Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
32,900 | Given that $E$ and $F$ are a chord of the circle $C: x^{2}+y^{2}-2x-4y+1=0$, and $CE\bot CF$, $P$ is the midpoint of $EF$. When the chord $EF$ moves on the circle $C$, there exist two points $A$ and $B$ on the line $l: x-y-3=0$ such that $∠APB≥\frac{π}{2}$ always holds. Determine the minimum value of the length of segment $AB$. | 4\sqrt{2} | 8.59375 |
32,901 | Given the function $f(x)=x^3+ax^2+bx+1$, its derivative $f'(x)$ satisfies $f'(1)=2a$, $f'(2)=-b$, where $a$ and $b$ are constants.
(I) Find the values of $a$ and $b$;
(II) Let $g(x)=f'(x)e^{-x}$, find the extreme values of the function $g(x)$. | 15e^{-3} | 21.875 |
32,902 | Let $A$ be the greatest possible value of a product of positive integers that sums to $2014$ . Compute the sum of all bases and exponents in the prime factorization of $A$ . For example, if $A=7\cdot 11^5$ , the answer would be $7+11+5=23$ . | 677 | 55.46875 |
32,903 | Into how many parts can a plane be divided by four lines? Consider all possible cases and make a drawing for each case. | 11 | 42.96875 |
32,904 | Given a circle $⊙O$ with equation $x^{2}+y^{2}=6$, let $P$ be a moving point on the circle. Draw a line $PM$ perpendicular to the x-axis at point $M$, and let $N$ be a point on $PM$ such that $\overrightarrow{PM}= \sqrt {2} \overrightarrow{NM}$.
(I) Find the equation for the trajectory of point $N$, denoted as curve $C$.
(II) Given points $A(2,1)$, $B(3,0)$, if a line passing through point $B$ intersects curve $C$ at points $D$ and $E$, determine whether the sum of slopes $k_{AD}+k_{AE}$ is a constant value. If it is, find this value; if not, explain why. | -2 | 5.46875 |
32,905 | How many distinct numbers can you get by multiplying two or more distinct members of the set $\{1, 2, 3, 7, 13\}$, or taking any number to the power of another (excluding power 1) from the same set? | 23 | 0 |
32,906 | A right cylinder with a height of 5 inches has a radius of 3 inches. A cone with a height of 3 inches and the same radius is placed on top of the cylinder. What is the total surface area of the combined object, in square inches? Express your answer in terms of \(\pi\). | 30\pi + 9\sqrt{2}\pi | 0 |
32,907 | The NIMO problem writers have invented a new chess piece called the *Oriented Knight*. This new chess piece has a limited number of moves: it can either move two squares to the right and one square upward or two squares upward and one square to the right. How many ways can the knight move from the bottom-left square to the top-right square of a $16\times 16$ chess board?
*Proposed by Tony Kim and David Altizio* | 252 | 25 |
32,908 | In the land of Chaina, people pay each other in the form of links from chains. Fiona, originating from Chaina, has an open chain with $2018$ links. In order to pay for things, she decides to break up the chain by choosing a number of links and cutting them out one by one, each time creating $2$ or $3$ new chains. For example, if she cuts the $1111$ th link out of her chain first, then she will have $3$ chains, of lengths $1110$ , $1$ , and $907$ . What is the least number of links she needs to remove in order to be able to pay for anything costing from $1$ to $2018$ links using some combination of her chains?
*2018 CCA Math Bonanza Individual Round #10* | 10 | 61.71875 |
32,909 | Given two vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with an angle of $\frac {2\pi}{3}$ between them, $|\overrightarrow {a}|=2$, $|\overrightarrow {b}|=3$, let $\overrightarrow {m}=3\overrightarrow {a}-2\overrightarrow {b}$ and $\overrightarrow {n}=2\overrightarrow {a}+k\overrightarrow {b}$:
1. If $\overrightarrow {m} \perp \overrightarrow {n}$, find the value of the real number $k$;
2. Discuss whether there exists a real number $k$ such that $\overrightarrow {m} \| \overrightarrow {n}$, and explain the reasoning. | \frac{4}{3} | 21.875 |
32,910 | Kevin Kangaroo starts at 0 on a number line and aims to reach the point 2, but his hopping strategy alternates. On odd-numbered hops, he covers $\frac{1}{2}$ of the distance to his goal, while on even-numbered hops, he covers $\frac{1}{4}$ of the distance remaining to the goal. Determine the total distance Kevin hops after six hops. Express your answer as a fraction. | \frac{485}{256} | 11.71875 |
32,911 | The distances from point \( P \), which lies inside an equilateral triangle, to its vertices are 3, 4, and 5. Find the area of the triangle. | 9 + \frac{25\sqrt{3}}{4} | 0.78125 |
32,912 | Given the numbers \(-2, -1, 0, 1, 2\), arrange them in some order. Compute the difference between the largest and smallest possible values that can be obtained using the iterative average procedure. | 2.125 | 3.125 |
32,913 | Find the maximum value of the function $y=\frac{x}{{{e}^{x}}}$ on the interval $[0,2]$.
