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 |
|---|---|---|---|---|---|---|
What is the greatest common factor of 180, 240, and 300? | 60 | 1 | 2,447.375 | 2,447.375 | -1 | |
Given a regular pyramid V-ABCD with a base edge length of 4 and a lateral edge length of $\sqrt{13}$, its surface area is ______. | 40 | 0.9375 | 3,875.3125 | 3,999.333333 | 2,015 | |
Points $A$, $B$, $C$, and $D$ are located on $\overline{AB}$ such that $AB = 3AD = 6BC$. If a point is selected at random on $\overline{AB}$, what is the probability that it is between $C$ and $D$? Express your answer as a common fraction.
[asy]
draw((0,0)--(12,.0000000001));
dot((0,0)); dot((4,0)); dot((10,0)); dot(... | \frac{1}{2} | 1 | 4,542.5625 | 4,542.5625 | -1 | |
Select 5 people from 3 orthopedic doctors, 4 neurosurgeons, and 5 internists to form an earthquake relief medical team. The number of ways to select such that there is at least one doctor from each specialty is (answer in digits). | 590 | 0.625 | 5,786.6875 | 4,343.5 | 8,192 | |
In a circular arrangement of 101 natural numbers, it is known that among any 5 consecutive numbers, there are at least two even numbers. What is the minimum number of even numbers that can be among the listed numbers? | 41 | 0.125 | 8,066.875 | 7,191 | 8,192 | |
Solve for $x$: $$5^{x + 4} = 125^x.$$ | x = 2 | 1 | 1,514.75 | 1,514.75 | -1 | |
We define the polynomial $$ P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x. $$ Find the largest prime divisor of $P (2)$ . | 61 | 0.25 | 7,254.1875 | 4,440.75 | 8,192 | |
The third quartile of the data $13$, $11$, $12$, $15$, $16$, $18$, $21$, $17$ is ______. | 17.5 | 0.25 | 2,034.9375 | 628 | 2,503.916667 | |
In right triangle $ABC$ with $\angle B = 90^\circ$, we have $$2\sin A = 3\cos A.$$What is $\sin A$? | \frac{3\sqrt{13}}{13} | 0 | 1,871.9375 | -1 | 1,871.9375 | |
How many positive integers less than 10,000 have at most three different digits? | 4119 | 0 | 7,971.1875 | -1 | 7,971.1875 | |
A box contains $3$ shiny pennies and $4$ dull pennies. One by one, pennies are drawn at random from the box and not replaced. If the probability is $a/b$ that it will take more than four draws until the third shiny penny appears and $a/b$ is in lowest terms, then $a+b=$ | 66 | 1. **Identify the Total Number of Combinations**:
The box contains 3 shiny pennies (denoted as 1) and 4 dull pennies (denoted as 0). The total number of ways to arrange these pennies is given by the binomial coefficient $\binom{7}{3}$, which counts the number of ways to choose 3 positions for shiny pennies out of 7.... | 0.375 | 6,864.6875 | 4,652.5 | 8,192 |
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties: \begin{enumerate} \item[(a)] $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$... | 29 | The minimum value of $T$ is 29. Write $t_{n+1} = t_0+T$ and define $s_k = t_k-t_{k-1}$ for $1\leq k\leq n+1$. On $[t_{k-1},t_k]$, we have $f'(t) = k(t-t_{k-1})$ and so $f(t_k)-f(t_{k-1}) = \frac{k}{2} s_k^2$. Thus if we define \[ g(s_1,\ldots,s_{n+1}) = \sum_{k=1}^{n+1} ks_k^2, \] then we want to minimize $\sum_{k=1}^{... | 0 | 8,192 | -1 | 8,192 |
Which of the following integers is equal to a perfect square: $2^{3}$, $3^{5}$, $4^{7}$, $5^{9}$, $6^{11}$? | 4^{7} | Since $4 = 2^{2}$, then $4^{7} = (2^{2})^{7} = 2^{14} = (2^{7})^{2}$, which means that $4^{7}$ is a perfect square. We can check, for example using a calculator, that the square root of each of the other four choices is not an integer, and so each of these four choices cannot be expressed as the square of an integer. | 0.0625 | 1,589.375 | 1,291 | 1,609.266667 |
Given that \\(AB\\) is a chord passing through the focus of the parabola \\(y^{2} = 4\sqrt{3}x\\), and the midpoint \\(M\\) of \\(AB\\) has an x-coordinate of \\(2\\), calculate the length of \\(AB\\. | 4 + 2\sqrt{3} | 0.3125 | 7,540.25 | 6,710.6 | 7,917.363636 | |
Let $a$, $b$, and $c$ be positive integers with $a\ge$ $b\ge$ $c$ such that
$a^2-b^2-c^2+ab=2011$ and
$a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.
