problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
Let $ABCD$ be a cyclic quadrilateral with sides $AB$, $BC$, $CD$, and $DA$. The side lengths are distinct integers less than $10$ and satisfy $BC + CD = AB + DA$. Find the largest possible value of the diagonal $BD$.
A) $\sqrt{93}$
B) $\sqrt{\frac{187}{2}}$
C) $\sqrt{\frac{191}{2}}$
D) $\sqrt{100}$ | \sqrt{\frac{191}{2}} | 0 | 8,192 | -1 | 8,192 | |
There are $20$ students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are $10$ students taking yoga, $13$ taking bridge, and $9$ taking painting. There are $9$ students taking at l... | 3 | Let's define the variables for the Venn Diagram:
- $a$: Number of students taking exactly Bridge and Yoga.
- $b$: Number of students taking exactly Bridge and Painting.
- $c$: Number of students taking all three classes (Yoga, Bridge, and Painting).
- $d$: Number of students taking exactly Yoga and Painting.
We are gi... | 0.875 | 3,975.1875 | 3,372.785714 | 8,192 |
$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD.$ The crease is $EF,$ where $E$ is on $AB$ and $F$ is on $CD.$ The dimensions $AE=8, BE=17,$ and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n,$ where $m$ and $n$ are relatively prime posi... | 293 | 0.75 | 4,465.875 | 3,432.833333 | 7,565 | |
Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$. | -1/9 | 1. **Identify the function and equation:** Given the function $f\left(\frac{x}{3}\right) = x^2 + x + 1$, we need to find the sum of all values of $z$ for which $f(3z) = 7$.
2. **Substitute and form a new equation:** Since $f\left(\frac{x}{3}\right) = x^2 + x + 1$, substituting $x = 9z$ (because $\frac{x}{3} = 3z$ impl... | 0 | 2,683.8125 | -1 | 2,683.8125 |
A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense? | 26 | 1. **Calculate the total distance driven**: The driver travels at a speed of $60$ miles per hour for $2$ hours. Therefore, the total distance driven is:
\[
60 \text{ miles/hour} \times 2 \text{ hours} = 120 \text{ miles}
\]
2. **Calculate the gasoline used**: The car's fuel efficiency is $30$ miles per gallon... | 1 | 1,275.5625 | 1,275.5625 | -1 |
Let $a$, $b$, and $c$ be the sides of a triangle with angles $\alpha$, $\beta$, and $\gamma$ opposite them respectively. Suppose $a^2 + b^2 = 9c^2$. Find the value of
\[\frac{\tan \gamma}{\tan \alpha + \tan \beta}.\] | -1 | 0 | 8,192 | -1 | 8,192 | |
In the diagram, $\triangle PQR$ is isosceles with $PQ = PR = 39$ and $\triangle SQR$ is equilateral with side length 30. The area of $\triangle PQS$ is closest to: | 75 | 0.5 | 7,409.125 | 6,684.375 | 8,133.875 | |
Given that $y$ is a multiple of $42522$, what is the greatest common divisor of $g(y)=(3y+4)(8y+3)(14y+9)(y+17)$ and $y$? | 102 | 0 | 6,692.125 | -1 | 6,692.125 | |
Express the sum of $0.\overline{123}+0.\overline{0123}+0.\overline{000123}$ as a common fraction. | \frac{123 \times 1000900}{999 \times 9999 \times 100001} | 0 | 8,192 | -1 | 8,192 | |
Three tenths plus four thousandths is equal to | 0.304 | 1 | 2,788.25 | 2,788.25 | -1 | |
A television station is set to broadcast 6 commercials in a sequence, which includes 3 different business commercials, 2 different World Expo promotional commercials, and 1 public service commercial. The last commercial cannot be a business commercial, and neither the World Expo promotional commercials nor the public s... | 36 | 0 | 8,192 | -1 | 8,192 | |
In the game of set, each card has four attributes, each of which takes on one of three values. A set deck consists of one card for each of the 81 possible four-tuples of attributes. Given a collection of 3 cards, call an attribute good for that collection if the three cards either all take on the same value of that att... | 25272 | In counting the number of sets of 3 cards, we first want to choose which of our two attributes will be good and which of our two attributes will not be good. There are $\binom{4}{2}=6$ such choices. Now consider the two attributes which are not good, attribute X and attribute Y . Since these are not good, some value sh... | 0 | 8,192 | -1 | 8,192 |
If $$log_{4}(a+4b)=log_{2}2 \sqrt {ab}$$, find the minimum value of $a+b$. | \frac{9}{4} | 0.9375 | 4,856.5 | 4,634.133333 | 8,192 | |
The sum of the first 3 terms of a geometric sequence $\{a_n\}$ is 13, and the sum of the first 6 terms is 65. Find $S_{12}$. | 1105 | 0.4375 | 6,683.1875 | 4,743.285714 | 8,192 | |
Given the function $f(x)= \sqrt {3}\sin 2x+2\cos ^{2}x-1$.