A) When $x=1$, $y=\frac{1}{e}$
B) When $x=2$, $y=\frac{2}{{{e}^{2}}}$
C) When $x=0$, $y=0$
D) When $x=\frac{1}{2}$, $y=\frac{1}{2\sqrt{e}}$ | \frac{1}{e} | 31.25 |
32,914 | A diagonal from a vertex of a polygon forms a triangle with the two adjacent sides, so the number of triangles formed by the polygon is at least the number of sides of the polygon. However, the number of triangles cannot exceed the number of ways to choose 2 sides from the polygon, which is given by the combination formula $\binom{n}{2}=\frac{n(n-1)}{2}$. If the number of triangles is at most 2021, determine the number of sides of the polygon. | 2023 | 13.28125 |
32,915 | The diagram shows five circles of the same radius touching each other. A square is drawn so that its vertices are at the centres of the four outer circles. What is the ratio of the area of the shaded parts of the circles to the area of the unshaded parts of the circles? | 2:3 | 0 |
32,916 | Given $\tan ( \frac {π}{4}+x)=- \frac {1}{2}$, find the value of $\tan 2x$ (Part 1) and simplify the trigonometric expression $\sqrt { \frac {1+\sin x}{1-\sin x}}+ \sqrt { \frac {1-\sin x}{1+\sin x}}$ if $x$ is an angle in the second quadrant. Then, find its value (Part 2). | 2\sqrt {10} | 0 |
32,917 | Round 1278365.7422389 to the nearest hundred. | 1278400 | 100 |
32,918 | How many non-empty subsets \( S \) of \( \{1, 2, 3, \ldots, 12\} \) have the following two properties?
1. No two consecutive integers belong to \( S \).
2. If \( S \) contains \( k \) elements, then \( S \) contains no number less than \( k \). | 128 | 0.78125 |
32,919 | Given the quadratic function $f(x)=x^{2}-x+k$, where $k\in\mathbb{Z}$, if the function $g(x)=f(x)-2$ has two distinct zeros in the interval $(-1, \frac{3}{2})$, find the minimum value of $\frac{[f(x)]^{2}+2}{f(x)}$. | \frac{81}{28} | 35.15625 |
32,920 | Given vectors $\overrightarrow{a} = (1, 2)$, $\overrightarrow{b} = (x, 1)$,
1. If $\langle \overrightarrow{a}, \overrightarrow{b} \rangle$ forms an acute angle, find the range of $x$.
2. Find the value of $x$ when $(\overrightarrow{a}+2\overrightarrow{b}) \perp (2\overrightarrow{a}-\overrightarrow{b})$. | \frac{7}{2} | 83.59375 |
32,921 | In triangle ABC, the lengths of the sides opposite to angles A, B, and C are a, b, and c, respectively. The area of triangle ABC is given by $\frac{\sqrt{3}}{6}b(b + c - a\cos C)$.
1. Find angle A.
2. If b = 1 and c = 3, find the value of $\cos(2C - \frac{\pi}{6})$. | -\frac{4\sqrt{3}}{7} | 15.625 |
32,922 | Which of the following is equal to \(\frac{4}{5}\)? | 0.8 | 1.5625 |
32,923 | In triangle $DEF$, the side lengths are $DE = 15$, $EF = 20$, and $FD = 25$. A rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. Letting $WX = \lambda$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \gamma \lambda - \delta \lambda^2.\]
Then the coefficient $\gamma = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 16 | 15.625 |
32,924 | Let \( M = \{1, 2, \ldots, 10\} \), and let \( A_1, A_2, \ldots, A_n \) be distinct non-empty subsets of \( M \). For \( i \neq j \), the intersection \( A_i \cap A_j \) contains at most two elements. Find the maximum value of \( n \). | 175 | 76.5625 |
32,925 | Given the function $f(x)=\cos^4x+2\sin x\cos x-\sin^4x$
$(1)$ Determine the parity, the smallest positive period, and the intervals of monotonic increase for the function $f(x)$.
