Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
14,900 | If 8 is added to the square of 5, the result is divisible by: | 11 | 15.625 |
14,901 | For a natural number \( N \), if at least five of the natural numbers from 1 to 9 can divide \( N \) evenly, then \( N \) is called a "Five-Divisible Number." Find the smallest "Five-Divisible Number" that is greater than 2000. | 2004 | 99.21875 |
14,902 | Given points \( A(4,0) \) and \( B(2,2) \) are inside the ellipse \( \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 \), and \( M \) is a point on the ellipse, find the maximum value of \( |MA| + |MB| \). | 10 + 2\sqrt{10} | 7.03125 |
14,903 | Given a cube $ABCD$-$A\_1B\_1C\_1D\_1$ with edge length $1$, point $M$ is the midpoint of $BC\_1$, and $P$ is a moving point on edge $BB\_1$. Determine the minimum value of $AP + MP$. | \frac{\sqrt{10}}{2} | 3.125 |
14,904 | Consider a four-digit natural number with the following property: if we swap its first two digits with the second two digits, we get a four-digit number that is 99 less.
How many such numbers are there in total, and how many of them are divisible by 9? | 10 | 65.625 |
14,905 | Pascal's Triangle's interior numbers are defined beginning from the third row. Calculate the sum of the cubes of the interior numbers in the fourth row. Following that calculation, if the sum of the cubes of the interior numbers of the fifth row is 468, find the sum of the cubes of the interior numbers of the sixth row... | 14750 | 3.125 |
14,906 | Find an eight-digit palindrome that is a multiple of three, composed of the digits 0 and 1, given that all its prime divisors only use the digits 1, 3, and %. (Palindromes read the same forwards and backwards, for example, 11011). | 10111101 | 33.59375 |
14,907 | Twenty-eight 4-inch wide square posts are evenly spaced with 4 feet between adjacent posts to enclose a rectangular field. The rectangle has 6 posts on each of the longer sides (including the corners). What is the outer perimeter, in feet, of the fence? | 112 | 0 |
14,908 | A rabbit escapes and runs 100 steps ahead before a dog starts chasing it. The rabbit can cover 8 steps in the same distance that the dog can cover in 3 steps. Additionally, the dog can run 4 steps in the same time that the rabbit can run 9 steps. How many steps must the dog run at least to catch up with the rabbit? | 240 | 4.6875 |
14,909 | A trapezoid is divided into seven strips of equal width. What fraction of the trapezoid's area is shaded? Explain why your answer is correct. | 4/7 | 13.28125 |
14,910 | Consider triangle \(ABC\) where \(BC = 7\), \(CA = 8\), and \(AB = 9\). \(D\) and \(E\) are the midpoints of \(BC\) and \(CA\), respectively, and \(AD\) and \(BE\) meet at \(G\). The reflection of \(G\) across \(D\) is \(G'\), and \(G'E\) meets \(CG\) at \(P\). Find the length \(PG\). | \frac{\sqrt{145}}{9} | 0.78125 |
14,911 | Let $ABC$ be a triangle with area $K$ . Points $A^*$ , $B^*$ , and $C^*$ are chosen on $AB$ , $BC$ , and $CA$ respectively such that $\triangle{A^*B^*C^*}$ has area $J$ . Suppose that \[\frac{AA^*}{AB}=\frac{BB^*}{BC}=\frac{CC^*}{CA}=\frac{J}{K}=x\] for some $0<x<1$ . What is $x$ ?
*2019 CCA Math Bonan... | 1/3 | 3.125 |
14,912 | A pedestrian departed from point \( A \) to point \( B \). After walking 8 km, a second pedestrian left point \( A \) following the first pedestrian. When the second pedestrian had walked 15 km, the first pedestrian was halfway to point \( B \), and both pedestrians arrived at point \( B \) simultaneously. What is the ... | 40 | 17.1875 |
14,913 | In what ratio does the angle bisector of an acute angle of an isosceles right triangle divide the area of the triangle? | 1 : \sqrt{2} | 0.78125 |
14,914 | How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it? | 576 | 8.59375 |
14,915 | The image depicts a top-down view of a three-layered pyramid made of 14 identical cubes. Each cube is assigned a natural number in such a way that the numbers corresponding to the cubes in the bottom layer are all different, and the number on any other cube is the sum of the numbers on the four adjacent cubes from the ... | 64 | 1.5625 |
14,916 | A department needs to arrange a duty schedule for the National Day holiday (a total of 8 days) for four people: A, B, C, and D. It is known that:
- A and B each need to be on duty for 4 days.
