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 \( A B C \) be a triangle such that \( A B = 7 \), and let the angle bisector of \(\angle B A C \) intersect line \( B C \) at \( D \). If there exist points \( E \) and \( F \) on sides \( A C \) and \( B C \), respectively, such that lines \( A D \) and \( E F \) are parallel and divide triangle \( A B C \) into ... | 13 | 0 | 8,192 | -1 | 8,192 | |
The measures of angles $X$ and $Y$ are both positive, integer numbers of degrees. The measure of angle $X$ is a multiple of the measure of angle $Y$, and angles $X$ and $Y$ are supplementary angles. How many measures are possible for angle $X$? | 17 | 0.625 | 6,012.625 | 4,705 | 8,192 | |
Lucy surveyed a group of people about their knowledge of mosquitoes. To the nearest tenth of a percent, she found that $75.3\%$ of the people surveyed thought mosquitoes transmitted malaria. Of the people who thought mosquitoes transmitted malaria, $52.8\%$ believed that mosquitoes also frequently transmitted the commo... | 70 | 0.25 | 7,432.5625 | 5,385.75 | 8,114.833333 | |
In triangle \(ABC\), the heights are given: \(h_{a} = \frac{1}{3}\), \(h_{b} = \frac{1}{4}\), \(h_{c} = \frac{1}{5}\). Find the ratio of the angle bisector \(CD\) to the circumradius. | \frac{24\sqrt{2}}{35} | 0 | 6,724.3125 | -1 | 6,724.3125 | |
Topsoil costs $\$6$ per cubic foot. What is the cost, in dollars, of 5 cubic yards of topsoil? | 810 | 0.9375 | 1,276.8125 | 1,339.666667 | 334 | |
Call a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy? | 406 | 0 | 7,553.25 | -1 | 7,553.25 | |
Each of the $2500$ students at a university studies either Physics or Chemistry, and some study both. The number who study Physics is between $70\%$ and $75\%$ of the university population, and the number who study Chemistry is between $40\%$ and $45\%$. Let $m$ be the smallest number of students who could study both s... | 250 | 0.6875 | 5,473.1875 | 4,724.909091 | 7,119.4 | |
What is the slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (2, 3) and the midpoint of the segment with endpoints at (5, 0) and (6, 3)? Express your answer in simplest form. | 0 | 1 | 1,478.875 | 1,478.875 | -1 | |
Suppose $a, b, c, d$ are real numbers such that $$|a-b|+|c-d|=99 ; \quad|a-c|+|b-d|=1$$ Determine all possible values of $|a-d|+|b-c|$. | 99 | 99 If $w \geq x \geq y \geq z$ are four arbitrary real numbers, then $|w-z|+|x-y|=$ $|w-y|+|x-z|=w+x-y-z \geq w-x+y-z=|w-x|+|y-z|$. Thus, in our case, two of the three numbers $|a-b|+|c-d|,|a-c|+|b-d|,|a-d|+|b-c|$ are equal, and the third one is less than or equal to these two. Since we have a 99 and a 1, the third num... | 0 | 8,192 | -1 | 8,192 |
Two thirds of a pitcher is filled with orange juice and the remaining part is filled with apple juice. The pitcher is emptied by pouring an equal amount of the mixture into each of 6 cups. Calculate the percentage of the total capacity of the pitcher that each cup receives. | 16.67\% | 0.75 | 2,919.25 | 2,770.083333 | 3,366.75 | |
After a fair die with faces numbered 1 to 6 is rolled, the number on the top face is $x$. What is the most likely outcome? | x > 2 | With a fair die that has faces numbered from 1 to 6, the probability of rolling each of 1 to 6 is $\frac{1}{6}$. We calculate the probability for each of the five choices. There are 4 values of $x$ that satisfy $x>2$, so the probability is $\frac{4}{6}=\frac{2}{3}$. There are 2 values of $x$ that satisfy $x=4$ or $x=5$... | 0 | 355.5625 | -1 | 355.5625 |
The extensions of a telephone exchange have only 2 digits, from 00 to 99. Not all extensions are in use. By swapping the order of two digits of an extension in use, you either get the same number or the number of an extension not in use. What is the highest possible number of extensions in use?
