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shelly

Questions (247)

Let $e(x)$ be an even function and let $o(x)$ be an odd function, such that\[e(x) + o(x) = x^2 + x^3\] for all $x.$ Let $f(x) =
1. 1 answer
2. 57 views
simplify ((mn^3)^2)*(((2^-2)(m^-1)(n^3))^-2)
1. 1 answer
2. 45 views
Find the number of integers $n$ that satisfy both of the inequalities $4n + 3 < 253$ and $-7n + 5 < 24$.
1. 1 answer
2. 51 views
For what values of $x$ is the expression $\frac{\log(x^2-4x-5+2x)}{\sqrt{x-1}}$ defined? Express your answer in interval
1. 1 answer
2. 72 views
The diagram shows eight congruent squares inside a circle. Every shaded square has one vertex on the circle. What is the ratio
1. 1 answer
2. 99 views
In triangle $ABC,$ the angle bisector of $\angle BAC$ meets $\overline{BC}$ at $D.$ If $\angle BAC = 60^\circ,$ $\angle ABC =
1. 1 answer
2. 112 views
In the diagram, the circle is inscribed in the square. This means that the circle and the square share points $S$, $T$, $U$, and
1. 1 answer
2. 94 views
Triangle $MNO$ is an isosceles triangle with $MN = NO = 25\;\text{cm}$. A line segment, drawn from the midpoint of
1. 1 answer
2. 190 views
The vertices of triangle $ABC$ lie on the sides of equilateral triangle $DEF$, as shown. If $CD = 5$, $CE = BD = 3$, and $\angle
1. 1 answer
2. 125 views
In square ABCD, P is on BC such that BP = 4 and PC = 1, and Q is on line CD such that DQ = 1 and QC = 4. Find sin angle PAQ.
1. 1 answer
2. 47 views
In square ABCD, P is on BC such that BP = 4 and PC = 1, and Q is on CD such that DQ= 4 and QC = 1. Find sin angle PAQ
1. 1 answer
2. 91 views
In square ABCD, P is on BC such that BP = 4 and PC = 1, and Q is on line CD such that DQ = 4 and QC = 1. Find sin angle PBQ.
1. 1 answer
2. 64 views
Given that $A = (\sqrt{2008}+\sqrt{2009}),$ $B = (-\sqrt{2008}-\sqrt{2009}),$ $C = (\sqrt{2008}-\sqrt{200}),$ and $D =
1. 1 answer
2. 42 views
How many pairs of positive integers $(x,y)$ satisfy $x^2-y^2=5100$?
1. 1 answer
2. 77 views
Given the right triangles $ABC$ and $ABD$, what is the length of segment $BC$, in units?
1. 1 answer
2. 60 views
In right triangle $ABC$, $BD=CD+9$. If $AB=10$ and $BC=25$, what is $AD$
1. 1 answer
2. 138 views
Two men stand back-to-back and walk in opposite directions for $40$ yards each. Each of them then turns left and walks another
1. 1 answer
2. 85 views
Real numbers $a$, $b$, $c$, $x$, and $y$ satisfy\begin{align*} ax+by+c&=x+7,\\ a+bx+cy&=2x+6y-4,\\ ay+b+cx&=8x+5y-3,
1. 1 answer
2. 87 views
Given that (x+y+z)(xy+xz+yz)=25 and that x^3+y^3+z^3=18 for real numbers x, y, and z, what is the value of xyz?
1. 1 answer
2. 54 views
Syd chooses two different primes, both of which are greater than 10, and multiplies them. The resulting product is less than
1. 3 answers
2. 77 views
In the array below, in how many different ways can we start with the letter A and move from letter to letter (horizontally,
1. 1 answer
2. 109 views
All of the digits of a three-digit integer are distinct and non-zero. Furthermore, the three-digit integer is divisible by 19.
1. 1 answer
2. 93 views
All of the digits of a three-digit integer are distinct and non-zero. Furthermore, the three-digit integer is divisible by 17.
1. 1 answer
2. 91 views
How many integers from $1$ to $9$ are divisors of the five-digit number $24,519$?