- A cannot be on duty on the first day, and A and B cannot be on duty on the same day.
- C needs to be on duty for 3 days and c... | 700 | 0 |
14,917 | A sequence of numbers is written on the blackboard: \(1, 2, 3, \cdots, 50\). Each time, the first 4 numbers are erased, and the sum of these 4 erased numbers is written at the end of the sequence, creating a new sequence. This operation is repeated until there are fewer than 4 numbers remaining on the blackboard. Deter... | 755 | 0 |
14,918 | Given an arithmetic sequence $\{a\_n\}$, the sum of its first $n$ terms, $S\_n$, satisfies $S\_3=0$ and $S\_5=-5$. The sum of the first 2016 terms of the sequence $\{ \frac{1}{a_{2n-1}a_{2n+1}} \}$ is $\_\_\_\_\_\_\_\_.$ | -\frac{2016}{4031} | 19.53125 |
14,919 | If a passenger travels from Moscow to St. Petersburg by a regular train, it will take him 10 hours. If he takes the express train, which he has to wait for more than 2.5 hours, he will arrive 3 hours earlier than the regular train. Find the ratio of the speeds of the express train and the regular train, given that 2 ho... | 2.5 | 0 |
14,920 | Given that $x \in (1,5)$, find the minimum value of the function $y= \frac{2}{x-1}+ \frac{1}{5-x}$. | \frac{3+2 \sqrt{2}}{4} | 24.21875 |
14,921 | At an observation station $C$, the distances to two lighthouses $A$ and $B$ are $300$ meters and $500$ meters, respectively. Lighthouse $A$ is observed at $30^{\circ}$ north by east from station $C$, and lighthouse $B$ is due west of station $C$. Calculate the distance between the two lighthouses $A$ and $B$. | 700 | 21.09375 |
14,922 | Given that \(a\), \(b\), \(c\), and \(d\) are four positive prime numbers such that the product of these four prime numbers is equal to the sum of 55 consecutive positive integers, find the smallest possible value of \(a + b + c + d\). Note that the four numbers \(a\), \(b\), \(c\), and \(d\) are not necessarily distin... | 28 | 8.59375 |
14,923 | The length of a chord intercepted on the circle $x^2+y^2-2x+4y-20=0$ by the line $5x-12y+c=0$ is 8. Find the value(s) of $c$. | -68 | 49.21875 |
14,924 | Let \(A\) and \(G\) be two opposite vertices of a cube with unit edge length. What is the distance between the plane determined by the vertices adjacent to \(A\), denoted as \(S_{A}\), and the plane determined by the vertices adjacent to \(G\), denoted as \(S_{G}\)? | \frac{\sqrt{3}}{3} | 57.8125 |
14,925 | Find the number of ways that 2010 can be written as a sum of one or more positive integers in non-decreasing order such that the difference between the last term and the first term is at most 1. | 2010 | 3.90625 |
14,926 | We know about a convex pentagon that each side is parallel to one of its diagonals. What can be the ratio of the length of a side to the length of the diagonal parallel to it? | \frac{\sqrt{5} - 1}{2} | 49.21875 |
14,927 | The sum of the non-negative numbers \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}\) is 1. Let \(M\) be the maximum of the quantities \(a_{1} + a_{2} + a_{3}, a_{2} + a_{3} + a_{4}, a_{3} + a_{4} + a_{5}, a_{4} + a_{5} + a_{6}, a_{5} + a_{6} + a_{7}\).
