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 |
|---|---|---|---|---|---|---|
In right triangle $ABC$ the hypotenuse $\overline{AB}=5$ and leg $\overline{AC}=3$. The bisector of angle $A$ meets the opposite side in $A_1$. A second right triangle $PQR$ is then constructed with hypotenuse $\overline{PQ}=A_1B$ and leg $\overline{PR}=A_1C$. If the bisector of angle $P$ meets the opposite side in $P_... | \frac{3\sqrt{5}}{4} | To solve this problem, we need to find the lengths of $A_1B$ and $A_1C$ first, and then use these to find $PP_1$.
1. **Finding $BC$ in $\triangle ABC$:**
Since $\triangle ABC$ is a right triangle with hypotenuse $AB = 5$ and leg $AC = 3$, we can find the other leg $BC$ using the Pythagorean theorem:
\[
BC = \... | 0 | 5,356.125 | -1 | 5,356.125 |
Find the integer \(n\), such that \(-180 < n < 180\), for which \(\tan n^\circ = \tan 276^\circ.\) | 96 | 0.125 | 7,009.6875 | 7,548 | 6,932.785714 | |
Assume that the scores $X$ of 400,000 students in a math mock exam in Yunnan Province approximately follow a normal distribution $N(98,100)$. It is known that a student's score ranks among the top 9100 in the province. Then, the student's math score will not be less than ______ points. (Reference data: $P(\mu -\sigma\ ... | 118 | 0.5 | 5,459 | 4,524.875 | 6,393.125 | |
Given \( a, b, c \geq 0 \), \( t \geq 1 \), and satisfying
\[
\begin{cases}
a + b + c = \frac{1}{2}, \\
\sqrt{a + \frac{1}{2}(b - c)^{2}} + \sqrt{b} + \sqrt{c} = \frac{\sqrt{6t}}{2},
\end{cases}
\]
find \( a^{2t} + b^{2t} + c^{2t} \). | \frac{1}{12} | 0.125 | 7,810.0625 | 5,136.5 | 8,192 | |
Simplify the expression $\dfrac{\sin(2\pi-\alpha)\cos(\pi+\alpha)\cos(\frac{\pi}{2}+\alpha)\cos(\frac{11\pi}{2}-\alpha)}{\cos(\pi-\alpha)\sin(3\pi-\alpha)\sin(-\pi-\alpha)\sin(\frac{9\pi}{2}+\alpha)\tan(\pi+\alpha)}$. | -1 | 0.375 | 6,521.8125 | 5,138.166667 | 7,352 | |
Let $\overrightarrow{{e_1}}$ and $\overrightarrow{{e_2}}$ be non-collinear vectors. If $k\overrightarrow{{e_1}}+4\overrightarrow{{e_2}}$ and $\overrightarrow{{e_1}}+k\overrightarrow{{e_2}}$ are collinear and have opposite directions, then the value of $k$ is ____. | -2 | 1 | 1,881.5 | 1,881.5 | -1 | |
A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$ , $BC = 12$ , $CA = 13$ . The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively pri... | 61 | 0.3125 | 7,199.6875 | 5,425.6 | 8,006.090909 | |
In the sequence ${a_{n}}$, $a_{n+1}=\begin{cases} 2a_{n}\left(a_{n} < \frac{1}{2}\right) \\ 2a_{n}-1\left(a_{n}\geqslant \frac{1}{2}\right) \end{cases}$, if $a_{1}=\frac{4}{5}$, then the value of $a_{20}$ is $\_\_\_\_\_\_$. | \frac{2}{5} | 1 | 3,371.125 | 3,371.125 | -1 | |
Calculate the sum of the square of the binomial coefficients: $C_2^2+C_3^2+C_4^2+…+C_{11}^2$. | 220 | 0.3125 | 6,020.25 | 3,755 | 7,049.909091 | |
Let $T$ denote the value of the sum\[\sum_{n=0}^{432} (-1)^{n} {1500 \choose 3n}\]Determine the remainder obtained when $T$ is divided by $100$. | 66 | 0.125 | 7,485.1875 | 7,147 | 7,533.5 | |
Given that $a-b=3$, find the value of $1+2b-(a+b)$.