1. 1 answer
2. 63 views
At the AoPS office, mice vary inversely with cats, that is, $\text{mice}=\frac{k}{\text{cats}}$, for some value of $k$. When
1. 1 answer
2. 143 views
For how many values of $a$ is it true that:(1) $a$ is a positive integer such that $a \le 50$. (2) the quadratic equation $x^2 +
1. 1 answer
2. 96 views
Find the sum of all the values of $x$ that satisfy the system of equations $y=|x^2-6x+5|$ and $y=\frac{29}{4}-2x$.
1. 1 answer
2. 82 views
What is the greatest integer $x$ such that $|6x^2-47x+15-28x|$ is prime?
1. 1 answer
2. 39 views
In the Olympic women's skating competition, the gold medal goes to first place, silver to second, and bronze to third. If there
1. 1 answer
2. 87 views
All of the digits of a three-digit integer are distinct and non-zero. Furthermore, the three-digit integer is divisible by 13.
1. 1 answer
2. 86 views
If $\log_{b/a} a^2 b = 2$, where $a$ and $b$ are positive real numbers, then find $\log_{ab} b$.
1. 1 answer
2. 80 views
In right triangle $ABC$, we have $\angle BAC = 90^\circ$ and $D$ is the midpoint of $\overline{AC}$. If $AB = 7$ and $BC = 7
1. 1 answer
2. 72 views
The largest value of $x$ that satisfies $\sqrt{x+1}=x-2$ can be written as $\dfrac{a+\sqrt{b}}{c}$ where $c$ has no common
1. 1 answer
2. 81 views
The number $\sqrt{53+20\sqrt{7}+148}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has
1. 1 answer
2. 93 views
The graphs of $y=x^4$ and $y=7x^2-10-5x^2+19/2$ intersect at four points with $x$-coordinates $\pm \sqrt{m}$ and $\pm \sqrt{n}$,
1. 1 answer
2. 52 views
Find the number of values of $x$ for which the expression $\frac{x^2-9}{x^2 + 9} + \frac{1}{x}$ is undefined.
1. 1 answer
2. 66 views
The domain of the function $f(x) = \frac{x}{x^2 - 8x + 3}$ is the set of all real numbers with the exception of the values $x =
1. 9 answers
2. 70 views
Find the sum of all values of $x$ for which the expression $\frac{x-3}{x^2-10x+16}$ is undefined.
1. 1 answer
2. 68 views
Let $f(x) = ax + b$ for $0 \le x \le 1$, where $a$ and $b$ are constants such that $a < 0$. What is the inverse of $f(x)$ in
1. 3 answers
2. 112 views
The graphs of $y=x^4$ and $y=5x^2-6+4x^2$ intersect at four points with $x$-coordinates $\pm \sqrt{m}$ and $\pm \sqrt{n}$, where
1. 1 answer
2. 42 views
The difference between two positive integers is $6$ and their product is $40$. What is the sum of the integers?
1. 1 answer
2. 60 views
Real numbers $x$ and $y$ have an arithmetic mean of $18$ and a geometric mean of $\sqrt{47}$. Find $x^2+y^2$.
1. 1 answer
2. 45 views
The quadratic equation $ax^2+20x+c=4x$ has exactly one solution. If $a+c=20$, and $a<c$ find the ordered pair $(a,c)$.
1. 1 answer
2. 55 views
Right triangle $ABC$ has $AB = AC = 6$ cm. Circular arcs are drawn with centers at $A, B$ and $C,$ so that the arc centered at
1. 1 answer
2. 102 views
The equation $x^3 + 8x^2 - 4x + c = 0$ has three roots, one of which is the product of the other two. What is c?