What is $a$? | 253 | 1. **Combine the given equations**:
\[
a^2 - b^2 - c^2 + ab + a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc = 2011 - 1997
\]
Simplifying this, we get:
\[
2a^2 + 2b^2 + 2c^2 - 2ab - 2ac - 2bc = 14
\]
2. **Factor and simplify**:
\[
(a-b)^2 + (a-c)^2 + (b-c)^2 = 14
\]
Since $a \geq b \geq c$, the s... | 0.5625 | 6,158.375 | 4,576.666667 | 8,192 |
Given that $\sqrt {3}\sin x+\cos x= \frac {2}{3}$, find the value of $\tan (x+ \frac {7\pi}{6})$. | \frac{\sqrt{2}}{4} | 0 | 7,667.9375 | -1 | 7,667.9375 | |
Let $S$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form $0.abcabcabc\ldots=0.\overline{abc}$, where the digits $a$, $b$, and $c$ are not necessarily distinct. To write the elements of $S$ as fractions in lowest terms, how many different numerators are required?
| 660 | 0 | 7,972.9375 | -1 | 7,972.9375 | |
Given the sequence $\{a\_n\}$ that satisfies $a\_n-(-1)^{n}a\_{n-1}=n$ $(n\geqslant 2)$, and $S\_n$ is the sum of the first $n$ terms of the sequence, find the value of $S\_{40}$. | 440 | 0.1875 | 7,421.25 | 6,033.333333 | 7,741.538462 | |
Solve the following equations:
a) $\log _{1 / 5} \frac{2+x}{10}=\log _{1 / 5} \frac{2}{x+1}$;
b) $\log _{3}\left(x^{2}-4 x+3\right)=\log _{3}(3 x+21)$;
c) $\log _{1 / 10} \frac{2 x^{2}-54}{x+3}=\log _{1 / 10}(x-4)$;
d) $\log _{(5+x) / 3} 3=\log _{-1 /(x+1)} 3$. | -4 | 0.9375 | 5,096.6875 | 4,890.333333 | 8,192 | |
When programming a computer to print the first 10,000 natural numbers greater than 0: $1,2,3, \cdots, 10000$, the printer unfortunately has a malfunction. Each time it prints the digits 7 or 9, it prints $x$ instead. How many numbers are printed incorrectly? | 5904 | 0.0625 | 7,962.6875 | 4,965 | 8,162.533333 | |
If $2^{x-3}=4^2$, find $x$. | 7 | 1 | 2,436.5 | 2,436.5 | -1 | |
How many zeros are in the expansion of $999,\!999,\!999,\!998^2$? | 11 | 0.0625 | 8,187.9375 | 8,127 | 8,192 | |
Let \( S = \{1, 2, \ldots, 280\} \). Find the smallest natural number \( n \) such that every \( n \)-element subset of \( S \) contains 5 pairwise coprime numbers. | 217 | 0.0625 | 8,113.4375 | 6,935 | 8,192 | |
Six identical rectangles are arranged to form a larger rectangle \( ABCD \). The area of \( ABCD \) is 6000 square units. What is the length \( z \), rounded off to the nearest integer? | 32 | 0 | 8,116.25 | -1 | 8,116.25 | |
Find the smallest \( n \) such that whenever the elements of the set \(\{1, 2, \ldots, n\}\) are colored red or blue, there always exist \( x, y, z, w \) (not necessarily distinct) of the same color such that \( x + y + z = w \). | 11 | 0.375 | 7,995.25 | 7,667.333333 | 8,192 | |
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \). Additionally, find its height dropped from vertex \( A_{4} \) onto the face \( A_{1} A_{2} A_{3} \).
