Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
31,600 | Given that in quadrilateral ABCD, $\angle A : \angle B : \angle C : \angle D = 1 : 3 : 5 : 6$, express the degrees of $\angle A$ and $\angle D$ in terms of a common variable. | 144 | 22.65625 |
31,601 | In the Cartesian coordinate system, with the origin as the pole and the positive x-axis as the polar axis, the polar equation of line $l$ is $$ρ\cos(θ+ \frac {π}{4})= \frac { \sqrt {2}}{2}$$, and the parametric equation of curve $C$ is $$\begin{cases} x=5+\cos\theta \\ y=\sin\theta \end{cases}$$, (where $θ$ is the parameter).
(Ⅰ) Find the Cartesian equation of line $l$ and the general equation of curve $C$;
(Ⅱ) Curve $C$ intersects the x-axis at points $A$ and $B$, with $x_A < x_B$, $P$ is a moving point on line $l$, find the minimum perimeter of $\triangle PAB$. | 2+ \sqrt {34} | 0 |
31,602 | Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2025 \) and \( y = |x - a| + |x - b| + |x - c| \) has exactly one solution. Find the minimum possible value of \( c \). | 1013 | 17.96875 |
31,603 | In the triangular pyramid $A-BCD$, where $AB=AC=BD=CD=BC=4$, the plane $\alpha$ passes through the midpoint $E$ of $AC$ and is perpendicular to $BC$, calculate the maximum value of the area of the section cut by plane $\alpha$. | \frac{3}{2} | 0 |
31,604 | What is the sum of every third odd number between $100$ and $300$? | 6800 | 24.21875 |
31,605 | $14 N$ is a 5-digit number composed of 5 different non-zero digits, and $N$ is equal to the sum of all three-digit numbers that can be formed using any 3 of these 5 digits. Find all such 5-digit numbers $N$. | 35964 | 14.84375 |
31,606 | In a WeChat group, there are five people playing the red envelope game: A, B, C, D, and E. There are 4 red envelopes, each person can grab at most one, and all red envelopes must be grabbed. Among the 4 red envelopes, there are two 2-yuan envelopes, one 3-yuan envelope, and one 4-yuan envelope (envelopes with the same amount are considered the same). How many scenarios are there where both A and B grab a red envelope? (Answer with a number) | 36 | 53.125 |
31,607 | Compute the following expression:
\[
\frac{(1 + 15) \left( 1 + \dfrac{15}{2} \right) \left( 1 + \dfrac{15}{3} \right) \dotsm \left( 1 + \dfrac{15}{17} \right)}{(1 + 17) \left( 1 + \dfrac{17}{2} \right) \left( 1 + \dfrac{17}{3} \right) \dotsm \left( 1 + \dfrac{17}{15} \right)}.
\] | 496 | 0 |
31,608 | If $M = 1! \times 2! \times 3! \times 4! \times 5! \times 6! \times 7! \times 8! \times 9!$, calculate the number of divisors of $M$ that are perfect squares. | 672 | 82.8125 |
31,609 | Find the limit, when $n$ tends to the infinity, of $$ \frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k} $$ | \sqrt{3} | 87.5 |
31,610 | Given the function $f(x) = x^2 - 3x - 1$, find the derivative of $f(2)$ and $f'(1)$. | -1 | 98.4375 |
31,611 | Given $y = f(x) + x^2$ is an odd function, and $f(1) = 1$, find the value of $g(-1)$, where $g(x) = f(x) + 2$. | -1 | 100 |
31,612 | Let $a$ and $b$ be nonzero real numbers such that $\tfrac{1}{3a}+\tfrac{1}{b}=2011$ and $\tfrac{1}{a}+\tfrac{1}{3b}=1$ . What is the quotient when $a+b$ is divided by $ab$ ? | 1509 | 30.46875 |
31,613 | Given $f(x)= \begin{cases} 2a-(x+ \frac {4}{x}),x < a\\x- \frac {4}{x},x\geqslant a\\end{cases}$.