1. 1 answer
2. 55 views
Solve the inequality\[\dfrac{x+1}{x+2}>\dfrac{3x+4}{2x+9}+\dfrac{1}{3}.\]
1. 1 answer
2. 80 views
In triangle $ABC$, $\angle C = 90^\circ$, $\angle A = 40^\circ$, and $AC = 10$. Find the radius of the incircle of triangle
1. 1 answer
2. 48 views
A polynomial with integer coefficients is of the form\[28x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 1 = 360.\] Find the number of
1. 1 answer
2. 105 views
In a certain polynomial, all the coefficients are integers, and the constant coefficient is $100000$. All the roots are
1. 1 answer
2. 103 views
A circle with radius $5$ and center $(a,b)$ is tangent to the lines $y = 6$ and $y = x.$ Compute the largest possible value of
1. 1 answer
2. 43 views
Let $P$ be a point that varies on the parabola $y^2 = 2x.$ The tangents from $P$ to the circle $(x - 1)^2 + y^2 = 1$ intersect
1. 0 answers
2. 84 views
A right triangle has side lengths $3$, $4$, and $5.$ A circle is then drawn with each side as a diameter. Find the area of the
1. 1 answer
2. 93 views
The complete floor plan of a vacation cottage is shown. Both bedrooms have the same dimensions. What is the total area of the
1. 1 answer
2. 81 views
In the figure shown, adjacent sides of the parallelogram are $4$ cm and $6$ cm. The indicated angle is $25^\circ$. What is the
1. 3 answers
2. 115 views
Let $AC$ be a diameter of a circle $\omega$ of radius $1$, and let $D$ be the point on $AC$ such that $CD = \frac{1}{5}$. Let
1. 3 answers
2. 139 views
A pair of circles intersect in points $P$ and $Q$. Let $\overline{AB}$ be a segment passing through $P$ having one endpoint on
1. 1 answer
2. 209 views
In the diagram, four circles of radius $1$ with centers $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of
1. 1 answer
2. 90 views
Find the domain of the function $f(x) = \frac{\sqrt{x-2}}{\sqrt{5}} + \sqrt{(x - 1)(x + 2)(x + 4)}.
1. 1 answer
2. 93 views
Find the domain of the function $f(x) = \sqrt{6-x-x^2-2x^2}$.
1. 1 answer
2. 38 views
Find the largest real number $c$ such that $1$ is in the range of $f(x)=x^2-5x+c+x^2-8x$.
1. 1 answer
2. 56 views
The roots of the quadratic equation $3x^2+5x+k = 0$ are $\frac{-5\pm i\sqrt{131}}{6}$. What is $k$?
1. 1 answer
2. 76 views
The roots of the quadratic equation $x(x-3)=1+8x-5$ may be expressed in the form $\frac{a+\sqrt{b}}{c}$ and
1. 1 answer
2. 84 views
How many unique sums can be formed by adding any three different numbers from the set $\{4,6,8,10,12,14,16\}\,?$
1. 1 answer
2. 73 views
Consider the equilateral triangle $ABC$ with sides of length $8\sqrt{3}$ cm. A point in the interior of $ABC$ is said to be
1. 1 answer
2. 110 views
If $y = \frac{x + 1}{x^2 + 1 - x},$ and $x$ is any real number, then what is the sum of the maximum and minimum possible values
1. 1 answer
2. 71 views
Triangle $ABC$ has vertices $A(2, 3),$ $B(1, -8)$ and $C(-3, 2).$ The line containing the altitude through $A$ intersects the
1. 1 answer
2. 75 views
Find the area of the region enclosed by the graph of the equation $x^2-14x+3y+70=15+9y-y^2$ that lies below the line $y=x-3$.
1. 1 answer
2. 92 views
Find the product of all positive integer values of $c$ such that the quadratic equation $3x^2+7x+c=15x-10$ has two real roots.
1. 1 answer
2. 88 views
In cyclic quadrilateral PQRS, P/3 = Q/5 = R/8 + 60. Find the largest angle in quadrilateral PQRS, in degrees.
1. 1 answer
2. 99 views
The point $(x,y)$ in the coordinate has a distance of $6$ units from the $x$-axis, a distance of $15$ units from the point
1. 1 answer
2. 111 views
There is exactly one value of $x$ for which the distance from $(5,6)$ to $(3x-1,ax+5)$ is $4$. If $a \neq 0,$ what is $a$?
1. 1 answer
2. 75 views
Find the non-zero value of $c$ for which there is exactly one positive value of $b$ for which there is one solution to the
1. 1 answer
2. 52 views
If x, y, and z are positive integers such that 6xyz + 30xy + 18xz + 12yz + 10x + 17y + 5z =481, find x + y + z.