(a) Less than 45
(b) 45... | 55 | 0 | 5,986.9375 | -1 | 5,986.9375 | |
Let $A B C$ be a triangle, and let $D, E$, and $F$ be the midpoints of sides $B C, C A$, and $A B$, respectively. Let the angle bisectors of $\angle F D E$ and $\angle F B D$ meet at $P$. Given that $\angle B A C=37^{\circ}$ and $\angle C B A=85^{\circ}$, determine the degree measure of $\angle B P D$. | 61^{\circ} | Because $D, E, F$ are midpoints, we have $A B C \sim D E F$. Furthermore, we know that $F D \| A C$ and $D E \| A B$, so we have $$\angle B D F=\angle B C A=180-37-85=58^{\circ}$$ Also, $\angle F D E=\angle B A C=37^{\circ}$. Hence, we have $$\angle B P D=180^{\circ}-\angle P B D-\angle P D B=180^{\circ}-\frac{85^{\cir... | 0 | 8,184.3125 | -1 | 8,184.3125 |
Find a natural number of the form \( n = 2^{x} 3^{y} 5^{z} \), knowing that half of this number has 30 fewer divisors, a third has 35 fewer divisors, and a fifth has 42 fewer divisors than the number itself. | 2^6 * 3^5 * 5^4 | 0 | 5,887.1875 | -1 | 5,887.1875 | |
Given that $a, b, c, d, e, f, p, q$ are Arabic numerals and $b > c > d > a$, the difference between the four-digit numbers $\overline{c d a b}$ and $\overline{a b c d}$ is a four-digit number of the form $\overline{p q e f}$. If $\overline{e f}$ is a perfect square and $\overline{p q}$ is not divisible by 5, determine ... | 1983 | 0 | 8,192 | -1 | 8,192 | |
Two chess players, A and B, are in the midst of a match. Player A needs to win 2 more games to be the final winner, while player B needs to win 3 more games. If each player has a probability of $\frac{1}{2}$ to win any given game, then calculate the probability of player A becoming the final winner. | \frac{11}{16} | 0.375 | 6,762.125 | 4,379 | 8,192 | |
At time $t=0,$ a ball is thrown downward at 24 feet per second from a height of 160 feet above the ground. The equation $h = -16t^2 - 24t +160$ describes the height (in feet) of the ball. In how many seconds will the ball hit the ground? Express your answer as a decimal. | 2.5 | 1 | 3,171.875 | 3,171.875 | -1 | |
Given the function $y=f(x)$, for any $x \in \mathbb{R}$, it satisfies $f(x+2)=\frac{1}{f(x)}$. When $x \in (0, 2]$, $f(x)=x$.
(1) Find the analytical expression of $f(x)$ when $x \in (2, 4]$;
(2) If $f(m)=1$, find the value of $m$;
(3) Calculate the sum: $f(1)+f(2)+f(3)+...+f(2015)$. | \frac{4535}{2} | 0.25 | 7,853 | 6,836 | 8,192 | |
Consider a $4 \times 4$ grid of squares with 25 grid points. Determine the number of different lines passing through at least 3 of these grid points. | 32 | 0 | 8,192 | -1 | 8,192 | |
Vovochka adds three-digit numbers in a column in the following way: he does not carry over tens, but writes the sum of pairs of digits in the same place value position under each pair of digits, even if the result is a two-digit number. For example, for the sum \(248 + 208\), he would get the result 4416. Find the smal... | 1800 | 0 | 8,192 | -1 | 8,192 | |
Point $P$ is $\sqrt3$ units away from plane $A$ . Let $Q$ be a region of $A$ such that every line through $P$ that intersects $A$ in $Q$ intersects $A$ at an angle between $30^o$ and $60^o$ . What is the largest possible area of $Q$ ? | 8\pi | 0.75 | 6,423.5 | 5,834 | 8,192 | |
What is the value of $19^2-17^2+15^2-13^2+11^2-9^2+7^2-5^2+3^2-1^2?$ | 200 | 1 | 3,650.75 | 3,650.75 | -1 | |
Given $0\leqslant x\_0 < 1$, for all integers $n > 0$, let $x\_n= \begin{cases} 2x_{n-1}, & 2x_{n-1} < 1 \\ 2x_{n-1}-1, & 2x_{n-1} \geqslant 1 \end{cases}$. Find the number of $x\_0$ that makes $x\_0=x\_6$ true. | 64 | 0 | 7,168.75 | -1 | 7,168.75 | |
The graph of \[y^4 - 4x^4 = 2y^2 - 1\]is the union of the graphs of two different conic sections. Which two types of conic sections are they?
