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 |
|---|---|---|---|---|---|---|
Given the ellipse $E$: $\frac{x^{2}}{2}+y^{2}=1$ with its right focus $F$, two perpendicular lines passing through $F$ intersect with $E$ at points $A$, $C$ and $B$, $D$.
1. Can the quadrilateral $ABCD$ form a parallelogram? Please explain the reason.
2. Find the minimum value of $|AC|+|BD|$. | \frac{8 \sqrt{2}}{3} | 0 | 8,107.375 | -1 | 8,107.375 | |
Recall that the sum of the angles of a triangle is 180 degrees. In triangle $ABC$, angle $A$ is a right angle. Let $BM$ be the median of the triangle and $D$ be the midpoint of $BM$. It turns out that $\angle ABD = \angle ACD$. What are the measures of these angles? | 30 | 0.75 | 5,403.1875 | 5,033.75 | 6,511.5 | |
Given $ \frac {\pi}{2}\leqslant \beta\leqslant \alpha\leqslant \frac {3\pi}{4} $, $ \cos (\alpha-\beta) = \frac {12}{13} $, $ \sin (\alpha+\beta) = -\frac {3}{5} $, find the values of $ \sin 2\alpha $ and $ \cos 2\beta $. | -\frac{63}{65} | 0.5 | 6,811.6875 | 5,566.875 | 8,056.5 | |
Given that the Riemann function defined on the interval $\left[0,1\right]$ is: $R\left(x\right)=\left\{\begin{array}{l}{\frac{1}{q}, \text{when } x=\frac{p}{q} \text{(p, q are positive integers, } \frac{p}{q} \text{ is a reduced proper fraction)}}\\{0, \text{when } x=0,1, \text{or irrational numbers in the interval } (... | \frac{5}{3} | 0 | 5,283.0625 | -1 | 5,283.0625 | |
Let $F(z)=\dfrac{z+i}{z-i}$ for all complex numbers $z\neq i$, and let $z_n=F(z_{n-1})$ for all positive integers $n$. Given that $z_0=\dfrac{1}{137}+i$ and $z_{2002}=a+bi$, where $a$ and $b$ are real numbers, find $a+b$. | 275 | Iterating $F$ we get:
\begin{align*} F(z) &= \frac{z+i}{z-i}\\ F(F(z)) &= \frac{\frac{z+i}{z-i}+i}{\frac{z+i}{z-i}-i} = \frac{(z+i)+i(z-i)}{(z+i)-i(z-i)}= \frac{z+i+zi+1}{z+i-zi-1}= \frac{(z+1)(i+1)}{(z-1)(1-i)}\\ &= \frac{(z+1)(i+1)^2}{(z-1)(1^2+1^2)}= \frac{(z+1)(2i)}{(z-1)(2)}= \frac{z+1}{z-1}i\\ F(F(F(z))) &= \frac... | 0 | 8,192 | -1 | 8,192 |
Calculate the definite integral:
$$
\int_{1}^{e^{2}} \frac{\ln ^{2} x}{\sqrt{x}} \, dx
$$ | 24e - 32 | 0 | 4,648.75 | -1 | 4,648.75 | |
Six chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left. There is only a problem if two people sitting next to each other have the person on the right crossing t... | 1106 | 0 | 8,192 | -1 | 8,192 | |
The sum of the first 2015 digits of the decimal part of the repeating decimal \(0.0142857\) is $\qquad$ | 9065 | 0 | 4,824.375 | -1 | 4,824.375 | |
Let $a$ and $b$ be positive real numbers such that each of the equations $x^2 + ax + 2b = 0$ and $x^2 + 2bx + a = 0$ has real roots. Find the smallest possible value of $a + b.$ | 6 | 0.9375 | 4,723.25 | 4,492 | 8,192 | |
Moving only south and east along the line segments, how many paths are there from $A$ to $B$? [asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;
for(int i = 0; i <= 9; ++i)
if (i!=4 && i !=5)
draw((2i,0)--(2i,3));
for(int j = 0; j <= 3; ++j)
draw((0,j)--(18,j));
draw((2*4,0)--(2*4,1));
draw(... | 160 | 0 | 8,192 | -1 | 8,192 | |
The graphs of $y = x^3 - 3x + 2$ and $x + 4y = 4$ intersect in the points $(x_1,y_1),$ $(x_2,y_2),$ and $(x_3,y_3).$ If $x_1 + x_2 + x_3 = A$ and $y_1 + y_2 + y_3 = B,$ compute the ordered pair $(A,B).$ | (0,3) | 0.875 | 4,711.375 | 4,214.142857 | 8,192 | |
Square $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\overline{BC}$ and $\overline{CD},$ respectively, so that $\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\overline{AE}.$ The length of a side of this smaller square is $\frac{a-\sqrt... | 12 | 0.625 | 6,915.9375 | 6,368.5 | 7,828.333333 | |
Let $ABCDV$ be a regular quadrangular pyramid with $V$ as the apex. The plane $\lambda$ intersects the $VA$ , $VB$ , $VC$ and $VD$ at $M$ , $N$ , $P$ , $Q$ respectively. Find $VQ : QD$ , if $VM : MA = 2 : 1$ , $VN : NB = 1 : 1$ and $VP : PC = 1 : 2$ . | 2:1 | 0 | 7,144.4375 | -1 | 7,144.4375 | |
Given a triangle $ABC$ with sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively, and it is given that $(3b-c)\cos A = a\cos C$.