How small can \(M\) be? | 1/3 | 18.75 |
14,928 | Let \( S \) be a set of size 11. A random 12-tuple \((s_1, s_2, \ldots, s_{12})\) of elements of \( S \) is chosen uniformly at random. Moreover, let \(\pi: S \rightarrow S\) be a permutation of \( S \) chosen uniformly at random. The probability that \( s_{i+1} \neq \pi(s_i) \) for all \( 1 \leq i \leq 12 \) (where \(... | 1000000000004 | 0 |
14,929 | Let $M = 123456789101112\dots5354$ be the number that results from writing the integers from $1$ to $54$ consecutively. What is the remainder when $M$ is divided by $55$? | 44 | 6.25 |
14,930 | In triangle \( \triangle ABC \), the three interior angles \( \angle A, \angle B, \angle C \) satisfy \( \angle A = 3 \angle B = 9 \angle C \). Find the value of
\[ \cos A \cdot \cos B + \cos B \cdot \cos C + \cos C \cdot \cos A = \quad . \] | -1/4 | 0 |
14,931 | Given a set of data is multiplied by 2 and then reduced by 80 for each data point, resulting in a new set of data with an average of 1.2 and a variance of 4.4, determine the average and variance of the original data. | 1.1 | 45.3125 |
14,932 | Let $f(x)=\sin\left(2x+\frac{\pi}{3}\right)+\sqrt{3}\sin^2x-\sqrt{3}\cos^2x-\frac{1}{2}$.
$(1)$ Find the smallest positive period and the interval of monotonicity of $f(x)$;
$(2)$ If $x_0\in\left[\frac{5\pi}{12},\frac{2\pi}{3}\right]$ and $f(x_{0})=\frac{\sqrt{3}}{3}-\frac{1}{2}$, find the value of $\cos 2x_{0}$. | -\frac{3+\sqrt{6}}{6} | 0.78125 |
14,933 | What is the sum of the digits of the integer which is equal to \(6666666^{2} - 3333333^{2}\)? | 63 | 53.125 |
14,934 | A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters, given that the area of the cross is $810 \mathrm{~cm}^{2}$. | 36 | 7.03125 |
14,935 | Given square $ABCD$, points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ bisect $\angle ADC$. Calculate the ratio of the area of $\triangle DEF$ to the area of square $ABCD$. | \frac{1}{4} | 45.3125 |
14,936 | In some cells of a \(10 \times 10\) board, there are fleas. Every minute, the fleas jump simultaneously to an adjacent cell (along the sides). Each flea jumps strictly in one of the four directions parallel to the sides of the board, maintaining its direction as long as possible; otherwise, it changes to the opposite d... | 40 | 0.78125 |
14,937 | Three people, A, B, and C, start from point $A$ to point $B$. A starts at 8:00, B starts at 8:20, and C starts at 8:30. They all travel at the same speed. Ten minutes after C starts, the distance from A to point $B$ is exactly half the distance from B to point $B$. At this time, C is 2015 meters away from point $B$. Ho... | 2418 | 36.71875 |
14,938 | In triangle $MPQ$, a line parallel to side $MQ$ intersects side $MP$, the median $MM_1$, and side $PQ$ at points $D$, $E$, and $F$ respectively. It is known that $DE = 5$ and $EF = 7$. What is the length of $MQ$? | 17 | 1.5625 |
14,939 | Define a sequence of integers by $T_1 = 2$ and for $n\ge2$ , $T_n = 2^{T_{n-1}}$ . Find the remainder when $T_1 + T_2 + \cdots + T_{256}$ is divided by 255.