Given that $2^x=3$, find the value of $2^{2x-3}$. | \frac{9}{8} | 0.8125 | 1,030.6875 | 1,119.384615 | 646.333333 | |
In triangle $XYZ$, $XY=12$, $YZ=16$, and $XZ=20$. Point $M$ is on $\overline{XY}$, $N$ is on $\overline{YZ}$, and $O$ is on $\overline{XZ}$. Let $XM = p \cdot XY$, $YN = q \cdot YZ$, and $ZO = r \cdot XZ$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=3/4$ and $p^2+q^2+r^2=1/2$. The ratio of the area of trian... | 41 | 0 | 8,192 | -1 | 8,192 | |
Given a circle C with its center C on the positive x-axis and a radius of 5, the chord intercepted by the line $x-y+3=0$ has a length of $2\sqrt{17}$.
(1) Find the equation of circle C;
(2) Suppose the line $ax-y+5=0$ intersects circle C at points A and B, find the range of the real number $a$;
(3) Under the condition ... | \frac{3}{4} | 0.4375 | 7,156.4375 | 6,210.142857 | 7,892.444444 | |
Shift the graph of the function $f(x) = 2\sin(2x + \frac{\pi}{4})$ to the right by $\varphi (\varphi > 0)$ units, then shrink the x-coordinate of each point on the graph to half of its original value (the y-coordinate remains unchanged), and make the resulting graph symmetric about the line $x = \frac{\pi}{4}$. Determi... | \frac{3}{8}\pi | 0 | 6,855.0625 | -1 | 6,855.0625 | |
Given the equation \(x^2 + y^2 = 2(|x| + |y|)\), calculate the area of the region enclosed by its graph. | 2\pi | 0 | 8,192 | -1 | 8,192 | |
What is the area of the circle defined by $x^2-6x +y^2-14y +33=0$ that lies beneath the line $y=7$? | \frac{25\pi}{2} | 0.5625 | 3,120.125 | 2,212.222222 | 4,287.428571 | |
Evaluate: $5-7\left(8-3^2\right)4.$ | 33 | 1 | 2,786.4375 | 2,786.4375 | -1 | |
Tom adds up all the even integers from 2 to 600, inclusive. Lara adds up all the integers from 1 to 200, inclusive. What is Tom's sum divided by Lara's sum? | 4.5 | 0 | 6,712.8125 | -1 | 6,712.8125 | |
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and it is known that $a_2=1$, $S_4=8$. Find $a_5$ and $S_{10}$. | 80 | 1 | 2,091.3125 | 2,091.3125 | -1 | |
Given a set of sample data with $8$ numbers, the average is $8$, and the variance is $12$. Two unknown numbers are added to this set of sample data to form a new set of sample data. It is known that the average of the new sample data is $9$. Find the minimum value of the variance of the new sample data. | 13.6 | 0.0625 | 7,572.375 | 7,724 | 7,562.266667 | |
On a line passing through the center $O$ of a circle with radius 12, points $A$ and $B$ are chosen such that $OA=15$, $AB=5$, and $A$ lies between $O$ and $B$. Tangents are drawn from points $A$ and $B$ to the circle, with the points of tangency lying on the same side of the line $OB$. Find the area of triangle $ABC$, ... | 150/7 | 0.5625 | 6,458.875 | 5,356.666667 | 7,876 | |
Factor the following expression: $37a^2 +111a$. | 37a(a+3) | 1 | 1,249.4375 | 1,249.4375 | -1 | |
What is the smallest natural number that can be added to 40,317 to make it a palindrome? | 87 | 0.5625 | 6,464.375 | 7,009.222222 | 5,763.857143 | |
$3^3 + 3^3 + 3^3 =$ | $3^4$ | 1. **Identify the expression and simplify**: The given expression is $3^3 + 3^3 + 3^3$.
2. **Factor out the common term**: Notice that each term in the sum is $3^3$. We can factor out $3^3$ as follows:
\[
3^3 + 3^3 + 3^3 = 3 \times 3^3
\]
3. **Apply the exponent rule**: Recall the exponent rule $a^m \times a... | 0 | 1,408.6875 | -1 | 1,408.6875 |
Let the bisectors of the exterior angles at $B$ and $C$ of triangle $ABC$ meet at $D$. Then, if all measurements are in degrees, angle $BDC$ equals: | \frac{1}{2}(180-A) | 1. **Identify the exterior angle bisectors**: The bisectors of the exterior angles at $B$ and $C$ of triangle $ABC$ meet at point $D$. The exterior angle at $B$ is $180^\circ - B$ and its bisector divides it into two equal parts, each being $90^\circ - \frac{B}{2}$. Similarly, the bisector of the exterior angle at $C$ ... | 0 | 7,880.125 | -1 | 7,880.125 |
Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die? | \frac{5}{24} | 1. **Total Outcomes**: When three six-sided dice are rolled, each die has 6 possible outcomes. Therefore, the total number of outcomes when rolling three dice is:
\[
6 \times 6 \times 6 = 216
\]
2. **Favorable Outcomes**: We need to count the number of ways two dice can sum to the value of the third die. We c... | 0.4375 | 7,397.8125 | 6,376.714286 | 8,192 |
Find the number of degrees in the measure of angle $x$.
