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 \(x\) and \(y\) be real numbers such that \(2(x^3 + y^3) = x + y\). Find the maximum value of \(x - y\). | \frac{\sqrt{2}}{2} | 0 | 6,444 | -1 | 6,444 | |
Determine the value of the following expressions:
$(1)(2 \frac{7}{9})^{0.5}+0.1^{-2}+(2 \frac{10}{27})\,^{- \frac{2}{3}}-3π^{0}+ \frac{37}{48}$;
$(2)(-3 \frac{3}{8})\,^{- \frac{2}{3}}+(0.002)\,^{- \frac{1}{2}}-10( \sqrt{5}-2)^{-1}+( \sqrt{2}- \sqrt{3})^{0}$. | - \frac{167}{9} | 0.6875 | 5,524.25 | 4,839 | 7,031.8 | |
What is the measure of an angle, in degrees, if its supplement is six times its complement? | 72^{\circ} | 1 | 1,318.625 | 1,318.625 | -1 | |
$2\left(1-\frac{1}{2}\right) + 3\left(1-\frac{1}{3}\right) + 4\left(1-\frac{1}{4}\right) + \cdots + 10\left(1-\frac{1}{10}\right)=$ | 45 | 1. **Identify the General Term**:
Each term in the sequence is of the form $n\left(1-\frac{1}{n}\right)$ for $n$ ranging from $2$ to $10$. Simplifying the expression inside the parentheses:
\[
1 - \frac{1}{n} = \frac{n-1}{n}
\]
Multiplying this by $n$ gives:
\[
n \left(\frac{n-1}{n}\right) = n-1
... | 0.9375 | 3,110.1875 | 2,771.4 | 8,192 |
A certain teacher received $10$, $6$, $8$, $5$, $6$ letters from Monday to Friday, then the variance of this data set is $s^{2}=$____. | 3.2 | 0.1875 | 1,488.3125 | 584.333333 | 1,696.923077 | |
Let $x,$ $y,$ $z$ be positive real numbers such that
\[\left( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \right) + \left( \frac{y}{x} + \frac{z}{y} + \frac{x}{z} \right) = 8.\]Find the minimum value of
\[\left( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \right) \left( \frac{y}{x} + \frac{z}{y} + \frac{x}{z} \right).\] | 22 \sqrt{11} - 57 | 0 | 7,594.1875 | -1 | 7,594.1875 | |
What is the smallest positive integer \(n\) such that \(\frac{n}{n+51}\) is equal to a terminating decimal? | 74 | 0 | 7,326.875 | -1 | 7,326.875 | |
The distance from the intersection point of the diameter of a circle with a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius of the circle.
$$
AB = 18, EO = 7, AE = 2 BE, R = ?
$$ | 11 | 0.5 | 6,366.25 | 5,218.75 | 7,513.75 | |
In rectangle $ABCD$, we have $A=(6,-22)$, $B=(2006,178)$, $D=(8,y)$, for some integer $y$. What is the area of rectangle $ABCD$? | 40400 |
1. **Identify the coordinates and slopes**:
- Coordinates given: $A = (6, -22)$, $B = (2006, 178)$, $D = (8, y)$.
- Slope of $AB$, $m_1$:
\[
m_1 = \frac{178 - (-22)}{2006 - 6} = \frac{200}{2000} = \frac{1}{10}
\]
2. **Determine the slope of $AD$ and find $y$**:
- Slope of $AD$, $m_2$:
\[... | 0.9375 | 5,315.0625 | 5,123.266667 | 8,192 |
In the market supply of light bulbs, products from Factory A account for 70%, while those from Factory B account for 30%. The pass rate for Factory A's products is 95%, and the pass rate for Factory B's products is 80%. What is the probability of purchasing a qualified light bulb manufactured by Factory A? | 0.665 | 0.25 | 4,725.625 | 1,331.5 | 5,857 | |
For any set \( S \), let \( |S| \) represent the number of elements in set \( S \) and let \( n(S) \) represent the number of subsets of set \( S \). If \( A \), \( B \), and \( C \) are three finite sets such that:
(1) \( |A|=|B|=2016 \);
(2) \( n(A) + n(B) + n(C) = n(A \cup B \cup C) \),
then the maximum value of \... | 2015 | 0.125 | 7,775.6875 | 5,944.5 | 8,037.285714 | |
Along a straight alley, there are 400 streetlights placed at equal intervals, numbered consecutively from 1 to 400. Alla and Boris start walking towards each other from opposite ends of the alley at the same time but with different constant speeds (Alla from the first streetlight and Boris from the four-hundredth stree... | 163 | 0.75 | 4,580.6875 | 4,102.333333 | 6,015.75 | |