Vertices:
- \( A_{1}(-1, 2, 4) \)
- \( A_{2}(-1, -2, -4) \)
- \( A_{3}(3, 0, -1) \)
- \( A_{4}(7, -3, 1) \) | 24 | 0 | 5,330.625 | -1 | 5,330.625 | |
How many times should two dice be rolled so that the probability of getting two sixes at least once is greater than $1/2$? | 25 | 0.8125 | 5,263.4375 | 4,833.538462 | 7,126.333333 | |
A teacher received letters on Monday to Friday with counts of $10$, $6$, $8$, $5$, $6$ respectively. Calculate the standard deviation of this data set. | \dfrac {4 \sqrt {5}}{5} | 0 | 1,366.5 | -1 | 1,366.5 | |
Find the smallest positive integer that cannot be expressed in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a$, $b$, $c$, and $d$ are all positive integers. | 11 | 0 | 8,192 | -1 | 8,192 | |
Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not atte... | 849 | Let $q$ be the number of questions Beta takes on day 1 and $a$ be the number he gets right. Let $b$ be the number he gets right on day 2.
These inequalities follow: \[\frac{a}{q} < \frac{160}{300} = \frac{8}{15}\] \[\frac{b}{500-q} < \frac{140}{200} = \frac{7}{10}\] Solving for a and b and adding the two inequalities:... | 0.1875 | 8,071.75 | 7,550.666667 | 8,192 |
Let $x_1,$ $x_2,$ $\dots,$ $x_n$ be nonnegative real numbers such that $x_1 + x_2 + \dots + x_n = 1$ and
\[x_1^2 + x_2^2 + \dots + x_n^2 \le \frac{1}{100}.\]Find the smallest possible value of $n.$ | 100 | 0.9375 | 4,771.875 | 4,543.866667 | 8,192 | |
Let $f(x) = 4x - 9$ and $g(f(x)) = 3x^2 + 4x - 2.$ Find $g(-10).$ | \frac{-45}{16} | 0 | 4,062.125 | -1 | 4,062.125 | |
Given $a$ and $b$ are positive numbers such that $a^b = b^a$ and $b = 4a$, solve for the value of $a$. | \sqrt[3]{4} | 0.9375 | 3,342.3125 | 3,019 | 8,192 | |
How many tetrahedrons can be formed using the vertices of a regular triangular prism? | 12 | 0.1875 | 7,692.8125 | 5,859.666667 | 8,115.846154 | |
Suppose $656_7=3ab_{10}$, where $a$ and $b$ represent base-10 digits. Find $\frac{a\cdot b}{15}$. | 1 | 1 | 2,191.625 | 2,191.625 | -1 | |
Let $ABC$ be a triangle with incenter $I$, centroid $G$, and $|AC|>|AB|$. If $IG\parallel BC$, $|BC|=2$, and $\text{Area}(ABC)=3\sqrt{5}/8$, calculate $|AB|$. | \frac{9}{8} | 0.3125 | 7,325.6875 | 5,917.2 | 7,965.909091 | |
If \( \sqrt{\frac{3}{x} + 3} = \frac{5}{3} \), solve for \( x \). | -\frac{27}{2} | 1 | 2,532.9375 | 2,532.9375 | -1 | |
What value of $k$ will make $x^2 - 16x + k$ the square of a binomial? | 64 | 1 | 1,814.4375 | 1,814.4375 | -1 | |
Karen has seven envelopes and seven letters of congratulations to various HMMT coaches. If she places the letters in the envelopes at random with each possible configuration having an equal probability, what is the probability that exactly six of the letters are in the correct envelopes? | 0 | 0, since if six letters are in their correct envelopes the seventh is as well. | 1 | 1,471.9375 | 1,471.9375 | -1 |
Find the quotient when $x^5 + 7$ is divided by $x + 1.$ | x^4 - x^3 + x^2 - x + 1 | 1 | 4,560.25 | 4,560.25 | -1 | |
In the figure shown, segment $AB$ is parallel to segment $YZ$. If $AZ = 42$ units, $BQ = 12$ units, and $QY = 24$ units, what is the length of segment $QZ$? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
pair Y = (0,0), Z = (16,0), A = (0,8), B = (6,8);
draw(A--B--Y--Z--cycle);
label("$A... | 28 | 0.6875 | 5,245.875 | 4,400.727273 | 7,105.2 | |
Compute the sum of the geometric series $-1 -3-9-27 -81-243-729$. | -1093 | 1 | 2,446.0625 | 2,446.0625 | -1 | |