(I) Find the smallest positive period of $f(x)$:
(II) Find the maximum and minimum values of $f(x)$ in the interval $\[- \dfrac {π}{6}, \dfrac {π}{4}\]$. | -1 | 1 | 4,367 | 4,367 | -1 | |
Evaluate \(\left(d^d - d(d-2)^d\right)^d\) when \( d = 4 \). | 1358954496 | 0.3125 | 1,805.3125 | 4,614.8 | 528.272727 | |
Given the function $f(x)=4\cos(3x+\phi)(|\phi|<\frac{\pi}{2})$, its graph is symmetrical about the line $x=\frac{11\pi}{12}$. When $x_1,x_2\in(-\frac{7\pi}{12},-\frac{\pi}{12})$, $x_1\neq x_2$, and $f(x_1)=f(x_2)$, find $f(x_1+x_2)$. | 2\sqrt{2} | 0.375 | 7,951.6875 | 7,551.166667 | 8,192 | |
BoatsRUs built 8 canoes in January of this year. Each subsequent month, they increased production by tripling the number of canoes built in the previous month. Calculate the total number of canoes built by the end of August of this year. | 26240 | 0.1875 | 739.5 | 1,011.333333 | 676.769231 | |
Let $m$ and $n$ be positive integers satisfying the conditions
$\quad\bullet\ \gcd(m+n,210)=1,$
$\quad\bullet\ m^m$ is a multiple of $n^n,$ and
$\quad\bullet\ m$ is not a multiple of $n.$
Find the least possible value of $m+n.$ | 407 | Taking inspiration from $4^4 \mid 10^{10}$ we are inspired to take $n$ to be $p^2$, the lowest prime not dividing $210$, or $11 \implies n = 121$. Now, there are $242$ factors of $11$, so $11^{242} \mid m^m$, and then $m = 11k$ for $k \geq 22$. Now, $\gcd(m+n, 210) = \gcd(11+k,210) = 1$. Noting $k = 26$ is the minimal ... | 0 | 8,192 | -1 | 8,192 |
Let the base of the rectangular prism $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a rhombus with an area of $2 \sqrt{3}$ and $\angle ABC = 60^\circ$. Points $E$ and $F$ lie on edges $CC'$ and $BB'$, respectively, such that $EC = BC = 2FB$. What is the volume of the pyramid $A-BCFE$? | $\sqrt{3}$ | 0 | 7,867.8125 | -1 | 7,867.8125 | |
There are 6 balls of each of the four colors: red, blue, yellow, and green. Each set of 6 balls of the same color is numbered from 1 to 6. If 3 balls with different numbers are randomly selected, and these 3 balls have different colors and their numbers are not consecutive, the number of ways to do this is ______. | 96 | 0.4375 | 7,462.0625 | 6,523.571429 | 8,192 | |
Suppose
\[\frac{1}{x^3 - 2x^2 - 13x + 10} = \frac{A}{x+2} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$? | \frac{1}{9} | 0.4375 | 7,458.5625 | 6,515.571429 | 8,192 | |
Svetlana takes a triplet of numbers and transforms it by the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and smallest numbers in the triplet on the 1580th step of applying this rule, if the initial triplet of numbers was $\{80, 71, 20\}$? If the p... | 60 | 0.25 | 7,863.875 | 6,901.75 | 8,184.583333 | |
Given that rectangle ABCD has dimensions AB = 7 and AD = 8, and right triangle DCE shares the same height as rectangle side DC = 7 and extends horizontally from D towards E, and the area of the right triangle DCE is 28, find the length of DE. | \sqrt{113} | 0.0625 | 2,618.9375 | 3,537 | 2,557.733333 | |