Let $\pi$ be a permutation of the numbers from 2 through 2012. Find the largest possible value of $\log _{2} \pi(2) \cdot \log _{3} \pi(3) \cdots \log _{2012} \pi(2012)$. | 1 | Note that $$\begin{aligned} \prod_{i=2}^{2012} \log _{i} \pi(i) & =\prod_{i=2}^{2012} \frac{\log \pi(i)}{\log i} \\ & =\frac{\prod_{i=2}^{2012} \log \pi(i)}{\prod_{i=2}^{2012} \log i} \\ & =1 \end{aligned}$$ where the last equality holds since $\pi$ is a permutation of the numbers 2 through 2012. | 0.25 | 7,860.25 | 6,865 | 8,192 |
For a natural number \( N \), if at least five of the natural numbers from 1 to 9 can divide \( N \) evenly, then \( N \) is called a "Five-Divisible Number." Find the smallest "Five-Divisible Number" that is greater than 2000. | 2004 | 0.0625 | 7,816.3125 | 4,946 | 8,007.666667 | |
In a school there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$ . | 23 | 0 | 8,192 | -1 | 8,192 | |
What is the eighth term in the arithmetic sequence $\frac 23, 1, \frac 43, \dots$? Express your answer in simplest form. | 3 | 0.9375 | 2,145.3125 | 1,742.2 | 8,192 | |
Given the sequence $\{a\_n\}$ satisfying $a\_1=2$, $a\_2=6$, and $a_{n+2} - 2a_{n+1} + a\_n = 2$, find the value of $\left\lfloor \frac{2017}{a\_1} + \frac{2017}{a\_2} + \ldots + \frac{2017}{a_{2017}} \right\rfloor$, where $\lfloor x \rfloor$ represents the greatest integer not greater than $x$. | 2016 | 0.9375 | 4,814.75 | 4,645.133333 | 7,359 | |
The graph of the equation $10x + 270y = 2700$ is drawn on graph paper where each square represents one unit in each direction. A second line defined by $x + y = 10$ also passes through the graph. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below both graphs and entirely in the first qua... | 50 | 0 | 8,192 | -1 | 8,192 | |
Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$? | 18 | 1. **Assign Variables and Use the Angle Bisector Theorem**:
Let $AC = x$. By the Angle Bisector Theorem, we have:
\[
\frac{AB}{AC} = \frac{BD}{CD} \implies \frac{10}{x} = \frac{BD}{3}
\]
Solving for $BD$, we get:
\[
BD = \frac{10}{x} \times 3 = \frac{30}{x}
\]
2. **Apply the Triangle Inequality... | 0.9375 | 4,390 | 4,136.533333 | 8,192 |
Let $x, y, n$ be positive integers with $n>1$. How many ordered triples $(x, y, n)$ of solutions are there to the equation $x^{n}-y^{n}=2^{100}$ ? | 49 | Break all possible values of $n$ into the four cases: $n=2, n=4, n>4$ and $n$ odd. By Fermat's theorem, no solutions exist for the $n=4$ case because we may write $y^{4}+\left(2^{25}\right)^{4}=x^{4}$. We show that for $n$ odd, no solutions exist to the more general equation $x^{n}-y^{n}=2^{k}$ where $k$ is a positive ... | 0.0625 | 8,192 | 8,192 | 8,192 |
Point O is the circumcenter of acute triangle ABC, with AB=6, AC=2. Determine the value of $\overrightarrow {AO}\cdot ( \overrightarrow {AB}+ \overrightarrow {AC})$. | 20 | 0.6875 | 5,993.75 | 4,994.545455 | 8,192 | |
Given that a certain product requires $6$ processing steps, where $2$ of these steps must be consecutive and another $2$ steps cannot be consecutive, calculate the number of possible processing sequences. | 144 | 0.1875 | 7,652.6875 | 6,548 | 7,907.615385 | |
When a granary records the arrival of 30 tons of grain as "+30", determine the meaning of "-30". | -30 | 0.75 | 1,251.5 | 1,375.916667 | 878.25 | |
Find the number of five-digit positive integers, $n$, that satisfy the following conditions:
(a) the number $n$ is divisible by $5,$
(b) the first and last digits of $n$ are equal, and