(Write your answer as a list, with "C" for circle, "E" for ellipse, "H" for hyperbola, and "P" for parabola. For example, "C, H" if you think the graph consists of a circle and ... | \text{H, E} | 0 | 1,811.375 | -1 | 1,811.375 | |
Given the function $f(x)=\frac{1}{3}x^3-ax^2+(a^2-1)x+b$, where $(a,b \in \mathbb{R})$
(I) If $x=1$ is an extreme point of $f(x)$, find the value of $a$;
(II) If the equation of the tangent line to the graph of $y=f(x)$ at the point $(1, f(1))$ is $x+y-3=0$, find the maximum and minimum values of $f(x)$ on the interval... | -4 | 0.75 | 4,062.625 | 3,872.75 | 4,632.25 | |
Let $\mathbf{u}$ and $\mathbf{v}$ be unit vectors, and let $\mathbf{w}$ be a vector such that $\mathbf{u} \times \mathbf{v} + \mathbf{u} = \mathbf{w}$ and $\mathbf{w} \times \mathbf{u} = \mathbf{v}.$ Compute $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}).$ | 1 | 0.5625 | 6,410.4375 | 5,440.222222 | 7,657.857143 | |
Given a random variable $\xi \sim N(1, \sigma ^{2})$, $a \gt 0$, $b \gt 0$, if $P(\xi \leq a) = P(\xi \geq b)$, then the minimum value of $\frac{{4a+b}}{{ab}}$ is ______. | \frac{9}{2} | 0.9375 | 4,549.625 | 4,306.8 | 8,192 | |
In trapezoid $ABCD$, leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD}$, and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001}$, find $BC^2$.
| 110 | 0.875 | 4,739.1875 | 4,245.928571 | 8,192 | |
Let $f(x) = ax^7 + bx^3 + cx - 5.$ If $f(-7) = 7,$ then find $f(7).$ | -17 | 0.9375 | 2,250.75 | 2,312.666667 | 1,322 | |
Given that \\(y=f(x)+x^{2}\\) is an odd function, and \\(f(1)=1\\), if \\(g(x)=f(x)+2\\), then \\(g(-1)=\\) . | -1 | 1 | 1,633.5625 | 1,633.5625 | -1 | |
Given that Tian Ji's top horse is faster than the King of Qi's middle horse, and Tian Ji's middle horse is faster than the King of Qi's bottom horse, but Tian Ji's top horse is slower than the King of Qi's top horse, and Tian Ji's bottom horse is slower than the King of Qi's bottom horse, calculate the probability that... | \frac{2}{3} | 0.125 | 7,429.875 | 7,188.5 | 7,464.357143 | |
Mrs. Riley recorded this information from a recent test taken by all of her students. Using the data, what was the average percent score for these $100$ students?
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
100&7\\\hline
90&18\\\hline
80&35\\\hline
70&25\... | 77 | 0.0625 | 705.1875 | 4,001 | 485.466667 | |
For each positive integer $1 \leq m \leq 10$, Krit chooses an integer $0 \leq a_{m}<m$ uniformly at random. Let $p$ be the probability that there exists an integer $n$ for which $n \equiv a_{m}(\bmod m)$ for all $m$. If $p$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100... | 1540 | Tuples of valid $a_{m}$ correspond with residues $\bmod \operatorname{lcm}(1,2, \ldots, 10)$, so the answer is $$\frac{\operatorname{lcm}(1,2, \ldots, 10)}{10!}=\frac{2^{3} \cdot 3^{2} \cdot 5 \cdot 7}{2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 7}=\frac{1}{1440}$$ | 0 | 7,936 | -1 | 7,936 |
Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After a while, having sold the demanded portion of cheese to the next customer, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom t... | 10 | 0.375 | 6,959.4375 | 5,462.666667 | 7,857.5 | |
In trapezoid $ABCD$ , the diagonals intersect at $E$ , the area of $\triangle ABE$ is 72 and the area of $\triangle CDE$ is 50. What is the area of trapezoid $ABCD$ ? | 242 | 0.625 | 6,141.625 | 5,066 | 7,934.333333 | |
In quadrilateral \(ABCD\), \(AB = BC\), \(\angle A = \angle B = 20^{\circ}\), \(\angle C = 30^{\circ}\). The extension of side \(AD\) intersects \(BC\) at point... | 30 | 0 | 8,192 | -1 | 8,192 | |
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((1,0)--(1,0.2));
dr... | \frac{5}{9} | 1. **Understanding the Problem:**
The problem involves a large square whose sides are trisected, and an inscribed square is formed by connecting these trisection points. We need to find the ratio of the area of the inscribed square to the area of the large square.