In the tetrahedron S-ABC, the lateral edge SA is perpendicular to the plane ABC, and the base ABC is an equilateral triangle with a side length of $\sqrt{3}$. If SA = $2\sqrt{3}$, then the volume of the circumscribed sphere of the tetrahedron is \_\_\_\_\_\_. | \frac{32}{3}\pi | 0.6875 | 5,847.375 | 4,937.636364 | 7,848.8 | |
Let $x = 2001^{1002} - 2001^{-1002}$ and $y = 2001^{1002} + 2001^{-1002}.$ Find $x^2 - y^2.$ | -4 | 1 | 2,605.4375 | 2,605.4375 | -1 | |
Let \( [x] \) denote the greatest integer less than or equal to the real number \( x \). Consider a sequence \( \{a_n\} \) defined by \( a_1 = 1 \) and \( a_n = \left[\sqrt{n a_{n-1}}\right] \). Find the value of \( a_{2017} \). | 2015 | 0.625 | 7,107 | 6,456 | 8,192 | |
If the expression $x^2 + 9x + 20$ can be written as $(x + a)(x + b)$, and the expression $x^2 + 7x - 60$ can be written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$? | 14 | 0.6875 | 3,257.5625 | 2,985.454545 | 3,856.2 | |
A company's capital increases by a factor of two each year compared to the previous year after dividends have been paid, with a fixed dividend of 50 million yuan paid to shareholders at the end of each year. The company's capital after dividends were paid at the end of 2010 was 1 billion yuan.
(i) Find the capital of ... | 2017 | 0 | 5,346.625 | -1 | 5,346.625 | |
In triangle $ABC$, where $AB = 6$ and $AC = 10$. Let $M$ be a point on $BC$ such that $BM : MC = 2:3$. If $AM = 5$, what is the length of $BC$?
A) $7\sqrt{2.2}$
B) $5\sqrt{6.1}$
C) $10\sqrt{3.05}$
D) $15 - 3\sqrt{6.1}$ | 5\sqrt{6.1} | 0 | 6,290.5 | -1 | 6,290.5 | |
Given a sequence $\{a_{n}\}$ that satisfies ${a}_{1}+3{a}_{2}+9{a}_{3}+⋯+{3}^{n-1}{a}_{n}=\frac{n+1}{3}$, where the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is denoted as $S_{n}$, find the minimum value of the real number $k$ such that $S_{n} \lt k$ holds for all $n$. | \frac{5}{6} | 0.375 | 5,541.9375 | 5,850.333333 | 5,356.9 | |
Ms. Carr expands her reading list to 12 books and asks each student to choose any 6 books. Harold and Betty each randomly select 6 books from this list. Calculate the probability that there are exactly 3 books that they both select. | \frac{405}{2223} | 0 | 5,658.4375 | -1 | 5,658.4375 | |
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $7, n,$ and $n+2$ cents, $120$ cents is the greatest postage that cannot be formed. | 43 | 0 | 8,192 | -1 | 8,192 | |
Find the volume of the set of points $(x, y, z)$ satisfying $$\begin{array}{r} x, y, z \geq 0 \\ x+y \leq 1 \\ y+z \leq 1 \\ z+x \leq 1 \end{array}$$ | \frac{1}{4} | Without loss of generality, assume that $x \geq y$ - half the volume of the solid is on this side of the plane $x=y$. For each value of $c$ from 0 to $\frac{1}{2}$, the region of the intersection of this half of the solid with the plane $y=c$ is a trapezoid. The trapezoid has height $1-2 c$ and average base $\frac{1}{2... | 0 | 7,676.25 | -1 | 7,676.25 |