$(2)$ When $x\in\left[0, \frac{\pi}{2}\right]$, find the maximum and minimum values of the function $f(x)$. | -1 | 53.125 |
32,926 | Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with left focus $F$ and a chord perpendicular to the major axis of length $6\sqrt{2}$, a line passing through point $P(2,1)$ with slope $-1$ intersects $C$ at points $A$ and $B$, where $P$ is the midpoint of $AB$. Find the maximum distance from a point $M$ on ellipse $C$ to focus $F$. | 6\sqrt{2} + 6 | 15.625 |
32,927 | A pyramid is constructed using twenty cubical blocks: the first layer has 10 blocks arranged in a square, the second layer contains 6 blocks arranged in a larger square centered on the 10, the third layer has 3 blocks arranged in a triangle, and finally one block sits on top of the third layer. Each block in layers 2, 3, and 4 has a number assigned which is the sum of the blocks directly below it from the previous layer. If the blocks in the first layer are numbered consecutively from 1 to 10 in any order, what is the smallest possible number that could be assigned to the top block?
A) 45
B) 54
C) 63
D) 72
E) 81 | 54 | 6.25 |
32,928 | In a school there are \( n \) students, each with a different student number. Each student number is a positive factor of \( 60^{60} \), and the H.C.F. of any two student numbers is not a student number in the school. Find the greatest possible value of \( n \). | 3721 | 2.34375 |
32,929 | What digit $B$ will make the number $527B$ divisible by $5$? | 0 \text{ or } 5 | 65.625 |
32,930 | (The full score for this question is 8 points) Arrange 3 male students and 2 female students in a row,
(1) The number of all different arrangements;
(2) The number of arrangements where exactly two male students are adjacent;
(3) The number of arrangements where male students are of different heights and are arranged from tallest to shortest from left to right.
[Answer all questions with numbers] | 20 | 39.0625 |
32,931 | Given a connected simple graph \( G \) with a known number of edges \( e \), where each vertex has some number of pieces placed on it (each piece can only be placed on one vertex of \( G \)). The only operation allowed is when a vertex \( v \) has a number of pieces not less than the number of its adjacent vertices \( d \), you can choose \( d \) pieces from \( v \) and distribute them to the adjacent vertices such that each adjacent vertex gets one piece. If every vertex in \( G \) has a number of pieces less than the number of its adjacent vertices, no operations can be performed.
Find the minimum value of \( m \) such that there exists an initial placement of the pieces with a total of \( m \) pieces, allowing you to perform infinitely many operations starting from this placement. | e | 12.5 |
32,932 | A shooter has a probability of hitting the target of $0.8$ each time. Now, using the method of random simulation to estimate the probability that the shooter hits the target at least $3$ times out of $4$ shots: first, use a calculator to generate random integers between $0$ and $9$, where $0$, $1$ represent missing the target, and $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ represent hitting the target; since there are $4$ shots, every $4$ random numbers are grouped together to represent the results of $4$ shots. After randomly simulating, $20$ groups of random numbers were generated:
$5727$ $0293$ $7140$ $9857$ $0347$ $4373$ $8636$ $9647$ $1417$ $4698$
$0371$ $6233$ $2616$ $8045$ $6011$ $3661$ $9597$ $7424$ $6710$ $4281$
Based on this, estimate the probability that the shooter hits the target at least $3$ times out of $4$ shots. | 0.75 | 64.84375 |
32,933 | A "Kiwi" business owner has two types of "Kiwi" for promotion, type \\(A\\) and type \\(B\\). If you buy 2 pieces of type \\(A\\) "Kiwi" and 1 piece of type \\(B\\) "Kiwi", the total cost is 120 yuan; if you buy 3 pieces of type \\(A\\) "Kiwi" and 2 pieces of type \\(B\\) "Kiwi", the total cost is 205 yuan.
\\((1)\\) Let the price per piece of type \\(A\\) and type \\(B\\) "Kiwi" be \\(a\\) yuan and \\(b\\) yuan, respectively. Find the values of \\(a\\) and \\(b\\).