(1) Find the value of $\cos A$;
(2) If the area of $\triangle ABC$ is $S=2\sqrt{2}$, find the minimum value of the perimeter of $\triangle ABC$. | 2\sqrt{6}+2\sqrt{2} | 0.125 | 6,676 | 5,480 | 6,846.857143 | |
The number $2022$ has the following property: it is a multiple of $6$ and the sum of its digits is $6$. Such positive integers are called "auspicious numbers." Among all three-digit positive integers, the number of "auspicious numbers" is ____. | 12 | 0.8125 | 5,254.0625 | 4,576.076923 | 8,192 | |
There are some identical square pieces of paper. If a part of them is paired up to form rectangles with a length twice their width, the total perimeter of all the newly formed rectangles is equal to the total perimeter of the remaining squares. Additionally, the total perimeter of all shapes after pairing is 40 centime... | 280 | 0.375 | 6,228.0625 | 5,073.333333 | 6,920.9 | |
In right triangle $ABC$, $AB=10$, $AC=6$ and $BC=8$ units. What is the distance from $C$ to the midpoint of segment $AB$? | 5 | 0.9375 | 3,158 | 2,822.4 | 8,192 | |
Given $\sin a= \frac{ \sqrt{5}}{5}$, $a\in\left( \frac{\pi}{2},\pi\right)$, find:
$(1)$ The value of $\sin 2a$;
$(2)$ The value of $\tan \left( \frac{\pi}{3}+a\right)$. | 5 \sqrt{3}-8 | 0.6875 | 5,677.375 | 4,534.363636 | 8,192 | |
The sum of the house numbers on one side of a street from corner to corner is 117. What is the house number of the fifth house from the beginning of this section? | 13 | 0 | 2,092.9375 | -1 | 2,092.9375 | |
In $\triangle ABC$, $AC=5 \sqrt {2}$, $\cos C= \frac {3}{5}$, $B= \frac {\pi}{4}$.
(1) Find the length of $AB$;
(2) Find the area of $\triangle ABC$, denoted as $S_{\triangle ABC}$. | 28 | 0.625 | 5,982.875 | 5,086.5 | 7,476.833333 | |
The positive numbers \( x, y, \) and \( z \) are such that \( x + y + z = 5 \). What is the minimum value of the expression \( x^{2} + y^{2} + 2z^{2} - x^{2} y^{2} z \)? | -6 | 0.625 | 6,578.3125 | 5,610.1 | 8,192 | |
Find the domain of the expression $\frac{\sqrt{x-2}}{\sqrt{5-x}}$. | [2,5) | 0.9375 | 1,238 | 1,226.066667 | 1,417 | |
Compute
\[\left( 1 - \frac{1}{\cos 23^\circ} \right) \left( 1 + \frac{1}{\sin 67^\circ} \right) \left( 1 - \frac{1}{\sin 23^\circ} \right) \left( 1 + \frac{1}{\cos 67^\circ} \right).\] | 1 | 1 | 3,497.8125 | 3,497.8125 | -1 | |
The graph of $y=\frac{5x^2-9}{3x^2+5x+2}$ has a horizontal asymptote at $y=a$. What is $a$? | \frac53 | 1 | 2,227.8125 | 2,227.8125 | -1 | |
Find the number of positive integers $n,$ $1 \le n \le 2000,$ for which the polynomial $x^2 + 2x - n$ can be factored as the product of two linear factors with integer coefficients. | 45 | 0.0625 | 7,210.1875 | 8,192 | 7,144.733333 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it is given that $b^{2}=ac$ and $a^{2}+bc=c^{2}+ac$. Calculate the value of $\dfrac {c}{b\sin B}$. | \dfrac{2\sqrt{3}}{3} | 0 | 5,820.6875 | -1 | 5,820.6875 | |
The taxi fare in Metropolis City is $3.00 for the first $\frac{3}{4}$ mile and additional mileage charged at the rate $0.30 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the number of miles you can ride for $15. | 3.75 | 0.75 | 5,006.4375 | 3,944.583333 | 8,192 | |
Using the digits $1$, $2$, $3$, $5$, and $6$ exactly once, the five-digit positive integers are formed and arranged in ascending order. What is the $60^{\text{th}}$ integer in this list? | 32651 | 0.5 | 6,464.1875 | 5,778.125 | 7,150.25 | |
