task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | $A_1,A_2,\cdots,A_8$ are fixed points on a circle. Determine the smallest positive integer $n$ such that among any $n$ triangles with these eight points as vertices, two of them will have a common side. | n = 9 | 58 | 4 |
math | Given circle C: $(x+1)^2+y^2=4$ and a point outside the circle A$(1, 2\sqrt{3})$,
(1) If line $m$ passes through the origin $O$ and exactly three points on circle $C$ are at a distance of 1 from line $m$, find the equation of line $m$;
(2) If a line $l$ passing through $A$ is tangent to circle $C$ at points $D$ and... | x+\sqrt{3}y-1=0 | 132 | 11 |
math | Let $n = 2^{35}3^{17}$. How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$? | 594 | 42 | 3 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $2\sqrt{3}\cos^2\frac{C}{2} = \sin C + \sqrt{3} + 1$.
1. Find the size of angle $C$.
2. If $a=2\sqrt{3}$ and $c=2$, find $b$. | 2 \text{ or } 4 | 106 | 8 |
math | Shifting the graph of the function $f(x)=\sin (2x- \frac {\pi}{6})$ to the left by $\frac {\pi}{3}$ units to obtain the graph of the function $g(x)$, find the equation of one of the axes of symmetry of the graph of $g(x)$. | \frac {\pi}{2} | 68 | 7 |
math | Consider the sequence where $x_1+1=x_2+2=x_3+3=\cdots=x_{100}+100=x_1+x_2+x_3+\cdots+x_{100}+101$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{100}x_n$. | 51 | 91 | 2 |
math | $(1)$ Simplify $f(\alpha)=\frac{{\sin({\pi+\alpha})\sin({2\pi-\alpha})\cos({-\pi-\alpha})\cos({\frac{\pi}{2}+\alpha})}}{{\sin({3\pi+\alpha})\cos({\pi-\alpha})\cos({\frac{3\pi}{2}+\alpha})}}$;<br/>$(2)$ Find the value of $\frac{{2\cos{10}^\circ-\sin{20}^\circ}}{{\sin{70}^\circ}}$. | \sqrt{3} | 127 | 5 |
math | Let vectors $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ and $\mathbf{d}$ be such that $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{d}.$ There exists a scalar $k$ such that the equation
\[k (\mathbf{c} \times \mathbf{b}) + \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} = \mat... | 1 | 148 | 1 |
math | Given the function $f(x)=|x+3|+|x-a|$ $(a > 0)$.
$(1)$ When $a=4$, it is known that $f(x)=7$, find the range of $x$;
$(2)$ If the solution set of $f(x) \geqslant 6$ is $\{x|x\leqslant -4 \text{ or } x\geqslant 2\}$, find the value of $a$. | a=1 | 106 | 3 |
math | The first row of a triangle is given as:
$$
1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{1993}
$$
Each element of the following rows is calculated as the difference between two elements that are above it. The 1993rd row contains only one element. Find this element. | \frac{1}{1993} | 87 | 10 |
math | In the function $y=\dfrac{1}{x-1}$, determine the range of the independent variable $x$. | x\neq 1 | 26 | 6 |
math | In the land of Eldoria, a license plate consists of three uppercase letters followed by four digits. How many valid license plates can be created? | 175,760,000 | 29 | 11 |
math | A and B went to a discount store to buy goods, where the price of each item was only 8 yuan or 9 yuan. It is known that both of them bought the same number of items, and together they spent a total of 172 yuan. How many items of each price did they buy in total? | 8 \text{ items at 8 yuan, 12 items at 9 yuan} | 67 | 19 |
math | Find the domain of the function
$$
y=\sqrt{\frac{x^{2}-7 x+12}{x^{2}-2 x-3}}
$$
Solve the inequality:
$$
(0.4)^{x^{2}-1} > (0.6)^{x^{2}+6}
$$ | (-\infty, -1) \cup [4, \infty) | 69 | 17 |
math | Convert $\text{BF02}_{16}$ to a base 10 integer. | 48898 | 19 | 5 |
math | Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on side $AB$ with $AE=3\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?
