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Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Suppose $x-3$ and $y+3$ are multiples of $7$.
What is the smallest positive integer, $n,$ for which $x^2+xy+y^2+n$ is a multiple of $7$?
SOLUTION:Since $x-3$ is... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Simplify: $$\dfrac{\sqrt{338}}{\sqrt{288}}+\dfrac{\sqrt{150}}{\sqrt{96}}.$$Express your answer as a common fraction.
SOLUTION:We can simplify each square root fi... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Find the matrix $\mathbf{P}$ such that for any vector $\mathbf{v},$ $\mathbf{P} \mathbf{v}$ is the projection of $\mathbf{v}$ onto the vector $\begin{pmatrix} 2 ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:The parabolas defined by the equations $y=2x^2-4x+4$ and $y=-x^2-2x+4$ intersect at points $(a,b)$ and $(c,d)$, where $c\ge a$. What is $c-a$? Express your answe... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:If $m$ and $n$ are odd integers, how many terms in the expansion of $(m+n)^6$ are odd?
SOLUTION:By the binomial theorem, $(m+n)^6$ expands as \begin{align*}
\bin... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:If $A=4-3i$, $M=-4+i$, $S=i$, and $P=2$, find $A-M+S-P$.
SOLUTION:Adding real parts and imaginary parts separately, we have $(4-(-4)+0-2)+(-3-1+1-0)i=\boxed{6-3i... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:In triangle $ABC,$ $AB = 9,$ $BC = 10,$ and $AC = 11.$ If $D$ and $E$ are chosen on $\overline{AB}$ and $\overline{AC}$ so that $AD = 4$ and $AE = 7,$ then find... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Determine the number of ways to arrange the letters of the word RADII.
SOLUTION:First we count the arrangements if the two I's are unique, which is 5!. Then sinc... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\begin{align*}
a + b + c + d &= 6, \\
a^2 + b^2 + c^2 + d^2 &= 12.
\end{align*}Let $m$ and $M$ denote minimum an... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $z$ be a complex number satisfying $z^2 = 4z - 19 + 8i$. Given that $|z|$ is an integer, find $z.$
SOLUTION:We can write the given equation as
\[z^2 - 4z = -... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A square with side length 1 is rotated about one vertex by an angle of $\alpha,$ where $0^\circ < \alpha < 90^\circ$ and $\cos \alpha = \frac{4}{5}.$ Find the a... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let
\[\mathbf{A} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{\sqrt{3}}{2} & 0 & -\frac{1}{2} \\ 0 & -1 & 0 \\ \frac{1}{2} & 0 & \frac{\sqrt{3}}{2} ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $a,$ $b,$ and $t$ be real numbers such that $a + b = t.$ Find, in terms of $t,$ the minimum value of $a^2 + b^2.$
SOLUTION:By QM-AM,
\[\sqrt{\frac{a^2 + b^2... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:How many three-digit whole numbers have no 7's and no 9's as digits?
SOLUTION:In order to be a three-digit number, the first digit cannot be 0, so there are norm... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A palindrome is a number that reads the same forwards and backwards, such as 3003. How many positive four-digit integers are palindromes?
SOLUTION:Constructing p... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:The operation $\star$ is defined as $a \star b = a + \frac{a}{b}$. What is the value of $12 \star 3$?
SOLUTION:We have $12 \star 3 = 12+ \frac{12}{3}=12+4=\boxed... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $x,$ $y,$ $z$ be positive real numbers such that
\[\left( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \right) + \left( \frac{y}{x} + \frac{z}{y} + \frac{x}{z} \r... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:What is the smallest possible real value of $x^2 + 8x$?
SOLUTION:Completing the square, we get $x^2 + 8x = (x^2 + 8x + 16) - 16 = (x + 4)^2 - 16,$ so the smalles... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Find the quadratic function $f(x) = x^2 + ax + b$ such that
\[\frac{f(f(x) + x)}{f(x)} = x^2 + 1776x + 2010.\]
SOLUTION:We have that
\begin{align*}
f(f(x) + x) &... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:What is the least positive integer that can be added to 412 to yield a multiple of 3?
SOLUTION:Note that dividing 412 by 3 gives a quotient of 137 and a remainde... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A point $P$ is randomly selected from the rectangular region with vertices $(0,0), (2,0)$, $(2,1),(0,1)$. What is the probability that $P$ is closer to the origi... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:For real numbers $t,$ the point
\[(x,y) = (\cos^2 t, \sin^2 t)\]is plotted. All the plotted points lie on what kind of curve?