Let $T$ be the set of all ordered triples of integers $(b_1, b_2, b_3)$ with $1 \leq b_1, b_2, b_3 \leq 20$. Each ordered triple in $T$ generates a sequence according to the rule $b_n = b_{n-1} \cdot |b_{n-2} - b_{n-3}|$ for all $n \geq 4$. Find the number of such sequences for which $b_n = 0$ for some $n$. | 780 | 0 | 8,192 | -1 | 8,192 | |
If two factors of $2x^3-hx+k$ are $x+2$ and $x-1$, the value of $|2h-3k|$ is | 0 | 1. **Using the Remainder Theorem**: Given that $x+2$ and $x-1$ are factors of $2x^3 - hx + k$, we can use the Remainder Theorem to set up equations. The Remainder Theorem states that if $x - c$ is a factor of a polynomial $p(x)$, then $p(c) = 0$.
2. **Setting up equations**:
- For $x + 2$, substitute $x = -2$ into ... | 1 | 2,689.1875 | 2,689.1875 | -1 |
Points \(A = (2,8)\), \(B = (2,2)\), and \(C = (6,2)\) lie in the first quadrant and are vertices of triangle \(ABC\). Point \(D=(a,b)\) is also in the first quadrant, and together with \(A\), \(B\), and \(C\), forms quadrilateral \(ABCD\). The quadrilateral formed by joining the midpoints of \(\overline{AB}\), \(\over... | 14 | 0 | 7,311.0625 | -1 | 7,311.0625 | |
A modified octahedron consists of two pyramids, each with a pentagonal base, glued together along their pentagonal bases, forming a polyhedron with twelve faces. An ant starts at the top vertex and walks randomly to one of the five adjacent vertices in the middle ring. From this vertex, the ant walks again to another r... | \frac{1}{5} | 0.4375 | 7,478.75 | 6,715.142857 | 8,072.666667 | |
Four positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to the fourth integer. Thus the numbers $29, 23, 21$, and $17$ are obtained. One of the original integers is: | 21 | 1. Let the original integers be $a, b, c,$ and $d$. We are given the following equations based on the problem statement:
\[
\frac{a+b+c}{3} + d = 29
\]
\[
\frac{a+b+d}{3} + c = 23
\]
\[
\frac{a+c+d}{3} + b = 21
\]
\[
\frac{b+c+d}{3} + a = 17
\]
2. Add all four equations together:
... | 0.375 | 6,125.625 | 3,734.5 | 7,560.3 |
How many positive perfect squares less than $10^6$ are multiples of $24$? | 83 | The prime factorization of $24$ is $2^3\cdot3$. Thus, each square must have at least $3$ factors of $2$ and $1$ factor of $3$ and its square root must have $2$ factors of $2$ and $1$ factor of $3$. This means that each square is in the form $(12c)^2$, where $12 c$ is a positive integer less than $\sqrt{10^6}$. There ar... | 0.9375 | 5,104.25 | 4,898.4 | 8,192 |
Given the function \( f(x)=\sin \omega x+\cos \omega x \) where \( \omega > 0 \) and \( x \in \mathbb{R} \), if the function \( f(x) \) is monotonically increasing on the interval \( (-\omega, \omega) \) and the graph of the function \( y=f(x) \) is symmetric with respect to the line \( x=\omega \), determine the value... | \frac{\sqrt{\pi}}{2} | 0 | 7,928.0625 | -1 | 7,928.0625 | |
Freshmen go for a long walk in the suburbs after the start of school. They arrive at point \( A \) 6 minutes later than the originally planned time of 10:10, and they arrive at point \( C \) 6 minutes earlier than the originally planned time of 13:10. There is exactly one point \( B \) between \( A \) and \( C \) that ... | 11:40 | 0 | 8,121.9375 | -1 | 8,121.9375 | |
In a grain storage facility, the following are the amounts of grain (in tons) that were received or dispatched over a period of 6 days (where "+" indicates received and "-" indicates dispatched):
+26, -32, -15, +34, -38, -20.