2. **Visualizing the Grid:**
The large square is... | 0.875 | 4,766.8125 | 4,277.5 | 8,192 |
Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid? | 408 | The formed tetrahedron has pairwise parallel planar and oppositely equal length ($4\sqrt{13},15,17$) edges and can be inscribed in a parallelepiped (rectangular box) with the six tetrahedral edges as non-intersecting diagonals of the box faces. Let the edge lengths of the parallelepiped be $p,q,r$ and solve (by Pythago... | 0.25 | 7,762.625 | 6,474.5 | 8,192 |
In the rectangular coordinate system, the parametric equation of line $l$ is given by $\begin{cases}x=1+t\cos a \\ y=1+t\sin a\end{cases}$ ($t$ is the parameter). In the polar coordinate system, the equation of circle $C$ is $\rho =4\cos \theta$.
(1) Find the rectangular coordinate equation of circle $C$.
(2) If $P(1... | 2\sqrt{2} | 0.5 | 6,477 | 6,566.125 | 6,387.875 | |
The positive integers are arranged in rows and columns as shown below.
| Row 1 | 1 |
| Row 2 | 2 | 3 |
| Row 3 | 4 | 5 | 6 |
| Row 4 | 7 | 8 | 9 | 10 |
| Row 5 | 11 | 12 | 13 | 14 | 15 |
| Row 6 | 16 | 17 | 18 | 19 | 20 | 21 |
| ... |
More rows continue to list the positive integers in order, with each new row contai... | 16 | 0 | 7,003.0625 | -1 | 7,003.0625 | |
Let $a$ and $b$ be the roots of $k(x^2 - x) + x + 5 = 0.$ Let $k_1$ and $k_2$ be the values of $k$ for which $a$ and $b$ satisfy
\[\frac{a}{b} + \frac{b}{a} = \frac{4}{5}.\]Find
\[\frac{k_1}{k_2} + \frac{k_2}{k_1}.\] | 254 | 1 | 4,548.25 | 4,548.25 | -1 | |
Let the set \( S \) contain 2012 elements, where the ratio of any two elements is not an integer. An element \( x \) in \( S \) is called a "good element" if there exist distinct elements \( y \) and \( z \) in \( S \) such that \( x^2 \) divides \( y \cdot z \). Find the maximum possible number of good elements in \( ... | 2010 | 0 | 8,192 | -1 | 8,192 | |
Given an arithmetic sequence $\{a\_n\}$ where all terms are positive, the sum of the first $n$ terms is $S\_n$. If $S\_n=2$ and $S\_3n=14$, find $S\_6n$. | 126 | 0 | 5,834.75 | -1 | 5,834.75 | |
If $x\%$ of four-digit numbers have a repeated digit (the repeated digits do not need to be adjacent), then what is $x$? Express your answer as a decimal to the nearest tenth. | 49.6 | 0.875 | 3,004.875 | 2,587.357143 | 5,927.5 | |
Given that line l: x - y + 1 = 0 is tangent to the parabola C with focus F and equation y² = 2px (p > 0).
(I) Find the equation of the parabola C;
(II) The line m passing through point F intersects parabola C at points A and B. Find the minimum value of the sum of the distances from points A and B to line l. | \frac{3\sqrt{2}}{2} | 0 | 8,143.125 | -1 | 8,143.125 | |
Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\leq i<j\leq 100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordere... | 197 |
To determine the largest possible value of \( N \) over all possible choices of 100 distinct ordered pairs of nonnegative integers \((a_i, b_i)\), we analyze pairs \((i, j)\) such that \(1 \leq i < j \leq 100\) and \(|a_i b_j - a_j b_i| = 1\).
This problem is connected to finding integer solutions of the equation \(|... | 0.1875 | 8,072 | 7,552 | 8,192 |
Find the point of intersection of the asymptotes of the graph of
\[y = \frac{x^2 - 4x + 3}{x^2 - 4x + 4}.\] | (2,1) | 1 | 1,770.1875 | 1,770.1875 | -1 | |
Given the sequence \(\{a_n\}\) with the first term 2, and it satisfies
\[ 6 S_n = 3 a_{n+1} + 4^n - 1. \]
Find the maximum value of \(S_n\). | 35 | 0.125 | 7,239.0625 | 6,096.5 | 7,402.285714 | |
In an isosceles right triangle $ABC$ with right angle at $C$ and an area of $18$ square units, the rays trisecting $\angle ACB$ intersect $AB$ at points $D$ and $E$. Calculate the area of triangle $CDE$.