[asy]
import markers;
size (5cm,5cm);
pair A,B,C,D,F,H;
A=(0,0);
B=(5,0);
C=(9,0);
D=(3.8,7);
F=(2.3,7.2);
H=(5.3,7.2);
draw((4.2,6.1){up}..{right}(5.3,7.2));
draw((3.6,6.1){up}..{left}(2.3,7.2));
draw (A--B--C--D--A);
draw (B--D);
markangle(n=1,radius=8,C,B... | 82^\circ | 0 | 8,110.0625 | -1 | 8,110.0625 | |
For integers $a, b, c, d$, let $f(a, b, c, d)$ denote the number of ordered pairs of integers $(x, y) \in \{1,2,3,4,5\}^{2}$ such that $a x+b y$ and $c x+d y$ are both divisible by 5. Find the sum of all possible values of $f(a, b, c, d)$. | 31 | Standard linear algebra over the field $\mathbb{F}_{5}$ (the integers modulo 5). The dimension of the solution set is at least 0 and at most 2, and any intermediate value can also be attained. So the answer is $1+5+5^{2}=31$. This also can be easily reformulated in more concrete equation/congruence-solving terms, espec... | 0.25 | 8,046.5625 | 7,610.25 | 8,192 |
In triangle \(ABC\), points \(P\) and \(Q\) are taken on the base \(AC\) such that \(AP < AQ\). The lines \(BP\) and \(BQ\) divide the median \(AM\) into three equal parts. It is known that \(PQ = 3\).
Find \(AC\). | 10 | 0.625 | 4,154.8125 | 3,892.6 | 4,591.833333 | |
There are $2022$ equally spaced points on a circular track $\gamma$ of circumference $2022$. The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{... | 2042222 |
There are \(2022\) equally spaced points on a circular track \(\gamma\) of circumference \(2022\). The points are labeled \(A_1, A_2, \ldots, A_{2022}\) in some order, each label used once. Initially, Bunbun the Bunny begins at \(A_1\). She hops along \(\gamma\) from \(A_1\) to \(A_2\), then from \(A_2\) to \(A_3\), u... | 0 | 8,094.8125 | -1 | 8,094.8125 |
Evaluate $\frac{3+x(3+x)-3^2}{x-3+x^2}$ for $x=-2$. | 8 | 1 | 2,454.1875 | 2,454.1875 | -1 | |
In the eight-term sequence $A,B,C,D,E,F,G,H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. What is $A+H$? | 25 | 1 | 2,562.5625 | 2,562.5625 | -1 | |
Arrange positive integers that are neither perfect squares nor perfect cubes (excluding 0) in ascending order as 2, 3, 5, 6, 7, 10, ..., and determine the 1000th number in this sequence. | 1039 | 0.5 | 5,889.625 | 4,801.5 | 6,977.75 | |
An eight-digit integer is formed by repeating a positive four-digit integer. For example, 25,632,563 or 60,786,078 are integers of this form. What is the greatest common divisor of all eight-digit integers of this form? | 10001 | 0.3125 | 7,430 | 5,753.6 | 8,192 | |
Solve in prime numbers the equation $x^y - y^x = xy^2 - 19$. | (2, 3)(2, 7) |
To find the solutions of the equation \(x^y - y^x = xy^2 - 19\) in prime numbers, we will begin by analyzing possible small prime candidates, as powers of small primes often have manageable forms that can be verified manually.