A) $\frac{1}{16}$
B) $\frac{1}{8}$
C) $\frac{1}{4}$
D) $\frac{1}{2}$
E) $\frac{3}{4}$ | \frac{1}{8} | 109 | 7 |
math | The natural number \( A \) has the following property: the number \( 1 + 2 + \cdots + A \) can be written (in base 10) as the number \( A \) followed by three other digits. Find \( A \). | A = 1999 | 56 | 7 |
math | In the trapezoid \(ABCD\) with bases \(AD = 20\) and \(BC = 10\), circles constructed using sides \(AB\), \(BC\), and \(CD\) as diameters intersect at one point. The length of the diagonal \(AC\) is 18. Find the length of \(BD\). | 24 | 73 | 2 |
math | Suppose $P$ is the point $(5,3)$ and $Q$ is the point $(-3,6)$. Find point $T$ such that $Q$ is the midpoint of segment $\overline{PT}$. | (-11,9) | 49 | 6 |
math | Consider a function \( g \) that maps nonnegative integers to real numbers, with \( g(1) = 1 \), and for all nonnegative integers \( m \ge n \),
\[ g(m + n) + g(m - n) = \frac{g(3m) + g(3n)}{3} \]
Find the sum of all possible values of \( g(10) \). | 100 | 88 | 3 |
math | The three-digit number $2a3$ is added to the number $326$ to give the three-digit number $5b9$. If $5b9$ is divisible by 9, then $a+b$ equals | 6 | 49 | 1 |
math | Given \( f(x) = \max \left| x^3 - a x^2 - b x - c \right| \) for \( 1 \leq x \leq 3 \), find the minimum value of \( f(x) \) as \( a, b, \) and \( c \) range over all real numbers. | 1/4 | 74 | 3 |
math | Kelvin the frog lives in a pond with an infinite number of lily pads, numbered \(0,1,2,3,\) and so forth. Kelvin starts on lily pad 0 and jumps from pad to pad in the following manner: when on lily pad \(i\), he will jump to lily pad \((i+k)\) with probability \(\frac{1}{2^{k}}\) for \(k>0\). What is the probability th... | \frac{1}{2} | 116 | 7 |
math | Determine all pairs of positive integers \((n, k)\) such that
\[ n! + n = n^k \]
holds. | (2, 2), (3, 2), (5, 3) | 30 | 18 |
math |
Exactly half of the population of the island of Misfortune are hares, and the rest are rabbits. If a resident of Misfortune makes a statement, he sincerely believes what he says. However, hares are faithfully mistaken on average in one out of every four cases, and rabbits are faithfully mistaken on average in one out ... | 27/59 | 118 | 5 |
math | Given vectors **a** = (3, 5, -1), **b** = (2, 2, 3), **c** = (4, -1, -3), calculate the coordinates of the vector 2**a** - 3**b** + 4**c**. | (16, 0, -23) | 65 | 11 |
math | Given that $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|= \sqrt{2}$, and $(\overrightarrow{a}-\overrightarrow{b})$ is perpendicular to $\overrightarrow{a}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | 45^{\circ} | 69 | 6 |
math | The average of the five numbers 1, 2, 3, 4, and 5 is $\frac{15}{5} = 3$. Let the two selected numbers be $x$ and $y$. The average of the remaining three numbers is then $\frac{1 + 2 + 3 + 5 - x - y}{3} = 3$, which simplifies to $9 - x - y = 9$, or $x + y = 0$. | 2 | 103 | 1 |