(1) After these 6 days, did the amount of grain in the storage increase or decrease? By ho... | 825 | 0.125 | 1,846.125 | 2,785 | 1,712 | |
Vitya collects toy cars from the "Retro" series. The problem is that the total number of different models in the series is unknown — it's a big commercial secret. However, it is known that different cars are produced in equal quantities, so it can be assumed that all models are evenly and randomly distributed across di... | 58 | 0.0625 | 7,364.5625 | 7,245 | 7,372.533333 | |
Given that the slant height of a cone is 2, and its net is a semicircle, what is the area of the cross section of the axis of the cone? | \sqrt{3} | 0.875 | 3,287.9375 | 2,766.285714 | 6,939.5 | |
Given vectors $a$ and $b$ that satisfy $(a+2b)\cdot(5a-4b)=0$, and $|a|=|b|=1$, find the angle $\theta$ between $a$ and $b$. | \dfrac{\pi}{3} | 0.8125 | 1,731 | 1,769.230769 | 1,565.333333 | |
If $x$ is a number satisfying the equation $\sqrt[3]{x+9}-\sqrt[3]{x-9}=3$, then $x^2$ is between: | 75 \text{ and } 85 | 1. **Define Variables**: Let $a = \sqrt[3]{x + 9}$ and $b = \sqrt[3]{x - 9}$.
2. **Cubing the Equations**: Cubing these definitions, we get:
\[ a^3 = x + 9 \quad \text{and} \quad b^3 = x - 9. \]
Subtracting these equations gives:
\[ a^3 - b^3 = (x + 9) - (x - 9) = 18. \]
3. **Factorize the Difference of Cub... | 0 | 6,657.25 | -1 | 6,657.25 |
Formulate the Taylor series expansion for \( n=2 \) of the function \( f(x, y) = x^y \) near the point \( M_0(1,1) \) and approximately calculate \( 1.1^{1.02} \). | 1.102 | 0.6875 | 6,800.625 | 6,168.181818 | 8,192 | |
In triangle $PQR,$ $PQ = 24,$ $QR = 25,$ $PR = 7,$ and point $H$ is the intersection of the altitudes (the orthocenter). Points $P',$ $Q',$ and $R',$ are the images of $P,$ $Q,$ and $R,$ respectively, after a $180^\circ$ rotation about $H.$ What is the area of the union of the two regions enclosed by the triangles $PQR... | 84 | 0 | 7,536 | -1 | 7,536 | |
A spider is making a web between $n>1$ distinct leaves which are equally spaced around a circle. He chooses a leaf to start at, and to make the base layer he travels to each leaf one at a time, making a straight line of silk between each consecutive pair of leaves, such that no two of the lines of silk cross each other... | n 2^{n-2} | There are $n$ ways to choose a starting vertex, and at each vertex he has only two choices for where to go next: the nearest untouched leaf in the clockwise direction, and the nearest untouched leaf in the counterclockwise direction. For, if the spider visited a leaf which is not nearest in some direction, there are tw... | 0 | 6,721.4375 | -1 | 6,721.4375 |
Convert the binary number $101111011_{(2)}$ to its decimal equivalent. | 379 | 0.9375 | 5,025.75 | 5,322.133333 | 580 | |
Given in $\triangle ABC$, $AB= \sqrt {3}$, $BC=1$, and $\sin C= \sqrt {3}\cos C$, the area of $\triangle ABC$ is ______. | \frac { \sqrt {3}}{2} | 0 | 4,974.1875 | -1 | 4,974.1875 | |
Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m_{}$ and $n_{}$ are integers, find $m+n.$ | 5 | [asy] import three; // calculate intersection of line and plane // p = point on line // d = direction of line // q = point in plane // n = normal to plane triple lineintersectplan(triple p, triple d, triple q, triple n) { return (p + dot(n,q - p)/dot(n,d)*d); } // projection of point A onto line BC triple projectionofp... | 0 | 7,944.9375 | -1 | 7,944.9375 |
Given $f(\alpha)= \dfrac {\sin (\alpha- \dfrac {5\pi}{2})\cos ( \dfrac {3\pi}{2}+\alpha)\tan (\pi-\alpha)}{\tan (-\alpha-\pi)\sin (\pi-\alpha)}$.