(c) the sum of the digits of $n$ is divisible by $5.$ | 200 | The number takes a form of $5\text{x,y,z}5$, in which $5|(x+y+z)$. Let $x$ and $y$ be arbitrary digits. For each pair of $x,y$, there are exactly two values of $z$ that satisfy the condition of $5|(x+y+z)$. Therefore, the answer is $10\times10\times2=\boxed{200}$
~Shreyas S | 0.5625 | 5,466.0625 | 4,449.222222 | 6,773.428571 |
Let \( r_{1}, r_{2}, \cdots, r_{20} \) be the roots of the polynomial \( x^{20}-7x^{3}+1 \). If \(\frac{1}{r_{1}^{2}+1}+\frac{1}{r_{2}^{2}+1}+\cdots+\frac{1}{r_{20}^{2}+1} \) can be expressed in the form \( \frac{m}{n} \) (with \( m \) and \( n \) coprime), find the value of \( m+n \). | 240 | 0.3125 | 7,572.9375 | 6,438 | 8,088.818182 | |
Let \( D \) be the midpoint of the hypotenuse \( BC \) of the right triangle \( ABC \). On the leg \( AC \), a point \( M \) is chosen such that \(\angle AMB = \angle CMD\). Find the ratio \(\frac{AM}{MC}\). | 1:2 | 0 | 6,998.3125 | -1 | 6,998.3125 | |
Suppose \( a \) is an integer. A sequence \( x_1, x_2, x_3, x_4, \ldots \) is constructed with:
- \( x_1 = a \),
- \( x_{2k} = 2x_{2k-1} \) for every integer \( k \geq 1 \),
- \( x_{2k+1} = x_{2k} - 1 \) for every integer \( k \geq 1 \).
For example, if \( a = 2 \), then:
\[ x_1 = 2, \quad x_2 = 2x_1 = 4, \quad x_3 =... | 1409 | 0 | 8,192 | -1 | 8,192 | |
How many positive three-digit integers less than 700 have at least two digits that are the same and none of the digits can be zero? | 150 | 0.5 | 6,670.375 | 5,148.75 | 8,192 | |
How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$ ? | 87 | 0.0625 | 7,354.3125 | 7,060 | 7,373.933333 | |
Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2.$ Given that the length of the chord... | 405 | 0.0625 | 7,938.8125 | 7,473 | 7,969.866667 | |
Calculate the value of $\sqrt{\frac{\sqrt{81} + \sqrt{81}}{2}}$. | 3 | Calculating, $\sqrt{\frac{\sqrt{81} + \sqrt{81}}{2}} = \sqrt{\frac{9 + 9}{2}} = \sqrt{9} = 3$. | 1 | 541 | 541 | -1 |
Let $\theta$ be an acute angle, and let
\[\sin \frac{\theta}{2} = \sqrt{\frac{x - 1}{2x}}.\]Express $\tan \theta$ in terms of $x.$ | \sqrt{x^2 - 1} | 1 | 4,400.6875 | 4,400.6875 | -1 | |
The fare in Moscow with the "Troika" card in 2016 is 32 rubles for one trip on the metro and 31 rubles for one trip on ground transportation. What is the minimum total number of trips that can be made at these rates, spending exactly 5000 rubles? | 157 | 0.75 | 6,196.5625 | 5,531.416667 | 8,192 | |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a\cos B - b\cos A = c$, and $C = \frac{π}{5}$, calculate the value of $\angle B$. | \frac{3\pi}{10} | 0.625 | 4,948.625 | 3,607.8 | 7,183.333333 | |
The 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and the... | 790 | The rectangle is divided into three smaller rectangles with a width of 50 mm and a length of $\dfrac{80}{3}$mm. According to the Pythagorean Theorem (or by noticing the 8-15-17 Pythagorean triple), the diagonal of the rectangle is $\sqrt{50^2+\left(\frac{80}{3}\right)^2}=\frac{170}{3}$mm. Since that on the lace, there ... | 0 | 8,192 | -1 | 8,192 |
Find the sum of $111_4+323_4+132_4$. Express your answer in base $4$. | 1232_4 | 0.5625 | 5,891.0625 | 4,101.444444 | 8,192 | |
For real numbers \( x \) and \( y \) such that \( x + y = 1 \), determine the maximum value of the expression \( A(x, y) = x^4 y + x y^4 + x^3 y + x y^3 + x^2 y + x y^2 \). | \frac{7}{16} | 0.1875 | 8,168.875 | 8,068.666667 | 8,192 | |
Find $\sec \frac{5 \pi}{3}.$ | 2 | 1 | 2,233.9375 | 2,233.9375 | -1 | |
Given the function $f(x) = 2\sin x\cos x - 2\sin^2 x$
(1) Find the smallest positive period of the function $f(x)$.