(1) When $a=1$, if $f(x)=3$, then $x=$ \_\_\_\_\_\_;
(2) When $a\leqslant -1$, if $f(x)=3$ has three distinct real roots that form an arithmetic sequence, then $a=$ \_\_\_\_\_\_. | - \frac {11}{6} | 5.46875 |
31,614 | If the ratio of the legs of a right triangle is $1:3$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:
A) $1:3$
B) $1:9$
C) $3:1$
D) $9:1$ | 9:1 | 1.5625 |
31,615 | A square piece of paper has sides of length $120$. From each corner, a wedge is cut such that each of the two cuts for the wedge starts at a distance $10$ from the corner, and they meet on the diagonal at an angle of $45^{\circ}$. After the cuts, the paper is folded up along the lines joining the vertices of adjacent cuts, forming a tray with taped edges. Determine the height of this tray, which is the perpendicular distance from the base of the tray to the plane formed by the upper edges. | 5\sqrt{2} | 0.78125 |
31,616 | A cube \(ABCDA_1B_1C_1D_1\) has edge length 1. Point \(M\) is taken on the side diagonal \(A_1D\), and point \(N\) is taken on \(CD_1\), such that the line segment \(MN\) is parallel to the diagonal plane \(A_1ACC_1\). Find the minimum length of \(MN\). | \frac{\sqrt{3}}{3} | 11.71875 |
31,617 | Given that $y = f(x) + x^2$ is an odd function, and $f(1) = 1$, if $g(x) = f(x) + 2$, then $g(-1) = \_\_\_\_\_\_\_$. | -1 | 99.21875 |
31,618 | Construct a five-digit number without repeated digits using 0, 1, 2, 3, and 4, with the condition that even and odd digits must be adjacent to each other. Find the total number of such five-digit numbers. | 20 | 21.875 |
31,619 | In right triangle \( \triangle ABC \) where \(\angle ACB = 90^\circ\), \(CA = 3\), and \(CB = 4\), there is a point \(P\) inside \(\triangle ABC\) such that the sum of the distances from \(P\) to the three sides is \(\frac{13}{5}\). Find the length of the locus of point \(P\). | \frac{\sqrt{5}}{2} | 7.03125 |
31,620 | What is the value of $\frac{(2200 - 2096)^2}{121}$? | 89 | 1.5625 |
31,621 | In an acute-angled triangle $ABC$ , the point $O$ is the center of the circumcircle, and the point $H$ is the orthocenter. It is known that the lines $OH$ and $BC$ are parallel, and $BC = 4OH $ . Find the value of the smallest angle of triangle $ ABC $ .
(Black Maxim) | 30 | 71.09375 |
31,622 | A basketball player scored 18, 22, 15, and 20 points respectively in her first four games of a season. Her points-per-game average was higher after eight games than it was after these four games. If her average after nine games was greater than 19, determine the least number of points she could have scored in the ninth game. | 21 | 63.28125 |
31,623 | A triangle has altitudes of lengths 15, 21, and 35. Find its area. | 210 | 0.78125 |
31,624 | The cells of a $20 \times 20$ table are colored in $n$ colors such that for any cell, in the union of its row and column, cells of all $n$ colors are present. Find the greatest possible number of blue cells if:
(a) $n=2$;
(b) $n=10$. | 220 | 0.78125 |
31,625 | Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \ge 4. $$ What is the maximum possible size for the set $S?$ $$ \mathrm a. ~ 12\qquad \mathrm b.~14\qquad \mathrm c. ~16 \qquad \mathrm d. ~18 \qquad \mathrm e. 20 $$ | 16 | 36.71875 |
31,626 | In a WeChat group, five people, namely A, B, C, D, and E, are playing a game of grabbing red packets. There are $4$ red packets in total, each person can grab at most one, and all red packets will be grabbed. Among the $4$ red packets, there are two $2$ yuan packets, one $3$ yuan packet, and one $4$ yuan packet. (Assume that red packets with the same amount are considered the same.) The number of ways that both A and B can grab a red packet is _______ (answer with a number). | 36 | 43.75 |
31,627 | Chess piece called *skew knight*, if placed on the black square, attacks all the gray squares.
What is the largest number of such knights that can be placed on the $8\times 8$ chessboard without them attacking each other?