(1) Simplify $f(\alpha)$
(2) If $\cos (\alpha+ \dfrac {3\pi}{2})= \dfrac {1}{5}$ and $\alpha$ is an angle in the second quadrant, find the value of $f(\alpha)$. | \dfrac{2\sqrt{6}}{5} | 0 | 3,936.8125 | -1 | 3,936.8125 | |
A spinner is divided into 8 equal sectors numbered from 1 to 8. Jane and her sister each spin the spinner once. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her sister wins. What is the probability that Jane wins? | \frac{11}{16} | 0.5625 | 6,995.3125 | 6,064.555556 | 8,192 | |
The probability of snow for each of the next three days is $\frac{3}{4}$. What is the probability that it will not snow at all during the next three days? Express your answer as a common fraction. | \frac{1}{64} | 1 | 1,753.875 | 1,753.875 | -1 | |
Six cards numbered $1$ through $6$ are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving the remaining five cards in either ascending or descending order.
| 52 | 0 | 8,192 | -1 | 8,192 | |
If I choose four cards from a standard $52$-card deck, without replacement, what is the probability that I will end up with one card from each suit, in a sequential order (e.g., clubs, diamonds, hearts, spades)? | \frac{2197}{499800} | 0.1875 | 7,976.375 | 7,696 | 8,041.076923 | |
Steve's empty swimming pool will hold $24,000$ gallons of water when full. It will be filled by $4$ hoses, each of which supplies $2.5$ gallons of water per minute. How many hours will it take to fill Steve's pool? | 40 | 1. **Calculate the total water flow rate per minute**:
Each hose supplies $2.5$ gallons of water per minute. With $4$ hoses working together, the total water flow rate per minute is:
\[
4 \times 2.5 = 10 \text{ gallons per minute}
\]
2. **Convert the flow rate to gallons per hour**:
Since there are $60... | 0.875 | 2,430.1875 | 1,607.071429 | 8,192 |
Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example, $23$ and $37$ are "very prime" numbers, but $237$ and $357$ are not. Find the largest "prime" number (with justification!). | 373 | 0 | 8,192 | -1 | 8,192 | |
Using the property that in finding the limit of the ratio of two infinitesimals, they can be replaced with their equivalent infinitesimals (property II), find the following limits:
1) \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 3 x}\)
2) \(\lim _{x \rightarrow 0} \frac{\tan^{2} 2 x}{\sin ^{2} \frac{x}{3}}\)
3) \(\l... | \frac{3}{10} | 0.5625 | 5,857.4375 | 5,394.444444 | 6,452.714286 | |
The value of \((4 + 44 + 444) \div 4\) is: | 123 | 1 | 2,561.375 | 2,561.375 | -1 | |
The supermarket sold two types of goods, both for a total of 660 yuan. One item made a profit of 10%, while the other suffered a loss of 10%. Express the original total price of these two items using a formula. | 1333\frac{1}{3} | 0.0625 | 3,852.6875 | 4,369 | 3,818.266667 | |
A shooter hits the following scores in five consecutive shots: 9.7, 9.9, 10.1, 10.2, 10.1. The variance of this set of data is __________. | 0.032 | 0.25 | 6,625.9375 | 4,582 | 7,307.25 | |
There are 4 different digits that can form 18 different four-digit numbers arranged in ascending order. The first four-digit number is a perfect square, and the second-last four-digit number is also a perfect square. What is the sum of these two numbers? | 10890 | 0 | 8,192 | -1 | 8,192 | |
Find the number of distinct numbers in the list
\[\left\lfloor \frac{1^2}{1000} \right\rfloor, \ \left\lfloor \frac{2^2}{1000} \right\rfloor, \ \left\lfloor \frac{3^2}{1000} \right\rfloor, \ \dots, \ \left\lfloor \frac{1000^2}{1000} \right\rfloor.\] | 751 | 0 | 8,192 | -1 | 8,192 | |
Let $N$ be the product of all odd primes less than $2^4$. What remainder does $N$ leave when divided by $2^4$? | 7 | 1 | 3,115.4375 | 3,115.4375 | -1 | |
Calculate the whole number remainder when 987,670 is divided by 128. | 22 | 0.0625 | 812.9375 | 952 | 803.666667 | |
Let $\triangle ABC$ be an acute triangle with circumcircle $\omega,$ and let $H$ be the intersection of the altitudes of $\triangle ABC.$ Suppose the tangent to the circumcircle of $\triangle HBC$ at $H$ intersects $\omega$ at points $X$ and $Y$ with $HA=3,HX=2,$ and $HY=6.$ The area of $\triangle ABC$ can be written i... | 58 | Let $O$ be circumcenter of $ABC,$ let $R$ be circumradius of $ABC,$ let $\omega'$ be the image of circle $\omega$ over line $BC$ (the circumcircle of $HBC$).