\[
1.047. \left(\frac{\sqrt{561^{2} - 459^{2}}}{4 \frac{2}{7} \cdot 0.15 + 4 \frac{2}{7} : \frac{20}{3}} + 4 \sqrt{10}\right) : \frac{1}{3} \sqrt{40}
\] | 125 | 0.8125 | 4,277.625 | 3,872.692308 | 6,032.333333 | |
$A B C D E$ is a cyclic convex pentagon, and $A C=B D=C E . A C$ and $B D$ intersect at $X$, and $B D$ and $C E$ intersect at $Y$. If $A X=6, X Y=4$, and $Y E=7$, then the area of pentagon $A B C D E$ can be written as $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are integers, $c$ is positive, $b$ is square-free, and $\oper... | 2852 | Since $A C=B D, A B C D$ is an isosceles trapezoid. Similarly, $B C D E$ is also an isosceles trapezoid. Using this, we can now calculate that $C Y=D Y=D X-X Y=A X-X Y=2$, and similarly $B X=C X=3$. By applying Heron's formula we find that the area of triangle $C X Y$ is $\frac{3}{4} \sqrt{15}$. Now, note that $$[A B C... | 0 | 8,192 | -1 | 8,192 |
Three people, A, B, and C, start from point $A$ to point $B$. A starts at 8:00, B starts at 8:20, and C starts at 8:30. They all travel at the same speed. Ten minutes after C starts, the distance from A to point $B$ is exactly half the distance from B to point $B$. At this time, C is 2015 meters away from point $B$. Ho... | 2418 | 0 | 8,059.5625 | -1 | 8,059.5625 | |
The difference \(\sqrt{|40 \sqrt{2}-57|}-\sqrt{40 \sqrt{2}+57}\) is an integer. Find this number. | -10 | 0.8125 | 6,147.5625 | 5,995.538462 | 6,806.333333 | |
Given a six-digit phone number, how many different seven-digit phone numbers exist such that, by crossing out one digit, you obtain the given six-digit number? | 70 | 0.0625 | 7,601.875 | 8,048 | 7,572.133333 | |
Right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\overline{AC}$ of isosceles right triangle $ABC$ with leg length $1$, as shown, so that the two triangles have equal perimeters. What is $\sin(2\angle BAD)$? | \frac{7}{9} |
1. **Identify the length of hypotenuse $AC$ in $\triangle ABC$:**
Since $\triangle ABC$ is an isosceles right triangle with leg length $1$, by the Pythagorean Theorem, we have:
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{1^2 + 1^2} = \sqrt{2}
\]
2. **Equal perimeter condition:**
The perimeter of $\triangle ABC$... | 0.6875 | 6,367.75 | 5,637.545455 | 7,974.2 |
Sixteen 6-inch wide square posts are evenly spaced with 4 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence? | 56 | 0.4375 | 6,848.9375 | 6,135.714286 | 7,403.666667 | |
What is the sum of three consecutive even integers if the sum of the first and third integers is $128$? | 192 | 1 | 2,259.4375 | 2,259.4375 | -1 | |
For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}$, and $g(2020) = \text{the digit sum of }12_{\text{8}} = 3$. Let... | 151 | 0 | 8,192 | -1 | 8,192 | |
Find the absolute value of the difference of single-digit integers \( C \) and \( D \) such that in base \( 5 \):
$$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
& & & D & D & C_5 \\
& & & \mathbf{3} & \mathbf{2} & D_5 \\
& & + & C & \mathbf{2} & \mathbf{4_5} \\
\cline{2-6}
& & C & \mathbf{2} & \mathbf{3} & \mathbf{1_5} \\
\... | 1_5 | 0 | 7,853.4375 | -1 | 7,853.4375 | |
Let $ABC$ be a triangle with $|AB|=18$ , $|AC|=24$ , and $m(\widehat{BAC}) = 150^\circ$ . Let $D$ , $E$ , $F$ be points on sides $[AB]$ , $[AC]$ , $[BC]$ , respectively, such that $|BD|=6$ , $|CE|=8$ , and $|CF|=2|BF|$ . Let $H_1$ , $H_2$ , $H_3$ be the reflections of the orthocenter of triangle $AB... | 96 | 0.875 | 6,439.1875 | 6,300.785714 | 7,408 | |
