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Fiona

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Given a = 4^(1/3) + 2^(1/3) + 1,find the value of 3/a + 3/a² + 1/a³.
1. 1 answer
2. 9 views
Fill in the blanks, to complete the factorization:(a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = (a + ___)(a + ___)(a +
1. 1 answer
2. 17 views
A large parallelogram is divided into four small parallelograms asshown. The areas of three of the four small parallelograms are
1. 1 answer
2. 79 views
Let ABCD be a parallelogram. Extend BC past B to F, and let E be the intersection of AB and FD. If the areas of triangles BEF
1. 1 answer
2. 41 views
Discrete independent random variables X and Y are given by the following laws of distribution:X 0 3 Y - 2 - 1 2 P 0,3 0,7 P 0,2
1. 1 answer
2. 41 views
If A, B, and C represent three distinct digits from 1 to 9 and they satisfy the following equations, what is the value of the
1. 1 answer
2. 32 views
For $0 \leq x \leq 1,$ the function $f(x)$ satisfies the relations $f\left(\dfrac x{x+1}\right) = \dfrac{f(x)}2$ and $f(1-x) = 1
1. 1 answer
2. 45 views
Sophie's favorite number is a two-digit number. If she reverses the digits, the result is $45$ less than her favorite number.
1. 1 answer
2. 27 views
Anna has a collection of $120$ marbles, each of which is either red or blue. If Anna has $135$ more red marbles than blue
1. 1 answer
2. 20 views
Find all real numbers that satisfy\frac{1}{a^3 + 7} -7 = -\frac{a}{a^3 + 7}.
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2. 28 views
What is the smallest positive integer n such that \sqrt[4]{675 + n} is an integer?
1. 1 answer
2. 21 views
Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to 56/11 times the largest of
1. 1 answer
2. 42 views
In a right triangle △ABC, ∠ABC=90∘, AB=6, and AC=10. Point D lies on side BC,and AD is the altitude from vertex A to the
1. 1 answer
2. 27 views
In triangle PQR, let X be the intersection of the angle bisector of angle P with side QR, and let Y be the foot of the
1. 1 answer
2. 31 views
In the diagram below, each side of convex quadrilateral $ABCD$ is trisected. (For example, $AP = PQ = QB.$) The area of convex
1. 1 answer
2. 54 views
In trapezoid $EFGH,$ $\overline{EF} \parallel \overline{GH},$ and $P$ is the point on $\overline{EH}$ such that $EP:PH = 1:2$.
1. 1 answer
2. 60 views
In the diagram below, quadrilateral $BAED$ is a parallelogram, and $BD = BC$ and ∠DAB=24°. Find $\angle EAB$, in degrees.
1. 1 answer
2. 38 views
Let $ABC$ be a triangle. The side lengths of triangle $ABC$ are $AB = 15,$ $BC = 9,$ and $AC = 10.$ Find the length of the
1. 1 answer
2. 42 views
Compute1 + \frac{3}{4} + \frac{5}{4^2} + \frac{7}{4^3}
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2. 31 views
A geometric sequence has 400 terms. The first term is 1600 and the common ratio is $-\frac{1}{2}.$ How many terms of this
1. 1 answer
2. 46 views
Assume that $f(3) = 4$. Name a point that must be on the graph of y= f(2/5*x + 4) + f(3x - 1) .
1. 1 answer
2. 45 views
Let a(x) and b(x) be functions. Find (a\circ b)(3) - (b\circ a)(3) if $a(x) = 2x + 5$ and $b(x) = 4 - x^2$.
1. 1 answer
2. 39 views
Let $f(x)$ and $g(x)$ be functions. Find $c$ if(f \circ g)(x) = (g \circ f)(x) for all $x$, where $f(x) = 3x - 4$ and $g(x) = 5x
1. 1 answer
2. 34 views
Find the real number $k$ such that $f(x) = f(k - x)$ for all real numbers $x$, where\[f(x) = 3x^2 - 4x - x^2 + 11x.\]
1. 1 answer
2. 46 views
Let $x$ and $y$ be nonnegative real numbers. If $x^2 + 5y^2 = 30$, then find the maximum value of $x + y$.