math | Three distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. What is the probability that the smallest positive difference between any two of those numbers is $3$ or greater? Express your answer as a common fraction. | \frac{1}{14} | 72 | 8 |
math | Given the vector $\overrightarrow{a}=(\cos \theta,\sin \theta)$, where $\theta\in[0,\pi]$, and the vector $\overrightarrow{b}=(\sqrt{3},-1)$
$(1)$ If $\overrightarrow{a}\perp \overrightarrow{b}$, find the value of $\theta$;
$(2)$ If $|2 \overrightarrow{a}- \overrightarrow{b}| < m$ always holds, find the range of th... | m > 4 | 113 | 4 |
math | Points $P$, $Q$, $R$, and $S$ lie on a line, in that order. If $PQ = 3$ units, $QR = 7$ units, and the total distance $PS = 17$ units, what is the ratio of $PR$ to $QS$? | \frac{10}{7} | 67 | 8 |
math | Given circle O: $x^2+y^2=5$ and a fixed point A(4, 3), a tangent line PQ is drawn from a point P(a, b) outside circle O to circle O at the tangent point Q, satisfying $|PQ|=|PA|$
(1) Find the relationship between the real numbers a and b;
(2) Find the minimum length of segment PQ;
(3) If a circle P with center P i... | \left(x- \frac{12}{5}\right)^2+\left(y- \frac{9}{5}\right)^2=\left(3- \sqrt{5}\right)^2 | 118 | 42 |
math | Given the function $f(x) = 2^x + \ln x$, if $a_n = 0.1n$ ($n \in \mathbb{N}^*$), find the value of $n$ that minimizes $|f(a_n) - 2012|$. | 110 | 65 | 3 |
math | The base edge length of a right square prism is 3, and its height is 2. Then, its lateral area is , and its volume is . | 24, 18 | 38 | 6 |
math | In the coordinate plane, points whose x-coordinates and y-coordinates are both integers are called lattice points. For any natural number \( n \), let \( O \) be the origin and \( A_{n}(n, n+3) \) be a point. Denote by \( f(n) \) the number of lattice points on the line segment \( OA_{n} \) excluding the endpoints. Wha... | 1326 | 114 | 4 |
math | Chloe chooses a real number uniformly at random from the interval $[0, 1000]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0, 2x]$, where $x$ is the number Chloe chose. Find the probability that Laurent's number is greater than twice Chloe's number. | \frac{1}{4} | 71 | 7 |
math | Let $a$ be a fixed real number. Consider the equation $$ (x+2)^{2}(x+7)^{2}+a=0, x \in R $$ where $R$ is the set of real numbers. For what values of $a$ , will the equ have exactly one double-root? | a = -39.0625 | 74 | 11 |
math | Let $a$ and $b$ be real numbers such that
\[\frac{\sin a}{\cos b} + \frac{\sin b}{\cos a} = 2 \quad \text{and} \quad \frac{\cos a}{\sin b} + \frac{\cos b}{\sin a} = 4.\] Compute
\[\frac{\tan a}{\tan b} + \frac{\tan b}{\tan a}.\] | 2 | 100 | 1 |
math | Given propositions $p$ and $q$:
- Proposition $p$: The equation $\frac{x^2}{2} + \frac{y^2}{m} = 1$ represents an ellipse with foci on the $y$-axis.
- Proposition $q$: The function $f(x) = \frac{4}{3}x^3 - 2mx^2 + (4m - 3)x - m$ is strictly increasing on $(-\infty, +\infty)$.
