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40.3k
| problem
stringlengths 10
5.15k
| ground_truth
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float64 0
100
|
|---|---|---|---|
11,800 |
Let the function $f(x) = \cos(2x + \frac{\pi}{3}) + \sqrt{3}\sin(2x) + 2a$.
(1) Find the intervals of monotonic increase for the function $f(x)$.
(2) When $x \in [0, \frac{\pi}{4}]$, the minimum value of $f(x)$ is 0. Find the maximum value of $f(x)$.
|
\frac{1}{2}
| 11.71875 |
11,801 |
Find the perimeter of an equilateral triangle inscribed in a circle, given that a chord of this circle, equal to 2, is at a distance of 3 from its center.
|
3 \sqrt{30}
| 96.09375 |
11,802 |
It is given that \( k \) is a positive integer not exceeding 99. There are no natural numbers \( x \) and \( y \) such that \( x^{2} - k y^{2} = 8 \). Find the difference between the maximum and minimum possible values of \( k \).
|
96
| 10.15625 |
11,803 |
Let \( S-ABC \) be a triangular prism with the base being an isosceles right triangle \( ABC \) with \( AB \) as the hypotenuse, and \( SA = SB = SC = 2 \) and \( AB = 2 \). If \( S \), \( A \), \( B \), and \( C \) are points on a sphere centered at \( O \), find the distance from point \( O \) to the plane \( ABC \).
|
\frac{\sqrt{3}}{3}
| 23.4375 |
11,804 |
Given a circle circumscribed around triangle \(FDC\), a tangent \(FK\) is drawn such that \(\angle KFC = 58^\circ\). Points \(K\) and \(D\) lie on opposite sides of line \(FC\) as shown in the diagram. Find the acute angle between the angle bisectors of \(\angle CFD\) and \(\angle FCD\). Provide your answer in degrees.
|
61
| 37.5 |
11,805 |
Two people agreed to meet at a specific location between 12 PM and 1 PM. The condition is that the first person to arrive will wait for the second person for 15 minutes and then leave. What is the probability that these two people will meet if each of them chooses their moment of arrival at the agreed location randomly within the interval between 12 PM and 1 PM?
|
7/16
| 78.125 |
11,806 |
For what is the smallest $n$ such that there exist $n$ numbers within the interval $(-1, 1)$ whose sum is 0 and the sum of their squares is 42?
|
44
| 21.875 |
11,807 |
How many positive integers have a square less than 10,000,000?
|
3162
| 47.65625 |
11,808 |
Given that player A needs to win 2 more games and player B needs to win 3 more games, and the probability of winning each game for both players is $\dfrac{1}{2}$, calculate the probability of player A ultimately winning.
|
\dfrac{11}{16}
| 4.6875 |
11,809 |
Let \( A B C \) be an isosceles triangle with \( B \) as the vertex of the equal angles. Let \( F \) be a point on the bisector of \( \angle A B C \) such that \( (A F) \) is parallel to \( (B C) \). Let \( E \) be the midpoint of \([B C]\), and let \( D \) be the symmetric point of \( A \) with respect to \( F \). Calculate the ratio of the distances \( E F / B D \).
|
1/2
| 51.5625 |
11,810 |
If the positive real numbers \(a\) and \(b\) satisfy \(\frac{1}{a} + \frac{1}{b} \leq 2 \sqrt{2}\) and \((a - b)^2 = 4 (ab)^3\), then \(\log_a b =\) ?
|
-1
| 12.5 |
11,811 |
Given that $\tan \alpha =2$, find the value of $\frac{4\sin^{3} \alpha -2\cos \alpha }{5\cos \alpha +3\sin \alpha }$.
|
\frac{2}{5}
| 94.53125 |
11,812 |
(1) Solve the inequality $\log_{\frac{1}{2}}(x+2) > -3$
(2) Calculate: $(\frac{1}{8})^{\frac{1}{3}} \times (-\frac{7}{6})^{0} + 8^{0.25} \times \sqrt[4]{2} + (\sqrt[3]{2} \times \sqrt{3})^{6}$.