1. 1 answer
2. 60 views
Find the number of ordered pairs (m,n) of integers that satisfymn = 3m + 6n + 2m - 5n + 1.
1. 1 answer
2. 43 views
For a real number t, let M(t) denote the maximum value of f(x) = 4x^2 - x + t for 11 \le x \le 1. What is the smallest possible
1. 1 answer
2. 37 views
A regular a-gon, a regular b-gon, a regular c-gon, and a regular d-gon fit perfectly around a point. What is the largest
1. 1 answer
2. 32 views
In the triangular array below, n different prime numbers p_1, p_2, \dots, p_n are written in the bottom row. Then in each box,
1. 1 answer
2. 34 views
Carl writes eight consecutive three-digit numbers on a blackboard. Each number that Carl writes is divisible by 2, 3, 4, or 5.
1. 1 answer
2. 39 views
When a prime is divided by 60, the remainder is a composite number. When a second prime is divided by 60, the remainder is a
1. 1 answer
2. 48 views
How many three-digit numbers are equal to five times the sum of their digits?
1. 1 answer
2. 15 views
Given a sequence which contains the numbers 1 through 8, a move consists of moving one of the numbers to the front of the
1. 1 answer
2. 38 views
A sequence of ten letters has the following properties:* Each letter is A, B, C, D, or E. * The only letters that can come
1. 1 answer
2. 26 views
The map below displays the train network of Beastville, including the stations X and $Y,$ and the lines $a,$ $b,$ $c,$ $d,$ $e,$
1. 1 answer
2. 33 views
Each vertex of cube ABCDEFGH is colored either black or white. Find the number of colorings in which all four vertices of at
1. 1 answer
2. 27 views
Franklin the Fly lives at one corner of a cube of side length 1. He doesn't like to wander too far from home; in fact, Franklin
1. 1 answer
2. 29 views
In triangle ABC, D is a point on side \overline{BC} such that $DC = AB.$ Let $M$ be the midpoint of $\overline{BD},$ and let $N$
1. 1 answer
2. 35 views
In triangle ABO, M is the midpoint of \overline{AB}. Points C, D, and N lie on line segments $\overline{AO},$ $\overline{BO},$
1. 1 answer
2. 30 views
In the diagram below, ABCD is a square, DE = EF = FG = 2, GH = 1, HB = 5, and \angle DEF = \angle EFG = \angle FGH = \angle GHB
1. 1 answer
2. 44 views
Unit square $ABCD$ is divided into two triangles, which are congruent to each other, and two trapezoids, which are also
1. 1 answer
2. 40 views
Ten positive integers (not necessarily distinct) are arranged in a circle. An integer is colored red if it is equal to the
1. 1 answer
2. 49 views
Consider the set of all fractions of the form \frac{2^m}{3^n}, where m and n are integers such that 0 \le m \le n \le 3. For
1. 1 answer
2. 25 views
For certain real numbers a and b, the function f satisfies f(x) = ax + b + cx^2 and f(bx + a) = x + cx^2 for all real numbers x.
1. 1 answer
2. 39 views
Find the number of ordered triples (a,b,c) of positive integers that satisfya = \gcd(b,c) + 33 b = \gcd(a,c) + 25 c = \gcd(a,b)
1. 1 answer
2. 42 views
avatarLet a, b, c, d be distinct positive integers such that {lcm}(a, b, c, d) < 1000 and a + b + c + d = 1000. Find the largest
1. 1 answer
2. 43 views
Find the largest positive integer n for which there exist positive integers a, b, and c such that\gcd(a + 3b, b + 3c, c + 3a) =
1. 1 answer
2. 31 views
Let N be the number of ordered 6-tuples (a_1, a_2, a_3, a_4, a_5, a_6) of positive integers that satisfy\frac{1}{a_1} +
1. 1 answer
2. 62 views
Let a, b, and c be integers such that 7a + 4b = 3c. Find the largest integer that always divides $abc$.