Given in a cube ABCD-A1B1C1D1 with edge length 1, P is a moving point inside the cube (including the surface),
if $x + y + z = s$, and $0 \leq x \leq y \leq z \leq 1$, then the volume of the geometric body formed by all possible positions of point P is $\_\_\_\_\_\_\_\_\_\_$. | \frac{1}{6} | 0.0625 | 7,967.375 | 5,388 | 8,139.333333 | |
Given that a certain basketball player has a 50% chance of making each shot, we use a random simulation method to estimate the probability that the player makes exactly two out of four shots: First, we generate a random integer between 0 and 9 using a calculator, where 0, 1, 2, 3, and 4 represent a successful shot, and... | 0.35 | 0 | 6,741.625 | -1 | 6,741.625 | |
Let $(x, y)$ be a solution to the system of equations \[\begin{aligned} \lfloor x \rfloor + \{y\} &= 2.4, \\ \{x\} + \lfloor y \rfloor &= 5.1. \end{aligned} \]Compute $|x - y|.$ | 3.3 | 0.75 | 3,789 | 3,502.666667 | 4,648 | |
In right triangle $DEF$, where $DE=15$, $DF=9$, and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$? | 7.5 | 0.6875 | 3,641.5 | 3,633.909091 | 3,658.2 | |
Nine identical bowling balls weigh the same as five identical canoes. If four of the canoes weigh a total of 120 pounds, how many pounds does one bowling ball weigh? | \frac{50}{3} | 0.5625 | 2,499.25 | 2,578.666667 | 2,397.142857 | |
The equation ${{a}^{2}}{{x}^{2}}+(a+2){{y}^{2}}+2ax+a=0$ represents a circle. Find the possible values of $a$. | -1 | 1 | 3,610.125 | 3,610.125 | -1 | |
Let
$$
\begin{array}{c}
A=\left(\binom{2010}{0}-\binom{2010}{-1}\right)^{2}+\left(\binom{2010}{1}-\binom{2010}{0}\right)^{2}+\left(\binom{2010}{2}-\binom{2010}{1}\right)^{2} \\
+\cdots+\left(\binom{2010}{1005}-\binom{2010}{1004}\right)^{2}
\end{array}
$$
Determine the minimum integer \( s \) such that
$$
s A \geq \bin... | 2011 | 0 | 8,192 | -1 | 8,192 | |
In parallelogram $EFGH$, $EF = 5z + 5$, $FG = 4k^2$, $GH = 40$, and $HE = k + 20$. Determine the values of $z$ and $k$ and find $z \times k$. | \frac{7 + 7\sqrt{321}}{8} | 0 | 4,694.3125 | -1 | 4,694.3125 | |
The graph of $y = ax^2 + bx + c$ has a maximum value of 72, and passes through the points $(0, -1)$ and $(6, -1)$. Find $a + b + c$. | \frac{356}{9} | 1 | 4,499.6875 | 4,499.6875 | -1 | |
Let $D$ be the circle with equation $x^2 - 10y - 7 = -y^2 - 8x + 4$. Find the center $(a, b)$ and radius $r$ of $D$, and determine the value of $a + b + r$. | 1 + 2\sqrt{13} | 0.875 | 3,770.375 | 3,138.714286 | 8,192 | |
The value of $\log_{10}{579}$ is between the consecutive integers $a$ and $b$. Find $a+b$. | 5 | 0.9375 | 5,598 | 5,425.066667 | 8,192 | |
Using the four arithmetic operators and parentheses, find a way to combine the numbers 10, 10, 4, and 2 such that the result is 24. What is the arithmetic expression? | (2 + 4 \div 10) \times 10 | 0 | 3,036.75 | -1 | 3,036.75 | |
The number $2013$ is expressed in the form $2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$? | 2 | 1. **Prime Factorization of 2013**:
The prime factorization of $2013$ is $2013 = 3 \times 11 \times 61$.