(2) Let $\triangle ABC$ have internal angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$, respectively, and satisfy $f(A) = 0$, $c = 1$, $b = \sqrt{2}$. Find the area of $\triangle ABC$. | \frac{1}{2} | 0.875 | 5,493.125 | 5,107.571429 | 8,192 | |
Find the projection of the vector $\begin{pmatrix} 4 \\ 5 \end{pmatrix}$ onto the vector $\begin{pmatrix} 2 \\ 0 \end{pmatrix}.$ | \begin{pmatrix} 4 \\ 0 \end{pmatrix} | 1 | 2,483.1875 | 2,483.1875 | -1 | |
Jarris the triangle is playing in the \((x, y)\) plane. Let his maximum \(y\) coordinate be \(k\). Given that he has side lengths 6, 8, and 10 and that no part of him is below the \(x\)-axis, find the minimum possible value of \(k\). | 24/5 | 0.3125 | 7,823.375 | 7,012.4 | 8,192 | |
The graph of the polynomial
$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$
has five distinct $x$-intercepts, one of which is at $(0,0)$. Which of the following coefficients cannot be zero?
$\textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$ | \text{(D)} | 0 | 6,567.6875 | -1 | 6,567.6875 | |
A sphere with center $O$ has radius $10$. A right triangle with sides $8, 15,$ and $17$ is situated in 3D space such that each side is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?
- **A)** $\sqrt{84}$
- **B)** $\sqrt{85}$
- **C)** $\sqrt{89}$
- **D)** $\sqrt{91}$
- *... | \sqrt{91} | 0 | 7,331.625 | -1 | 7,331.625 | |
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 | Let $BC=x$ and $CD=y+\sqrt{11}$. From Pythagoras with $AD$, we obtain $x^2+y^2=1001$. Since $AC$ and $BD$ are perpendicular diagonals of a quadrilateral, then $AB^2+CD^2=BC^2+AD^2$, so we have \[\left(y+\sqrt{11}\right)^2+11=x^2+1001.\] Substituting $x^2=1001-y^2$ and simplifying yields \[y^2+\sqrt{11}y-990=0,\] and th... | 0.8125 | 5,197 | 4,505.846154 | 8,192 |
Distinct points $A, B, C, D$ are given such that triangles $A B C$ and $A B D$ are equilateral and both are of side length 10 . Point $E$ lies inside triangle $A B C$ such that $E A=8$ and $E B=3$, and point $F$ lies inside triangle $A B D$ such that $F D=8$ and $F B=3$. What is the area of quadrilateral $A E F D$ ? | \frac{91 \sqrt{3}}{4} | $\angle F B D+\angle A B F=\angle A B D=60^{\circ}$. Since $E B=B F=3$, this means that $E B F$ is an equilateral triangle of side length 3. Now we have $[A E F D]=[A E B D]-[E B F]-[F B D]=[A E B]+[A B D]-[E B F]-$ $[F B D]=[A B D]-[E B F]=\frac{\sqrt{3}}{4}\left(10^{2}-3^{2}\right)=\frac{91 \sqrt{3}}{4}$. | 0 | 8,002.0625 | -1 | 8,002.0625 |
Find $k$ where $2^k$ is the largest power of $2$ that divides the product \[2008\cdot 2009\cdot 2010\cdots 4014.\] | 2007 | 0.8125 | 5,942.4375 | 5,423.307692 | 8,192 | |
What number can be added to both the numerator and denominator of $\frac{3}{5}$ so that the resulting fraction will be equivalent to $\frac{5}{6}$? | 7 | 1 | 2,932 | 2,932 | -1 | |
Without using any tables, find the exact value of the product:
\[ P = \cos \frac{\pi}{15} \cos \frac{2\pi}{15} \cos \frac{3\pi}{15} \cos \frac{4\pi}{15} \cos \frac{5\pi}{15} \cos \frac{6\pi}{15} \cos \frac{7\pi}{15}. \] | 1/128 | 0.1875 | 7,680.25 | 5,462.666667 | 8,192 | |
Suppose two distinct numbers are chosen from between 6 and 20, inclusive. What is the probability that their product is even, or exactly one of the numbers is a prime? | \frac{94}{105} | 0.5625 | 6,921.6875 | 5,933.666667 | 8,192 | |
Calculate the argument of the sum:
\[ e^{5\pi i/36} + e^{11\pi i/36} + e^{17\pi i/36} + e^{23\pi i/36} + e^{29\pi i/36} \]
in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$. | \frac{17\pi}{36} | 0.3125 | 7,507.25 | 6,000.8 | 8,192 | |
For any number $y$, define the operations $\&y = 2(7-y)$ and $\&y = 2(y-7)$. What is the value of $\&(-13\&)$? | 66 | 0.1875 | 7,326.0625 | 6,794.333333 | 7,448.769231 | |