How many multiples of 5 are there between 5 and 205? | 41 | 0.1875 | 5,247.5625 | 6,037.666667 | 5,065.230769 | |
Grisha wrote 100 numbers on the board. Then he increased each number by 1 and noticed that the product of all 100 numbers did not change. He increased each number by 1 again, and again the product of all the numbers did not change, and so on. Grisha repeated this procedure $k$ times, and each of the $k$ times the produ... | 99 | 0 | 8,192 | -1 | 8,192 | |
0.8 + 0.02 | 0.82 | 0.9375 | 292.25 | 295.533333 | 243 | |
A rectangle has one side of length 5 and the other side less than 4. When the rectangle is folded so that two opposite corners coincide, the length of the crease is \(\sqrt{6}\). Calculate the length of the other side. | \sqrt{5} | 0.1875 | 7,389.3125 | 3,911 | 8,192 | |
In $\triangle ABC$, points $E$ and $F$ are on $AB$ and $BC$, respectively, such that $AE = BF$ and $BE = CF$. If $\angle BAC = 70^{\circ}$, what is the measure of $\angle ABC$? | 40^{\circ} | Since $AE = BF$ and $BE = CF$, then $AB = AE + BE = BF + CF = BC$. Therefore, $\triangle ABC$ is isosceles with $\angle BAC = \angle BCA = 70^{\circ}$. Since the sum of the angles in $\triangle ABC$ is $180^{\circ}$, then $\angle ABC = 180^{\circ} - \angle BAC - \angle BCA = 180^{\circ} - 70^{\circ} - 70^{\circ} = 40^{... | 0.8125 | 4,960.1875 | 4,214.384615 | 8,192 |
A rectangle in the coordinate plane has vertices at $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. Compute the radius $d$ to the nearest tenth such that the probability the point is within $d$ units from any lattice point is $\tfrac{1}{4}$. | 0.3 | 0.5 | 7,693.125 | 7,194.25 | 8,192 | |
Find the remainder when $x^3 - 3x + 5$ is divided by $x + 2.$ | 3 | 1 | 1,823.3125 | 1,823.3125 | -1 | |
Let \( f(x) = \frac{x + a}{x^2 + \frac{1}{2}} \), where \( x \) is a real number and the maximum value of \( f(x) \) is \( \frac{1}{2} \) and the minimum value of \( f(x) \) is \( -1 \). If \( t = f(0) \), find the value of \( t \). | -\frac{1}{2} | 0.875 | 6,265.875 | 5,990.714286 | 8,192 | |
Cynthia and Lynnelle are collaborating on a problem set. Over a $24$ -hour period, Cynthia and Lynnelle each independently pick a random, contiguous $6$ -hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during completely disjoint intervals of time. | 4/9 | 0.25 | 7,771.625 | 6,994.25 | 8,030.75 | |
There are 6 married couples at a party. At the start of the party, every person shakes hands once with every other person except his or her spouse. How many handshakes are there? | 60 | 0.9375 | 3,569.3125 | 3,261.133333 | 8,192 | |
Consider the sequence of six real numbers 60, 10, 100, 150, 30, and $x$ . The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of $x$ ? (The median of a sequence of six real numbers is the average of the two middle numbers after all the n... | 135 | 0.9375 | 4,421.9375 | 4,445.733333 | 4,065 | |
Three natural numbers are written on the board: two ten-digit numbers \( a \) and \( b \), and their sum \( a + b \). What is the maximum number of odd digits that could be written on the board? | 30 | 0 | 8,089.375 | -1 | 8,089.375 | |
Consider the function $g(x)$ satisfying
\[ g(xy) = 2g(x)g(y) \]
for all real numbers $x$ and $y$ and $g(0) = 2.$ Find $g(10)$. | \frac{1}{2} | 0.25 | 7,439.6875 | 5,182.75 | 8,192 | |
Given the function $f(x) = x^3 - 6x + 5, x \in \mathbb{R}$.