*Ray Li.* | 20 | 19.53125 |
14,940 | The numbers \(a, b, c, d\) belong to the interval \([-11.5, 11.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 552 | 1.5625 |
14,941 | The triangle \( \triangle ABC \) has side \( AC \) with length \( 24 \text{ cm} \) and a height from vertex \( B \) with length \( 25 \text{ cm} \). Side \( AB \) is divided into five equal parts, with division points labeled \( K, L, M, N \) from \( A \) to \( B \). Each of these points has a parallel line drawn to si... | 120 | 13.28125 |
14,942 | Given that \( P \) is a point on the hyperbola \( C: \frac{x^{2}}{4}-\frac{y^{2}}{12}=1 \), \( F_{1} \) and \( F_{2} \) are the left and right foci of \( C \), and \( M \) and \( I \) are the centroid and incenter of \(\triangle P F_{1} F_{2}\) respectively, if \( M I \) is perpendicular to the \( x \)-axis, then the r... | \sqrt{6} | 2.34375 |
14,943 | At the "Economics and Law" congress, a "Best of the Best" tournament was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. During one match, participants had to ask each other questions within a limited time and record correct answers. Each participant played with each other... | 105 | 0.78125 |
14,944 | There are several solid and hollow circles arranged in a certain pattern as follows: ●○●●○●●●○●○●●○●●●… Among the first 2001 circles, find the number of hollow circles. | 667 | 47.65625 |
14,945 | Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for any real numbers \( x \) and \( y \), \( f(2x) + f(2y) = f(x+y) f(x-y) \). Additionally, \( f(\pi) = 0 \) and \( f(x) \) is not identically zero. What is the period of \( f(x) \)? | 4\pi | 19.53125 |
14,946 | Given that the sequence $\left\{a_{n}\right\}$ has a period of 7 and the sequence $\left\{b_{n}\right\}$ has a period of 13, determine the maximum value of $k$ such that there exist $k$ consecutive terms satisfying
\[ a_{1} = b_{1}, \; a_{2} = b_{2}, \; \cdots , \; a_{k} = b_{k} \] | 91 | 10.9375 |
14,947 | The three-tiered "pyramid" shown in the image is built from $1 \mathrm{~cm}^{3}$ cubes and has a surface area of $42 \mathrm{~cm}^{2}$. We made a larger "pyramid" based on this model, which has a surface area of $2352 \mathrm{~cm}^{2}$. How many tiers does it have? | 24 | 3.125 |
14,948 | Triangle \( ABC \) is isosceles, and \( \angle ABC = x^\circ \). If the sum of the possible measures of \( \angle BAC \) is \( 240^\circ \), find \( x \). | 20 | 3.125 |
14,949 | The sequence $\{a_n\}$ satisfies $a_n+a_{n+1}=n^2+(-1)^n$. Find the value of $a_{101}-a_1$. | 5150 | 16.40625 |
14,950 | We inscribe a cone around a sphere of unit radius. What is the minimum surface area of the cone? | 8\pi | 3.125 |
14,951 | Determine one of the symmetry axes of the function $y = \cos 2x - \sin 2x$. | -\frac{\pi}{8} | 17.96875 |
14,952 | Given that point $P(-15a, 8a)$ is on the terminal side of angle $\alpha$, where $a \in \mathbb{R}$ and $a \neq 0$, find the values of the six trigonometric functions of $\alpha$. | -\frac{15}{8} | 0 |
14,953 | In an isosceles trapezoid \(ABCD\), the side \(AB\) and the shorter base \(BC\) are both equal to 2, and \(BD\) is perpendicular to \(AB\). Find the area of this trapezoid. | 3\sqrt{3} | 14.84375 |
14,954 | Let $p,$ $q,$ $r,$ $s$ be real numbers such that
\[\frac{(p - q)(r - s)}{(q - r)(s - p)} = \frac{3}{4}.\]Find the sum of all possible values of
\[\frac{(p - r)(q - s)}{(p - q)(r - s)}.\] | -1 | 4.6875 |
14,955 | Vasya wrote a note on a piece of paper, folded it in four, and wrote the inscription "MAME" on top. Then he unfolded the note, wrote something else, folded it again along the crease lines at random (not necessarily in the same way as before), and left it on the table with a random side facing up. Find the probability t... | 1/8 | 11.71875 |
14,956 | Segment \( BD \) is the median of an isosceles triangle \( ABC \) (\( AB = BC \)). A circle with a radius of 4 passes through points \( B \), \( A \), and \( D \), and intersects side \( BC \) at point \( E \) such that \( BE : BC = 7 : 8 \). Find the perimeter of triangle \( ABC \). | 20 | 3.90625 |
14,957 | From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$ . The point $ M \in (AE$ is such that $ M$ external to $ ABC$ , $ \angle AMB \equal{} 20 ^\circ$ and $ \angle AMC \equal{} 30 ^ \circ$ . What is the measure of the angle $ \angle... | 20 | 17.1875 |
14,958 | Let \( f(x) \) be a function with the property that \( f(x) + f\left(\frac{x-1}{3x-2}\right) = x \) for all real numbers \( x \) other than \( \frac{2}{3} \). What is the sum \( f(0) + f(1) + f(2) \)? | \frac{87}{40} | 9.375 |
14,959 | A covered rectangular soccer field of length 90 meters and width 60 meters is being designed. It must be illuminated by four floodlights, each hung at some point on the ceiling. Each floodlight illuminates a circle with a radius equal to the height at which it is hung. Determine the minimum possible height of the ceili... | 27.1 | 7.8125 |
14,960 | In the magical forest of Santa Claus, cedars grow one and a half times taller than firs and grow for 9 hours. Firs grow for 2 hours. Santa Claus planted cedar seeds at 12 o'clock and fir seeds at 2 o'clock in the afternoon. What time was it when the trees were of the same height? (The trees grow uniformly for the speci... | 15 | 28.125 |
14,961 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it satisfies $2b\sin \left(C+ \frac {\pi}{6}\right)=a+c$.