A) 4.0
B) 4.5
C) 5.0
D) 5.5 | 4.5 | 0 | 8,192 | -1 | 8,192 | |
The line $L_{1}$: $ax+(1-a)y=3$ and $L_{2}$: $(a-1)x+(2a+3)y=2$ are perpendicular to each other, find the values of $a$. | -3 | 0.125 | 5,888.6875 | 2,800 | 6,329.928571 | |
All natural numbers from 1 to 2017 inclusive were written in a row. How many times was the digit 7 written? | 602 | 0.1875 | 8,002.75 | 7,182.666667 | 8,192 | |
Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run? | 20\pi + 40 | To solve this problem, we need to calculate the total distance the aardvark runs along different segments of the circles and straight lines. We assume the path is symmetric and consists of arcs and radial segments.
1. **Arc of the larger circle (radius = 20 meters):**
The aardvark runs along a quarter of the circ... | 0 | 5,711.0625 | -1 | 5,711.0625 |
Compute $\cos 270^\circ$. | 0 | 1 | 2,565.3125 | 2,565.3125 | -1 | |
The sum of all the positive factors of integer $x$ is 24. If one of the factors is 3, what is the value of $x$? | 15 | 1 | 4,466.25 | 4,466.25 | -1 | |
This century will mark the 200th anniversary of the birth of the famous Russian mathematician Pafnuty Lvovich Chebyshev, a native of Kaluga province. The sum of the digits in the hundreds and thousands places of the year he was born is 3 times the sum of the digits in the units and tens places, and the digit in the ten... | 1821 | 0.4375 | 6,701.3125 | 4,784.714286 | 8,192 | |
(Lucas Numbers) The Lucas numbers are defined by $L_{0}=2, L_{1}=1$, and $L_{n+2}=L_{n+1}+L_{n}$ for every $n \geq 0$. There are $N$ integers $1 \leq n \leq 2016$ such that $L_{n}$ contains the digit 1 . Estimate $N$. | 1984 | ```
Answer: 1984
lucas_ones n = length . filter (elem '1') $ take (n + 1) lucas_strs
where
lucas = 2 : 1 : zipWith (+) lucas (tail lucas)
lucas_strs = map show lucas
main = putStrLn . show $ lucas_ones 2016
``` | 0 | 7,922.8125 | -1 | 7,922.8125 |
What number is one third of the way from $\frac14$ to $\frac34$? | \frac{5}{12} |
#### Step-by-step Explanation:
1. **Understanding the Problem:**
We need to find the number that is one third of the way from $\frac{1}{4}$ to $\frac{3}{4}$.
2. **Using the Concept of Weighted Average:**
The number one third of the way from $\frac{1}{4}$ to $\frac{3}{4}$ can be calculated using the formula for... | 1 | 3,137.9375 | 3,137.9375 | -1 |
Three squares, $ABCD$, $EFGH$, and $GHIJ$, each have side length $s$. Point $C$ is located at the midpoint of side $HG$, and point $D$ is located at the midpoint of side $EF$. The line segment $AJ$ intersects the line segment $GH$ at point $X$. Determine the ratio of the area of the shaded region formed by triangle $AX... | \frac{1}{3} | 0 | 8,192 | -1 | 8,192 | |
Calculate the number of multiples of 4 that are between 100 and 500. | 99 | 0.0625 | 590.3125 | 458 | 599.133333 | |
The function $f$ satisfies the condition $$ f (x + 1) = \frac{1 + f (x)}{1 - f (x)} $$ for all real $x$ , for which the function is defined. Determine $f(2012)$ , if we known that $f(1000)=2012$ . | 2012 | 0.9375 | 4,685.5 | 4,451.733333 | 8,192 | |
Find the greatest whole number that will satisfy this inequality: $4x-3 < 2 - x $. | 0 | 1 | 1,322.8125 | 1,322.8125 | -1 | |
Determine the minimum possible value of the sum \[\frac{a}{3b} + \frac{b}{5c} + \frac{c}{6a},\] where \( a, b, \) and \( c \) are positive real numbers. | \frac{3}{\sqrt[3]{90}} | 0 | 7,367.6875 | -1 | 7,367.6875 | |
In a regular \( n \)-gon, \( A_{1} A_{2} A_{3} \cdots A_{n} \), where \( n > 6 \), sides \( A_{1} A_{2} \) and \( A_{5} A_{4} \) are extended to meet at point \( P \). If \( \angle A_{2} P A_{4}=120^\circ \), determine the value of \( n \). | 18 | 0.125 | 7,524.625 | 7,402 | 7,542.142857 | |