1. 3 answers
2. 88 views
A sphere is inscribed in a cone with height $3$ and base radius $3$. What is the ratio of the volume of the sphere to the volume
1. 1 answer
2. 125 views
A sphere is inscribed in a cone with height $3$ and base radius $4$. What is the ratio of the volume of the sphere to the volume
1. 1 answer
2. 106 views
Rose has a spherical plum of radius $2$ and a spherical watermelon of radius $8$. She builds a glass sphere around the two
1. 1 answer
2. 126 views
A sphere of radius $4$ inches is inscribed in a cone with a base of radius $8$ inches. In inches, what is the height of the
1. 1 answer
2. 160 views
Let $S$ be the set of points $(x,y)$ in the coordinate plane that satisfy the inequalities\begin{cases} x &\ge 0 \\ y &\ge 0 \\
1. 1 answer
2. 83 views
Find the sum of all values of $k$ for which $x^2 + kx - 9x + 25 - 9$ is the square of a binomial.
1. 1 answer
2. 53 views
If $(10 - x)(x - 2) = 11$, then compute $(10 - x)^3 + (x - 2)^3$.
1. 1 answer
2. 39 views
An equilateral triangle of side $12$ centimeters is rotated about a side to form a solid. What is the number of square
1. 1 answer
2. 96 views
Let $a$ and $b$ be the roots of the quadratic equation $2x^2 - 7x + 2 = -x^2 + 4x + 9.$ Find $\frac{1}{a-1}+\frac{1}{b-1}.$
1. 1 answer
2. 83 views
Find all values of $q$ such that the quadratic equation $qx^2 + (2q - 9)x + (q - 8) = 0$ has two positive real roots.
1. 1 answer
2. 78 views
The quadratic equation $x^2-5x+t =3x$ has only positive integer roots. Find the average of all distinct possible values of $t$.
1. 1 answer
2. 87 views
The quadratic equation $x^2-5x+t =-x^2+3x$ has only positive integer roots. Find the average of all distinct possible values of
1. 1 answer
2. 84 views
Three points lie on the graph of the parabola $y = x^2$. The three points form an equilateral triangle. One of the vertices is
1. 1 answer
2. 96 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. 78 views
The surface area of a solid cylinder is found using the formula $SA = 2\pi r^2 + 2\pi rh$. The volume of a cone is found using
1. 1 answer
2. 178 views
The surface area of a solid cylinder is found using the formula $SA = 2\pi r^2 + 2\pi rh$. The volume of a cone is found using
1. 1 answer
2. 195 views
Find all values of $a$ such that $\frac{a-3}{\sqrt{a}} = -a \sqrt{a}$.
1. 1 answer
2. 57 views
Our badminton team has finished $75\%$ of its season. So far, we have won $32\%$ of the games we played. What percent of the
1. 1 answer
2. 91 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, so
1. 1 answer
2. 72 views
In triangle $ABC,$ the angle bisector of $\angle BAC$ meets $\overline{BC}$ at $D.$ If $\angle BAC = 60^\circ,$ $\angle ABC =
1. 1 answer
2. 122 views
The function $f(x),$ defined for $x > 0,$ is positive, differentiable, and decreasing. The function $f(x)$ has the following
1. 1 answer
2. 158 views
Given $m\geq 2$ denote by $b^{-1}$ the inverse of $b\pmod{m}$. That is, $b^{-1}$ is the residue for which $bb^{-1}\equiv
1. 1 answer
2. 87 views
Two quarter-circles are drawn inside a unit square, as shown. A circle is drawn that is tangent to the quarter-circles, and a
1. 1 answer
2. 116 views
The quadratic equation $x^2-5x+5=-4x+7$ has two roots. Find the absolute difference of these roots.
1. 1 answer
2. 80 views
Let $m$ be an integer. If the quadratic equation $x^2 + mx + 3m +1 = 0$ has one or more integer roots, then find the sum of all
1. 1 answer
2. 102 views
The real numbers x and y satisfy x (x+y) = 27 and (x+y) = 54. What is the value of (x+y)^2?
1. 1 answer
2. 84 views
The graph of the parabola $y = x^2 + 2bx + c$ is tangent to the line $y = 9x - 7$, where $c \neq 0$. What is $c-4b$?
1. 2 answers
2. 75 views

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