(I) Find the magnitude of angle $B$;
(II) If point $M$ is the midpoint of $BC$, and $AM=AC=2$, find the value of $a$. | \frac {4 \sqrt {7}}{7} | 0 |
14,962 | In triangle $DEF$, points $D'$, $E'$, and $F'$ are on the sides $EF$, $FD$, and $DE$, respectively. Given that $DD'$, $EE'$, and $FF'$ are concurrent at the point $P$, and that $\frac{DP}{PD'}+\frac{EP}{PE'}+\frac{FP}{PF'}=94$, find $\frac{DP}{PD'}\cdot \frac{EP}{PE'}\cdot \frac{FP}{PF'}$. | 92 | 0.78125 |
14,963 |
Neznaika does not know about multiplication and exponentiation operations. However, he is good at addition, subtraction, division, and square root extraction, and he knows how to use parentheses. While practicing, Neznaika chose three numbers 20, 2, and 2, and formed the expression:
$$
\sqrt{(2+20): 2} .
$$
Can he u... | 20 + 10\sqrt{2} | 0 |
14,964 | A person's age at the time of their death was one 31st of their birth year. How old was this person in 1930? | 39 | 35.9375 |
14,965 | Two trains are moving towards each other on parallel tracks - one with a speed of 60 km/h and the other with a speed of 80 km/h. A passenger sitting in the second train noticed that the first train passed by him in 6 seconds. What is the length of the first train? | 233.33 | 69.53125 |
14,966 | Given the function $f(x)=a\ln x-x^{2}+1$.
(I) If the tangent line of the curve $y=f(x)$ at $x=1$ is $4x-y+b=0$, find the values of the real numbers $a$ and $b$;
(II) Discuss the monotonicity of the function $f(x)$. | -4 | 2.34375 |
14,967 | On the board, two sums are written:
$$
\begin{array}{r}
1+22+333+4444+55555+666666+7777777+ \\
+88888888+999999999
\end{array}
$$
and
$9+98+987+9876+98765+987654+9876543+$
$+98765432+987654321$
Determine which of them is greater (or if they are equal). | 1097393685 | 0 |
14,968 | Compute the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.
$$
y = e^{x} + e, \ln \sqrt{3} \leq x \leq \ln \sqrt{15}
$$ | 2 + \frac{1}{2} \ln \left( \frac{9}{5} \right) | 0 |
14,969 | In a Cartesian coordinate plane, the "rectilinear distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as $d(P, Q) = \left|x_{1} - x_{2}\right| + \left|y_{1} - y_{2}\right|$. If point $C(x, y)$ has an equal "rectilinear distance" to points $A(1, 3)$ and $B(6, 9)$, where the ... | 5(\sqrt{2} + 1) | 0 |
14,970 | Let \(x, y, z\) be positive real numbers such that \(xyz = 1\). Find the maximum value of
\[
\frac{x^2y}{x+y} + \frac{y^2z}{y+z} + \frac{z^2x}{z+x}.
\] | \frac{3}{2} | 95.3125 |
14,971 | The probability of inducing cerebrovascular disease by smoking 5 cigarettes in one hour is 0.02, and the probability of inducing cerebrovascular disease by smoking 10 cigarettes in one hour is 0.16. An employee of a certain company smoked 5 cigarettes in one hour without inducing cerebrovascular disease. Calculate the ... | \frac{6}{7} | 0 |
14,972 | Given an ellipse $$C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$$ with eccentricity $$\frac{\sqrt{3}}{2}$$, and the distance from its left vertex to the line $x + 2y - 2 = 0$ is $$\frac{4\sqrt{5}}{5}$$.