(A) Line
(B) Circle
(C) Parabola
... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:What is the remainder when $333^{333}$ is divided by $11$?
SOLUTION:We use the property that $a \equiv b \pmod{m}$ implies $a^c \equiv b^c \pmod{m}$.
$333 \equi... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:After traveling 50 miles by taxi, Ann is charged a fare of $\$120$. Assuming the taxi fare is directly proportional to distance traveled, how much would Ann be c... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Given a point $P$ on a triangular piece of paper $ABC,\,$ consider the creases that are formed in the paper when $A, B,\,$ and $C\,$ are folded onto $P.\,$ Let u... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:What is the domain of the function $f(x)=\log_2(\log_3(\log_4x))$?
SOLUTION:In order for the given function to have a real value, $\log_3(\log_4x)>0$ (since the ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:If Anna flips 8 coins, what is the probability that she gets more heads than tails?
SOLUTION:We could solve this problem using casework, but using a little bit o... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:If
\[x + \sqrt{x^2 - 1} + \frac{1}{x - \sqrt{x^2 - 1}} = 20,\]then find
\[x^2 + \sqrt{x^4 - 1} + \frac{1}{x^2 + \sqrt{x^4 - 1}}.\]
SOLUTION:Rationalizing the den... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Square ABCD has its center at $(8,-8)$ and has an area of 4 square units. The top side of the square is horizontal. The square is then dilated with the dilation ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Compute the domain of the real-valued function $$f(x)=\sqrt{3-\sqrt{5-\sqrt{x}}}.$$
SOLUTION:For the contents of the innermost square root to be nonnegative, we ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Given that $x+y = 10$ and $2x+y = 13$, evaluate $x^2-y^2$.
SOLUTION:Subtracting the first given equation from the second equation, we have $2x+y-(x+y)=13-10 \Rig... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Suppose that $a$ is a multiple of 4 and $b$ is a multiple of 8. Which of the following statements are true?
A. $a+b$ must be even.
B. $a+b$ must be a multiple o... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Suppose that there are two congruent triangles $\triangle ABC$ and $\triangle ACD$ such that $AB = AC = AD,$ as shown in the following diagram. If $\angle BAC = ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 4 units to the numerical value of its perimeter,... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Find $h(x)$, with terms in order of decreasing degree, if \[9x^3-3x+1+h(x)=3x^2-5x+3.\]
SOLUTION:This equation is solved by \[h(x)=(3x^2-5x+3)-(9x^3-3x+1)=\boxe... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:The center of the circle with equation $x^2+y^2=4x+12y-39$ is the point $(h,k)$. What is $h+k$?
SOLUTION:We will complete the square to determine the standard fo... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A parabola with equation $y = x^2 + bx + c$ passes through the points $(2,3)$ and $(4,3)$. What is $c$?
SOLUTION:Substitute $(2,3)$ and $(4,3)$ into the equatio... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:The diameter $AB$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and line $ED$ is per... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $\mathbf{v} = \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix}.$ The columns of a matrix are $\mathbf... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:What is the value of $K$ in the equation $16^3\times8^3=2^K$?
SOLUTION:Simplify \[
16^3\times 8^3=(2^4)^3\times(2^3)^3=2^{12}\times2^{9}=2^{21}.
\] Then $2^{21}=... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Marcus has two numbers, $a$ and $b$. When he divides $a$ by 45 he gets a remainder of 37. When he divides $b$ by 30 he gets a remainder of $9$. What remainder... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A zealous geologist is sponsoring a contest in which entrants have to guess the age of a shiny rock. He offers these clues: the age of the rock is formed from th... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $\alpha$ and $\beta$ be real numbers. Find the minimum value of
\[(2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2.\]
SOLUTION:L... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:If $y=kx^{\frac{1}{4}}$ and $y=3\sqrt{2}$ at $x=81$, what is the value of $y$ at $x=4$?
SOLUTION:First we have to solve for $k$ where $3\sqrt{2}=k\cdot81^{\frac{... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A teacher has a class with $24$ students in it. If she wants to split the students into equal groups of at most $10$ students each, what is the least number of g... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Diana is three years younger than Eduardo. Eduardo is four years older than Chad. Faye is three years older than Chad. Diana is 14 years old. How many years old ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $k, a_2, a_3$ and $k, b_2, b_3$ be nonconstant geometric sequences with different common ratios. If \[a_3-b_3=2(a_2-b_2),\]then what is the sum of the comm... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:How many even perfect square factors does $2^4 \cdot 7^9$ have?