2. **Form of the Expression**:
We need to express $2013$ in the form $\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!}$, where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positiv... | 0 | 8,192 | -1 | 8,192 |
There exists a positive real number $x$ such that $ \cos (\arctan (x)) = x $. Find the value of $x^2$. | \frac{-1 + \sqrt{5}}{2} | 0 | 2,707.1875 | -1 | 2,707.1875 | |
What is the product of the [real](https://artofproblemsolving.com/wiki/index.php/Real) [roots](https://artofproblemsolving.com/wiki/index.php/Root) of the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$? | 20 | 0.875 | 4,861.25 | 4,601 | 6,683 | |
Given 2414 cards, each with a unique natural number from 1 to 2414. We need to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done? | 29112 | 0 | 8,099.8125 | -1 | 8,099.8125 | |
Determine all real numbers $a$ such that the inequality $|x^{2}+2 a x+3 a| \leq 2$ has exactly one solution in $x$. | 1,2 | Let $f(x)=x^{2}+2 a x+3 a$. Note that $f(-3 / 2)=9 / 4$, so the graph of $f$ is a parabola that goes through $(-3 / 2,9 / 4)$. Then, the condition that $|x^{2}+2 a x+3 a| \leq 2$ has exactly one solution means that the parabola has exactly one point in the strip $-1 \leq y \leq 1$, which is possible if and only if the ... | 0.0625 | 7,344.0625 | 8,192 | 7,287.533333 |
What is the average (mean) number of hamburgers eaten per student if 12 students ate 0 hamburgers, 14 students ate 1 hamburger, 8 students ate 2 hamburgers, 4 students ate 3 hamburgers, and 2 students ate 4 hamburgers? | 1.25 | The mean number of hamburgers eaten per student equals the total number of hamburgers eaten divided by the total number of students. 12 students each eat 0 hamburgers. This is a total of 0 hamburgers eaten. 14 students each eat 1 hamburger. This is a total of 14 hamburgers eaten. 8 students each eat 2 hamburgers. This ... | 0.375 | 458.75 | 411.333333 | 487.2 |
What is the median of the first ten positive integers? Express your answer as a decimal to the nearest tenth. | 5.5 | 1 | 1,479.5 | 1,479.5 | -1 | |
Find the numerical value of
\[\frac{\sin 18^\circ \cos 12^\circ + \cos 162^\circ \cos 102^\circ}{\sin 22^\circ \cos 8^\circ + \cos 158^\circ \cos 98^\circ}.\] | 1 | 0.6875 | 5,441.25 | 4,190.909091 | 8,192 | |
There are 4 different points \( A, B, C, D \) on two non-perpendicular skew lines \( a \) and \( b \), where \( A \in a \), \( B \in a \), \( C \in b \), and \( D \in b \). Consider the following two propositions:
(1) Line \( AC \) and line \( BD \) are always skew lines.
(2) Points \( A, B, C, D \) can never be the fo... | (1)(2) | 0 | 6,839.6875 | -1 | 6,839.6875 | |
The lines $x = \frac{1}{4}y + a$ and $y = \frac{1}{4}x + b$ intersect at the point $(1,2)$. What is $a + b$? | \frac{9}{4} | 1. **Given Equations and Intersection Point**: We are given the equations of two lines:
- $x = \frac{1}{4}y + a$
- $y = \frac{1}{4}x + b$
These lines intersect at the point $(1,2)$.
2. **Substitute the Intersection Point into Each Equation**:
- Substituting $x = 1$ and $y = 2$ into the first equation:
... | 1 | 1,980.75 | 1,980.75 | -1 |
A square of side length $1$ and a circle of radius $\frac{\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square? | \frac{2\pi}{9} - \frac{\sqrt{3}}{3} | 1. **Identify the geometric setup**: We have a square with side length $1$ and a circle with radius $\frac{\sqrt{3}}{3}$ sharing the same center. We need to find the area inside the circle but outside the square.