The lateral sides of a right trapezoid are 10 and 8. The diagonal of the trapezoid, drawn from the vertex of the acute angle, bisects this angle. Find the area of the trapezoid. | 104 | 0 | 7,347.4375 | -1 | 7,347.4375 | |
Given points A, B, and C on the surface of a sphere O, the height of the tetrahedron O-ABC is 2√2 and ∠ABC=60°, with AB=2 and BC=4. Find the surface area of the sphere O. | 48\pi | 0.625 | 6,225.1875 | 5,159.2 | 8,001.833333 | |
Find the number of $x$-intercepts on the graph of $y = \sin \frac{1}{x}$ (evaluated in terms of radians) in the interval $(0.0001, 0.001).$ | 2865 | 0.625 | 5,287.1875 | 3,692.4 | 7,945.166667 | |
Consider the largest solution to the equation \[\log_{10x^2} 10 + \log_{100x^3} 10 = -2.\]Find the value of $\frac{1}{x^{12}},$ writing your answer in decimal representation. | 10000000 | 0.5 | 6,462.4375 | 4,732.875 | 8,192 | |
Factor the following expression: $45x+30$. | 15(3x+2) | 0.9375 | 719.3125 | 745.266667 | 330 | |
The graph of $xy = 1$ is a hyperbola. Find the distance between the foci of this hyperbola. | 4 | 0.75 | 5,585.25 | 4,716.333333 | 8,192 | |
Place the terms of the sequence $\{2n-1\}$ ($n\in\mathbb{N}^+$) into brackets according to the following pattern: the first bracket contains the first term, the second bracket contains the second and third terms, the third bracket contains the fourth, fifth, and sixth terms, the fourth bracket contains the seventh term... | 1251 | 0.125 | 7,691.75 | 5,064 | 8,067.142857 | |
Let $x = \sqrt{\frac{\sqrt{53}}{2} + \frac{3}{2}}.$ There exist unique positive integers $a,$ $b,$ $c$ such that
\[x^{100} = 2x^{98} + 14x^{96} + 11x^{94} - x^{50} + ax^{46} + bx^{44} + cx^{40}.\]Find $a + b + c.$ | 157 | 0 | 8,192 | -1 | 8,192 | |
Draw a square of side length 1. Connect its sides' midpoints to form a second square. Connect the midpoints of the sides of the second square to form a third square. Connect the midpoints of the sides of the third square to form a fourth square. And so forth. What is the sum of the areas of all the squares in this infi... | 2 | The area of the first square is 1, the area of the second is $\frac{1}{2}$, the area of the third is $\frac{1}{4}$, etc., so the answer is $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots=2$. | 1 | 3,358.5625 | 3,358.5625 | -1 |
Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100? | 20201 | 1. **Identify the pattern**: We observe that the number of unit squares in figures $0$, $1$, $2$, and $3$ are $1$, $5$, $13$, and $25$ respectively. We need to find a formula that describes this sequence.
2. **Recognize the sequence type**: The sequence appears to be quadratic because the differences between consecuti... | 1 | 2,954.5 | 2,954.5 | -1 |
Find the coefficient of \(x^5\) in the expansion of \(\left(1+2x+3x^2+4x^3\right)^5\). | 1772 | 0.0625 | 7,918.6875 | 5,650 | 8,069.933333 | |
Given a triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. The area of the triangle is given by $S= \frac{a^{2}+b^{2}-c^{2}}{4}$ and $\sin A= \frac{3}{5}$.
1. Find $\sin B$.
2. If side $c=5$, find the area of $\triangle ABC$, denoted as $S$. | \frac{21}{2} | 0.75 | 5,130.375 | 4,109.833333 | 8,192 | |
In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $b=2$, $c=2\sqrt{2}$, and $C=\frac{\pi}{4}$, find the area of $\Delta ABC$. | \sqrt{3} +1 | 0 | 6,278.0625 | -1 | 6,278.0625 | |
In the number $74982.1035$ the value of the place occupied by the digit 9 is how many times as great as the value of the place occupied by the digit 3? | 100,000 | 1. **Identify the place values**:
- The digit $9$ in the number $74982.1035$ is in the hundreds place, which means it represents $9 \times 100 = 900$.