Let $P$ be the image of the reflection of $H$ over line $BC, P$ lies on circle $\omega.$ Let $M$ be the midpoint of $XY.$ Then $P$ lies on $\omega, OA = O'H, OA ... | 0 | 8,192 | -1 | 8,192 |
Given two points $A(-2,0)$ and $B(0,2)$, and point $C$ is any point on the circle $x^{2}+y^{2}-2x=0$, find the minimum area of $\triangle ABC$. | 3 - \sqrt{2} | 0.625 | 7,400.6875 | 6,925.9 | 8,192 | |
Let \( X \), \( Y \), and \( Z \) be nonnegative integers such that \( X+Y+Z = 15 \). What is the maximum value of \[ X\cdot Y\cdot Z + X\cdot Y + Y\cdot Z + Z\cdot X ? \] | 200 | 0.875 | 6,623.9375 | 6,399.928571 | 8,192 | |
Given vectors $\overrightarrow{a}=(\cos \alpha,\sin \alpha)$, $\overrightarrow{b}=(\cos x,\sin x)$, $\overrightarrow{c}=(\sin x+2\sin \alpha,\cos x+2\cos \alpha)$, where $(0 < \alpha < x < \pi)$.
$(1)$ If $\alpha= \frac {\pi}{4}$, find the minimum value of the function $f(x)= \overrightarrow{b} \cdot \overrightarrow{c}... | - \frac { \sqrt {3}}{5} | 0 | 7,249.3125 | -1 | 7,249.3125 | |
The segment connecting the centers of two intersecting circles is divided by their common chord into segments of 4 and 1. Find the length of the common chord, given that the radii of the circles are in the ratio $3:2$. | 2 \sqrt{11} | 0.75 | 5,234 | 4,248 | 8,192 | |
The extensions of sides \(AD\) and \(BC\) of a convex quadrilateral \(ABCD\) intersect at point \(M\), and the extensions of sides \(AB\) and \(CD\) intersect at point \(O\). Segment \(MO\) is perpendicular to the angle bisector of \(\angle AOD\). Find the ratio of the areas of triangle \(AOD\) and quadrilateral \(ABC... | 2:1 | 0 | 8,192 | -1 | 8,192 | |
Solve for \(x\): \[\frac{x-60}{3} = \frac{5-3x}{4}.\] | \frac{255}{13} | 1 | 2,656 | 2,656 | -1 | |
The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 5 \\ 2 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} -\frac{5}{2} \\ -1 \end{pmatrix}\]lie on a line. Enter the equation of this line in the form "$y = mx + b$". | y = -\frac{5}{2} x - \frac{29}{4} | 0.875 | 4,118.125 | 3,536.142857 | 8,192 | |
Consider pairs $(f,g)$ of functions from the set of nonnegative integers to itself such that
[list]
[*]$f(0) \geq f(1) \geq f(2) \geq \dots \geq f(300) \geq 0$
[*]$f(0)+f(1)+f(2)+\dots+f(300) \leq 300$
[*]for any 20 nonnegative integers $n_1, n_2, \dots, n_{20}$, not necessarily distinct, we have $$g(n_1+n_2+\dots+n_{... | 115440 |
Consider pairs \((f, g)\) of functions from the set of nonnegative integers to itself such that:
- \(f(0) \geq f(1) \geq f(2) \geq \dots \geq f(300) \geq 0\),
- \(f(0) + f(1) + f(2) + \dots + f(300) \leq 300\),
- for any 20 nonnegative integers \(n_1, n_2, \dots, n_{20}\), not necessarily distinct, we have \(g(n_1 + n... | 0 | 8,192 | -1 | 8,192 |
There are 8 students arranged in two rows, with 4 people in each row. If students A and B must be arranged in the front row, and student C must be arranged in the back row, then the total number of different arrangements is ___ (answer in digits). | 5760 | 0.4375 | 7,117.75 | 5,875.714286 | 8,083.777778 | |
Calculate the value of the expression: $3 - 7 + 11 - 15 + 19 - \cdots - 59 + 63 - 67 + 71$. | -36 | 0.4375 | 6,628.625 | 4,742.571429 | 8,095.555556 | |
Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O,$ and that the ratio of the volume of $O$ to that of $C$ is $\frac mn,$ where $m$ and $n$ are relatively prime integers, find $m+n.$
| 11 | 0.5625 | 7,246.8125 | 6,511.666667 | 8,192 | |
Given the function $y=a^{x+4}+2$ with $a \gt 0$ and $a \gt 1$, find the value of $\sin \alpha$ if the terminal side of angle $\alpha$ passes through a point on the graph of the function. | \frac{3}{5} | 0.25 | 7,439 | 5,781.25 | 7,991.583333 | |
How can you cut a 5 × 5 square with straight lines so that the resulting pieces can be assembled into 50 equal squares? It is not allowed to leave unused pieces or to overlap them. | 50 | 0.125 | 8,009.5625 | 6,732.5 | 8,192 | |