|
\frac{221}{2}
| 77.34375 |
11,813 |
There is a basket of apples. If Class A shares the apples such that each person gets 3 apples, 10 apples remain. If Class B shares the apples such that each person gets 4 apples, 11 apples remain. If Class C shares the apples such that each person gets 5 apples, 12 apples remain. How many apples are there in the basket at least?
|
67
| 85.9375 |
11,814 |
Let the set \( A = \{0, 1, 2, \ldots, 9\} \). The collection \( \{B_1, B_2, \ldots, B_k\} \) is a family of non-empty subsets of \( A \). When \( i \neq j \), the intersection \( B_i \cap B_j \) has at most two elements. Find the maximum value of \( k \).
|
175
| 69.53125 |
11,815 |
In a right trapezoid \(ABCD\), the sum of the lengths of the bases \(AD\) and \(BC\) is equal to its height \(AB\). In what ratio does the angle bisector of \(\angle ABC\) divide the lateral side \(CD\)?
|
1:1
| 24.21875 |
11,816 |
During an underwater archaeology activity, a diver needs to dive 50 meters to the seabed to carry out archaeological work. The oxygen consumption includes the following three aspects:
1. The average descent speed is $x$ meters per minute, and the oxygen consumption per minute is $\frac{1}{100}x^2$ liters;
2. The diver's working time on the seabed ranges from at least 10 minutes to a maximum of 20 minutes, with an oxygen consumption of 0.3 liters per minute;
3. When returning to the surface, the average speed is $\frac{1}{2}x$ meters per minute, with an oxygen consumption of 0.32 liters per minute. The total oxygen consumption of the diver in this archaeological activity is $y$ liters.
(1) If the working time on the seabed is 10 minutes, express $y$ as a function of $x$;
(2) If $x \in [6,10]$ and the working time on the seabed is 20 minutes, find the range of the total oxygen consumption $y$;
(3) If the diver carries 13.5 liters of oxygen, what is the maximum number of minutes the diver can stay underwater? (Round the result to the nearest integer).
|
18
| 17.96875 |
11,817 |
Let $u$ and $v$ be real numbers satisfying the inequalities $2u + 3v \le 10$ and $4u + v \le 9.$ Find the largest possible value of $u + 2v$.
|
6.1
| 0 |
11,818 |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $a = b\cos C + c\sin B$.
(1) Find angle $B$.
(2) If $b = 4$, find the maximum area of triangle $ABC$.
|
4\sqrt{2} + 4
| 12.5 |
11,819 |
Given the hyperbola $$\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$$ (a > 0, b > 0), a circle with center at point (b, 0) and radius a is drawn. The circle intersects with one of the asymptotes of the hyperbola at points M and N, and ∠MPN = 90°. Calculate the eccentricity of the hyperbola.
|
\sqrt{2}
| 48.4375 |
11,820 |
Determine the sum of all positive integers \( N < 1000 \) for which \( N + 2^{2015} \) is divisible by 257.
|
2058
| 17.1875 |
11,821 |
Given that the circumcenter of triangle $ABC$ is $O$, and $2 \overrightarrow{O A} + 3 \overrightarrow{O B} + 4 \overrightarrow{O C} = 0$, determine the value of $\cos \angle BAC$.
|
\frac{1}{4}
| 29.6875 |
11,822 |
Find the remainder when \(5x^4 - 9x^3 + 3x^2 - 7x - 30\) is divided by \(3x - 9\).
|
138
| 22.65625 |
11,823 |
How many multiples of 4 are between 70 and 300?