*Proposed by Arsenii Nikolaiev* | 32 | 64.84375 |
31,628 | If there exists a permutation $a_{1}, a_{2}, \cdots, a_{n}$ of $1, 2, \cdots, n$ such that $k + a_{k}$ $(k = 1, 2, \cdots, n)$ are all perfect squares, then $n$ is called a "good number". Among the set $\{11, 13, 15, 17, 19\}$, which numbers are "good numbers" and which are not? Explain your reasoning. | 11 | 2.34375 |
31,629 | In the pattern of numbers shown, every row begins with a 1 and ends with a 2. Each of the numbers, not on the end of a row, is the sum of the two numbers located immediately above and to the right, and immediately above and to the left. For example, in the fourth row the 9 is the sum of the 4 and the 5 in the third row. If this pattern continues, the sum of all of the numbers in the thirteenth row is: | 12288 | 17.96875 |
31,630 | In triangle $ABC$, where the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, it is given that $2 \sqrt {3}ac\sin B = a^{2} + b^{2} - c^{2}$.
$(1)$ Determine the size of angle $C$;
$(2)$ If $b\sin (\pi - A) = a\cos B$ and $b= \sqrt {2}$, find the area of $\triangle ABC$. | \frac{\sqrt {3} + 1}{4} | 0 |
31,631 | In a plane Cartesian coordinate system, points where both the vertical and horizontal coordinates are integers are called lattice points. The number of lattice points $(x, y)$ satisfying the inequality $(|x|-1)^{2}+(|y|-1)^{2}<2$ is: | 16 | 81.25 |
31,632 | Given real numbers $a$ and $b$ satisfying the equation $\sqrt{(a-1)^2} + \sqrt{(a-6)^2} = 10 - |b+3| - |b-2|$, find the maximum value of $a^2 + b^2$. | 45 | 37.5 |
31,633 | It is known that each side and diagonal of a regular polygon is colored in one of exactly 2018 different colors, and not all sides and diagonals are the same color. If a regular polygon contains no two-colored triangles (i.e., a triangle whose three sides are precisely colored with two colors), then the coloring of the polygon is called "harmonious." Find the largest positive integer $N$ such that there exists a harmonious coloring of a regular $N$-gon. | 2017^2 | 0 |
31,634 |
In a shooting contest, 8 targets are arranged in two columns with 3 targets and one column with 2 targets. The rules are:
- The shooter can freely choose which column to shoot at.
- He must attempt the lowest target not yet hit.
a) If the shooter ignores the second rule, in how many ways can he choose only 3 positions out of the 8 distinct targets to shoot?
b) If the rules are followed, in how many ways can the 8 targets be hit? | 560 | 22.65625 |
31,635 | Three $1 \times 1 \times 1$ cubes are joined face to face in a single row and placed on a table, and have a total of 11 exposed $1 \times 1$ faces. Determine the number of exposed $1 \times 1$ faces when sixty $1 \times 1 \times 1$ cubes are joined face to face in a single row and placed on a table. | 182 | 18.75 |
31,636 | On Jessie's 10th birthday, in 2010, her mother said, "My age is now five times your age." In what year will Jessie's mother be able to say, "My age is now 2.5 times your age," on Jessie's birthday? | 2027 | 64.84375 |
31,637 | Suppose point $P$ is on the curve represented by the equation $\sqrt{(x-5)^2+y^2} - \sqrt{(x+5)^2+y^2} = 6$, and $P$ is also on the line $y=4$. Determine the x-coordinate of point $P$. | -3\sqrt{2} | 3.90625 |
31,638 | Find the smallest positive integer $M$ such that the three numbers $M$, $M+1$, and $M+2$, one of them is divisible by $3^2$, one of them is divisible by $5^2$, and one is divisible by $7^2$. | 98 | 1.5625 |
31,639 | Consider the function defined piecewise by
\[ f(x) = \left\{ \begin{aligned}
2x + 1 & \quad \text{if } x < 1 \\
x^2 & \quad \text{if } x \ge 1
\end{aligned} \right. \]
Determine the value of \( f^{-1}(-3) + f^{-1}(-1) + f^{-1}(1) + f^{-1}(3) + f^{-1}(9) \). | 1 + \sqrt{3} | 70.3125 |
31,640 | Let the roots of the polynomial $f(x) = x^6 + 2x^3 + 1$ be denoted as $y_1, y_2, y_3, y_4, y_5, y_6$. Let $h(x) = x^3 - 3x$. Find the product $\prod_{i=1}^6 h(y_i)$. | 676 | 49.21875 |
31,641 | On the board we write a series of $n$ numbers, where $n \geq 40$ , and each one of them is equal to either $1$ or $-1$ , such that the following conditions both hold:
(i) The sum of every $40$ consecutive numbers is equal to $0$ .