If the parabola defined by $y = ax^2 + 6$ is tangent to the line $y = x,$ then calculate the constant $a.$ | \frac{1}{24} | 1 | 1,855.1875 | 1,855.1875 | -1 | |
$ABCD$ is a rectangle. $E$ is a point on $AB$ between $A$ and $B$ , and $F$ is a point on $AD$ between $A$ and $D$ . The area of the triangle $EBC$ is $16$ , the area of the triangle $EAF$ is $12$ and the area of the triangle $FDC$ is 30. Find the area of the triangle $EFC$ . | 38 | 0.6875 | 5,689.8125 | 4,771.818182 | 7,709.4 | |
Let rectangle $A B C D$ have lengths $A B=20$ and $B C=12$. Extend ray $B C$ to $Z$ such that $C Z=18$. Let $E$ be the point in the interior of $A B C D$ such that the perpendicular distance from $E$ to \overline{A B}$ is 6 and the perpendicular distance from $E$ to \overline{A D}$ is 6 . Let line $E Z$ intersect $A B$... | 72 | Draw the line parallel to \overline{A D}$ through $E$, intersecting \overline{A B}$ at $F$ and \overline{C D}$ at $G$. It is clear that $X F E$ and $Y G E$ are congruent, so the area of $A X Y D$ is equal to that of $A F G D$. But $A F G D$ is simply a 12 by 6 rectangle, so the answer must be 72 . (Note: It is also pos... | 0.125 | 6,658.0625 | 4,965.5 | 6,899.857143 |
Two water particles fall freely in succession from a $300 \mathrm{~m}$ high cliff. The first one has already fallen $\frac{1}{1000} \mathrm{~mm}$ when the second one starts to fall.
How far apart will the two particles be at the moment when the first particle reaches the base of the cliff? (The result should be calcul... | 34.6 | 0 | 8,192 | -1 | 8,192 | |
A rectangular box has a volume of 108 cubic feet. How many cubic yards are in the volume of this box? | 4 | 1 | 1,440.8125 | 1,440.8125 | -1 | |
Find $1+2\cdot3-4+5.$ | 8 | 1 | 1,528.5 | 1,528.5 | -1 | |
The chart below gives the air distance in miles between selected world cities. If two different cities from the chart are chosen at random, what is the probability that the distance between them is less than $7000$ miles? Express your answer as a common fraction.
\begin{tabular}{|c|c|c|c|c|}
\hline
& Bangkok & Cape To... | \frac{2}{3} | 0.875 | 1,753.9375 | 1,906.714286 | 684.5 | |
Given that \( M \) is the midpoint of the height \( D D_{1} \) of a regular tetrahedron \( ABCD \), find the dihedral angle \( A-M B-C \) in radians. | \frac{\pi}{2} | 0 | 8,192 | -1 | 8,192 | |
The centers of faces $ABCD$ and $ADD'A'$ of the cube $ABCD-A'B'C'D'$ are $M$ and $N$ respectively. Find the sine of the angle between the skew lines $MN$ and $BD'$. | \frac{\sqrt{3}}{3} | 0 | 4,630.0625 | -1 | 4,630.0625 | |
What is the average of all the integer values of $N$ such that $\frac{N}{84}$ is strictly between $\frac{4}{9}$ and $\frac{2}{7}$? | 31 | 0.9375 | 3,672.5625 | 3,633.466667 | 4,259 | |
Find $d$, given that $\lfloor d\rfloor$ is a solution to \[3x^2 + 19x - 70 = 0\] and $\{d\} = d - \lfloor d\rfloor$ is a solution to \[4x^2 - 12x + 5 = 0.\] | -8.5 | 0 | 8,192 | -1 | 8,192 | |
Square $ABCD$ has an area of $256$ square units. Point $E$ lies on side $\overline{BC}$ and divides it in the ratio $3:1$. Points $F$ and $G$ are the midpoints of $\overline{AE}$ and $\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has an area of $48$ square units, what is the area of triangle $GCD$? | 48 | 0.0625 | 7,950.4375 | 5,991 | 8,081.066667 | |
What is the sum of all of the odd divisors of $180$? | 78 | 1 | 1,750.5 | 1,750.5 | -1 | |
Gru and the Minions plan to make money through cryptocurrency mining. They chose Ethereum as one of the most stable and promising currencies. They bought a system unit for 9499 rubles and two graphics cards for 31431 rubles each. The power consumption of the system unit is 120 W, and for each graphics card, it is 125 W... | 165 | 0.0625 | 6,186.3125 | 4,680 | 6,286.733333 | |