Find the range of values for $m$ if the pro... | [1, 2] | 132 | 6 |
math | How many sides can a convex polygon have if all its diagonals are of the same length? | 5 | 19 | 1 |
math | There are two candles of different compositions but equal lengths. One can burn out in 3 hours, and the other in 4 hours. It is required that by 4 o'clock in the afternoon, the remaining part of one candle is exactly twice the remaining part of the other. The question is, when should these two candles be lit? | 1:36\ \text{PM} | 69 | 10 |
math | Find the number of zeros of the function $y= \frac {1}{3}x^{3}-x^{2}-3x+9$. | 2 | 31 | 1 |
math | Definition: The ratio $k$ of the length of the base of an isosceles triangle to the length of the height on the base is called the "characteristic value" of this isosceles triangle. If the perimeter of isosceles $\triangle ABC$ is $100$ cm and $AB=18$ cm, then the "characteristic value" $k$ of this triangle is $\_\_\_\... | \frac{9}{20} | 95 | 8 |
math | Given a function $y=f(x)$ ($x \in \mathbb{R}$) that satisfies $f(x+2)=f(x)$, and when $x \in [-1, 1]$, $f(x) = |x|$. The function $y=g(x)$ is defined on $\mathbb{R}$ as an odd function, and when $x \in (0, +\infty)$, $g(x) = \log_{3}x$. Find the number of intersection points between the graphs of $y=f(x)$ and $y=g(x)$. | 4 | 123 | 1 |
math | Given the expression $3^{3+3^{3}}$, determine the number of distinct values that can be obtained by changing the order of operations and parentheses. | 3 | 32 | 1 |
math | A small town has fewer than 6000 inhabitants. We know that there are $10\%$ more girls than boys among the children, and $15\%$ more men than women among the adults. There are $20\%$ more children than adults in the town.
How many people live in the town? | 3311 | 71 | 4 |
math | A supermarket sells two models of appliances, model A and model B, with purchase prices of $180$ yuan per unit and $160$ yuan per unit, respectively. The table below shows the sales situation in the past two weeks (purchase price and selling price remain unchanged, profit $=$ selling price $-$ purchase price):
| Sales... | 1800 | 265 | 4 |
math | A store is selling an assortment of 250 pencils for a discounted price of $\$25.50$ after applying a 15% discount. Shipping costs an additional $\$8.10$. Including the charges for both the pencils and the shipping, what is the average cost, in cents, for each pencil? Express your answer rounded to the nearest whole num... | 15 | 79 | 2 |
math | Given a square with its four vertices and center, determine the probability that the distance between any two of these 5 points is not less than the length of the square's side. | \frac{3}{5} | 36 | 7 |
math | Let $a$, $b$, and $c \in \mathbb{R}$. Among the original proposition "If $a > b$, then $ac^{2} > bc^{2}$" and its converse, inverse, and contrapositive propositions, determine the total number of true propositions. | 2 | 62 | 1 |
math | Calculate the last three digits of $11^{30}$. | 801 | 14 | 3 |
math | Let $b_1, b_2, \dots$ be a sequence where $b_1=5$, $b_2=7$, and $b_n = \frac{b_{n-1}}{b_{n-2}}$ for each positive integer $n \geq 3$. What is $b_{2023}$? | \frac{1}{7} | 76 | 7 |
math | Find $PQ$ in the triangle below.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (sqrt(3),0);
R = (0,2);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$P$",P,S);
label("$Q$",Q,S);
label("$R$",R,N);
label("$18\sqrt{3}$",R/2,W);
label("$30^\circ$",(1.25,0),N);
[/asy] | 54 | 130 | 2 |
math | Equilateral triangle $ABC$ and rectangle $BCDE$ are coplanar, with $BC = DE = 2 \times CD$. What is the number of degrees in the measure of angle $CAD$? | \angle CAD = 15^\circ | 44 | 9 |
math | The numbers \(2 \overline{ac} + 1\) and \(3 \overline{ac} + 1\) are perfect squares. Find \(\overline{ac}\). | 40 | 40 | 2 |
math | Given a cashier mistakenly counts $y$ half-dollar coins as $1$ dollar bills and $y$ $5$ dollar bills as $10$ dollar bills, calculate how much the cashier should adjust the total cash amount to account for these mistakes. | 5.50y | 51 | 5 |
math | 1. Find the intercept form of the equation of the line passing through points A(1, 2) and B$(-\frac{1}{2}, 1)$.