\\((2)\\) The cost price of each piece of type \\(B\\) "Kiwi" is 40 yuan. According to market research: if sold at the unit price found in \\((1)\\), the "Kiwi" business owner sells 100 pieces of type \\(B\\) "Kiwi" per day; if the selling price increases by 1 yuan, the daily sales volume of type \\(B\\) "Kiwi" decreases by 5 pieces. Find the selling price per piece that maximizes the daily profit of type \\(B\\) "Kiwi", and what is the maximum profit? | 1125 | 51.5625 |
32,934 | Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a>0$, $b>0$) with its left focus $F$ and right vertex $A$, upper vertex $B$. If the distance from point $F$ to line $AB$ is $\frac{5\sqrt{14}}{14}b$, find the eccentricity of the ellipse. | \frac{2}{3} | 25 |
32,935 | Given a sequence $\{a_n\}$ where $a_n = n$, for each positive integer $k$, in between $a_k$ and $a_{k+1}$, insert $3^{k-1}$ twos (for example, between $a_1$ and $a_2$, insert three twos, between $a_2$ and $a_3$, insert $3^1$ twos, between $a_3$ and $a_4$, insert $3^2$ twos, etc.), to form a new sequence $\{d_n\}$. Let $S_n$ denote the sum of the first $n$ terms of the sequence $\{d_n\}$. Find the value of $S_{120}$. | 245 | 0 |
32,936 | In the arithmetic sequence $\{a_n\}$, $a_{10} < 0$, $a_{11} > 0$ and $a_{11} > |a_{10}|$. If the sum of the first $n$ terms of $\{a_n\}$, denoted as $S_n$, is less than $0$, the maximum value of $n$ is ____. | 19 | 57.8125 |
32,937 | A, B, C, and D obtained the top four positions (without ties) in the school, and they made the following statements:
A: "I am neither first nor second."
B: "My position is adjacent to C's position."
C: "I am neither second nor third."
D: "My position is adjacent to B's position."
It is known that A, B, C, and D respectively obtained the places $A, B, C, D$. Determine the four-digit number $\overrightarrow{\mathrm{ABCD}}$. | 4213 | 17.1875 |
32,938 | Given circle $O$, points $E$ and $F$ are on the same side of diameter $\overline{AB}$, $\angle AOE = 60^\circ$, and $\angle FOB = 90^\circ$. Calculate the ratio of the area of the smaller sector $EOF$ to the area of the circle. | \frac{1}{12} | 42.1875 |
32,939 | Given that Let \\(S_{n}\\) and \\(T_{n}\\) be the sums of the first \\(n\\) terms of the arithmetic sequences \\(\{a_{n}\}\\) and \\(\{b_{n}\}\\), respectively, and \\( \frac {S_{n}}{T_{n}}= \frac {n}{2n+1} (n∈N^{*})\\), determine the value of \\( \frac {a_{6}}{b_{6}}\\). | \frac{11}{23} | 78.90625 |
32,940 | Let $I$ be the incenter of $\triangle ABC$ , and $O$ be the excenter corresponding to $B$ . If $|BI|=12$ , $|IO|=18$ , and $|BC|=15$ , then what is $|AB|$ ? | 24 | 6.25 |
32,941 | Given the function $f(x)=\sin 2x-2\cos^2x$ $(x\in\mathbb{R})$.
- (I) Find the value of $f\left( \frac{\pi}{3}\right)$;
- (II) When $x\in\left[0, \frac{\pi}{2}\right]$, find the maximum value of the function $f(x)$ and the corresponding value of $x$. | \frac{3\pi}{8} | 17.1875 |
32,942 | Given a sequence ${a_n}$ defined by $|a_{n+1}|+|a_n|=3$ for any positive integer $n$ and $a_1=2$, find the minimum value of the sum $S_{2019}$ of its first 2019 terms. | -3025 | 0 |
32,943 | Given Elmer’s new car provides a 70% better fuel efficiency and uses a type of fuel that is 35% more expensive per liter, calculate the percentage by which Elmer will save money on fuel costs if he uses his new car for his journey. | 20.6\% | 0 |
32,944 | A ray starting from point $A(-4,1)$ reflects off the line $l_{1}: x-y+3=0$ and the reflected ray passes through point $B(-3,2)$. Find the slope of the line containing the reflected ray. | -3 | 34.375 |
32,945 | Let $a$ and $b$ be positive integers such that all but $2009$ positive integers are expressible in the form $ma + nb$ , where $m$ and $n$ are nonnegative integers. If $1776 $ is one of the numbers that is not expressible, find $a + b$ . | 133 | 20.3125 |
32,946 | How many $5$ -digit numbers $N$ (in base $10$ ) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$ ? (The leading digit of $N$ cannot be zero.)