In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's inbox. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day, and the boss deliver... | 704 | Re-stating the problem for clarity, let $S$ be a set arranged in increasing order. At any time an element can be appended to the end of $S$, or the last element of $S$ can be removed. The question asks for the number of different orders in which all of the remaining elements of $S$ can be removed, given that $8$ had be... | 0 | 7,828.9375 | -1 | 7,828.9375 |
A point \( D \) is marked on the altitude \( BH \) of triangle \( ABC \). Line \( AD \) intersects side \( BC \) at point \( E \), and line \( CD \) intersects side \( AB \) at point \( F \). It is known that \( BH \) divides segment \( FE \) in the ratio \( 1:3 \), starting from point \( F \). Find the ratio \( FH:HE ... | 1:3 | 0 | 8,192 | -1 | 8,192 | |
From a class, 5 students were selected to measure their heights (in cm), and the data obtained were 160, 162, 159, 160, 159. The variance $s^2$ of this group of data is \_\_\_\_\_\_. | \frac{6}{5} | 0.125 | 3,973.8125 | 5,226.5 | 3,794.857143 | |
Given $\sqrt{2 + \frac{2}{3}} = 2\sqrt{\frac{2}{3}}, \sqrt{3 + \frac{3}{8}} = 3\sqrt{\frac{3}{8}}, \sqrt{4 + \frac{4}{15}} = 4\sqrt{\frac{4}{15}}\ldots$, if $\sqrt{6 + \frac{a}{b}} = 6\sqrt{\frac{a}{b}}$ (where $a,b$ are real numbers), please deduce $a = \_\_\_\_$, $b = \_\_\_\_$. | 35 | 0.875 | 2,993.3125 | 2,250.642857 | 8,192 | |
It can be shown that there exists a unique polynomial $P$ in two variables such that for all positive integers $m$ and $n$, $$P(m, n)=\sum_{i=1}^{m} \sum_{j=1}^{n}(i+j)^{7}$$ Compute $P(3,-3)$. | -2445 | Note that for integers $m>0, n>1$, $$P(m, n)-P(m, n-1)=\sum_{i=1}^{m}(i+n)^{7}=(n+1)^{7}+(n+2)^{7}+(n+3)^{7}$$ for all real $n$. Moreover, $P(3,1)-P(3,0)=P(3,1) \Longrightarrow P(3,0)=0$. Then $$\begin{aligned} P(3,-3) & =P(3,0)-\left(1^{7}+2^{7}+3^{7}\right)-\left(0^{7}+1^{7}+2^{7}\right)-\left((-1)^{7}+0^{7}+1^{7}\ri... | 0 | 7,682 | -1 | 7,682 |
A Senate committee consists of 10 Republicans and 8 Democrats. In how many ways can we form a subcommittee that has at most 5 members, including exactly 3 Republicans and at least 2 Democrats? | 10080 | 0.0625 | 3,059.4375 | 2,514 | 3,095.8 | |
Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such t... | 336 | Since $\triangle AFG$ is isosceles, $AF = FG$, and since $\triangle AEF$ is equilateral, $AF = EF$. Thus, $EF = FG$, and since these triangles share an altitude, they must have the same area.
Drop perpendiculars from $E$ and $F$ to line $BC$; call the meeting points $P$ and $Q$, respectively. $\triangle BEP$ is clearl... | 0 | 8,192 | -1 | 8,192 |
The sum of the squares of four consecutive positive integers is 9340. What is the sum of the cubes of these four integers? | 457064 | 0 | 8,192 | -1 | 8,192 | |
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6 \times 5$. What is the missing number in the top row? | 4 | Let's analyze the problem step by step, using the given information and setting up equations accordingly.
1. **Understanding the structure**: The number in each box is the product of the numbers in the two boxes that touch it in the row above. We are given that $30 = 6 \times 5$.
2. **Setting up the equations**: Let ... | 0 | 7,497.125 | -1 | 7,497.125 |
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle? | 200 | 1. **Identify the dimensions of the rectangle**: Given that the circle is inscribed in the rectangle, the diameter of the circle is equal to the width of the rectangle. Since the radius of the circle is $5$, the diameter is $2 \times 5 = 10$. Therefore, the width of the rectangle is $10$.