2. Find the equation of the line with a slope of $\frac{4}{3}$ that forms a triangle with the coordinate axes having an area of 4. | y = \frac{4}{3}x \pm \frac{4\sqrt{6}}{3} | 71 | 24 |
math | If three distinct members of the set $\{ 2, 4, 8, 10, 15, 20, 25, 50 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 200? Express your answer as a common fraction. | \frac{3}{28} | 69 | 8 |
math | A square with area $9$ is inscribed in a rectangle which has one side length double the other. One vertex of the square divides one of the shorter sides of the rectangle into two segments, one of length \( a \), and the other of length \( b \). The area of the rectangle is 18. Calculate the value of \( ab \). | 0 | 74 | 1 |
math | Andrei, Boris, and Valentin participated in a 1 km race (assume that each of them ran at a constant speed). Andrei was 100 meters ahead of Boris at the finish line. Boris was 50 meters ahead of Valentin at the finish line. What was the distance between Andrei and Valentin at the moment when Andrei finished? | 145 \ \text{meters} | 76 | 10 |
math | Given that $\sin(\pi+\theta)=\frac{1}{4}$, find the value of $\frac{\cos(\pi+\theta)}{\cos\theta[\cos(\pi+\theta)-1]}+\frac{\sin(\frac{\pi}{2}-\theta)}{\cos(\theta+2\pi)\cos(\pi+\theta)+\cos(-\theta)}$. | 32 | 79 | 2 |
math | Cara is sitting at a rectangular table with her seven friends. How many different pairs of people could Cara be sitting between, considering that she must sit at one of the long sides of the table with three friends on each side of her? | 6 | 48 | 1 |
math | Given the sequence $\{a\_n\}$ that satisfies $\log\_3 a\_n + 1 = \log\_3 a\_{n+1}$ for all $n \in \mathbb{N}^*$, and $a\_2 + a\_4 + a\_6 = 9$, determine the value of $\log\_3 (a\_5 + a\_7 + a\_9)$. | 5 | 88 | 1 |
math | Solve the equations:
1) \(x|x|=4\)
2) \((x-3)^{2}=1-\pi\)
3) \(\left|-1-x^{2}\right|=5\)
4) \(x^{2}-3x+x^{-1}=x^{-1}\)
5) \((\sqrt{x-2})^{4}=|x|-x\)
6) \(x(x-5)=x^{2}-5|x|\). | [0, \infty) | 99 | 7 |
math | In the Cartesian coordinate plane, a moving point $M(x,y)$ is on the left side of the $y$-axis, and the difference between the distance from point $M$ to a fixed point $F(-1,0)$ and the distance to the $y$-axis is $1$.
$(1)$ Find the equation of the trajectory $C$ of the moving point $M$;
$(2)$ If a line $l$ passi... | 8 | 139 | 1 |
math | Suppose that we have a right triangle $DEF$ with the right angle at $E$ such that $DF = \sqrt{85}$ and $DE = 7.$ A circle is drawn with its center on $DE$ such that the circle is tangent to $DF$ and $EF.$ If $Q$ is the point where the circle and side $DF$ meet, then what is $FQ$? | 6 | 87 | 1 |
math | Regular polygons with $4, 5, 7,$ and $9$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. How many points inside the circle do two of their sides from different polygons intersect?
A) $56$
B) $58$
C) $60$
D) $62$
E) $64$ | 58 | 92 | 2 |
math | Yan is positioned between his home and a stadium. He has two choices to reach the stadium: either he can walk directly there, or he can walk back home and then ride his bicycle to the stadium. Riding his bicycle is 5 times faster than walking. Both options take the same amount of time. What is the ratio of Yan's distan... | \frac{2}{3} | 80 | 7 |
math | Given lines $l_{1}$: $(m+2)x+(m+3)y-5=0$ and $l_{2}$: $6x+(2m-1)y=5$. Find the real number values of $m$ that satisfy the following conditions:
$(1) l_{1}//l_{2}$; $(2) l_{1} \perp l_{2}$. | m=-1 \text{ or } m=-\frac{9}{2} | 84 | 17 |
math | Let $A$, $B$, $C$ and $D$ be the vertices of a regular tetrahedron each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the v... | 182 | 125 | 3 |
math | Among $n$ new students, any group of 3 students has 2 students who know each other, and any group of 4 students has 2 students who do not know each other. Find the maximum value of $n$. | 8 | 48 | 1 |
math | Given an arithmetic sequence $\{a_n\}$ such that $\frac{a_{11}}{a_{10}} < -1$, and the sum of its first $n$ terms, $S_n$, has a maximum value, find the value of $n$ when $S_n$ reaches its minimum positive value. | n = 19 | 68 | 5 |
math | If \( a \) is the remainder when \( 2x^3 - 3x^2 + x - 1 \) is divided by \( x+1 \), find \( a \).