*Based on a proposal by Yannick Yao* | 256 | 60.15625 |
32,947 | Given $\sin\left(\frac{\pi}{3}+\frac{\alpha}{6}\right)=-\frac{3}{5}$, $\cos\left(\frac{\pi}{12}-\frac{\beta}{2}\right)=-\frac{12}{13}$, $-5\pi < \alpha < -2\pi$, $-\frac{11\pi}{6} < \beta < \frac{\pi}{6}$,
Find the value of $\sin \left(\frac{\alpha }{6}+\frac{\beta }{2}+\frac{\pi }{4}\right)$. | \frac{16}{65} | 11.71875 |
32,948 | In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $B$ and $D$ coincide, forming the pentagon $ABEFC$. What is the length of segment $EF$? Express your answer in simplest radical form. | 4\sqrt{5} | 23.4375 |
32,949 | Three $1 \times 1 \times 1$ cubes are joined face to face in a single row and placed on a table. The cubes have a total of 11 exposed $1 \times 1$ faces. If sixty $1 \times 1 \times 1$ cubes are joined face to face in a single row and placed on a table, how many $1 \times 1$ faces are exposed? | 182 | 14.0625 |
32,950 | The probability that a light bulb lasts more than 1000 hours is 0.2. Determine the probability that 1 out of 3 light bulbs fails after 1000 hours of use. | 0.096 | 53.90625 |
32,951 | In an institute, there are truth-tellers, who always tell the truth, and liars, who always lie. One day, each of the employees made two statements:
1) There are fewer than ten people in the institute who work more than I do.
2) In the institute, at least one hundred people have a salary greater than mine.
It is known that the workload of all employees is different, and their salaries are also different. How many people work in the institute? | 110 | 6.25 |
32,952 | Given that Asha's study times were 40, 60, 50, 70, 30, 55, 45 minutes each day of the week and Sasha's study times were 50, 70, 40, 100, 10, 55, 0 minutes each day, find the average number of additional minutes per day Sasha studied compared to Asha. | -3.57 | 1.5625 |
32,953 | How many four-digit positive integers $x$ satisfy $3874x + 481 \equiv 1205 \pmod{31}$? | 290 | 89.84375 |
32,954 | Let \( A(2,0) \) be a fixed point on the plane, \( P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right) \) be a moving point. When \( t \) changes from \( 15^{\circ} \) to \( 45^{\circ} \), the area swept by the line segment \( AP \) is ______. | \frac{\pi}{6} | 19.53125 |
32,955 | For some positive integers $c$ and $d$, the product \[\log_c(c+1) \cdot \log_{c+1} (c+2) \dotsm \log_{d-2} (d-1) \cdot\log_{d-1} d\]contains exactly $930$ terms, and its value is $3.$ Compute $c+d.$ | 1010 | 76.5625 |
32,956 | On the eve of the 2010 Guangzhou Asian Games, a 12-person tour group took a commemorative photo near a venue of the Asian Games. They initially stood in a formation with 4 people in the front row and 8 people in the back row. Now, the photographer plans to keep the order of the front row unchanged, and move 2 people from the back row to the front row, ensuring that the 2 moved people are not adjacent in the front row. The number of different ways to adjust their positions is _____. (Answer in numerals) | 560 | 14.0625 |
32,957 | Two 8-sided dice are tossed (each die has faces numbered 1 to 8). What is the probability that the sum of the numbers shown on the dice is either a prime or a multiple of 4? | \frac{39}{64} | 13.28125 |
32,958 | Rohan wants to cut a piece of string into nine pieces of equal length. He marks his cutting points on the string. Jai wants to cut the same piece of string into only eight pieces of equal length. He marks his cutting points on the string. Yuvraj then cuts the string at all the cutting points that are marked. How many pieces of string does Yuvraj obtain?