(ii) The sum of every $42$ consecutive numbers is not equal to $0$ .
We denote by $S_n$ the sum of the $n$ numbers of the board. Find the maximum possible value of $S_n$ for all possible values of $n$ . | 20 | 57.8125 |
31,642 | A sphere is inscribed in a right cone with base radius $15$ cm and height $30$ cm. The radius of the sphere can be expressed as $b\sqrt{d} - g$ cm, where $g = b + 6$. What is the value of $b + d$? | 12.5 | 0.78125 |
31,643 | Four vertices of a rectangle include the points $(2, 3)$, $(2, 15)$, and $(13, 3)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 13)^2 + (y - 3)^2 = 16$? | 4\pi | 86.71875 |
31,644 | Determine the total number of distinct, natural-number factors for the number $4^5 \cdot 5^2 \cdot 6^3 \cdot 7!$. | 864 | 79.6875 |
31,645 | Given that the function $g(x)$ satisfies
\[ g(x + g(x)) = 5g(x) \]
for all $x$, and $g(1) = 5$. Find $g(26)$. | 125 | 39.84375 |
31,646 | From the numbers $1, 2, \cdots, 10$, a number $a$ is randomly selected, and from the numbers $-1, -2, \cdots, -10$, a number $b$ is randomly selected. What is the probability that $a^{2} + b$ is divisible by 3? | 0.3 | 0 |
31,647 | The circle inscribed in a right trapezoid divides its larger lateral side into segments of lengths 1 and 4. Find the area of the trapezoid. | 18 | 7.03125 |
31,648 | Given that Star lists the whole numbers $1$ through $30$ once and Emilio copies those numbers replacing each occurrence of the digit $3$ by the digit $2$, find the difference between Star's and Emilio's total sums. | 13 | 3.125 |
31,649 | Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads? | \dfrac{9}{16} | 3.90625 |
31,650 | Mr. Zhang knows that there are three different levels of bus service from location A to location B in the morning: good, average, and poor. However, he does not know their exact schedule. His plan is as follows: He will not board the first bus he sees but will take the second one if it's more comfortable than the first one; otherwise, he will wait for the third bus. What are the probabilities that Mr. Zhang ends up on a good bus and on a poor bus, respectively? | \frac{1}{6} | 0.78125 |
31,651 | Given a sphere resting on a flat surface and a 1.5 m tall post, the shadow of the sphere is 15 m and the shadow of the post is 3 m, determine the radius of the sphere. | 7.5 | 42.1875 |
31,652 | Rectangle $EFGH$ has area $2016$. An ellipse with area $2016\pi$ passes through $E$ and $G$ and has foci at $F$ and $H$. What is the perimeter of the rectangle? | 8\sqrt{1008} | 0 |
31,653 | The moisture content of freshly cut grass is $60\%$, and the moisture content of hay is $15\%$. How much hay will be obtained from one ton of freshly cut grass? | 470.588 | 0 |
31,654 | Miki's father is saving money in a piggy bank for the family's vacation, adding to it once a week. Miki counts and notes how much money has accumulated every week and looks for patterns in the growth. Let $P_{n}$ denote the amount in the piggy bank in the $n$-th week (in forints). Here are a few observations:
(1) $P_{5} = 2P_{3}$,
(2) $P_{8} = P_{3} + 100$,
(3) $P_{9} = P_{4} + P_{7}$.
"The amount of forints has always been even, but it has never been divisible by 3."
"The number of forints today is a perfect square, and I also noticed that dad increases the deposit each week by the same amount that the third deposit exceeded the second deposit; thus the contents of our piggy bank will never be a perfect square again."