|
57
| 7.03125 |
11,824 |
The distance from $A$ to $B$ is covered 3 hours and 12 minutes faster by a passenger train compared to a freight train. In the time it takes the freight train to travel from $A$ to $B$, the passenger train covers 288 km more. If the speed of each train is increased by $10 \mathrm{~km} / \mathrm{h}$, the passenger train will cover the distance from $A$ to $B$ 2 hours and 24 minutes faster than the freight train. Determine the distance from $A$ to $B$.
|
360
| 37.5 |
11,825 |
Using two red, three blue, and four green small cubes, calculate the number of different towers that can be built using eight of these cubes.
|
1260
| 32.03125 |
11,826 |
In some 16 cells of an $8 \times 8$ board, rooks are placed. What is the minimum number of pairs of rooks that can attack each other in this configuration?
|
16
| 49.21875 |
11,827 |
On the game show $\text{\emph{Wheel of Fortune Redux}}$, you see the following spinner. Given that each region is of the same area, what is the probability that you will earn exactly $\$3200$ in your first three spins? The spinner includes the following sections: $"\$2000"$, $"\$300"$, $"\$700"$, $"\$1500"$, $"\$500"$, and "$Bankrupt"$. Express your answer as a common fraction.
|
\frac{1}{36}
| 33.59375 |
11,828 |
In 60 chandeliers (each with 4 shades), the shades need to be replaced. Each electrician takes 5 minutes to replace one shade. A total of 48 electricians will be working. No more than one shade can be replaced in a chandelier at the same time. What is the minimum time required to replace all the shades in all the chandeliers?
|
25
| 53.125 |
11,829 |
A point \(A_{1}\) is taken on the side \(AC\) of triangle \(ABC\), and a point \(C_{1}\) is taken on the extension of side \(BC\) beyond point \(C\). The length of segment \(A_{1}C\) is 85% of the length of side \(AC\), and the length of segment \(BC_{1}\) is 120% of the length of side \(BC\). What percentage of the area of triangle \(ABC\) is the area of triangle \(A_{1}BC_{1}\)?
|
102
| 55.46875 |
11,830 |
In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. If the angles \( A \), \( B \), and \( C \) form a geometric progression, and \( b^{2} - a^{2} = ac \), then the radian measure of angle \( B \) is equal to ________.
|
\frac{2\pi}{7}
| 14.84375 |
11,831 |
Given the function \( f(x) = 4 \pi \arcsin x - (\arccos(-x))^2 \), find the difference between its maximum value \( M \) and its minimum value \( m \). Specifically, calculate \( M - m \).
|
3\pi^2
| 62.5 |
11,832 |
Find all real numbers \( x \) that satisfy the equation
$$
\frac{x-2020}{1} + \frac{x-2019}{2} + \cdots + \frac{x-2000}{21} = \frac{x-1}{2020} + \frac{x-2}{2019} + \cdots + \frac{x-21}{2000},
$$
and simplify your answer(s) as much as possible. Justify your solution.
|
2021
| 65.625 |
11,833 |
The sequence $\{a_n\}$ satisfies $a_n=13-3n$, $b_n=a_n⋅a_{n+1}⋅a_{n+2}$, $S_n$ is the sum of the first $n$ terms of $\{b_n\}$. Find the maximum value of $S_n$.
|
310
| 27.34375 |
11,834 |
Given $f(x) = 2\cos^{2}x + \sqrt{3}\sin2x + a$, where $a$ is a real constant, find the value of $a$, given that the function has a minimum value of $-4$ on the interval $\left[0, \frac{\pi}{2}\right]$.
|
-4
| 49.21875 |
11,835 |
Given $\tan(\theta-\pi)=2$, find the value of $\sin^2\theta+\sin\theta\cos\theta-2\cos^2\theta$.
|
\frac{4}{5}
| 82.8125 |
11,836 |
If $C_{n}^{2}A_{2}^{2} = 42$, find the value of $\dfrac{n!}{3!(n-3)!}$.