2. **Calculate the diagonal of the square**: The diagonal of the square can be calculated using the Pythag... | 0 | 7,982.6875 | -1 | 7,982.6875 |
Given that a light bulb is located $15$ centimeters below the ceiling in Bob's living room, the ceiling is $2.8$ meters above the floor, Bob is $1.65$ meters tall and can reach $55$ centimeters above his head, and Bob standing on a chair can just reach the light bulb, calculate the height of the chair, in centimeters. | 45 | 0.9375 | 1,617.9375 | 1,652.8 | 1,095 | |
The "Tiao Ri Method", invented by mathematician He Chengtian during the Southern and Northern Dynasties of China, is an algorithm for finding a more accurate fraction to represent a numerical value. Its theoretical basis is as follows: If the deficient approximate value and the excess approximate value of a real number... | \frac{22}{7} | 0.0625 | 7,430.125 | 3,755 | 7,675.133333 | |
If $y = \displaystyle\frac{1}{3x+1}$, what is the value of $x$ when $y = 1$? | 0 | 1 | 1,884.75 | 1,884.75 | -1 | |
Express as a common fraction: $\cfrac{ \frac{2}{5}+\frac{3}{4} }{ \frac{4}{9}+\frac{1}{6}}$. | \frac{207}{110} | 0.9375 | 3,341.8125 | 3,018.466667 | 8,192 | |
Calculate the volume of tetrahedron PQRS with edge lengths PQ = 4, PR = 5, PS = 6, QR = 3, QS = √37, and RS = 7. | 10.25 | 0 | 8,142.75 | -1 | 8,142.75 | |
Three of the four vertices of a rectangle are $(5, 11)$, $(16, 11)$ and $(16, -2)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 5)^2 + (y + 2)^2 = 9$? Express your answer in terms of $\pi$. | \frac94\pi | 0.4375 | 5,776.125 | 5,187 | 6,234.333333 | |
Given that acute angles $\alpha$ and $\beta$ satisfy $\sin\alpha=\frac{4}{5}$ and $\cos(\alpha+\beta)=-\frac{12}{13}$, determine the value of $\cos \beta$. | -\frac{16}{65} | 0 | 7,920.3125 | -1 | 7,920.3125 | |
What is the value of $\sqrt{15 - 6\sqrt{6}} + \sqrt{15 + 6\sqrt{6}}$? | 6 | 1 | 2,120 | 2,120 | -1 | |
A sequence consists of $2010$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2010$ terms is $5307$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum? | 2151 | 0.3125 | 7,960.5625 | 7,451.4 | 8,192 | |
Let $x$ be a positive real number. Find the maximum possible value of $$\frac{x^2+2-\sqrt{x^4+4}}{x}.$$ | 2\sqrt{2}-2 | 0.375 | 7,303.75 | 5,823.333333 | 8,192 | |
It is now 3:00:00 PM, as read on a 12-hour digital clock. In 315 hours, 58 minutes, and 16 seconds, the time will be $X:Y:Z$. What is the value of $X + Y + Z$? | 77 | 0 | 7,699.9375 | -1 | 7,699.9375 | |
The expression $2 + \sqrt{2} + \frac{1}{2 + \sqrt{2}} + \frac{1}{\sqrt{2} - 2}$ equals: | 2 | 1. **Simplify $\frac{1}{2 + \sqrt{2}}$:**
To remove the radical from the denominator, multiply the numerator and denominator by the conjugate of the denominator:
\[
\frac{1}{2 + \sqrt{2}} \cdot \frac{2 - \sqrt{2}}{2 - \sqrt{2}} = \frac{2 - \sqrt{2}}{(2 + \sqrt{2})(2 - \sqrt{2})}.
\]
Using the differenc... | 1 | 3,007.125 | 3,007.125 | -1 |
A semicircle with radius 2021 has diameter $AB$ and center $O$. Points $C$ and $D$ lie on the semicircle such that $\angle AOC < \angle AOD = 90^{\circ}$. A circle of radius $r$ is inscribed in the sector bounded by $OA$ and $OC$ and is tangent to the semicircle at $E$. If $CD=CE$, compute $\lfloor r \rfloor$. | 673 | We are given $$m \angle EOC = m \angle COD$$ and $$m \angle AOC + m \angle COD = 2m \angle EOC + m \angle COD = 90^{\circ}$$ So $m \angle EOC = 30^{\circ}$ and $m \angle AOC = 60^{\circ}$. Letting the radius of the semicircle be $R$, we have $$(R-r) \sin \angle AOC = r \Rightarrow r = \frac{1}{3} R$$ so $$\lfloor r \rf... | 0.1875 | 7,863.4375 | 6,443 | 8,191.230769 |
For some constants \( c \) and \( d \), let
\[ g(x) = \left\{
\begin{array}{cl}
cx + d & \text{if } x < 3, \\
10 - 2x & \text{if } x \ge 3.