Which week does Miki's last observation refer to, and is Miki's prediction correct? | 18 | 0 |
31,655 | Given vectors $\overrightarrow {a}$ = (4, 3) and $\overrightarrow {b}$ = (-1, 2), with $\overrightarrow {m}$ = $\overrightarrow {a}$ - $λ \overrightarrow {b}$ and $\overrightarrow {n}$ = 2$\overrightarrow {a}$ + $\overrightarrow {b}$, find the values of $λ$ such that $\overrightarrow {m}$ is perpendicular to $\overrightarrow {n}$ and $\overrightarrow {m}$ is parallel to $\overrightarrow {n}$. | -\frac{1}{2} | 85.9375 |
31,656 | A cuckoo clock rings "cuckoo" every hour, with the number of rings corresponding to the hour shown by the hour hand (e.g., at 7:00, it rings 7 times). One morning, Maxim approached the clock at 9:05 and started moving the minute hand until 7 hours had passed. How many times did the clock ring "cuckoo" during this period? | 43 | 0.78125 |
31,657 | Define \[P(x) =(x-1^2)(x-2^2)\cdots(x-50^2).\] How many integers $n$ are there such that $P(n)\leq 0$? | 1300 | 12.5 |
31,658 | The increasing sequence of positive integers $a_1, a_2, a_3, \dots$ follows the rule:
\[a_{n+2} = a_{n+1} + a_n\]
for all $n \geq 1$. If $a_6 = 50$, find $a_7$. | 83 | 6.25 |
31,659 | Given that in the expansion of $(1-2x)^{n} (n \in \mathbb{N^*})$, the coefficient of $x^{3}$ is $-80$, find the sum of all the binomial coefficients in the expansion. | 32 | 10.15625 |
31,660 | Estimate the product $(.331)^3$. | 0.037 | 9.375 |
31,661 | Given points A (-3, 5) and B (2, 15), find a point P on the line $l: 3x - 4y + 4 = 0$ such that $|PA| + |PB|$ is minimized. The minimum value is \_\_\_\_\_\_. | 5\sqrt{13} | 30.46875 |
31,662 | Find the biggest positive integer $n$ such that $n$ is $167$ times the amount of it's positive divisors. | 2004 | 35.15625 |
31,663 | The function $f_n (x)\ (n=1,2,\cdots)$ is defined as follows.
\[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\]
Evaluate
\[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\] | e^{1/2} | 46.09375 |
31,664 | Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six?
Clarification: If $A$ is a friend of $B$ then $B$ is a friend of $A$ . | 13 | 25.78125 |
31,665 | A rectangle was cut into three rectangles, two of which have dimensions 9 m x 12 m and 10 m x 15 m. What is the maximum possible area of the original rectangle? Express your answer in square meters. | 330 | 3.125 |
31,666 | It is currently 3:15:15 PM on a 12-hour digital clock. After 196 hours, 58 minutes, and 16 seconds, what will the time be in the format $A:B:C$? What is the sum $A + B + C$? | 52 | 4.6875 |
31,667 | Find the largest possible subset of {1, 2, ... , 15} such that the product of any three distinct elements of the subset is not a square. | 10 | 9.375 |
31,668 | $(1)$ Calculate: $\sqrt{12}-(-\frac{1}{2})^{-1}-|\sqrt{3}+3|+(2023-\pi)^0$<br/>$(2)$ Simplify the algebraic expression $\frac{3x-8}{x-1}-\frac{x+1}{x}÷\frac{{x}^{2}-1}{{x}^{2}-3x}$, and select appropriate integers from $0 \lt x\leqslant 3$ to substitute and find the value. | -1 | 9.375 |
31,669 | Determine the number of arrangements of the letters a, b, c, d, e in a sequence such that neither a nor b is adjacent to c. | 36 | 17.1875 |
31,670 | 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 |
31,671 | Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$ . | 10 | 0 |
31,672 | Calculate the force with which water presses on a dam, whose cross-section has the shape of an isosceles trapezoid. The density of water is $\rho=1000 \, \text{kg} / \text{m}^{3}$, and the acceleration due to gravity $g$ is $10 \, \text{m} / \text{s}^{2}$.
Hint: The pressure at depth $x$ is $\rho g x$.