If \( b \ \text{cm}^2 \) is the total surface area of a cube of side \( (8+a) \ \text{cm} \), find \( b \).
One ball is taken at random from a bag containing \( b+4 \) red balls and \( 2b-2 \) white balls.
If \(\sin \th... | -1 | 161 | 2 |
math | Given that the line $l_{1}$: $x-my+1=0$ passes through the fixed point $A$, and the line $l_{2}$: $mx+y-m+3=0$ passes through the fixed point $B$. The lines $l_{1}$ and $l_{2}$ intersect at point $P$. Find $|PA|^{2}+|PB|^{2}$. | 13 | 87 | 2 |
math | In $\bigtriangleup ABC$, $E$ is a point on side $\overline{AB}$, and $D$ is a point on side $\overline{BC}$ such that $BD=DE=EC$. Let $\angle BDE$ be $90^{\circ}$. Determine the degree measure of $\angle AED$. | 45^\circ | 71 | 4 |
math | Consider the sequence $ (x_n)_{n\ge 0}$ where $ x_n\equal{}2^{n}\minus{}1\ ,\ n\in \mathbb{N}$ . Determine all the natural numbers $ p$ for which:
\[ s_p\equal{}x_0\plus{}x_1\plus{}x_2\plus{}...\plus{}x_p\]
is a power with natural exponent of $ 2$ . | p = 2 | 102 | 4 |
math | Given $x \in (-1, 3)$, determine the range of the function $y = (x - 2)^2$. | [0, 9) | 29 | 6 |
math | If the inequality log<sub>a</sub>x + x - 4 > 0 (a > 0 and a ≠ 1) has a solution in the interval (0, 2), then the range of values for the real number a is _____. | (0, 1) \cup (1, \sqrt{2}) | 54 | 16 |
math | Find the area of a trapezoid with bases 4 and 7 and side lengths 4 and 5.
| 22 | 25 | 2 |
math | David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. Determine the distance fr... | 210 | 87 | 3 |
math | Determine all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for any \( x, y \in \mathbf{R} \), the following equation holds:
\[ f(x) \cdot f[y \cdot f(x)-1] = x^{2} f(y) - f(x). \] | f(x) = 0 \quad \text{and} \quad f(x) = x | 75 | 20 |
math | Given a wooden cube with edges of length $4$ meters and square holes, each with a side of $1$ meter and a depth of $1$ meter, centered on each face and parallel to the edges of the cube, calculate the entire surface area including the inside in square meters. | 114 | 59 | 3 |
math | A student must choose a program of five courses from a list of courses consisting of English, Algebra, Geometry, History, Art, Latin, and Biology. This program must contain English and at least two mathematics courses. In how many ways can this program be chosen? | 6 | 53 | 1 |
math | Determine the area enclosed by the graph of the equation $x^2 + y^2 = 2|x| + 2|y|$.
A) $4\pi$
B) $2\pi + 8$
C) $8$
D) $2\pi + 4$
E) $2\pi + 2\sqrt{2}$ | 8 | 77 | 1 |
math | Let $x_1, x_2, \dots, x_{50}$ be real numbers such that $x_1 + x_2 + \dots + x_{50} = 0$ and
\[
\frac{x_1}{1+x_1} + \frac{x_2}{1+x_2} + \dots + \frac{x_{50}}{1+x_{50}} = 1.