(Ⅰ) Find the equation of ellipse C;
(Ⅱ) Suppose line $l$ intersects ellipse C at points A and B. If the ci... | \frac{4}{5} | 21.875 |
14,973 | What is the probability that two trainees were born on the same day (not necessarily the same year)? Note: There are 62 trainees. | 99.59095749 | 0 |
14,974 | Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left(a > b > 0\right)$ that passes through the point $(0,1)$, and its eccentricity is $\frac{\sqrt{3}}{2}$.
$(1)$ Find the standard equation of the ellipse $E$;
$(2)$ Suppose a line $l: y = \frac{1}{2}x + m$ intersects the ellipse $E$ at points $A$ and $C$. ... | \frac{\sqrt{10}}{2} | 3.90625 |
14,975 | Given a sequence of 15 zeros and ones, determine the number of sequences where all the zeros are consecutive. | 121 | 14.0625 |
14,976 | In a \(7 \times 7\) table, some cells are black while the remaining ones are white. In each white cell, the total number of black cells located with it in the same row or column is written; nothing is written in the black cells. What is the maximum possible sum of the numbers in the entire table? | 168 | 20.3125 |
14,977 | Inside a circle, 16 radii of this circle and 10 circles with the same center as the circle are drawn. Into how many regions do the radii and circles divide the circle? | 176 | 41.40625 |
14,978 | Find the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$ , where $a$ , $ b$ , $c$ , $d$ are non-negative integers.
| 11 | 82.03125 |
14,979 | Divide every natural number with at least two digits by the sum of its digits! When will the quotient be the largest, and when will it be the smallest? | 1.9 | 10.9375 |
14,980 | In the acute-angled triangle \(ABC\), it is known that \(\sin (A+B)=\frac{3}{5}\), \(\sin (A-B)=\frac{1}{5}\), and \(AB=3\). Find the area of \(\triangle ABC\). | \frac{6 + 3\sqrt{6}}{2} | 0 |
14,981 | Dani wrote the integers from 1 to \( N \). She used the digit 1 fifteen times. She used the digit 2 fourteen times.
What is \( N \) ? | 41 | 0 |
14,982 | The side of the base of a regular quadrilateral pyramid \( \operatorname{ABCDP} \) (with \( P \) as the apex) is \( 4 \sqrt{2} \), and the angle between adjacent lateral faces is \( 120^{\circ} \). Find the area of the cross-section of the pyramid by a plane passing through the diagonal \( BD \) of the base and paralle... | 4\sqrt{6} | 3.90625 |
14,983 | Given an ellipse $$C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ and a hyperbola $$\frac {x^{2}}{4-v}+ \frac {y^{2}}{1-v}=1 (1<v<4)$$ share a common focus. A line $l$ passes through the right vertex B of the ellipse C and intersects the parabola $y^2=2x$ at points P and Q, with $OP \perpendicular OQ$.
(Ⅰ) Find the... | \frac {1}{2} | 29.6875 |
14,984 | A blind box refers to a toy box where consumers cannot know the specific product style in advance. A certain brand has launched two blind box sets. Set $A$ contains $4$ different items, including a small rabbit toy. Set $B$ contains $2$ different items, with a $50\%$ chance of getting a small rabbit toy.