Given:
\[ a = 6.6 \, \text{m}, \quad b = 10.8 \, \text{m}, \quad h = 4.0 \, \text{m} \] | 640000 | 7.03125 |
31,673 | How many natural numbers between 200 and 400 are divisible by 8? | 26 | 45.3125 |
31,674 | Arrange 3 volunteer teachers to 4 schools, with at most 2 people per school. How many different distribution plans are there? (Answer with a number) | 60 | 47.65625 |
31,675 | 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 | 22.65625 |
31,676 | Points $A$ , $B$ , and $O$ lie in the plane such that $\measuredangle AOB = 120^\circ$ . Circle $\omega_0$ with radius $6$ is constructed tangent to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$ . For all $i \ge 1$ , circle $\omega_i$ with radius $r_i$ is constructed such that $r_i < r_{i - 1}$ and $\omega_i$ is tangent to $\overrightarrow{OA}$ , $\overrightarrow{OB}$ , and $\omega_{i - 1}$ . If
\[
S = \sum_{i = 1}^\infty r_i,
\]
then $S$ can be expressed as $a\sqrt{b} + c$ , where $a, b, c$ are integers and $b$ is not divisible by the square of any prime. Compute $100a + 10b + c$ .
*Proposed by Aaron Lin* | 233 | 0.78125 |
31,677 | If $\frac{1}{(2n-1)(2n+1)}=\frac{a}{2n-1}+\frac{b}{2n+1}$ holds for any natural number $n$, then $a=$______, $b=______. | -\frac{1}{2} | 7.8125 |
31,678 | Let $T = (1+i)^{19} - (1-i)^{19}$, where $i=\sqrt{-1}$. Determine $|T|$. | 512\sqrt{2} | 2.34375 |
31,679 | On a chemistry quiz, there were $7y$ questions. Tim missed $2y$ questions. What percent of the questions did Tim answer correctly? | 71.43\% | 19.53125 |
31,680 | 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} | 1.5625 |
31,681 | The radius of the circumcircle of the acute-angled triangle \(ABC\) is 1. It is known that on this circumcircle lies the center of another circle passing through the vertices \(A\), \(C\), and the orthocenter of triangle \(ABC\). Find \(AC\). | \sqrt{3} | 28.125 |
31,682 | What is the largest four-digit negative integer congruent to $1 \pmod{17}?$ | -1002 | 23.4375 |
31,683 | Given that point \(Z\) moves on \(|z| = 3\) in the complex plane, and \(w = \frac{1}{2}\left(z + \frac{1}{z}\right)\), where the trajectory of \(w\) is the curve \(\Gamma\). A line \(l\) passes through point \(P(1,0)\) and intersects the curve \(\Gamma\) at points \(A\) and \(B\), and intersects the imaginary axis at point \(M\). If \(\overrightarrow{M A} = t \overrightarrow{A P}\) and \(\overrightarrow{M B} = s \overrightarrow{B P}\), find the value of \(t + s\). | -\frac{25}{8} | 9.375 |
31,684 | Let $f(n)$ denote the largest odd factor of $n$ , including possibly $n$ . Determine the value of
\[\frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048},\]
rounded to the nearest integer. | 1365 | 90.625 |
31,685 | Given a fixed point $C(2,0)$ and a line $l: x=8$ on a plane, $P$ is a moving point on the plane, $PQ \perp l$, with the foot of the perpendicular being $Q$, and $\left( \overrightarrow{PC}+\frac{1}{2}\overrightarrow{PQ} \right)\cdot \left( \overrightarrow{PC}-\frac{1}{2}\overrightarrow{PQ} \right)=0$.
(1) Find the trajectory equation of the moving point $P$;
(2) If $EF$ is any diameter of circle $N: x^{2}+(y-1)^{2}=1$, find the maximum and minimum values of $\overrightarrow{PE}\cdot \overrightarrow{PF}$. | 12-4\sqrt{3} | 40.625 |
31,686 | In a new diagram, the grid is composed of squares. The grid is segmented into various levels that step upwards as you move to the right. Determine the area of the shaded region in the following configuration:
- The grid dimensions are 15 units wide and 5 units tall.