A 15
B 16
C 17
D 18
E 19 | 16 | 60.9375 |
32,959 | A bag contains $5$ small balls of the same shape and size, with $2$ red balls and $3$ white balls. Three balls are randomly drawn from the bag.<br/>$(1)$ Find the probability that exactly one red ball is drawn;<br/>$(2)$ Let the random variable $X$ represent the number of red balls drawn. Find the distribution of the random variable $X$. | \frac{3}{10} | 6.25 |
32,960 | Given a sphere O with a radius of 2, a cone is inscribed in the sphere O. When the volume of the cone is maximized, find the radius of the sphere inscribed in the cone. | \frac{4(\sqrt{3} - 1)}{3} | 44.53125 |
32,961 | In triangle $ABC$, $AB=15$ and $AC=8$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and the incenter $I$ of triangle $ABC$ is on the segment $AD$. Let the midpoint of $AD$ be $M$. Find the ratio of $IP$ to $PD$ where $P$ is the intersection of $AI$ and $BM$. | 1:1 | 1.5625 |
32,962 | Find the maximum of the expression
$$
|| \ldots|| x_{1}-x_{2}\left|-x_{3}\right|-\ldots\left|-x_{2023}\right|,
$$
where \( x_{1}, x_{2}, \ldots, x_{2023} \) are distinct natural numbers between 1 and 2023. | 2022 | 26.5625 |
32,963 | Consider a triangle with vertices at $(2, 1)$, $(4, 7)$, and $(7, 3)$. This triangle is reflected about the line $y=4$. Find the area of the union of the original and the reflected triangles. | 26 | 34.375 |
32,964 | In quadrilateral \( \square ABCD \), point \( M \) lies on diagonal \( BD \) with \( MD = 3BM \). Let \( AM \) intersect \( BC \) at point \( N \). Find the value of \( \frac{S_{\triangle MND}}{S_{\square ABCD}} \). | 1/8 | 37.5 |
32,965 | (1) Given a sequence $\{a_n\}$ that satisfies $a_1a_2…a_n=n+1$, find $a_3=$
(2) Let $\overrightarrow{e_1}, \overrightarrow{e_2}$ be unit vectors, where $\overrightarrow{a}=2\overrightarrow{e_1}+\overrightarrow{e_2}, \overrightarrow{b}=\overrightarrow{e_2}$, and $\overrightarrow{a} \cdot \overrightarrow{b}=2$, find $|\overrightarrow{a}|=$
(3) For an arithmetic sequence with the first term $-24$, starting from the 9th term, the terms become positive. Find the range of the common difference $d$.
(4) In the sequence $\{a_n\}$, if the point $(n,a_n)$ lies on a fixed line $l$ passing through the point $(5,3)$, find the sum of the first 9 terms of the sequence, $S_9=$
(5) In rectangle $ABCD$, where $AB=2AD=2$, if $P$ is a moving point on $DC$, find the minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}-\overrightarrow{PA} \cdot \overrightarrow{BC}=$
(6) In $\triangle ABC$, where angle $A$ is the largest and angle $C$ is the smallest, and $A=2C$, $a+c=2b$, find the ratio $a:b:c=$ | 6:5:4 | 13.28125 |
32,966 | As shown in the diagram, \( D \), \( E \), and \( F \) are points on the sides \( BC \), \( CA \), and \( AB \) of \(\triangle ABC\), respectively, and \( AD \), \( BE \), \( CF \) intersect at point \( G \). Given that the areas of \(\triangle BDG\), \(\triangle CDG\), and \(\triangle AEG\) are 8, 6, and 14 respectively, find the area of \(\triangle ABC\). | 63 | 0 |
32,967 | How many divisors of $9!$ are multiples of $10$? | 70 | 82.8125 |
32,968 | In triangle $DEF$, $DE = 6$, $EF = 8$, and $FD = 10$. Point $Q$ is randomly selected inside triangle $DEF$. What is the probability that $Q$ is closer to $D$ than it is to either $E$ or $F$? | \frac{1}{2} | 10.15625 |
32,969 | A permutation $a_1,a_2,\cdots ,a_6$ of numbers $1,2,\cdots ,6$ can be transformed to $1,2,\cdots,6$ by transposing two numbers exactly four times. Find the number of such permutations. | 360 | 78.125 |
32,970 | In a tournament, any two players play against each other. Each player earns one point for a victory, 1/2 for a draw, and 0 points for a loss. Let \( S \) be the set of the 10 lowest scores. We know that each player obtained half of their score by playing against players in \( S \).
a) What is the sum of the scores of the players in \( S \)?
b) Determine how many participants are in the tournament.