|
35
| 79.6875 |
11,837 |
You are in a completely dark room with a drawer containing 10 red, 20 blue, 30 green, and 40 khaki socks. What is the smallest number of socks you must randomly pull out in order to be sure of having at least one of each color?
|
91
| 53.90625 |
11,838 |
Given an arithmetic sequence $\\{a_{n}\\}$, let $S_{n}$ denote the sum of its first $n$ terms. If $a_{4}=-12$ and $a_{8}=-4$:
$(1)$ Find the general term formula for the sequence;
$(2)$ Find the minimum value of $S_{n}$ and the corresponding value of $n$.
|
-90
| 3.125 |
11,839 |
The base of a rectangular parallelepiped is a square with a side length of \(2 \sqrt{3}\). The diagonal of a lateral face forms an angle of \(30^\circ\) with the plane of an adjacent lateral face. Find the volume of the parallelepiped.
|
72
| 16.40625 |
11,840 |
Calculate: $\sqrt[3]{27}+|-\sqrt{2}|+2\sqrt{2}-(-\sqrt{4})$.
|
5 + 3\sqrt{2}
| 63.28125 |
11,841 |
Find the largest odd natural number that cannot be expressed as the sum of three distinct composite numbers.
|
17
| 80.46875 |
11,842 |
What is the smallest three-digit positive integer which can be written in the form \( p q^{2} r \), where \( p, q \), and \( r \) are distinct primes?
|
126
| 67.96875 |
11,843 |
The domain of the function $y=\sin x$ is $[a,b]$, and its range is $\left[-1, \frac{1}{2}\right]$. Calculate the maximum value of $b-a$.
|
\frac{4\pi}{3}
| 45.3125 |
11,844 |
Given that one third of the students go home on the school bus, one fifth go home by automobile, one eighth go home on their bicycles, and one tenth go home on scooters, what fractional part of the students walk home?
|
\frac{29}{120}
| 77.34375 |
11,845 |
With all angles measured in degrees, the product $\prod_{k=1}^{22} \sec^2(4k)^\circ=p^q$, where $p$ and $q$ are integers greater than 1. Find the value of $p+q$.
|
46
| 50 |
11,846 |
Calculate $\lim _{n \rightarrow \infty}\left(\sqrt[3^{2}]{3} \cdot \sqrt[3^{3}]{3^{2}} \cdot \sqrt[3^{4}]{3^{3}} \ldots \sqrt[3^{n}]{3^{n-1}}\right)$.
|
\sqrt[4]{3}
| 42.1875 |
11,847 |
$K, L, M$ and $N$ are points on sides $AB, BC, CD$ and $DA$ , respectively, of the unit square $ABCD$ such that $KM$ is parallel to $BC$ and $LN$ is parallel to $AB$ . The perimeter of triangle $KLB$ is equal to $1$ . What is the area of triangle $MND$ ?
|
1/4
| 21.09375 |
11,848 |
In an equilateral triangle \(ABC\), a point \(P\) is chosen such that \(AP = 10\), \(BP = 8\), and \(CP = 6\). Find the area of this triangle.
|
36 + 25\sqrt{3}
| 0 |
11,849 |
An archipelago consists of \( N \geq 7 \) islands. Any two islands are connected by no more than one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are necessarily two that are connected by a bridge. What is the maximum value that \( N \) can take?
|
36
| 7.03125 |
11,850 |
Find the number by which the three numbers 480608, 508811, and 723217 when divided will give the same remainder.
|
79
| 3.125 |
11,851 |
Given that \(\frac{810 \times 811 \times 812 \times \cdots \times 2010}{810^{n}}\) is an integer, find the maximum value of \(n\).
|
150
| 63.28125 |
11,852 |
For a real number \( x \), let \( [x] \) denote the greatest integer that does not exceed \( x \). For a certain integer \( k \), there exist exactly 2008 positive integers \( n_{1}, n_{2}, \cdots, n_{2008} \), such that \( k=\left[\sqrt[3]{n_{1}}\right]=\left[\sqrt[3]{n_{2}}\right]=\cdots=\left[\sqrt[3]{n_{2008}}\right] \), and \( k \) divides \( n_{i} \) (for \( i = 1, 2, \cdots, 2008 \)). What is the value of \( k \)?