**Step 1: Try small primes for \(x\) and \(y\) and verify conditions.**
Since \(x\) and \(... | 0 | 8,074.75 | -1 | 8,074.75 |
Calculate the corrected average score and variance of a class of 50 students, given that the original average was 70 and the original variance was 102, after two students' scores were corrected from 50 to 80 and from 90 to 60. | 90 | 0.875 | 3,536.625 | 2,904.928571 | 7,958.5 | |
Given the polar equation of curve $C_1$ is $\rho^2=\frac {2}{3+\cos2\theta}$, establish a rectangular coordinate system with the pole O as the origin and the polar axis as the positive direction of the x-axis. After stretching all the x-coordinates of points on curve $C_1$ to twice their original values and shortening ... | \frac{13\sqrt{2}}{4} | 0 | 7,067.375 | -1 | 7,067.375 | |
For how many positive integers $k$ do the lines with equations $9x+4y=600$ and $kx-4y=24$ intersect at a point whose coordinates are positive integers? | 7 | Suppose that $k$ is a fixed, but unknown, positive integer. Suppose also that the lines with equations $9x+4y=600$ and $kx-4y=24$ intersect at the point with positive integer coordinates $(x, y)$. Since $9x+4y=600$ and $kx-4y=24$, adding these equations, we get $9x+kx=624$ and so $(9+k)x=624$. Since $x$ and $y$ are to ... | 0.4375 | 6,672.3125 | 4,718.428571 | 8,192 |
An isosceles triangle has side lengths 8 cm, 8 cm and 10 cm. The longest side of a similar triangle is 25 cm. What is the perimeter of the larger triangle, in centimeters? | 65 | 1 | 1,407.3125 | 1,407.3125 | -1 | |
In triangle $XYZ$, $\angle Z=90^\circ$, $XZ=3$ and $YZ=4$. Points $W$ and $V$ are on $\overline{XY}$ and $\overline{YZ}$, respectively, and $\angle WVZ=90^\circ$. If $WV=2$, then what is the length of $WY$? | \frac{10}{3} | 1 | 3,736.125 | 3,736.125 | -1 | |
At breakfast, lunch, and dinner, Joe randomly chooses with equal probabilities either an apple, an orange, or a banana to eat. On a given day, what is the probability that Joe will eat at least two different kinds of fruit? | \frac{8}{9} | 0.875 | 3,894.5 | 3,280.571429 | 8,192 | |
Aerith timed herself solving a contest and noted the time both as days:hours:minutes:seconds and in seconds. For example, if she spent 1,000,000 seconds, she recorded it as 11:13:46:40 and 1,000,000 seconds. Bob subtracts these numbers, ignoring punctuation. In this case, he computes:
\[ 11134640 - 1000000 = 10134640 ... | 40 | 0.125 | 7,551.75 | 5,965 | 7,778.428571 | |
Given the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) with asymptotic equations $y = \pm \sqrt{3}x$, and $O$ as the origin, the point $M(\sqrt{5}, \sqrt{3})$ lies on the hyperbola.
$(1)$ Find the equation of the hyperbola $C$.
$(2)$ If a line $l$ intersects the hyperbola at points $P$ and ... | 24 | 0.0625 | 7,675.4375 | 7,749 | 7,670.533333 | |
Given positive real numbers \(a, b, c\) satisfy \(2(a+b)=ab\) and \(a+b+c=abc\), find the maximum value of \(c\). | \frac{8}{15} | 0.9375 | 4,289.5625 | 4,029.4 | 8,192 | |
Expanding $(1+0.2)^{1000}_{}$ by the binomial theorem and doing no further manipulation gives
${1000 \choose 0}(0.2)^0+{1000 \choose 1}(0.2)^1+{1000 \choose 2}(0.2)^2+\cdots+{1000 \choose 1000}(0.2)^{1000}$ $= A_0 + A_1 + A_2 + \cdots + A_{1000},$ where $A_k = {1000 \choose k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$. For w... | 166 | We know that $A_k$ will increase as $k$ increases until certain $k=m$, where $A_0 < A_1 < A_2 < \dots < A_{m-2} < A_{m-1} < A_m$ and
$A_m > A_{m+1} > A_{m+2} > \dots > A_{1000}.$
Next, to change $A_{k-1}$ to $A_k$, we multiply $A_{k-1}$ by $\frac{1000-k+1}{5k}$. It follows that the numerator must be greater than the
... | 0.4375 | 6,647.6875 | 5,116.285714 | 7,838.777778 |
Find the matrix that corresponds to projecting onto the vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}.$ | \begin{pmatrix} 4/13 & -6/13 \\ -6/13 & 9/13 \end{pmatrix} | 0 | 4,069.1875 | -1 | 4,069.1875 | |