(1) Find the equation of the tangent line to the function $f(x)$ at $x = 1$;
(2) Find the extreme values of $f(x)$ in the interval $[-2, 2]$. | 5 - 4\sqrt{2} | 0.75 | 3,540.125 | 3,360.833333 | 4,078 | |
Multiply $2$ by $54$. For each proper divisor of $1,000,000$, take its logarithm base $10$. Sum these logarithms to get $S$, and find the integer closest to $S$. | 141 | 0.875 | 4,820.5625 | 4,338.928571 | 8,192 | |
A quadrilateral $ABCD$ has a right angle at $\angle ABC$ and satisfies $AB = 12$ , $BC = 9$ , $CD = 20$ , and $DA = 25$ . Determine $BD^2$ .
.
| 769 | 0.125 | 7,805 | 5,774 | 8,095.142857 | |
Find the sum of the digits in the answer to
$\underbrace{9999\cdots 99}_{94\text{ nines}} \times \underbrace{4444\cdots 44}_{94\text{ fours}}$
where a string of $94$ nines is multiplied by a string of $94$ fours. | 846 | 1. **Identify the pattern**: We start by observing the multiplication of smaller strings of nines and fours:
- $9 \times 4 = 36$, and the sum of the digits is $3 + 6 = 9$.
- $99 \times 44 = 4356$, and the sum of the digits is $4 + 3 + 5 + 6 = 18$.
2. **Generalize the pattern**: We notice that the sum of the digi... | 0.6875 | 6,641.625 | 5,936.909091 | 8,192 |
Convert the binary number $110101_{(2)}$ to decimal. | 53 | 1 | 2,956 | 2,956 | -1 | |
Define the sequence \(\{a_n\}\) where \(a_n = n^3 + 4\) for \(n \in \mathbf{N}_+\). Let \(d_n = \gcd(a_n, a_{n+1})\), which is the greatest common divisor of \(a_n\) and \(a_{n+1}\). Find the maximum value of \(d_n\). | 433 | 0.4375 | 7,197.125 | 5,918 | 8,192 | |
Three students, A, B, and C, are playing badminton with the following rules:<br/>The player who loses two games in a row will be eliminated. Before the game, two players are randomly selected to play against each other, while the third player has a bye. The winner of each game will play against the player with the bye ... | \frac{7}{16} | 0 | 8,192 | -1 | 8,192 | |
Ana's monthly salary was $2000$ in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was | 1920 | 1. **Calculate the salary after the raise in June:**
Ana's initial salary in May is $2000. A 20\% raise means her salary is increased by $2000 \times 20\% = $400. Therefore, her new salary in June becomes:
\[
2000 + 400 = 2400
\]
Alternatively, this can be calculated directly by multiplying her original ... | 1 | 1,762.9375 | 1,762.9375 | -1 |
The total in-store price for a blender is $\textdollar 129.95$. A television commercial advertises the same blender for four easy payments of $\textdollar 29.99$ and a one-time shipping and handling charge of $\textdollar 14.95$. Calculate the number of cents saved by purchasing the blender through the television adver... | 496 | 1 | 2,874.125 | 2,874.125 | -1 | |
Let $ y_0$ be chosen randomly from $ \{0, 50\}$ , let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$ , let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$ , and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$ . (In each choice, the possible outcomes are equally likely to occur.) Let $ P... | 107 | 0.5625 | 6,123 | 5,175 | 7,341.857143 | |