1. 1 answer
2. 26 views
Xavier and Yvonne meet one day at a cafe. For any day that Xavier goes to the cafe, the probability that Xavier goes the next
1. 1 answer
2. 25 views
Find the number of subsets S of \{0, 1, 2, 3, \dots, 15\} that have the following property: If n is in S, and m \equiv n + 1
1. 1 answer
2. 28 views
Let A and B be two fixed points in the plane such that the distance between them is $1.$ Point $P$ is chosen at random on the
1. 1 answer
2. 39 views
In the $4 \times 4$ grid of points below, each point is one unit away from its closest neighbor. In how many ways are there to
1. 1 answer
2. 34 views
Five workers $A,$ $B,$ $C,$ $D,$ and $E$ at a coffee stand are working out a schedule for the next $10$ days. The schedule must
1. 1 answer
2. 28 views
The sides of convex quadrilateral ABCD are AB = 8, BC = 5, $CD = 5,$ and $DA = 8.$ Diagonals $\overline{AC}$ and $\overline{BD}$
1. 1 answer
2. 44 views
Let H be the orthocenter of acute triangle ABC, and let M be the midpoint of \overline{AC}. Ray $\overrightarrow{MH}$ intersects
1. 1 answer
2. 29 views
Triangle ABC has a right angle at B. Point D lies on side \overline{AC} such that CD = 6. The circle with diameter
1. 1 answer
2. 27 views
In right triangle ABC, \angle ACB = 90^\circ, and the legs are a = BC and b = AC. Let $S$ and $T$ be points on $\overline{AC}$
1. 1 answer
2. 28 views
In tetrahedron ABCD, AB = AC = AD = 12 and BC = BD = CD = 12. There is a sphere that is tangent to all six edges of the
1. 1 answer
2. 35 views
Let the roots of z^{99} = 1 be z_k = x_k + iy_k for 1 \le k \le 99, where $x_k$ and $y_k$ are real. Let $P(z)$ be the monic
1. 1 answer
2. 14 views
Find the number of positive integers n \le 10000 that satisfy\lfloor \log_4 n \rfloor + \lfloor \log_8 n \rfloor + \lfloor
1. 1 answer
2. 37 views
Let a and b be integers such that the polynomialx^4 + ax^3 + bx^2 + ax + 1 = 0 has four distinct positive real roots. Find the
1. 1 answer
2. 29 views
Let $d$ be a function taking the positive integers to the nonnegative integers such that $d(p) = 1$ for any prime $p,$ andd(ab)
1. 1 answer
2. 52 views
Let r, s, and t be the real roots of(x - \sqrt[3]{13})(x - \sqrt[3]{53})(x - \sqrt[3]{103}) = 0. Compute r^3 + s^3 + t^3.
1. 1 answer
2. 26 views
Expand (x + 2)(3x^2 + 12) + (x - 2).
1. 1 answer
2. 23 views
Let P be the smallest prime such that there exist positive integers A and B satisfying A^2 + P^3 = B^4.Find all possible
1. 1 answer
2. 26 views
Jeri finds a pile of money with at least 200. If she puts $50$ of the pile in her left pocket, gives away $\frac{1}{8}$ of the
1. 1 answer
2. 34 views
The line below has a slope of $\frac{2}{5}.$ Find the $x$-intercept.The line passes through (0,0).
1. 1 answer
2. 47 views
Compute the length of the segment tangent from the origin to the circle that passes through the points (3, 4), (6, 8) and (5,
1. 1 answer
2. 34 views
an integer between 500 and 999, inclusive, is chosen at random. what is the probability that it is an even integer whose digits
1. 1 answer
2. 47 views
Five workers have been hired to complete a job. If one additional worker is hired, they could complete the job $8$ days earlier.
1. 1 answer
2. 29 views
Three runners, Dirk, Edith, and Foley all start at the same time for a $24$ km race, and each of them runs at a constant speed.