2. **Use the given ratio to f... | 1 | 3,085 | 3,085 | -1 |
A square with sides of 10 inches is shown. If $P$ is a point such that the segments $\overline{PA}$, $\overline{PB}$, and $\overline{PC}$ are equal in length, and segment $\overline{PC}$ is perpendicular to segment $\overline{GD}$, what is the area, in square inches, of triangle $APB$? Here, $G$ is the midpoint of side... | \frac{75}{4} | 0.3125 | 6,905.4375 | 5,188.6 | 7,685.818182 | |
Find the coefficient of the third term and the constant term in the expansion of $\left(x^3 + \frac{2}{3x^2}\right)^5$. | \frac{80}{27} | 1 | 2,905.0625 | 2,905.0625 | -1 | |
Given a triangular pyramid $S-ABC$ with the base being an isosceles right triangle with $AB$ as the hypotenuse, and $SA = SB = SC = 2$, $AB = 2$, let points $S$, $A$, $B$, and $C$ all lie on the surface of a sphere centered at $O$. What is the distance from point $O$ to the plane $ABC$?
| \frac{\sqrt{3}}{3} | 0 | 5,482.1875 | -1 | 5,482.1875 | |
The cost of five pencils and one pen is $\$2.50$, and the cost of one pencil and two pens is $\$1.85$. What is the cost of two pencils and one pen? | 1.45 | 1 | 2,085.0625 | 2,085.0625 | -1 | |
Uri buys two burgers and a soda for $\$2.10$, and Gen buys a burger and two sodas for $\$2.40$. How many cents does a soda cost? | 90 | 1 | 1,959.25 | 1,959.25 | -1 | |
Circular arcs of radius 3 inches form a continuous pattern as shown. What is the area, in square inches, of the shaded region in a 2-foot length of this pattern? Each arc completes half of a circle. | 18\pi | 0.375 | 6,212.125 | 4,892.333333 | 7,004 | |
During the National Day military parade, the marching sequences of three formations, namely A, B, and C, pass by the viewing stand in a certain order. If the order is randomly arranged, the probability that B passes before both A and C is ( ). | \frac{1}{3} | 0.375 | 6,684.4375 | 4,534.833333 | 7,974.2 | |
In triangle \( \triangle ABC \), given that \( \frac{\cos A}{\sin B} + \frac{\cos B}{\sin A} = 2 \) and the perimeter of \( \triangle ABC \) is 12, find the maximum possible area of the triangle. | 36(3 - 2\sqrt{2}) | 0 | 8,192 | -1 | 8,192 | |
The line with equation $y=2x-6$ is translated upwards by 4 units. What is the $x$-intercept of the resulting line? | 1 | The line with equation $y=2 x-6$ has slope 2. When this line is translated, the slope does not change. The line with equation $y=2 x-6$ has $y$-intercept -6. When this line is translated upwards by 4 units, its $y$-intercept is translated upwards by 4 units and so becomes -2. This means that the new line has equation $... | 1 | 1,908.3125 | 1,908.3125 | -1 |
A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two hour chase, the top sail of the privateer is carried away; she can now make only $17$ miles while the merchant... | $5\text{:}30\text{ p.m.}$ | 1. **Initial Setup and Relative Speed Calculation:**
The privateer starts at position $0$ and the merchantman starts at position $10$ miles away. The privateer chases the merchantman at a speed of $11$ mph, while the merchantman is moving away at $8$ mph. The relative speed of the privateer with respect to the merch... | 0 | 7,898.125 | -1 | 7,898.125 |
Compute $\tan(-405^\circ)$. | -1 | 0.9375 | 2,746.375 | 2,383.333333 | 8,192 | |
Each square in an $8 \times 8$ grid is to be painted either white or black. The goal is to ensure that for any $2 \times 3$ or $3 \times 2$ rectangle selected from the grid, there are at least two adjacent squares that are black. What is the minimum number of squares that need to be painted black in the grid? | 24 | 0 | 8,192 | -1 | 8,192 | |
Given an ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) whose left focus $F_1$ coincides with the focus of the parabola $y^2 = -4x$, and the eccentricity of ellipse $E$ is $\frac{\sqrt{2}}{2}$. A line $l$ with a non-zero slope passes through point $M(m,0)$ ($m > \frac{3}{4}$) and intersects the elli... | \frac{\sqrt{2}}{2} | 0 | 8,061.9375 | -1 | 8,061.9375 |
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