|
668
| 17.96875 |
11,853 |
Gretchen has ten socks, two of each color: red, blue, green, yellow, and purple. She randomly draws six socks. What is the probability that she ends up with exactly two pairs, each of a different color?
|
\frac{4}{7}
| 16.40625 |
11,854 |
A fair die, numbered 1, 2, 3, 4, 5, 6, is thrown three times. The numbers obtained are recorded sequentially as $a$, $b$, and $c$. The probability that $a+bi$ (where $i$ is the imaginary unit) is a root of the equation $x^{2}-2x+c=0$ is $\_\_\_\_\_\_$.
|
\frac{1}{108}
| 72.65625 |
11,855 |
What is the sum of all the two-digit primes that are greater than 20 but less than 80 and are still prime when their two digits are interchanged?
|
291
| 0 |
11,856 |
Specify a six-digit number $N$ consisting of distinct digits such that the numbers $2N$, $3N$, $4N$, $5N$, and $6N$ are permutations of its digits.
|
142857
| 91.40625 |
11,857 |
Given that five volunteers are randomly assigned to conduct promotional activities in three communities, A, B, and C, at least 2 volunteers are assigned to community A, and at least 1 volunteer is assigned to each of communities B and C, calculate the number of different arrangements.
|
80
| 28.125 |
11,858 |
At the CleverCat Academy, there are three skills that the cats can learn: jump, climb, and hunt. Out of the cats enrolled in the school:
- 40 cats can jump.
- 25 cats can climb.
- 30 cats can hunt.
- 10 cats can jump and climb.
- 15 cats can climb and hunt.
- 12 cats can jump and hunt.
- 5 cats can do all three skills.
- 6 cats cannot perform any of the skills.
How many cats are in the academy?
|
69
| 78.125 |
11,859 |
Several different positive integers are written on a blackboard. The product of the smallest two of them is 16. The product of the largest two of them is 225. What is the sum of all the integers written on the blackboard?
|
44
| 59.375 |
11,860 |
A steamboat, 2 hours after departing from dock $A$, stops for 1 hour and then continues its journey at a speed that is 0.8 times its initial speed. As a result, it arrives at dock $B$ 3.5 hours late. If the stop had occurred 180 km further, and all other conditions remained the same, the steamboat would have arrived at dock $B$ 1.5 hours late. Find the distance $AB$.
|
270
| 42.1875 |
11,861 |
The sequence $(a_{n})$ is defined by the following relations: $a_{1}=1$, $a_{2}=3$, $a_{n}=a_{n-1}-a_{n-2}+n$ (for $n \geq 3$). Find $a_{1000}$.
|
1002
| 86.71875 |
11,862 |
Inside the square $A B C D$, a point $P$ is chosen such that the distances from $P$ to vertices $A$, $B$, and $C$ are in the ratio $A P: B P: C P=1: 2: 3$. What is the measure of angle $A P B$?
|
135
| 14.84375 |
11,863 |
Let $p,$ $q,$ $r,$ and $s$ be the roots of \[x^4 + 10x^3 + 20x^2 + 15x + 6 = 0.\] Find the value of \[\frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}.\]
|
\frac{10}{3}
| 91.40625 |
11,864 |
Given that $α+β= \frac {π}{3}$ and $tanα+tanβ=2$, find the value of $cos(α-β)$.