How many ways are there to color every integer either red or blue such that \(n\) and \(n+7\) are the same color for all integers \(n\), and there does not exist an integer \(k\) such that \(k, k+1\), and \(2k\) are all the same color? | 6 | It suffices to color the integers from 0 through 6 and do all arithmetic mod 7. WLOG, say that 0 is red (we'll multiply by 2 in the end). Then 1 must be blue because \((0,0,1)\) can't be monochromatic. 2 must be red because \((1,2,2)\) can't be monochromatic. Then we have two cases for what 3 is: Case 1: 3 is red. Then... | 0 | 8,166.6875 | -1 | 8,166.6875 |
Eight hockey teams are competing against each other in a single round to advance to the semifinals. What is the minimum number of points that guarantees a team advances to the semifinals? | 11 | 0 | 7,370.625 | -1 | 7,370.625 | |
Remove all perfect squares from the sequence of positive integers \(1, 2, 3, \cdots\) to get a new sequence, and calculate the 2003rd term of this new sequence. | 2047 | 0 | 5,672.0625 | -1 | 5,672.0625 | |
Let $ABCD$ be a trapezium with $AB// DC, AB = b, AD = a ,a<b$ and $O$ the intersection point of the diagonals. Let $S$ be the area of the trapezium $ABCD$ . Suppose the area of $\vartriangle DOC$ is $2S/9$ . Find the value of $a/b$ . | \frac{2 + 3\sqrt{2}}{7} | 0 | 7,183.875 | -1 | 7,183.875 | |
A regular dodecahedron is projected orthogonally onto a plane, and its image is an $n$-sided polygon. What is the smallest possible value of $n$ ? | 6 | We can achieve 6 by projecting onto a plane perpendicular to an edge of the dodecaheron. Indeed, if we imagine viewing the dodecahedron in such a direction, then 4 of the faces are projected to line segments (namely, the two faces adjacent to the edge and the two opposite faces), and of the remaining 8 faces, 4 appear ... | 0.125 | 7,007.5 | 5,004 | 7,293.714286 |
What percent of the positive integers less than or equal to $100$ have no remainders when divided by $5?$ | 20 | 1 | 1,530.6875 | 1,530.6875 | -1 | |
Given the parabola $y^{2}=4x$, and the line $l$: $y=- \frac {1}{2}x+b$ intersects the parabola at points $A$ and $B$.
(I) If the $x$-axis is tangent to the circle with $AB$ as its diameter, find the equation of the circle;
(II) If the line $l$ intersects the negative semi-axis of $y$, find the maximum area of $\triangl... | \frac {32 \sqrt {3}}{9} | 0 | 7,046.8125 | -1 | 7,046.8125 | |
Calculate the monotonic intervals of $F(x)=\int_{0}^{x}{(t^{2}+2t-8)dt}$ for $x > 0$.
(1) Determine the monotonic intervals of $F(x)$.
(2) Find the maximum and minimum values of the function $F(x)$ on the interval $[1,2]$. | -\frac{28}{3} | 1 | 3,248 | 3,248 | -1 | |
If \( a \) and \( b \) are positive numbers such that \( a^b = b^a \) and \( b = 4a \), then find the value of \( a \). | \sqrt[3]{4} | 1 | 3,237.25 | 3,237.25 | -1 | |
Let $d_1 = a^2 + 3^a + a \cdot 3^{(a+1)/2}$ and $d_2 = a^2 + 3^a - a \cdot 3^{(a+1)/2}$. If $1 \le a \le 300$, how many integral values of $a$ are there such that $d_1 \cdot d_2$ is a multiple of $7$? | 43 | 0 | 8,192 | -1 | 8,192 | |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $90^{\circ}$, if vector $\overrightarrow{c}$ satisfies $|\overrightarrow{c}- \overrightarrow{a} \overrightarrow{b}|=2$, calculate the maximum value of $|\overrightarrow{c}|$. | 2 + \sqrt{2} | 0 | 4,375.125 | -1 | 4,375.125 | |
Given that the domains of functions f(x) and g(x) are both R, and f(x) + g(2-x) = 5, g(x) - f(x-4) = 7. If the graph of y=g(x) is symmetric about the line x=2, g(2) = 4, find the sum of the values of f(k) for k from 1 to 22. | -24 | 0.375 | 7,050 | 6,337 | 7,477.8 | |
Trapezoid $ABCD$ has base $AB = 20$ units and base $CD = 30$ units. Diagonals $AC$ and $BD$ intersect at $X$. If the area of trapezoid $ABCD$ is $300$ square units, what is the area of triangle $BXC$? | 72 | 0.8125 | 5,945 | 5,604.769231 | 7,419.333333 | |
Given $f(\alpha)= \frac{\sin (\alpha-3\pi)\cos (2\pi-\alpha)\cdot\sin (-\alpha+ \frac{3}{2}\pi)}{\cos (-\pi-\alpha)\sin (-\pi-\alpha)}$.
$(1)$ Simplify $f(\alpha)$.
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos \left(\alpha- \frac{3}{2}\pi\right)= \frac{1}{5}$, find the value of $f(\alpha)$. | \frac{2\sqrt{6}}{5} | 0 | 4,291.8125 | -1 | 4,291.8125 | |
On the Cartesian plane, find the number of integer coordinate points (points where both x and y are integers) that satisfy the following system of inequalities:
\[
\begin{cases}
y \leq 3x, \\
y \geq \frac{1}{3}x, \\
x + y \leq 100.