Note: Each player plays only once with each opponent. | 25 | 14.0625 |
32,971 | Consider a circle centered at $O$ . Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are of distance $2$ apart such that $AC < AD$ . We can write $\tan \angle BOD =\frac{a}{b}$ , where $a, b$ are positive integers such that gcd $(a, b) = 1$ . Compute $a + b$ . | 113 | 4.6875 |
32,972 | How many ordered triplets $(a, b, c)$ of positive integers such that $30a + 50b + 70c \leq 343$ . | 30 | 98.4375 |
32,973 | Evaluate the product $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{6561} \cdot \frac{19683}{1}$. | 243 | 36.71875 |
32,974 | The number of different arrangements possible for 6 acts, if the original sequence of the 4 acts remains unchanged. | 30 | 9.375 |
32,975 | Zhang Hua, Li Liang, and Wang Min each sent out x, y, z New Year's cards, respectively. If it is known that the least common multiple of x, y, z is 60, the greatest common divisor of x and y is 4, and the greatest common divisor of y and z is 3, then how many New Year's cards did Zhang Hua send out? | 20 | 36.71875 |
32,976 | A Mediterranean polynomial has only real roots and it is of the form
\[ P(x) = x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \] with real coefficients $a_0\ldots,a_7$ . Determine the largest real number that occurs as a root of some Mediterranean polynomial.
*(Proposed by Gerhard Woeginger, Austria)* | 11 | 20.3125 |
32,977 | Let \( a_1, a_2, \dots \) be a sequence of positive real numbers such that
\[ a_n = 7a_{n-1} - 2n \] for all \( n > 1 \). Find the smallest possible value of \( a_1 \). | \frac{13}{18} | 14.84375 |
32,978 | A student, Liam, wants to earn a total of 30 homework points. For earning the first four homework points, he has to do one homework assignment each; for the next four points, he has to do two homework assignments each; and so on, such that for every subsequent set of four points, the number of assignments he needs to do increases by one. What is the smallest number of homework assignments necessary for Liam to earn all 30 points? | 128 | 7.03125 |
32,979 | Find the distance between the foci of the ellipse and its eccentricity when defined by the equation:
\[\frac{x^2}{16} + \frac{y^2}{9} = 8.\] | \frac{\sqrt{7}}{4} | 69.53125 |
32,980 | Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and
\[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots
\]
Determine real number $ a$ such that if $ x_1 > a$ , then the sequence is monotonically decreasing, and if $ 0 < x_1 < a$ , then the sequence is not monotonic. | 8^{1/8} | 8.59375 |
32,981 | When five students are lining up to take a photo, and two teachers join in, with the order of the five students being fixed, calculate the total number of ways for the two teachers to stand in line with the students for the photo. | 42 | 46.875 |
32,982 | Given the following conditions:①$\left(2b-c\right)\cos A=a\cos C$, ②$a\sin\ \ B=\sqrt{3}b\cos A$, ③$a\cos C+\sqrt{3}c\sin A=b+c$, choose one of these three conditions and complete the solution below.<br/>Question: In triangle $\triangle ABC$, with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, satisfying ______, and $c=4$, $b=3$.<br/>$(1)$ Find the area of $\triangle ABC$;<br/>$(2)$ If $D$ is the midpoint of $BC$, find the cosine value of $\angle ADC$.<br/>Note: If multiple conditions are chosen and answered separately, the first answer will be scored. | \frac{7\sqrt{481}}{481} | 2.34375 |
32,983 | In Class 5A, a survey was conducted to find out which fruits the students like. It turned out that:
- 13 students like apples,
- 11 students like plums,
- 15 students like peaches, and
- 6 students like melons.
A student can like more than one fruit. Each student who likes plums also likes either apples or peaches (but not both). Each student who likes peaches also likes either plums or melons (but not both).