SOLUTION:A positive integer is a factor of $2^4\cdot7^9$ if and only if its prime factorization i... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A boy has the following seven coins in his pocket: $2$ pennies, $2$ nickels, $2$ dimes, and $1$ quarter. He takes out two coins, records the sum of their values,... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:When a polynomial $p(x)$ is divided by $x + 1,$ the remainder is 5. When $p(x)$ is divided by $x + 5,$ the remainder is $-7.$ Find the remainder when $p(x)$ is... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Solve for $r$: $r = \displaystyle\frac{\sqrt{5^2+12^2}}{\sqrt{16+9}}$. Express as a common fraction.
SOLUTION:We have \[r = \displaystyle\frac{\sqrt{5^2+12^2}}{... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Pizzas are sized by diameter. What percent increase in area results if Chantel's pizza increases from a 10-inch pizza to a 12-inch pizza?
SOLUTION:The area of th... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:For some constants $a$ and $c,$
\[\begin{pmatrix} a \\ -1 \\ c \end{pmatrix} \times \begin{pmatrix} 7 \\ 3 \\ 5 \end{pmatrix} = \begin{pmatrix} -11 \\ -16 \\ 25 ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A fair coin is flipped 7 times. What is the probability that at least 5 of the flips come up heads?
SOLUTION:First, we count the number of total outcomes. Each... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Juan rolls a fair regular octahedral die marked with the numbers 1 through 8. Then Amal rolls a fair six-sided die. What is the probability that the product of t... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $\mathcal{C}$ be the hyperbola $y^2 - x^2 = 1$. Given a point $P_0$ on the $x$-axis, we construct a sequence of points $(P_n)$ on the $x$-axis in the follow... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $P$ be the point on line segment $\overline{AB}$ such that $AP:PB = 3:2.$ Then
\[\overrightarrow{P} = t \overrightarrow{A} + u \overrightarrow{B}\]for some ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 8 and 20 units, respectively, and the altitude is 12 units. Points $E$ and $F$ are the midp... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:What is the slope of the line determined by any two solutions to the equation $\frac{2}{x}+\frac{3}{y} = 0$? Express your answer as a common fraction.
SOLUTION:W... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A ball bounces back up $\frac{2}{3}$ of the height from which it falls. If the ball is dropped from a height of $243$ cm, after how many bounces does the ball fi... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A professor is assigning grades to a class of 10 students. As a very kind professor, he only gives out A's, B's, and C's. How many ways can the professor assign ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of this box. What percent of the original volume ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:What is the sum of all the solutions of $x = \left|2x-|60-2x|\right|$?
SOLUTION:We take cases on the sign of $60-2x.$ If $60-2x \ge 0,$ then the equation becomes... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Below is the graph of $y = a \sin bx$ for some constants $a < 0$ and $b > 0.$ Find $a.$
[asy]import TrigMacros;
size(400);
real g(real x)
{
return (-2*sin(x... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Amy works for 36 hours per week for 10 weeks during the summer, making $\$3000$. If she works for 30 weeks during the school year at the same rate of pay and nee... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Ivan rents a car for $\$$25 a day and $\$$0.20 a mile. If he rents it for 4 days and drives it 400 miles, how many dollars does he pay?
SOLUTION:The cost of rent... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:The sum of two positive integers $a$ and $b$ is 1001. What is the largest possible value of $\gcd(a,b)$?
SOLUTION:Note that $\gcd(a,b)$ divides both $a$ and $b$... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Simplify the following expression:
$$5x + 6 - x + 12$$
SOLUTION:Rearranging and grouping, we obtain $(5x - x) + (6 + 12) = \boxed{4x + 18}$.
<<<Examples ends
Q... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:What is $4\cdot 6+8\cdot 3-28\div 2$?
SOLUTION:Recall that multiplication and division should be done in order from left to right before addition and subtraction... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$?
SOLUTION:Multiplying... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Compute $\dbinom{5}{3}$.
SOLUTION:$$\dbinom{5}{3} = \dfrac{5!}{3!2!}=\dfrac{(5\times 4)(3\times 2\times 1)}{(3\times 2\times 1)(2\times 1)}=\dfrac{5\times 4}{2\t... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Compute $2011 \cdot 2012 \cdot 2013 \cdot 2014$ modulo 5.