|
\frac { \sqrt {3}-1}{2}
| 0 |
11,865 |
Find the greatest negative value of the expression \( x - y \) for all pairs of numbers \((x, y)\) satisfying the equation
$$
(\sin x + \sin y)(\cos x - \cos y) = \frac{1}{2} + \sin(x - y) \cos(x + y)
$$
|
-\frac{\pi}{6}
| 22.65625 |
11,866 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c \sin A = \sqrt{3}a \cos C$ and $(a-c)(a+c)=b(b-c)$, find the period and the monotonically increasing interval of the function $f(x) = 2 \sin x \cos (\frac{\pi}{2} - x) - \sqrt{3} \sin (\pi + x) \cos x + \sin (\frac{\pi}{2} + x) \cos x$. Also, find the value of $f(B)$.
|
\frac{5}{2}
| 70.3125 |
11,867 |
Given the complex number \( z \) satisfying
$$
\left|\frac{z^{2}+1}{z+\mathrm{i}}\right|+\left|\frac{z^{2}+4 \mathrm{i}-3}{z-\mathrm{i}+2}\right|=4,
$$
find the minimum value of \( |z - 1| \).
|
\sqrt{2}
| 10.15625 |
11,868 |
Given that the function $f(x+1)$ is an odd function, and the function $f(x-1)$ is an even function, and $f(0) = 2$, determine the value of $f(4)$.
|
-2
| 6.25 |
11,869 |
How many ways are there to choose 6 different numbers from $1, 2, \cdots, 49$, where at least two numbers are consecutive?
|
\binom{49}{6} - \binom{44}{6}
| 0 |
11,870 |
Given a dart board is a regular hexagon divided into regions, the center of the board is another regular hexagon formed by joining the midpoints of the sides of the larger hexagon, and the dart is equally likely to land anywhere on the board, find the probability that the dart lands within the center hexagon.
|
\frac{1}{4}
| 89.84375 |
11,871 |
In a round-robin chess tournament, only grandmasters and masters participated. The number of masters was three times the number of grandmasters, and the total points scored by the masters was 1.2 times the total points scored by the grandmasters.
How many people participated in the tournament? What can be said about the tournament's outcome?
|
12
| 17.96875 |
11,872 |
The sum of all real roots of the equation $|x^2-3x+2|+|x^2+2x-3|=11$ is .
|
\frac{5\sqrt{97}-19}{20}
| 0 |
11,873 |
In $\triangle ABC$ , $AB = 40$ , $BC = 60$ , and $CA = 50$ . The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$ . Find $BP$ .
*Proposed by Eugene Chen*
|
40
| 33.59375 |
11,874 |
The teacher fills some numbers into the circles in the diagram below (each circle can and must only contain one number). The sum of the three numbers in each of the left and right closed loops is 30, and the sum of the four numbers in each of the top and bottom closed loops is 40. If the number in circle $X$ is 9, then the number in circle $Y$ is $\qquad$
|
11
| 12.5 |
11,875 |
Given a convex quadrilateral \(ABCD\) with \(X\) as the midpoint of the diagonal \(AC\), it turns out that \(CD \parallel BX\). Find \(AD\) if it is known that \(BX = 3\), \(BC = 7\), and \(CD = 6\).
|
14
| 16.40625 |
11,876 |
For \( n \in \mathbf{Z}_{+}, n \geqslant 2 \), let
\[
S_{n}=\sum_{k=1}^{n} \frac{k}{1+k^{2}+k^{4}}, \quad T_{n}=\prod_{k=2}^{n} \frac{k^{3}-1}{k^{3}+1}
\]
Then, \( S_{n} T_{n} = \) .
|
\frac{1}{3}
| 28.90625 |
11,877 |
On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained by adding the smallest non-zero digit of its decimal representation to the previous number. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th place in this sequence?
|
3001
| 42.1875 |
11,878 |
How many right-angled triangles can Delia make by joining three vertices of a regular polygon with 18 sides?
|
144
| 64.0625 |
11,879 |
Place parentheses and operation signs in the sequence 22222 so that the result is 24.