$(1)$ Individ... | \frac{1}{3} | 6.25 |
14,985 | Find the value of $\frac{\frac{1}{2}-\frac{1}{3}}{\frac{1}{3}-\frac{1}{4}} \times \frac{\frac{1}{4}-\frac{1}{5}}{\frac{1}{5}-\frac{1}{6}} \times \frac{\frac{1}{6}-\frac{1}{7}}{\frac{1}{7}-\frac{1}{8}} \times \ldots \times \frac{\frac{1}{2004}-\frac{1}{2005}}{\frac{1}{2005}-\frac{1}{2006}} \times \frac{\frac{1}{2006}-\f... | 1004 | 49.21875 |
14,986 | Two mathematics teachers administer a geometry test, assessing the ability to solve problems and knowledge of theory for each 10th-grade student. The first teacher spends 5 and 7 minutes per student, and the second teacher spends 3 and 4 minutes per student. What is the minimum time needed to assess 25 students? | 110 | 0 |
14,987 | In the diagram, \(ABCD\) is a parallelogram. \(E\) is on side \(AB\), and \(F\) is on side \(DC\). \(G\) is the intersection point of \(AF\) and \(DE\), and \(H\) is the intersection point of \(CE\) and \(BF\). Given that the area of parallelogram \(ABCD\) is 1, \(\frac{\mathrm{AE}}{\mathrm{EB}}=\frac{1}{4}\), and the ... | \frac{7}{92} | 0 |
14,988 | 120 granite slabs weighing 7 tons each and 80 slabs weighing 9 tons each have been stockpiled at the quarry. A railroad platform can hold up to 40 tons. What is the minimum number of platforms required to transport all the slabs? | 40 | 0 |
14,989 | Let $a, b, c, d$ be the four roots of $X^{4}-X^{3}-X^{2}-1$. Calculate $P(a)+P(b)+P(c)+P(d)$, where $P(X) = X^{6}-X^{5}-X^{4}-X^{3}-X$. | -2 | 25.78125 |
14,990 | The school plans to schedule six leaders to be on duty from May 1st to May 3rd, with each leader on duty for one day and two leaders scheduled each day. Given that Leader A cannot be on duty on May 2nd and Leader B cannot be on duty on May 3rd, determine the number of different ways to arrange the duty schedule. | 42 | 57.03125 |
14,991 |
For the four-digit number $\overline{abcd}$ (where $1 \leq a \leq 9$ and $0 \leq b, c, d \leq 9$):
- If $a > b$, $b < c$, and $c > d$, then $\overline{abcd}$ is called a $P$-type number;
- If $a < b$, $b > c$, and $c < d$, then $\overline{abcd}$ is called a $Q$-type number.
Let $N(P)$ and $N(Q)$ denote the number of ... | 285 | 99.21875 |
14,992 | Among the positive integers less than $10^{4}$, how many positive integers $n$ are there such that $2^{n} - n^{2}$ is divisible by 7? | 2857 | 67.96875 |
14,993 | In a store, there are 21 white and 21 purple shirts hanging in a row. Find the smallest $k$ such that, regardless of the initial order of the shirts, it is possible to remove $k$ white and $k$ purple shirts, so that the remaining white shirts hang consecutively and the remaining purple shirts also hang consecutively. | 10 | 60.9375 |
14,994 | It takes person A 1 minute and 20 seconds to complete a lap, and person B meets person A every 30 seconds. Determine the time it takes for person B to complete a lap. | 48 | 80.46875 |
14,995 |
Today is 17.02.2008. Natasha noticed that in this date, the sum of the first four digits is equal to the sum of the last four digits. When will this coincidence happen for the last time this year? | 25.12.2008 | 3.90625 |
14,996 | Given the linear function \( y = ax + b \) and the hyperbolic function \( y = \frac{k}{x} \) (where \( k > 0 \)) intersect at points \( A \) and \( B \), with \( O \) being the origin. If the triangle \( \triangle OAB \) is an equilateral triangle with an area of \( \frac{2\sqrt{3}}{3} \), find the value of \( k \). | \frac{2}{3} | 3.125 |
14,997 | Positive real numbers \( a, b, c \) are in a geometric progression \((q \neq 1)\), and \( \log _{a} b, \log _{b} c, \log _{c} a \) are in an arithmetic progression. Find the common difference \( d \). | -\frac{3}{2} | 9.375 |
14,998 | Calculate the definite integral:
$$
\int_{0}^{\pi / 4} \frac{7+3 \operatorname{tg} x}{(\sin x+2 \cos x)^{2}} d x
$$ | 3 \ln \left(\frac{3}{2}\right) + \frac{1}{6} | 7.03125 |
14,999 | Four vehicles were traveling on the highway at constant speeds: a car, a motorcycle, a scooter, and a bicycle. The car passed the scooter at 12:00, encountered the bicyclist at 14:00, and met the motorcyclist at 16:00. The motorcyclist met the scooter at 17:00 and caught up with the bicyclist at 18:00.
At what time d... | 15:20 | 1.5625 |
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