Jitka hiked a trail. After hiking 60% of the length of the trail, she had 8 km left to go. What is the length of the trail? | 20 \text{ km} | After Jitka hiked 60% of the trail, 40% of the trail was left, which corresponds to 8 km. This means that 10% of the trail corresponds to 2 km. Therefore, the total length of the trail is \( 10 \times 2 = 20 \text{ km} \). | 0.6875 | 382.8125 | 386.818182 | 374 |
In how many ways can the number 1024 be factored into three natural factors such that the first factor is divisible by the second, and the second is divisible by the third? | 14 | 0.25 | 6,848.25 | 5,291.75 | 7,367.083333 | |
Twelve points are spaced around a $3 \times 3$ square at intervals of one unit. Two of the 12 points are chosen at random. Find the probability that the two points are one unit apart. | \frac{2}{11} | 0.4375 | 7,137.6875 | 6,083.571429 | 7,957.555556 | |
Given that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and $a_{1}=4$, $a_{n}+a_{n+1}=4n+2$ for $n\in \mathbb{N}^{*}$, calculate the maximum value of $n$ that satisfies $S_{n} \lt 2023$. | 44 | 0.25 | 8,033.625 | 7,558.5 | 8,192 | |
You have 2 six-sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 fives. You pick a die and roll it. Because of some secret magnetic attraction of the unfair die, you have a 75% chance of picking the unfair die and a 25% chance of picking the fair die. If you roll a three, what is the prob... | 1/7 | 0.875 | 3,581 | 2,922.285714 | 8,192 | |
Find the midsegment (median) of an isosceles trapezoid, if its diagonal is 25 and its height is 15. | 20 | 0.5 | 6,247.75 | 4,952.875 | 7,542.625 | |
Calculate the numerical value by listing.<br/>Select $5$ people from $8$ people including $A$, $B$, and $C$ to line up.<br/>$(1)$ If $A$ must be included, how many ways are there to line up?<br/>$(2)$ If $A$, $B$, and $C$ are not all included, how many ways are there to line up?<br/>$(3)$ If $A$, $B$, and $C$ are all i... | 4440 | 0.0625 | 7,566.4375 | 6,731 | 7,622.133333 | |
Let $a_1 = \sqrt 7$ and $b_i = \lfloor a_i \rfloor$ , $a_{i+1} = \dfrac{1}{b_i - \lfloor b_i \rfloor}$ for each $i\geq i$ . What is the smallest integer $n$ greater than $2004$ such that $b_n$ is divisible by $4$ ? ( $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$ ) | 2005 | 0.1875 | 7,739.3125 | 7,437 | 7,809.076923 | |
Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes? | 67 | 1. **Identify the total distance and total time**: Sam drove a total of $96$ miles in $90$ minutes. To convert minutes into hours, we divide by $60$:
\[
90 \text{ minutes} = \frac{90}{60} \text{ hours} = 1.5 \text{ hours}
\]
2. **Calculate the overall average speed**: The average speed for the entire trip ca... | 1 | 1,700.0625 | 1,700.0625 | -1 |
In a large bag of decorative ribbons, $\frac{1}{4}$ are yellow, $\frac{1}{3}$ are purple, $\frac{1}{8}$ are orange, and the remaining 45 ribbons are silver. How many of the ribbons are orange? | 19 | 0 | 8,185.9375 | -1 | 8,185.9375 | |
The greatest prime number that is a divisor of $16,385$ can be deduced similarly, find the sum of the digits of this greatest prime number. | 19 | 0 | 2,843.625 | -1 | 2,843.625 | |
Given $\sin \alpha + \cos \beta = \frac{1}{3}$ and $\sin \beta - \cos \alpha = \frac{1}{2}$, find $\sin (\alpha-\beta)=$ ______. | - \frac{59}{72} | 0.8125 | 7,021.75 | 6,751.692308 | 8,192 | |
The sum of the first 1000 terms of a geometric sequence is 300. The sum of the first 2000 terms is 570. Find the sum of the first 3000 terms. | 813 | 0.6875 | 5,443 | 4,193.454545 | 8,192 | |
The diagram shows a large triangle divided into squares and triangles. Let \( S \) be the number of squares of any size in the diagram and \( T \) be the number of triangles of any size in the diagram. What is the value of \( S \times T \)?