|
(2+2+2) \times (2+2)
| 11.71875 |
11,880 |
Let the real numbers \(a_{1}, a_{2}, \cdots, a_{100}\) satisfy the following conditions: (i) \(a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{100} \geqslant 0\); (ii) \(a_{1}+a_{2} \leqslant 100\); (iii) \(a_{3}+a_{4} + \cdots + a_{100} \leqslant 100\). Find the maximum value of \(a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2}\) and the values of \(a_{1}, a_{2}, \cdots, a_{100}\) when the maximum value is reached.
|
10000
| 73.4375 |
11,881 |
We have $2022$ $1s$ written on a board in a line. We randomly choose a strictly increasing sequence from ${1, 2, . . . , 2022}$ such that the last term is $2022$ . If the chosen sequence is $a_1, a_2, ..., a_k$ ( $k$ is not fixed), then at the $i^{th}$ step, we choose the first a $_i$ numbers on the line and change the 1s to 0s and 0s to 1s. After $k$ steps are over, we calculate the sum of the numbers on the board, say $S$ . The expected value of $S$ is $\frac{a}{b}$ where $a, b$ are relatively prime positive integers. Find $a + b.$
|
1012
| 78.125 |
11,882 |
Translate the function $f(x) = \sin 2x + \sqrt{3}\cos 2x$ to the left by $\varphi$ ($\varphi > 0$) units. If the resulting graph is symmetric about the y-axis, then the minimum value of $\varphi$ is \_\_\_\_\_.
|
\frac{\pi}{12}
| 71.09375 |
11,883 |
In the diagram, \(ABCD\) is a square with a side length of \(8 \, \text{cm}\). Point \(E\) is on \(AB\) and point \(F\) is on \(DC\) so that \(\triangle AEF\) is right-angled at \(E\). If the area of \(\triangle AEF\) is \(30\%\) of the area of \(ABCD\), what is the length of \(AE\)?
|
4.8
| 49.21875 |
11,884 |
Calculate the degree of ionization using the formula:
$$
\alpha=\sqrt{ } K_{\mathrm{HCN}} \mathrm{C}
$$
Given values:
$$
\alpha_{\text {ion }}=\sqrt{ }\left(7,2 \cdot 10^{-10}\right) / 0,1=\sqrt{ } 7,2 \cdot 10^{-9}=8,5 \cdot 10^{-5}, \text{ or } 8,5 \cdot 10^{-5} \cdot 10^{2}=0,0085\%
$$
Alternatively, if the concentration of ions is known, you can calculate $\alpha$ as:
$$
\mathrm{C} \cdot \alpha=[\mathrm{H}^{+}]=[\mathrm{CN}^{-}], [\mathrm{H}^{+}]=[\mathrm{CN}^{-}]=8,5 \cdot 10^{-6} \text{ mol/L}
$$
Then:
$$
\alpha_{\text{ion }}=8,5 \cdot 10^{-6}, 0,1=8,5 \cdot 10^{-5} \text{ or } 8,5 \cdot 10^{-5} \cdot 10^{2}=0,0085\%
$$
|
0.0085
| 95.3125 |
11,885 |
Among the 9 natural numbers $1,2,3, \cdots, 9$, if 3 numbers are chosen, let $x$ be the number of pairs of adjacent numbers among the chosen 3 numbers (for example, if the 3 chosen numbers are $1,2,3$, there are 2 pairs of adjacent numbers: 1,2 and 2,3, so the value of $x$ is 2). What is the expected value of $x$?
|
2/3
| 13.28125 |
11,886 |
Determine how many integers $n$ between 1 and 15 (inclusive) result in a fraction $\frac{n}{30}$ that has a repeating decimal.
|
10
| 72.65625 |
11,887 |
In the book "Nine Chapters on the Mathematical Art," a tetrahedron with all four faces being right-angled triangles is called a "biēnào." Given that tetrahedron $ABCD$ is a "biēnào," $AB\bot $ plane $BCD$, $BC\bot CD$, and $AB=\frac{1}{2}BC=\frac{1}{3}CD$. If the volume of this tetrahedron is $1$, then the surface area of its circumscribed sphere is ______.