Given $$|\vec{a}|=3, |\vec{b}|=2$$. If $$\vec{a} \cdot \vec{b} = -3$$, then the angle between $$\vec{a}$$ and $$\vec{b}$$ is \_\_\_\_\_\_. | \frac{2}{3}\pi | 0 | 1,235.4375 | -1 | 1,235.4375 | |
A facility has 7 consecutive parking spaces, and there are 3 different models of cars to be parked. If it is required that among the remaining 4 parking spaces, exactly 3 are consecutive, then the number of different parking methods is \_\_\_\_\_\_. | 72 | 0 | 8,086.5625 | -1 | 8,086.5625 | |
On the extension of side $AD$ of rhombus $ABCD$, point $K$ is taken beyond point $D$. The lines $AC$ and $BK$ intersect at point $Q$. It is known that $AK=14$ and that points $A$, $B$, and $Q$ lie on a circle with a radius of 6, the center of which belongs to segment $AA$. Find $BK$. | 20 | 0 | 8,074.3125 | -1 | 8,074.3125 | |
Let $L(m)$ be the $x$ coordinate of the left end point of the intersection of the graphs of $y=x^2-6$ and $y=m$, where $-6<m<6$. Let $r=[L(-m)-L(m)]/m$. Then, as $m$ is made arbitrarily close to zero, the value of $r$ is: | \frac{1}{\sqrt{6}} | 1. **Identify the intersection points**: The intersection points of the graphs of $y = x^2 - 6$ and $y = m$ are given by solving the equation:
\[ x^2 - 6 = m \]
\[ x^2 = m + 6 \]
\[ x = \pm \sqrt{m + 6} \]
Thus, the $x$-coordinates of the intersection points are $\pm \sqrt{m + 6}$.
2. **Determine $L(m)$ an... | 0 | 4,134.4375 | -1 | 4,134.4375 |
In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline... | 21 | 0.875 | 5,368.9375 | 5,444.071429 | 4,843 | |
The cost of 60 copies of the first volume and 75 copies of the second volume is 2700 rubles. In reality, the total payment for all these books was only 2370 rubles because a discount was applied: 15% off the first volume and 10% off the second volume. Find the original price of these books. | 20 | 0.5625 | 4,627.625 | 3,468.111111 | 6,118.428571 | |
How many distinct five-digit positive integers are there such that the product of their digits equals 16? | 15 | 0 | 8,125.6875 | -1 | 8,125.6875 | |
Let the hyperbola $C:\frac{x^2}{a^2}-y^2=1\;(a>0)$ intersect the line $l:x+y=1$ at two distinct points $A$ and $B$.
$(1)$ Find the range of real numbers for $a$.
$(2)$ If the intersection point of the line $l$ and the $y$-axis is $P$, and $\overrightarrow{PA}=\frac{5}{12}\overrightarrow{PB}$, find the value of the ... | a = \frac{17}{13} | 0.625 | 6,711.0625 | 5,822.5 | 8,192 | |
Let $P$ be an interior point of triangle $ABC$ . Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right) $$ taking into consideration all possible choices of triangle $ABC$ and o... | \frac{2}{\sqrt{3}} | 0 | 8,139.5625 | -1 | 8,139.5625 | |
Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $\frac {m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find ... | 515 | This can be solved quickly and easily with generating functions.
Let $x^n$ represent flipping $n$ heads.
The generating functions for these coins are $(1+x)$,$(1+x)$,and $(3+4x)$ in order.