What is the minimum number of students that can be in Class 5A? | 22 | 0.78125 |
32,984 | Let $ABC$ be triangle such that $|AB| = 5$ , $|BC| = 9$ and $|AC| = 8$ . The angle bisector of $\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\widehat{CAB}$ meets $BC$ at $Y$ . Let $Z$ be the intersection of lines $XY$ and $AC$ . What is $|AZ|$ ? | 10 | 3.90625 |
32,985 | In the regular tetrahedron \(ABCD\), points \(E\) and \(F\) are on edges \(AB\) and \(AC\) respectively, such that \(BE = 3\) and \(EF = 4\), and \(EF\) is parallel to face \(BCD\). What is the area of \(\triangle DEF\)? | 2\sqrt{33} | 0.78125 |
32,986 | A sealed bottle, which contains water, has been constructed by attaching a cylinder of radius \(1 \mathrm{~cm}\) to a cylinder of radius \(3 \mathrm{~cm}\). When the bottle is right side up, the height of the water inside is \(20 \mathrm{~cm}\). When the bottle is upside down, the height of the liquid is \(28 \mathrm{~cm}\). What is the total height, in cm, of the bottle? | 29 | 2.34375 |
32,987 | Let $V$ be a 10-dimensional real vector space and $U_1,U_2$ two linear subspaces such that $U_1 \subseteq U_2, \dim U_1 =3, \dim U_2=6$ . Let $\varepsilon$ be the set of all linear maps $T: V\rightarrow V$ which have $T(U_1)\subseteq U_1, T(U_2)\subseteq U_2$ . Calculate the dimension of $\varepsilon$ . (again, all as real vector spaces) | 67 | 33.59375 |
32,988 | Assume that the probability of a certain athlete hitting the bullseye with a dart is $40\%$. Now, the probability that the athlete hits the bullseye exactly once in two dart throws is estimated using a random simulation method: first, a random integer value between $0$ and $9$ is generated by a calculator, where $1$, $2$, $3$, and $4$ represent hitting the bullseye, and $5$, $6$, $7$, $8$, $9$, $0$ represent missing the bullseye. Then, every two random numbers represent the results of two throws. A total of $20$ sets of random numbers were generated in the random simulation:<br/>
| $93$ | $28$ | $12$ | $45$ | $85$ | $69$ | $68$ | $34$ | $31$ | $25$ |
|------|------|------|------|------|------|------|------|------|------|
| $73$ | $93$ | $02$ | $75$ | $56$ | $48$ | $87$ | $30$ | $11$ | $35$ |
Based on this estimation, the probability that the athlete hits the bullseye exactly once in two dart throws is ______. | 0.5 | 90.625 |
32,989 | A child builds towers using identically shaped cubes of different colors. Determine the number of different towers with a height of 6 cubes that can be built with 3 yellow cubes, 3 purple cubes, and 2 orange cubes (Two cubes will be left out). | 350 | 0.78125 |
32,990 | A bowl contained 320 grams of pure white sugar. Mixture \( Y \) was formed by taking \( x \) grams of the white sugar out of the bowl, adding \( x \) grams of brown sugar to the bowl, and then mixing uniformly. In Mixture \( Y \), the ratio of the mass of the white sugar to the mass of the brown sugar, expressed in lowest terms, was \( w: b \). Mixture \( Z \) is formed by taking \( x \) grams of Mixture \( Y \) out of the bowl, adding \( x \) grams of brown sugar to the bowl, and then mixing uniformly. In Mixture \( Z \), the ratio of the mass of the white sugar to the mass of the brown sugar is \( 49: 15 \). The value of \( x+w+b \) is: | 48 | 14.84375 |
32,991 | If $100^a = 4$ and $100^b = 5,$ then find $20^{(1 - a - b)/(2(1 - b))}.$ | \sqrt{20} | 0 |
32,992 | A mix of blue paint and yellow paint is required such that the ratio of blue paint to yellow paint is 5 to 3. If Julia plans to create 45 cans of this mix and each can contains the same volume, determine how many cans of blue paint will Julia need. | 28 | 88.28125 |
32,993 | Consider a pentagonal prism with seven faces, fifteen edges, and ten vertices. One of its faces will be used as the base for a new pyramid. Calculate the maximum value of the sum of the number of exterior faces, vertices, and edges of the combined solid (prism and pyramid). | 42 | 26.5625 |
32,994 | Suppose that $f(x)$ and $g(x)$ are functions which satisfy the equations $f(g(x)) = 2x^2$ and $g(f(x)) = x^4$ for all $x \ge 1$. If $g(4) = 16$, compute $[g(2)]^4$. | 16 | 36.71875 |
32,995 | Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | 97.65625 |
32,996 | A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by 1 and reduces the number of rows by 2, there are exactly enough places in the new formation for each band member. What is the largest number of members the band could have? | 98 | 94.53125 |
32,997 | What is the degree of the polynomial $(4 +5x^3 +100 +2\pi x^4 + \sqrt{10}x^4 +9)$? | 4 | 99.21875 |
32,998 | Evaluate $\left\lceil3\left(6-\frac12\right)\right\rceil$. | 17 | 100 |
32,999 | Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work? | 6 | 99.21875 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.