|
14\pi
| 52.34375 |
11,888 |
Let \( a, b, c \) be pairwise distinct positive integers such that \( a+b, b+c \) and \( c+a \) are all square numbers. Find the smallest possible value of \( a+b+c \).
|
55
| 43.75 |
11,889 |
Find the smallest value of \(a\) for which the sum of the squares of the roots of the equation \(x^{2}-3ax+a^{2}=0\) is equal to \(0.28\).
|
-0.2
| 95.3125 |
11,890 |
From the set \(\left\{-3, -\frac{5}{4}, -\frac{1}{2}, 0, \frac{1}{3}, 1, \frac{4}{5}, 2\right\}\), two numbers are drawn without replacement. Find the concept of the two numbers being the slopes of perpendicular lines.
|
3/28
| 34.375 |
11,891 |
In the rectangular coordinate system xOy, it is known that 0 < α < 2π. Point P, with coordinates $(1 - \tan{\frac{\pi}{12}}, 1 + \tan{\frac{\pi}{12}})$, lies on the terminal side of angle α. Determine the value of α.
|
\frac{\pi}{3}
| 67.96875 |
11,892 |
Define \( n! = 1 \times 2 \times \ldots \times n \), for example \( 5! = 1 \times 2 \times 3 \times 4 \times 5 \). If \(\frac{n! \times (n+1)!}{2}\) (where \( \mathbf{n} \) is a positive integer and \( 1 \leq n \leq 100 \)) is a perfect square, what is the sum of all such \( \mathbf{n} \)?
|
273
| 79.6875 |
11,893 |
Find the maximum constant \( k \) such that for \( x, y, z \in \mathbb{R}_+ \), the following inequality holds:
$$
\sum \frac{x}{\sqrt{y+z}} \geqslant k \sqrt{\sum x},
$$
where " \( \sum \) " denotes a cyclic sum.
|
\sqrt{\frac{3}{2}}
| 1.5625 |
11,894 |
To celebrate her birthday, Ana is going to prepare pear and apple pies. In the market, an apple weighs $300 \text{ g}$ and a pear weighs $200 \text{ g}$. Ana's bag can hold a maximum weight of $7 \text{ kg}$. What is the maximum number of fruits she can buy to make pies with both types of fruits?
|
34
| 1.5625 |
11,895 |
Find the smallest positive integer $M$ such that both $M$ and $M^2$ end in the same sequence of three digits $xyz$ when written in base $10$, where $x$ is not zero.
|
376
| 97.65625 |
11,896 |
How many non-similar regular 1200-pointed stars are there, considering the definition of a regular $n$-pointed star provided in the original problem?
|
160
| 75.78125 |
11,897 |
There are two alloys of copper and zinc. In the first alloy, there is twice as much copper as zinc, and in the second alloy, there is five times less copper than zinc. In what ratio should these alloys be combined to obtain a new alloy in which zinc is twice as much as copper?
|
1 : 2
| 15.625 |
11,898 |
Given that \( z_{1} \) and \( z_{2} \) are complex numbers with \( \left|z_{1}\right|=3 \), \( \left|z_{2}\right|=5 \), and \( \left|z_{1}+z_{2}\right|=7 \), find the value of \( \arg \left(\left(\frac{z_{2}}{z_{1}}\right)^{3}\right) \).
|
\pi
| 79.6875 |
11,899 |
At 7:00 AM, Xiaoming leaves his house and heads to school at a speed of 52 meters per minute. When he reaches the school, the hour and minute hands on his watch are positioned symmetrically around the number 7 on the clock. It's known that Xiaoming walked for less than an hour. How far is the distance between Xiaoming's house and the school, in meters?
|
1680
| 23.4375 |
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