\end{array}
\right.\]
The function \( g \) has the property that \( g(g(x)) = x \) for all \( x \). What is \( c + d \)? | 4.5 | 0 | 8,192 | -1 | 8,192 | |
Each of two teams, Team A and Team B, sends 7 players in a predetermined order to participate in a Go contest. The players from both teams compete sequentially starting with Player 1 from each team. The loser of each match is eliminated, and the winner continues to compete with the next player from the opposing team. T... | 3432 | 0 | 7,867.875 | -1 | 7,867.875 | |
The slope angle of the tangent line to the curve $y=x\cos x$ at $x=0$ is what angle? | \frac{\pi}{4} | 0.9375 | 1,610.625 | 1,627.733333 | 1,354 | |
Given the areas of the three squares in the figure, what is the area of the interior triangle? [asy]
/* AMC8 2003 #6 Problem */
draw((0,0)--(12,0)--(12,5)--cycle);
draw((12,0)--(17,0)--(17,5)--(12,5));
draw((0,0)--(0,-12)--(12,-12)--(12,0));
draw((0,0)--(-5,12)--(7,17)--(12,5));
draw((11,0)--(11,1)--(12,1));
label("169... | 30 | 0.75 | 4,655.125 | 3,476.166667 | 8,192 | |
A five-digit number has one of its digits crossed out, and the resulting four-digit number is added to the original number. The sum is 54321. Find the original number. | 49383 | 0.1875 | 7,815.9375 | 6,186.333333 | 8,192 | |
What is the probability of spinning the spinner pictured and getting a prime number? Express your answer as a common fraction. [asy]
import olympiad; defaultpen(linewidth(0.8)); size(100); dotfactor=4;
draw(Circle((0,0),1));
string[] labels = {"3","6","1","4","5","2"};
for(int i = 0; i < 6; ++i){
label(labels[i],0.7*d... | \frac{1}{2} | 1 | 887.875 | 887.875 | -1 | |
Determine the least possible value of \((x+2)(x+3)(x+4)(x+5) + 2024\) where \(x\) is a real number. | 2023 | 0.75 | 6,050.1875 | 5,336.25 | 8,192 | |
Calculate the sum $C_{3}^{2}+C_{4}^{2}+C_{5}^{2}+\ldots+C_{19}^{2}$. | 1139 | 0.75 | 4,761.4375 | 3,824.916667 | 7,571 | |
Given that construction teams A and B each have a certain number of workers. If team A lends 90 workers to team B, then team B's total number of workers will be twice that of team A. If team B lends a certain number of workers to team A, then team A's total number of workers will be 6 times that of team B. How many wor... | 153 | 0.875 | 5,178.1875 | 5,075.285714 | 5,898.5 | |
Fix positive integers $k,n$. A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have at least one colour of candy in common, f... | n(3k) |
Fix positive integers \( k \) and \( n \). Consider a candy vending machine that has many different colors of candy, with \( 2n \) candies of each color. A couple of kids each buys from the vending machine 2 candies of different colors. We are to find the maximum number of kids such that for any \( k+1 \) kids, there ... | 0 | 8,192 | -1 | 8,192 |
Calculate $46_8 - 27_8$ and express your answer in base 8. | 17_8 | 1 | 2,354.125 | 2,354.125 | -1 | |
Given two integers $ m,n$ satisfying $ 4 < m < n.$ Let $ A_{1}A_{2}\cdots A_{2n \plus{} 1}$ be a regular $ 2n\plus{}1$ polygon. Denote by $ P$ the set of its vertices. Find the number of convex $ m$ polygon whose vertices belongs to $ P$ and exactly has two acute angles. | (2n + 1) \left[ \binom{n}{m - 1} + \binom{n + 1}{m - 1} \right] |
Given two integers \( m \) and \( n \) satisfying \( 4 < m < n \), let \( A_1A_2\cdots A_{2n+1} \) be a regular \( 2n+1 \) polygon. Denote by \( P \) the set of its vertices. We aim to find the number of convex \( m \)-gons whose vertices belong to \( P \) and have exactly two acute angles.
Notice that if a regular \... | 0 | 8,192 | -1 | 8,192 |
Let $A$ be a subset of $\{1, 2, \dots , 1000000\}$ such that for any $x, y \in A$ with $x\neq y$ , we have $xy\notin A$ . Determine the maximum possible size of $A$ . | 999001 | 0 | 8,022.625 | -1 | 8,022.625 | |
Sindy writes down the positive integers less than 200 in increasing order, but skips the multiples of 10. She then alternately places + and - signs before each of the integers, yielding an expression $+1-2+3-4+5-6+7-8+9-11+12-\cdots-199$. What is the value of the resulting expression? | -100 | Group the numbers into $(1-2+3-4+\ldots+18-19)+(21-22+\ldots+38-39)+\ldots+(181-182+\ldots+198-199)$. We can easily show that each group is equal to -10, and so the answer is -100. | 0 | 8,006.375 | -1 | 8,006.375 |
Compute the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.