A) 30
B) 35
C) 48
D) 70
E) 100 | 70 | 0 | 5,749.1875 | -1 | 5,749.1875 | |
Cara is sitting at a round table with her eight friends. How many different pairs of friends could Cara be potentially sitting between? | 28 | 0 | 5,719.3125 | -1 | 5,719.3125 | |
Sasha wrote down numbers from one to one hundred, and Misha erased some of them. Among the remaining numbers, 20 contain the digit one, 19 contain the digit two, and 30 contain neither one nor two. How many numbers did Misha erase? | 33 | 0 | 8,009.625 | -1 | 8,009.625 | |
Let $a, b, c$ be the three roots of $p(x)=x^{3}+x^{2}-333 x-1001$. Find $a^{3}+b^{3}+c^{3}$. | 2003 | We know that $x^{3}+x^{2}-333 x-1001=(x-a)(x-b)(x-c)=x^{3}-(a+b+c) x^{2}+(a b+b c+c a) x-a b c$. Also, $(a+b+c)^{3}-3(a+b+c)(a b+b c+c a)+3 a b c=a^{3}+b^{3}+c^{3}$. Thus, $a^{3}+b^{3}+c^{3}=(-1)^{3}-3(-1)(-333)+3 \cdot 1001=2003$. | 0.9375 | 3,632.375 | 3,328.4 | 8,192 |
Modify the constants in the problem: Let's alter the probability such that when a coin is flipped four times, the probability of having exactly two heads is $\frac{1}{12}$. Given that the coin's probability of landing on heads is less than $\frac{1}{2}$, find the probability that the coin lands on heads.
**A)** $\frac{... | \frac{12 - \sqrt{96 + 48\sqrt{2}}}{24} | 0 | 8,192 | -1 | 8,192 | |
The school committee has organized a "Chinese Dream, My Dream" knowledge speech competition. There are 4 finalists, and each contestant can choose any one topic from the 4 backup topics to perform their speech. The number of scenarios where exactly one of the topics is not selected by any of the 4 contestants is ______... | 324 | 0 | 7,121.75 | -1 | 7,121.75 | |
An equilateral triangle $ABC$ has an area of $27\sqrt{3}$. The rays trisecting $\angle BAC$ intersect side $BC$ at points $D$ and $E$. Find the area of $\triangle ADE$. | 3\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
On one side of the acute angle \(A\), points \(P\) and \(Q\) are marked such that \(AP = 4\), \(AQ = 12\). On the other side, points \(M\) and \(N\) are marked at distances of 6 and 10 from the vertex. Find the ratio of the areas of triangles \(MNO\) and \(PQO\), where \(O\) is the intersection point of the lines \(MQ\... | 1:5 | 0 | 6,402.3125 | -1 | 6,402.3125 | |
What is the sum of all two-digit positive integers whose squares end with the digits 25? | 495 | 1 | 3,588.5625 | 3,588.5625 | -1 | |
The lateral surface area of a regular triangular pyramid is 3 times the area of its base. The area of the circle inscribed in the base is numerically equal to the radius of this circle. Find the volume of the pyramid. | \frac{2 \sqrt{6}}{\pi^3} | 0 | 3,909.9375 | -1 | 3,909.9375 | |
The sequence is defined recursively:
\[ x_{0} = 0, \quad x_{n+1} = \frac{(n^2 + n + 1) x_{n} + 1}{n^2 + n + 1 - x_{n}}. \]
Find \( x_{8453} \). | 8453 | 0.875 | 3,350.5625 | 2,658.928571 | 8,192 |
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