- The digit $3$ in the number $74982.1035$ is in the thousandths place, which means it represents $3 \times 0.001 = 0.003$.
2. **Calculate the ratio of the place... | 0 | 1,599.625 | -1 | 1,599.625 |
In a regular tetrahedron \( ABCD \) with side length \( \sqrt{2} \), it is known that \( \overrightarrow{AP} = \frac{1}{2} \overrightarrow{AB} \), \( \overrightarrow{AQ} = \frac{1}{3} \overrightarrow{AC} \), and \( \overrightarrow{AR} = \frac{1}{4} \overrightarrow{AD} \). If point \( K \) is the centroid of \( \triangl... | \frac{1}{36} | 0 | 8,192 | -1 | 8,192 | |
Yangyang leaves home for school. If she walks 60 meters per minute, she arrives at school at 6:53. If she walks 75 meters per minute, she arrives at school at 6:45. What time does Yangyang leave home? | 6:13 | 0.125 | 808.125 | 853.5 | 801.642857 | |
In 2003, the average monthly rainfall in Mathborough was $41.5\text{ mm.}$ In 2004, the average monthly rainfall in Mathborough was $2\text{ mm}$ more than in 2003. What was the total amount of rain that fell in Mathborough in 2004? | 522 | 1 | 1,494.0625 | 1,494.0625 | -1 | |
For $\pi \le \theta < 2\pi$, let
\begin{align*} P &= \frac12\cos\theta - \frac14\sin 2\theta - \frac18\cos 3\theta + \frac{1}{16}\sin 4\theta + \frac{1}{32} \cos 5\theta - \frac{1}{64} \sin 6\theta - \frac{1}{128} \cos 7\theta + \cdots \end{align*}
and
\begin{align*} Q &= 1 - \frac12\sin\theta -\frac14\cos 2\theta + \... | 36 | Use sum to product formulas to rewrite $P$ and $Q$
\[P \sin\theta\ + Q \cos\theta\ = \cos \theta\ - \frac{1}{4}\cos \theta + \frac{1}{8}\sin 2\theta + \frac{1}{16}\cos 3\theta - \frac{1}{32}\sin 4\theta + ...\]
Therefore, \[P \sin \theta - Q \cos \theta = -2P\]
Using \[\frac{P}{Q} = \frac{2\sqrt2}{7}, Q = \frac{7}{2\... | 0 | 7,969.6875 | -1 | 7,969.6875 |
Mr. Lee V. Soon starts his morning commute at 7:00 AM to arrive at work by 8:00 AM. If he drives at an average speed of 30 miles per hour, he is late by 5 minutes, and if he drives at an average speed of 70 miles per hour, he is early by 4 minutes. Find the speed he needs to maintain to arrive exactly at 8:00 AM. | 32.5 | 0 | 7,909.625 | -1 | 7,909.625 | |
Select 4 students from 5 female and 4 male students to participate in a speech competition.
(1) If 2 male and 2 female students are to be selected, how many different selections are there?
(2) If at least 1 male and 1 female student must be selected, and male student A and female student B cannot be selected at the... | 99 | 0.3125 | 6,881.25 | 5,745.2 | 7,397.636364 | |
What is the sum of all of the possibilities for Sam's number if Sam thinks of a 5-digit number, Sam's friend Sally tries to guess his number, Sam writes the number of matching digits beside each of Sally's guesses, and a digit is considered "matching" when it is the correct digit in the correct position? | 526758 | We label the digits of the unknown number as vwxyz.
Since vwxyz and 71794 have 0 matching digits, then $v \neq 7$ and $w \neq 1$ and $x \neq 7$ and $y \neq 9$ and $z \neq 4$.
Since vwxyz and 71744 have 1 matching digit, then the preceding information tells us that $y=4$.
Since $v w x 4 z$ and 51545 have 2 matchin... | 0 | 6,902.5625 | -1 | 6,902.5625 |
Given the set \( S = \{1, 2, \cdots, 2005\} \), and a subset \( A \subseteq S \) such that the sum of any two numbers in \( A \) is not divisible by 117, determine the maximum value of \( |A| \). | 1003 | 0 | 8,192 | -1 | 8,192 | |
Given six thin wooden sticks, the two longer ones are $\sqrt{3}a$ and $\sqrt{2}a$, while the remaining four are of length $a$. If they are used to form a triangular prism, find the cosine of the angle between the lines containing the two longer edges. | \frac{\sqrt{6}}{3} | 0 | 7,895.1875 | -1 | 7,895.1875 | |
For every $n$ the sum of $n$ terms of an arithmetic progression is $2n + 3n^2$. The $r$th term is: | 6r - 1 | 1. **Identify the formula for the sum of the first $n$ terms**: Given that the sum of the first $n$ terms of an arithmetic progression is $S_n = 2n + 3n^2$.