$$
y = e^{x} + e, \ln \sqrt{3} \leq x \leq \ln \sqrt{15}
$$ | 2 + \frac{1}{2} \ln \left( \frac{9}{5} \right) | 0.25 | 7,222.6875 | 5,672.75 | 7,739.333333 | |
Consider the function \( g(x) = \left\{ \begin{aligned} x-3 & \quad \text{ if } x < 5 \\ \sqrt{x-1} & \quad \text{ if } x \ge 5 \end{aligned} \right. \). Find the value of \( g^{-1}(-6) + g^{-1}(-5) + \dots + g^{-1}(4) + g^{-1}(5) \). | 58 | 0 | 5,280.75 | -1 | 5,280.75 | |
A number from the set $\{30, 31, 32, \ldots, 500\}$ is randomly selected. What is the probability that the number is greater than 100 but less than or equal to 200? Express your answer as a common fraction. | \frac{100}{471} | 0.9375 | 2,706.1875 | 2,340.466667 | 8,192 | |
There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 2 \le 0.\]How many integer values of $b$ are possible? | 2 | 0 | 8,192 | -1 | 8,192 | |
The owner of an individual clothing store purchased 30 dresses for $32 each. The selling price of the 30 dresses varies for different customers. Using $47 as the standard price, any excess amount is recorded as positive and any shortfall is recorded as negative. The results are shown in the table below:
| Number Sold ... | 472 | 0.0625 | 617.8125 | 789 | 606.4 | |
A convex polyhedron $P$ has $26$ vertices, $60$ edges, and $36$ faces, $24$ of which are triangular and $12$ of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does $P$ have? | 241 | Every pair of vertices of the polyhedron determines either an edge, a face diagonal or a space diagonal. We have ${26 \choose 2} = \frac{26\cdot25}2 = 325$ total line segments determined by the vertices. Of these, $60$ are edges. Each triangular face has $0$ face diagonals and each quadrilateral face has $2$, so there ... | 0.5625 | 6,292.5625 | 4,815.222222 | 8,192 |
The sum of two numbers is 50 and their difference is 6. What is their product? | 616 | 1 | 1,718.375 | 1,718.375 | -1 | |
Calculate $3.5 \times 0.3 + 1.2 \times 0.4$. | 1.53 | 1 | 281.1875 | 281.1875 | -1 | |
The sum of the numerical coefficients in the binomial $(2a+2b)^7$ is $\boxed{32768}$. | 16384 | 0.0625 | 7,835.5625 | 6,370 | 7,933.266667 | |
If $\tan \alpha = 8$ and $\tan \beta = 7,$ then find $\tan (\alpha - \beta).$ | \frac{1}{57} | 0.9375 | 2,804.4375 | 2,445.266667 | 8,192 | |
The function $f$ is graphed below. Each small box has width and height 1.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickd... | 6 | 1 | 2,873.8125 | 2,873.8125 | -1 | |
Rectangle ABCD has AB = 4 and BC = 3. Segment EF is constructed through B such that EF is perpendicular to DB, and A and C lie on DE and DF, respectively. Find the length of EF. | \frac{125}{12} | 0.8125 | 5,328.5 | 4,667.692308 | 8,192 | |
Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if ne... | 1/2013 | 0.125 | 7,779.5 | 4,892 | 8,192 | |
The function \( f(x) = \max \left\{\sin x, \cos x, \frac{\sin x + \cos x}{\sqrt{2}}\right\} \) (for \( x \in \mathbb{R} \)) has a maximum value and a minimum value. Find the sum of these maximum and minimum values. | 1 - \frac{\sqrt{2}}{2} | 0 | 6,839.5625 | -1 | 6,839.5625 | |
Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle B DC = 90^o$ . Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$ , respectively, with $P$ lying in between $B$ and $Q$ . If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles ... | 799 | 0.375 | 6,775.25 | 4,414 | 8,192 | |
Let $f : Q \to Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$ . If $f(2004) = 2547$ , find $f(2547)$ . | 2547 | 0 | 8,192 | -1 | 8,192 |
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