\end{cases}
\] | 2551 | 0 | 8,069.8125 | -1 | 8,069.8125 | |
In a quadrilateral $WXYZ$, the angles satisfy $\angle W = 3\angle X = 4\angle Y = 6\angle Z$. Determine the exact degree measure of $\angle W$. | \frac{1440}{7} | 0.9375 | 2,673.5625 | 2,702 | 2,247 | |
An archipelago consists of \( N \geq 7 \) islands. Any two islands are connected by no more than one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are necessarily two that are connected by a bridge. What is the maximum value that \( N \) can take? | 36 | 0.5625 | 5,934.625 | 4,423.444444 | 7,877.571429 | |
For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $500$ that are neither $3$-nice nor $5$-nice. | 266 | 0.25 | 8,037.4375 | 7,727.25 | 8,140.833333 | |
Given that in a class test, $15\%$ of the students scored $60$ points, $50\%$ scored $75$ points, $20\%$ scored $85$ points, and the rest scored $95$ points, calculate the difference between the mean and median score of the students' scores on this test. | 2.75 | 0.8125 | 4,238.625 | 4,345.076923 | 3,777.333333 | |
In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units? | 2 | 1. **Identify the fixed elements and set up the problem**: Given that points $A$ and $B$ are $10$ units apart, we can place them at coordinates $(0,0)$ and $(10,0)$ respectively without loss of generality. This simplifies the problem to finding a point $C$ such that the perimeter of $\triangle ABC$ is $50$ units and th... | 0 | 7,164.5 | -1 | 7,164.5 |
Teacher Xixi and teacher Shanshan are teachers in the senior and junior classes of a kindergarten, respectively. Teacher Xixi prepared a large bag of apples to distribute to her students, giving exactly 3 apples to each child; teacher Shanshan prepared a large bag of oranges to distribute to her students, giving exactl... | 72 | 0.0625 | 6,425.375 | 5,447 | 6,490.6 | |
In the rectangular coordinate system xOy, there is a line l₁: x = 2, and a curve C: {x = 2cosϕ, y = 2 + 2sinϕ} (where ϕ is a parameter). With O as the pole and the non-negative half-axis of the x-axis as the polar axis, establish a polar coordinate system. The polar coordinates of point M are $(3, \frac {π}{6})$.
1. F... | \frac {3 \sqrt {3}}{2} | 0 | 5,124.1875 | -1 | 5,124.1875 | |
Find the number of different arrangements for a class to select 6 people to participate in two volunteer activities, with each activity accommodating no more than 4 people. | 50 | 0.4375 | 7,003.1875 | 5,474.714286 | 8,192 | |
If I have a $5\times5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and each row of the board contains no more than one pawn? | 14400 | 1 | 4,464.375 | 4,464.375 | -1 | |
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. Given that $(a+c)^2 = b^2 + 2\sqrt{3}ac\sin C$.
1. Find the measure of angle B.
2. If $b=8$, $a>c$, and the area of triangle ABC is $3\sqrt{3}$, find the value of $a$. | 5 + \sqrt{13} | 0.125 | 8,048.8125 | 7,046.5 | 8,192 | |
Let N = $69^5 + 5 \cdot 69^4 + 10 \cdot 69^3 + 10 \cdot 69^2 + 5 \cdot 69 + 1$. How many positive integers are factors of $N$? | 216 | 1. **Rewriting the Expression**: Let $a = 69$. Then, the expression for $N$ becomes:
\[
N = a^5 + 5a^4 + 10a^3 + 10a^2 + 5a + 1
\]
2. **Recognizing the Binomial Expansion**: The expression can be recognized as the expansion of $(a+1)^5$ using the binomial theorem:
\[
(a+1)^5 = \sum_{k=0}^{5} \binom{5}{k... | 0.8125 | 2,944.1875 | 1,733.153846 | 8,192 |
Let \(ABCD\) be a rectangle, and let \(E\) and \(F\) be points on segment \(AB\) such that \(AE = EF = FB\). If \(CE\) intersects the line \(AD\) at \(P\), and \(PF\) intersects \(BC\) at \(Q\), determine the ratio of \(BQ\) to \(CQ\). | 1/3 | 0.75 | 5,689.875 | 5,393.833333 | 6,578 | |
If: (1) \( a, b, c, d \) are all elements of \( \{1,2,3,4\} \); (2) \( a \neq b, b \neq c, c \neq d, d \neq a \); (3) \( a \) is the smallest value among \( a, b, c, d \),
then, how many different four-digit numbers \( \overline{abcd} \) can be formed? | 24 | 0.0625 | 7,890.875 | 5,737 | 8,034.466667 | |
Find the largest constant $C$ so that for all real numbers $x$, $y$, and $z$,