\]
Find the value of
\[
\frac{x_1^2}{1+x_1} + \frac{x_2^2}{1+x_2} + \dots + \frac{x_{50}^2}{1+x_{50}}.
\] | 1 | 152 | 1 |
math | Given $|a| = 4$, $|b| = 5$, and $|a + b| = -(a + b)$, find the value of $2a - b$. | 2a - b = 13 \text{ or } -3 | 41 | 15 |
math | Given that a certain shooter shoots $10$ times in a row, with the number of hits and their frequencies shown in the table:
| Number of Hits | $7$ | $8$ | $9$ | $10$ |
|----------------|-----|-----|-----|------|
| Frequency | $2$ | $3$ | $4$ | $1$ |
calculate the median and mode of this data set. | 8.5, 9 | 93 | 6 |
math | Emily paid for a $\$2$ sandwich using 50 coins consisting of pennies, nickels, and dimes, and received no change. How many dimes did Emily use? | 10 | 39 | 2 |
math | Given a square with side length \(a\), its corners have been cut off to form a regular octagon. Determine the area of this octagon. | 2a^2(\sqrt{2} - 1) | 31 | 13 |
math | Given the function \( g_1 \) defined on the positive integers by \( g_1(1) = 1 \) and \( g_1(n) = (p_1-1)^{e_1-1} (p_2-1)^{e_2-1} \cdots (p_k-1)^{e_k-1} \), where \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \) is the prime factorization of \( n > 1 \), and for every \( m \geq 2 \), let \( g_m(n) = g_1(g_{m-1}(n)) \), d... | 0 | 214 | 1 |
math | Let \( a_{1}, a_{2}, \ldots, a_{n} \) be integers \( (n > 1) \) satisfying \( a_{1} + a_{2} + \cdots + a_{n} = a_{1} a_{2} \cdots a_{n} = 2005 \). Find the smallest possible value of \( n \). | 5 | 85 | 1 |
math | Given acute triangle $\triangle ABC$ in plane $P$ , a point $Q$ in space is defined such that $\angle AQB = \angle BQC = \angle CQA = 90^\circ.$ Point $X$ is the point in plane $P$ such that $QX$ is perpendicular to plane $P$ . Given $\angle ABC = 40^\circ$ and $\angle ACB = 75^\circ,$ find $\angle ... | 140^\circ | 120 | 5 |
math | The angle that has the same terminal side as 315° is what angle? | -45^{\circ} | 18 | 7 |
math | Given the quadratic equation in \( x \) given by \( a(1+\mathrm{i}) x^{2} + \left(1+a^{2} \mathrm{i}\right) x + a^{2} + \mathrm{i} = 0 \) has real roots, find the value of the real number \( a \). | a = -1 | 69 | 4 |
math | Given that a function $f(x)$ is defined on the real numbers $\mathbb{R}$ and is odd, and $f(3)=0$, and when $x > 0$, the inequality $f(x) > -xf'(x)$ holds, determine the number of zeros of the function $g(x)=xf(x)+\log|x+1|$. | 3 | 76 | 1 |
math | The boys from two sixth-grade classes made 123 chairs for kindergartens in the school workshops. How many boys were working, and how many chairs did each boy make if they made them equally? | 41 \text{ boys, each making } 3 \text{ chairs} | 43 | 17 |
math | The equation of line $l$ which passes through the focus of the parabola $y^2=2x$ and is parallel to the line $3x-2y+5=0$. | 6x - 4y - 3 = 0 | 42 | 12 |
math | Given $(0.7^{1.3})^m < (1.3^{0.7})^m$, find the range of the real number $m$. | (-\infty,0) | 35 | 7 |
math | Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$ . Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$ . | \frac{77}{2} | 67 | 8 |
math | The complex number $(1+i)(-i)$ equals $-i+i^2$. | 1-i | 18 | 2 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.