$ positive divisors, $n$ has exactly $10$ positive divisors, and $mn$
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2. 28 views
Consider the graph of r = theta for theta > 0 given in radians, also known as the Archimedean spiral. Prove that the consecutive
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2. 52 views
All sacks of sugar have the same weight. All sacks of flour also have the same weight, but not necessarily the same as the
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2. 45 views
In triangle ABC, points D and F are on AB, E is on AC, such that DE is parallel to BC and EF is parallel to CD. If AF = 7 and DF
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2. 25 views
How many digits are in the whole-number representation of 10^100 - 9^100?
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2. 30 views
In triangle PQR, M is the midpoint of PQ .Let X be the point on QR such that PX bisects QPR and let the perpendicular bisector
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2. 45 views
Simon has built a gazebo, whose shape is a regular heptagon, with a side length of 3 units. He has also built a walkway around
1. 1 answer
2. 45 views
What is the arithmetic mean of all of the positive two-digit integers with the property that the integer is equal to the sum of
1. 1 answer
2. 68 views
Simplify sqrt((sqrt(5) - sqrt(2))/(sqrt(5) + sqrt(2))) + sqrt((sqrt(5) + sqrt(2))/(sqrt(5) - sqrt(2)))
1. 1 answer
2. 26 views
In quadrilateral BCED, sides BD and CE are extended past B and C, respectively, to meet at point A. If BD = 18, BC = 8, CE = 2,
1. 1 answer
2. 45 views

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