- The shaded region fills up from the bottom to a height of 2 units for the first 4 units of width, then rises to fill up to 3 units height until the 9th unit of width, then continues up to 4 units height until the 13th unit of width, and finally fills up to 5 units height until the 15th unit of width.
- An unshaded triangle is formed on the rightmost side, with a base of 15 units along the bottom and a height of 5 units. | 37.5 | 13.28125 |
31,687 | Given sets $A=\{x|x^{2}+2x-3=0,x\in R\}$ and $B=\{x|x^{2}-\left(a+1\right)x+a=0,x\in R\}$.<br/>$(1)$ When $a=2$, find $A\cap C_{R}B$;<br/>$(2)$ If $A\cup B=A$, find the set of real numbers for $a$. | \{1\} | 0.78125 |
31,688 | Alice's password consists of a two-digit number, followed by a symbol from the set {$!, @, #, $, %}, followed by another two-digit number. Calculate the probability that Alice's password consists of an even two-digit number followed by one of {$, %, @}, and another even two-digit number. | \frac{3}{20} | 68.75 |
31,689 | Given a hyperbola with its left and right foci being $F_1$ and $F_2$ respectively, and the length of chord $AB$ on the left branch passing through $F_1$ is 5. If $2a=8$, calculate the perimeter of $\triangle ABF_2$. | 26 | 71.09375 |
31,690 | Given a positive sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, if both $\{a_n\}$ and $\{\sqrt{S_n}\}$ are arithmetic sequences with the same common difference, calculate $S_{100}$. | 2500 | 35.9375 |
31,691 | Evaluate the expression \(\dfrac{\sqrt[4]{7}}{\sqrt[6]{7}}\). | 7^{\frac{1}{12}} | 4.6875 |
31,692 | In the sum $K A N$ each letter stands for a different digit.
$$
\frac{+G A}{R O O}
$$
What is the answer to the subtraction $R N$ ?
$$
\underline{-K G}
$$ | 11 | 18.75 |
31,693 | Alice wants to compare the percentage increase in area when her pizza size increases first from an 8-inch pizza to a 10-inch pizza, and then from the 10-inch pizza to a 14-inch pizza. Calculate the percent increase in area for both size changes. | 96\% | 23.4375 |
31,694 | Given a sequence $\{a_{n}\}$ such that $a_{1}+2a_{2}+\cdots +na_{n}=n$, and a sequence $\{b_{n}\}$ such that ${b_{m-1}}+{b_m}=\frac{1}{{{a_m}}}({m∈N,m≥2})$. Find:<br/>
$(1)$ The general formula for $\{a_{n}\}$;<br/>
$(2)$ The sum of the first $20$ terms of $\{b_{n}\}$. | 110 | 31.25 |
31,695 | A taxi has a starting fare of 10 yuan. After exceeding 10 kilometers, for every additional kilometer, the fare increases by 1.50 yuan (if the increase is less than 1 kilometer, it is rounded up to 1 kilometer; if the increase is more than 1 kilometer but less than 2 kilometers, it is rounded up to 2 kilometers, etc.). Now, traveling from A to B costs 28 yuan. If one walks 600 meters from A before taking a taxi to B, the fare is still 28 yuan. If one takes a taxi from A, passes B, and goes to C, with the distance from A to B equal to the distance from B to C, how much is the taxi fare? | 61 | 49.21875 |
31,696 | A rectangular floor that is $12$ feet wide and $19$ feet long is tiled with rectangular tiles that are $1$ foot by $2$ feet. Find the number of tiles a bug visits when walking from one corner to the diagonal opposite corner. | 30 | 57.03125 |
31,697 | Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that at least 25 cents worth of coins come up heads? | \frac{3}{4} | 0.78125 |
31,698 | How many integers are there in $\{0,1, 2,..., 2014\}$ such that $C^x_{2014} \ge C^{999}{2014}$ ?
Note: $C^{m}_{n}$ stands for $\binom {m}{n}$ | 17 | 60.9375 |
31,699 | Given that cos(15°+α) = $\frac{3}{5}$, where α is an acute angle, find: $$\frac{tan(435° -α)+sin(α-165° )}{cos(195 ° +α )\times sin(105 ° +α )}$$. | \frac{5}{36} | 3.90625 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.