The product is $3+10x+11x^2+4x^3$. ($ax^n$ means there are $a$ ways to get $n$ heads, eg there are $10$ ways to get $1$ head, an... | 0.6875 | 6,299.625 | 5,761.363636 | 7,483.8 |
In a square $ABCD$ with side length $4$, find the probability that $\angle AMB$ is an acute angle. | 1-\dfrac{\pi}{8} | 0.625 | 5,897.1875 | 5,102.1 | 7,222.333333 | |
A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(3,0)$, $(3,2)$, and $(0,2)$. What is the probability that $x < y$? | \dfrac{1}{3} | 0.9375 | 4,293.4375 | 4,033.533333 | 8,192 | |
Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either ascending or descending order. | 74 | 0 | 8,192 | -1 | 8,192 | |
To the eight-digit number 20222023, append one digit to the left and one digit to the right so that the resulting ten-digit number is divisible by 72. Determine all possible solutions. | 3202220232 | 0.5 | 6,479.6875 | 5,728.125 | 7,231.25 | |
What is the probability, expressed as a decimal, of drawing one marble which is either green or white from a bag containing 4 green, 3 white, and 8 black marbles? | 0.4667 | 0.75 | 1,938.5 | 783.833333 | 5,402.5 | |
The value of $ 21!$ is $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$ , where $ a$ , $ b$ , and $ c$ are digits. What is the value of $ 100a \plus{} 10b \plus{} c$ ? | 709 | 0.375 | 7,528.8125 | 6,511 | 8,139.5 | |
In a certain school, there are $3$ times as many boys as girls and $9$ times as many girls as teachers. Using the letters $b, g, t$ to represent the number of boys, girls, and teachers, respectively, then the total number of boys, girls, and teachers can be represented by the expression | \frac{37b}{27} | 1. **Identify relationships**:
Given that there are $3$ times as many boys as girls, we can write:
\[
b = 3g
\]
Also, there are $9$ times as many girls as teachers, so:
\[
g = 9t \quad \text{or equivalently} \quad t = \frac{g}{9}
\]
2. **Express total population in terms of $g$**:
The total... | 0 | 3,171.5625 | -1 | 3,171.5625 |
If $x=3$, $y=2x$, and $z=3y$, what is the value of $z$? | 18 | Since $x=3$ and $y=2x$, then $y=2 \cdot 3=6$. Since $y=6$ and $z=3y$, then $z=3 \cdot 6=18$. | 1 | 250.75 | 250.75 | -1 |
Given two geometric sequences $\{a_n\}$ and $\{b_n\}$, satisfying $a_1=a$ ($a>0$), $b_1-a_1=1$, $b_2-a_2=2$, and $b_3-a_3=3$.
(1) If $a=1$, find the general formula for the sequence $\{a_n\}$.
(2) If the sequence $\{a_n\}$ is unique, find the value of $a$. | \frac{1}{3} | 0 | 8,192 | -1 | 8,192 | |
Given that angle $A$ is an internal angle of a triangle and $\cos A= \frac{3}{5}$, find $\tan A=$ \_\_\_\_\_\_ and $\tan (A+ \frac{\pi}{4})=$ \_\_\_\_\_\_. | -7 | 1 | 2,248.625 | 2,248.625 | -1 | |
Let $f(n)$ be the largest prime factor of $n^{2}+1$. Compute the least positive integer $n$ such that $f(f(n))=n$. | 89 | Suppose $f(f(n))=n$, and let $m=f(n)$. Note that we have $mn \mid m^{2}+n^{2}+1$. First we find all pairs of positive integers that satisfy this condition, using Vieta root jumping. Suppose $m^{2}+n^{2}+1=kmn$, for some positive integer $k$. Considering this as a quadratic in $m$, let the other root (besides $m$) be $m... | 0 | 8,082.4375 | -1 | 8,082.4375 |
Given two parallel lines \\(l_{1}\\) and \\(l_{2}\\) passing through points \\(P_{1}(1,0)\\) and \\(P_{2}(0,5)\\) respectively, and the distance between \\(l_{1}\\) and \\(l_{2}\\) is \\(5\\), then the slope of line \\(l_{1}\\) is \_\_\_\_\_\_. | \dfrac {5}{12} | 0.3125 | 8,055.25 | 7,754.4 | 8,192 | |
Calculate the definite integral:
$$
\int_{0}^{\pi / 4} \frac{5 \operatorname{tg} x+2}{2 \sin 2 x+5} d x
$$ | \frac{1}{2} \ln \left(\frac{14}{5}\right) | 0.125 | 5,916.1875 | 4,992.5 | 6,048.142857 | |
An ordered pair $(a, c)$ of integers, each of which has an absolute value less than or equal to 6, is chosen at random. What is the probability that the equation $ax^2 - 3ax + c = 0$ will not have distinct real roots both greater than 2?
A) $\frac{157}{169}$ B) $\frac{167}{169}$ C) $\frac{147}{169}$ D) $\frac{160}{1... | \frac{167}{169} | 0 | 8,192 | -1 | 8,192 |
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