\[x^2 + y^2 + z^3 + 1 \ge C(x + y + z).\] | \sqrt{2} | 0 | 8,010.3125 | -1 | 8,010.3125 | |
Given that the cosine value of the vertex angle of an isosceles triangle equals $\dfrac{4}{5}$, calculate the sine value of the base angle of this triangle. | \dfrac{2\sqrt{3}}{5} | 0 | 4,761.375 | -1 | 4,761.375 | |
Let $S_{n}$ be the sum of the first $n$ terms of a geometric sequence $\{a_{n}\}$. If $S_{4}=-5$ and $S_{6}=21S_{2}$, calculate the value of $S_{8}$. | -85 | 0.75 | 5,447.125 | 4,939.166667 | 6,971 | |
A certain school club has 10 members, and two of them are put on duty each day from Monday to Friday. Given that members A and B must be scheduled on the same day, and members C and D cannot be scheduled together, the total number of different possible schedules is (▲). Choices:
A) 21600
B) 10800
C) 7200
D) 5400 | 5400 | 0 | 8,192 | -1 | 8,192 | |
Given \\((a+b-c)(a+b+c)=3ab\\) and \\(c=4\\), the maximum area of \\(\Delta ABC\\) is \_\_\_\_\_\_\_. | 4\sqrt{3} | 0.8125 | 6,134.75 | 5,660 | 8,192 | |
Let $A = \{1, 2, 3, 4, 5, 6, 7\}$, and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$. | 399 | Any such function can be constructed by distributing the elements of $A$ on three tiers.
The bottom tier contains the constant value, $c=f(f(x))$ for any $x$. (Obviously $f(c)=c$.)
The middle tier contains $k$ elements $x\ne c$ such that $f(x)=c$, where $1\le k\le 6$.
The top tier contains $6-k$ elements such that $... | 0 | 8,023.9375 | -1 | 8,023.9375 |
Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular. If $DP= 18$ and $EQ = 24$, then what is ${DF}$? | 8\sqrt{13} | 0.375 | 7,479.5625 | 6,348.5 | 8,158.2 | |
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
[asy]
size(100);
pair A,... | 12 | To solve this problem, we will use Burnside's Lemma, which states that the number of distinct colorings, up to symmetry, is the average number of colorings fixed by each group action. The group actions in this case are the symmetries of a hexagon, which include rotations and reflections.
#### Step 1: Identify the symm... | 0 | 8,192 | -1 | 8,192 |
The minimum positive period and maximum value of the function $f\left(x\right)=\sin \frac{x}{3}+\cos \frac{x}{3}$ are respectively $3\pi$ and $\sqrt{2}$. | \sqrt{2} | 0.625 | 6,269.25 | 5,115.6 | 8,192 | |
The product of two 2-digit numbers is $5488$. What is the smaller of the two numbers? | 56 | 1 | 4,027 | 4,027 | -1 | |
In triangle $ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given vectors $\overrightarrow{m} = (\cos A, \sin A)$ and $\overrightarrow{n} = (\cos B, -\sin B)$, and $|\overrightarrow{m} - \overrightarrow{n}| = 1$.
(1) Find the degree measure of angle $C$;
(2) If $c=3$, find t... | \frac{3\sqrt{3}}{4} | 0 | 5,156.3125 | -1 | 5,156.3125 | |
$\log p+\log q=\log(p+q)$ only if: | p=\frac{q}{q-1} | 1. **Start with the given equation and apply logarithmic properties:**
\[
\log p + \log q = \log(p+q)
\]
Using the property that $\log a + \log b = \log(ab)$, we can rewrite the left side:
\[
\log(pq) = \log(p+q)
\]
2. **Since the logarithms are equal, their arguments must be equal:**
\[
pq ... | 0 | 3,753.5 | -1 | 3,753.5 |
Given the word 'ARROW', find the probability that a random arrangement of its letters will have both R's next to each other. | \frac{2}{5} | 0.75 | 4,612.875 | 4,327.166667 | 5,470 | |
The product of the roots of the equation \((x-4)(x-2)+(x-2)(x-6)=0\) is | 10 | Since the two terms have a common factor, then we factor and obtain \((x-2)((x-4)+(x-6))=0\). This gives \((x-2)(2x-10)=0\). Therefore, \(x-2=0\) (which gives \(x=2\)) or \(2x-10=0\) (which gives \(x=5\)). Therefore, the two roots of the equation are \(x=2\) and \(x=5\). Their product is 10. | 1 | 2,389.1875 | 2,389.1875 | -1 |
For what value of $x$ does $3^{3x^{2} - 8x + 5} = 3^{3x^{2} + 5x - 6}$? | \frac{11}{13} | 1 | 2,466.125 | 2,466.125 | -1 | |
Given the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$, its foci are equal to the minor axis length of the ellipse $Ω:x^{2}+ \frac{y^{2}}{4}=1$, and the major axis lengths of C and Ω are equal.