SOLUTION:Using the properties of modular arithmetic, $2011 \cdot 2012 \cdot 2013 \cdot 2014 \equiv 1 \c... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:The function $f(x)$ satisfies
\[f(xy) = \frac{f(x)}{y}\]for all positive real numbers $x$ and $y.$ If $f(30) = 20,$ find $f(40).$
SOLUTION:Setting $x = 30$ and ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Find the domain of the rational function $g(x) = \frac{x^3-2x^2+4x+3}{x^2-4x+3}$. Express your answer as a union of intervals.
SOLUTION:Let $p(x) = x^2-4x+3$. A ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 120^\circ,$ $\overline{AB... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:What is the area of the region enclosed by the graph of the equation $x^2-14x+3y+70=21+11y-y^2$ that lies below the line $y=x-3$?
SOLUTION:The equation can be re... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of l... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:There are 30 people in my math class. 12 of them have cool dads, 15 of them have cool moms, and 9 of them have cool dads and cool moms. How many people have mo... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Compute
\[\frac{5}{3^2 \cdot 7^2} + \frac{9}{7^2 \cdot 11^2} + \frac{13}{11^2 \cdot 15^2} + \dotsb.\]
SOLUTION:The $n$th term of the series is given by
\[\frac{4... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:The ratio of the areas of two squares is $\frac{192}{80}$. After rationalizing the denominator, the ratio of their side lengths can be expressed in the simplifie... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:License plates from different states follow different alpha-numeric formats, which dictate which characters of a plate must be letters and which must be numbers.... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A $\emph{convex}$ polygon is a polygon in which every interior angle is less than 180 degrees. A $\emph{diagonal}$ of a convex polygon is a line segment that co... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $ABCD$ be an isosceles trapezoid with bases $AB=92$ and $CD=19$. Suppose $AD=BC=x$ and a circle with center on $\overline{AB}$ is tangent to segments $\overl... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:The graph of $xy = 1$ is a hyperbola. Find the distance between the foci of this hyperbola.
SOLUTION:From the graph of $xy = 1,$ we can tell that the foci will ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 17.\]
SOLUTION:Let $n = \lfloor x \rfloor$ and $a = \{x\}.$ Then, we hav... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Andrew's grandfather's age is twelve times Andrew's age. If Andrew's grandfather was 55 years old when Andrew was born, how many years old is Andrew now?
SOLUTIO... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Six cards numbered $1$ through $6$ are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A polynomial $p(x)$ is called self-centered if it has integer coefficients and $p(100) = 100.$ If $p(x)$ is a self-centered polynomial, what is the maximum numb... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $P(x) = 0$ be the polynomial equation of least possible degree, with rational coefficients, having $\sqrt[3]{7} + \sqrt[3]{49}$ as a root. Compute the produ... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$.
SOLUTION:We can substitute $x=y-1$ to obtain a polynomi... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Let $f(x)$ be a polynomial such that
\[f(x^2 + 1) = x^4 + 4x^2.\]Find $f(x^2 - 1).$
SOLUTION:Let $y = x^2 + 1.$ Then $x^2 = y - 1,$ and $x^4 = y^2 - 2y + 1,$ so... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:How many of the 200 smallest positive integers are congruent to 1 (mod 9)?
SOLUTION:An integer congruent to 1 (mod 9) can be written in the form $9n + 1$ for som... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:How many two-digit numbers are there in which the tens digit is greater than the ones digit?
SOLUTION:In addition to two-digit numbers, consider the one-digit nu... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:A square has area $\frac14$. What is the side length of the square?
SOLUTION:If $n$ is the side length of the square, we have $n=\sqrt{\frac14}$, so $n^2=\frac1... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Consider a sequence $x_1,$ $x_2,$ $x_3,$ $\dots$ defined by
\begin{align*}
x_1 &= \sqrt[3]{3}, \\
x_2 &= (\sqrt[3]{3})^{\sqrt[3]{3}},
\end{align*}and in general,... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside the plane of $\triangle ABC$ and are on opposite sides of the plane. Furthermor... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:Six distinct integers are picked at random from $\{1,2,3,\ldots,10\}$. What is the probability that, among those selected, the second smallest is $3$?
$\textbf{(... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:The ellipse whose equation is
\[\frac{x^2}{25} + \frac{y^2}{9} = 1\]is graphed below. The chord $\overline{AB}$ passes through a focus $F$ of the ellipse. If $... |
Given a mathematics or algebraic problem, Think step by step and determine the answer. Simplify your answer as much as possible.
Examples starts>>>
QUESTION:One of the following four-digit numbers is not divisible by 4: 3544, 3554, 3564, 3572, 3576. What is the product of the units digit and the tens digit of that nu... |
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