1. 1 answer
2. 47 views
Find all values of $a$ such that $\frac{a-3}{\sqrt{a}} = a$.
1. 1 answer
2. 26 views
Ruth has a beaker containing a solution of $800$ mL of acid and $200$ mL of water. She thinks the solution is a little strong,
1. 1 answer
2. 31 views
Ruth has a beaker containing a solution of $800$ mL of acid and $200$ mL of water. She thinks the solution is a little strong,
1. 1 answer
2. 31 views
A plane flies from Penthaven to Jackson and then back to Penthaven. When there is no wind, the round trip takes $5$ hours and
1. 1 answer
2. 22 views
All members of our painting team paint at the same rate. If $20$ members can paint a $1500$ square foot wall in $30$ minutes,
1. 1 answer
2. 33 views
All members of our painting team paint at the same rate. If $12$ members can paint a specific wall in $12$ minutes, then how
1. 1 answer
2. 34 views
I drove to the beach at a rate of 40 miles per hour. If I had driven at a rate of 30 miles per hour instead, then I would have
1. 1 answer
2. 27 views
I have to paint one side of a wall. The wall is $6$ meters tall and $60$ meters long. What is the area of the wall, in square
1. 1 answer
2. 33 views
Alice and Bob each have a certain amount of money. If Alice receives n dollars from Bob, then she will have 3 times as much
1. 1 answer
2. 34 views
In a store window, there was a box of berries having a total weight of $200$ kg. The berries were $98\%$ water, by weight. After
1. 1 answer
2. 32 views
At a math convention, a poll is taken about cable news. It turns out that $5$ out of every $8$ people are bored by cable news.
1. 1 answer
2. 31 views
Calculate \arccos \sqrt{\cfrac{1+\sqrt{\cfrac{1+\sqrt{\cfrac{1-\sqrt{\cfrac{1+\cfrac{\sqrt{3}}{2}}{2}}}{2}}}{2}}}{2}}. As usual,
1. 1 answer
2. 46 views
I need help solving the problem: How many points of the form (x,y) , where both coordinates are positive integers, lie below the
1. 1 answer
2. 40 views
In the SuperLottery, three balls are drawn (at random, without replacement) from white balls numbered from $1$ to $10,$ and one
1. 1 answer
2. 48 views
In the SuperLottery, three balls are drawn (at random, without replacement) from white balls numbered from $1$ to $10,$ and one
1. 1 answer
2. 62 views
Each vertex of cube a b c d e f g h is colored either black or white. Find the number of colorings in which all four vertices of
1. 1 answer
2. 26 views
Find the number of $7$-digit positive integers, where the sum of the digits is divisible by $3.$
1. 1 answer
2. 33 views
Expand (sqrt(x) + 3x)*4 + (sqrt(x) - 3x)*4.
1. 1 answer
2. 21 views
Fill in the blanks with positive integers:(3 + sqrt(6))*3 = ___ + ___ * sqrt(6)
1. 1 answer
2. 23 views
Given a sequence which contains the numbers $1$ through $8,$ a move consists of moving one of the numbers to the front of the
1. 1 answer
2. 38 views
Given that x^n - 1/x^n is expressible as a polynomial in x - 1/x with real coefficient only if n is an odd positive integers,
1. 1 answer
2. 28 views
avatarFind the number of solutions to x_1 + x_2 + x_3 + x_4 + x_5 \le 1 in nonnegative integers.
1. 1 answer
2. 29 views
In how many ways can you distribute $8$ indistinguishable balls among $6$ distinguishable boxes, if at least three of the boxes
1. 1 answer
2. 34 views
Miyu is giving out $8$ identical chocolates to her $5$ friends, including Dhruv. All possible distributions are equally likely.
1. 1 answer
2. 27 views
Fill in the blanks with integers to get an identity.(n - k + 2)^2 = __ n^2 + ___ k^2 + ___ n + ___k + ___ nk + ___
1. 1 answer
2. 27 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. 38 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.What
1. 1 answer
2. 45 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.What
1. 1 answer
2. 33 views

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