(1) Find the equation of ellipse C;
(2) Let $F_1$, $F_2$ be the left and right foci of ellipse C, respective... | \sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
You flip a fair coin which results in heads ( $\text{H}$ ) or tails ( $\text{T}$ ) with equal probability. What is the probability that you see the consecutive sequence $\text{THH}$ before the sequence $\text{HHH}$ ? | \frac{7}{8} | 0 | 7,751.9375 | -1 | 7,751.9375 | |
In $\triangle ABC$, the internal angles $A$, $B$, and $C$ satisfy the equation $$2(\tan B + \tan C) = \frac{\tan B}{\cos C} + \frac{\tan C}{\cos B}$$. Find the minimum value of $\cos A$. | \frac{1}{2} | 0.375 | 7,449.8125 | 6,212.833333 | 8,192 | |
Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6)$, and the product of the radii is $68$. The x-axis and the line $y = mx$, where $m > 0$, are tangent to both circles. It is given that $m$ can be written in the form $a\sqrt {b}/c$, where $a$, $b$, and $c$ are positive integ... | 282 | Let the smaller angle between the $x$-axis and the line $y=mx$ be $\theta$. Note that the centers of the two circles lie on the angle bisector of the angle between the $x$-axis and the line $y=mx$. Also note that if $(x,y)$ is on said angle bisector, we have that $\frac{y}{x}=\tan{\frac{\theta}{2}}$. Let $\tan{\frac{\t... | 0.5 | 7,332.5 | 6,473 | 8,192 |
The four hydrogen atoms in the methane molecule $\mathrm{CH}_{4}$ are located at the vertices of a regular tetrahedron with edge length 1. The carbon atom $C$ is located at the center of the tetrahedron $C_{0}$. Let the four hydrogen atoms be $H_{1}, H_{2}, H_{3}, H_{4}$. Then $\sum_{1 \leq i < j \leq 4} \overrightarro... | -2 | 0 | 6,318.875 | -1 | 6,318.875 | |
How many zeros are at the end of the product $s(1) \cdot s(2) \cdot \ldots \cdot s(100)$, where $s(n)$ denotes the sum of the digits of the natural number $n$? | 19 | 0.0625 | 8,084.4375 | 7,986 | 8,091 | |
Given that a school has 5 top students and 3 teachers, where each teacher mentors no more than 2 students, calculate the number of different mentorship arrangements possible. | 90 | 0.3125 | 7,357.625 | 5,522 | 8,192 | |
In $\triangle ABC$, $a=1$, $B=45^{\circ}$, $S_{\triangle ABC}=2$, find the diameter of the circumcircle of $\triangle ABC$. | 5 \sqrt {2} | 0 | 4,344.0625 | -1 | 4,344.0625 | |
In $\triangle ABC$, the sides opposite to angles $A, B, C$ are denoted as $a, b, c$, respectively, and $a=1, A=\frac{\pi}{6}$.
(Ⅰ) When $b=\sqrt{3}$, find the magnitude of angle $C$;
(Ⅱ) Find the maximum area of $\triangle ABC$. | \frac{2+ \sqrt{3}}{4} | 0 | 7,485.5625 | -1 | 7,485.5625 | |
A company buys an assortment of 150 pens from a catalog for \$15.00. Shipping costs an additional \$5.50. Furthermore, they receive a 10% discount on the total price due to a special promotion. What is the average cost, in cents, for each pen? | 12 | 0.1875 | 505.1875 | 548 | 495.307692 | |
In rectangle \(ABCD\), \(\overline{AB}=30\) and \(\overline{BC}=15\). Let \(E\) be a point on \(\overline{CD}\) such that \(\angle CBE=45^\circ\) and \(\triangle ABE\) is isosceles. Find \(\overline{AE}.\) | 15 | 0 | 4,169.0625 | -1 | 4,169.0625 |
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