2. **Expression for the $r$th term**: The $r$th term of an arithmetic sequence can be found by subtracting the sum of the first $r-1$ terms from the sum of the fi... | 1 | 2,288.0625 | 2,288.0625 | -1 |
How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits? | 90 | 0.875 | 3,080.3125 | 2,607.857143 | 6,387.5 | |
Let $a_{n+1} = \frac{4}{7}a_n + \frac{3}{7}a_{n-1}$ and $a_0 = 1$ , $a_1 = 2$ . Find $\lim_{n \to \infty} a_n$ . | 1.7 | 0 | 4,006.625 | -1 | 4,006.625 | |
What integer $n$ satisfies $0 \leq n < 201$ and $$200n \equiv 144 \pmod {101}~?$$ | 29 | 0.625 | 6,033.9375 | 4,851.1 | 8,005.333333 | |
In the game of preference, each of the three players is dealt 10 cards, and two cards are placed in the kitty. How many different arrangements are possible in this game? (Consider possible distributions without accounting for which 10 cards go to each specific player.) | \frac{32!}{(10!)^3 \cdot 2! \cdot 3!} | 0 | 6,936.5 | -1 | 6,936.5 | |
In an isosceles trapezoid \(ABCD\), \(AB\) is parallel to \(CD\), \(AB = 6\), \(CD = 14\), \(\angle AEC\) is a right angle, and \(CE = CB\). What is \(AE^2\)? | 84 | 0.3125 | 7,607.9375 | 6,990.8 | 7,888.454545 | |
Eight congruent copies of the parabola \( y = x^2 \) are arranged symmetrically around a circle such that each vertex is tangent to the circle, and each parabola is tangent to its two neighbors. Find the radius of the circle. Assume that one of the tangents to the parabolas corresponds to the line \( y = x \tan(45^\cir... | \frac{1}{4} | 0 | 8,192 | -1 | 8,192 | |
Let $f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}.$ For constant $a$ such that $\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12$ , evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.$ | \frac{1}{2} | 0.0625 | 6,984.5625 | 6,188 | 7,037.666667 | |
Both $a$ and $b$ are positive integers and $b > 1$. When $a^b$ is the greatest possible value less than 399, what is the sum of $a$ and $b$? | 21 | 0.125 | 8,157.4375 | 7,915.5 | 8,192 | |
$A B C$ is a right triangle with $\angle A=30^{\circ}$ and circumcircle $O$. Circles $\omega_{1}, \omega_{2}$, and $\omega_{3}$ lie outside $A B C$ and are tangent to $O$ at $T_{1}, T_{2}$, and $T_{3}$ respectively and to $A B, B C$, and $C A$ at $S_{1}, S_{2}$, and $S_{3}$, respectively. Lines $T_{1} S_{1}, T_{2} S_{2... | \frac{\sqrt{3}+1}{2} | Let $[P Q R]$ denote the area of $\triangle P Q R$. The key to this problem is following fact: $[P Q R]=\frac{1}{2} P Q \cdot P R \sin \angle Q P R$. Assume that the radius of $O$ is 1. Since $\angle A=30^{\circ}$, we have $B C=1$ and $A B=\sqrt{3}$. So $[A B C]=\frac{\sqrt{3}}{2}$. Let $K$ denote the center of $O$. No... | 0 | 8,192 | -1 | 8,192 |
For $y=\frac{x+2}{5x-7}$, at what $x$-value is there a vertical asymptote? | \frac{7}{5} | 1 | 1,139.5625 | 1,139.5625 | -1 | |
Each cell of a $3 \times 3$ grid is labeled with a digit in the set $\{1,2,3,4,5\}$. Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from 1 to 5 is recorded at least once. | 2664 | We perform casework by placing the entries from largest to smallest. - The grid must have exactly one 5 since an entry equal to 5 will be the maximum in its row and in its column. We can place this in 9 ways. - An entry equal to 4 must be in the same row or column as the 5; otherwise, it will be recorded twice, so we o... | 0 | 7,867.5625 | -1 | 7,867.5625 |
Fill the numbers 1 to 6 into the six boxes in the image. The smallest result you can get is ______ | 342 | 0 | 7,863.9375 | -1 | 7,863.9375 | |
In a labor and technical competition among five students: A, B, C, D, and E, the rankings from first to fifth place were determined. When A and B asked about their results, the respondent told A, "Unfortunately, both you and B did not win the championship"; and told B, "You certainly are not the worst." Based on these ... | 36 | 0.0625 | 7,049.5625 | 5,347 | 7,163.066667 | |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 72$. | \frac{1}{72} | 0 | 7,840.25 | -1 | 7,840.25 | |
$A$ and $B$ are on a circle of radius $20$ centered at $C$ , and $\angle ACB = 60^\circ$ . $D$ is chosen so that $D$ is also on the circle, $\angle ACD = 160^\circ$ , and $\angle DCB = 100^\circ$ . Let $E$ be the intersection of lines $AC$ and $BD$ . What is $DE$ ? | 20 | 0.0625 | 8,192 | 8,192 | 8,192 | |
In the binomial expansion of $(x-1)^n$ ($n\in\mathbb{N}_{+}$), if only the 5th binomial coefficient is the largest, find the constant term in the binomial expansion of $(2\sqrt{x}-\frac{1}{\sqrt{x}})^n$. | 1120 | 0.875 | 4,017.5 | 3,421.142857 | 8,192 | |
Reading material: For non-zero real numbers $a$ and $b$, if the value of the fraction $\frac{(x-a)(x-b)}{x}$ with respect to $x$ is zero, then the solutions are $x_{1}=a$ and $x_{2}=b$. Also, because $\frac{(x-a)(x-b)}{x}=\frac{{x}^{2}-(a+b)x+ab}{x}=x+\frac{ab}{x}-\left(a+b\right)$, the solutions to the equation $x+\fr... | 32 | 0.3125 | 7,083.125 | 4,643.6 | 8,192 | |
In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth? | 2\frac{2}{3} | 1. **Define the variables:**
Let $f$ represent the value of one fish, $l$ represent the value of a loaf of bread, and $r$ represent the value of a bag of rice.
2. **Set up the equations based on the given trades:**
- According to the problem, three fish can be traded for two loaves of bread. This can be written ... | 0 | 2,621.1875 | -1 | 2,621.1875 |
A circle is inscribed in quadrilateral $EFGH$, tangent to $\overline{EF}$ at $R$ and to $\overline{GH}$ at $S$. Given that $ER=24$, $RF=31$, $GS=40$, and $SH=29$, find the square of the radius of the circle. | 945 | 0 | 8,192 | -1 | 8,192 | |
Square corners, 5 units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is | 500 | 1. **Identify the dimensions of the original cardboard sheet**: The original sheet is $20$ units by $30$ units.
2. **Calculate the area of the original sheet**:
\[
\text{Area}_{\text{original}} = 20 \times 30 = 600 \text{ square units}
\]
3. **Determine the size of the squares removed**: Each square removed... | 0.75 | 4,055.125 | 3,852.5 | 4,663 |
If $x-y=15$ and $xy=4$, what is the value of $x^2+y^2$? | 233 | 1 | 2,583.6875 | 2,583.6875 | -1 | |
In a kennel with 60 dogs, 9 dogs like watermelon, 48 dogs like salmon, and 5 like both salmon and watermelon. How many dogs in the kennel will not eat either? | 8 | 1 | 1,295.6875 | 1,295.6875 | -1 | |
Jenny places a total of 18 red Easter eggs in several green baskets and a total of 24 orange Easter eggs in some blue baskets. Each basket contains the same number of eggs and there are at least 4 eggs in each basket. How many eggs did Jenny put in each basket? | 6 | 1 | 1,465.375 | 1,465.375 | -1 | |
Darren has borrowed $100$ clams from Ethan at a $10\%$ simple daily interest. Meanwhile, Fergie has borrowed $150$ clams from Gertie at a $5\%$ simple daily interest. In how many days will Darren and Fergie owe the same amounts, assuming that they will not make any repayments in that time period? | 20\text{ days} | 1 | 2,153.875 | 2,153.875 | -1 | |
Circles centered at $A$ and $B$ each have radius 2, as shown. Point $O$ is the midpoint of $\overline{AB}$, and $OA=2\sqrt{2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $\overline{EF}$ is a common tangent. What is the area of the shaded region $ECODF$?
[asy]unitsiz... | 8\sqrt{2}-4-\pi | 0 | 8,084.4375 | -1 | 8,084.4375 |
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