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40.3k
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5.15k
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100
|
|---|---|---|---|
10,800 |
In the Cartesian coordinate system $(xOy)$, let $l$ be a line with an angle of inclination $\alpha$ and parametric equations $\begin{cases} x = 3 + t\cos\alpha \\ y = t\sin\alpha \end{cases}$ (where $t$ is a parameter). The line $l$ intersects the curve $C$: $\begin{cases} x = \frac{1}{\cos\theta} \\ y = \tan\theta \end{cases}$ (where $\theta$ is a parameter) at two distinct points $A$ and $B$.
1. If $\alpha = \frac{\pi}{3}$, find the rectangular coordinates of the midpoint of the line segment $AB$.
2. If the slope of line $l$ is 2 and it passes through the known point $P(3, 0)$, find the value of $|PA| \cdot |PB|$.
|
\frac{40}{3}
| 17.1875 |
10,801 |
In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} x=4+ \frac { \sqrt {2}}{2}t \\ y=3+ \frac { \sqrt {2}}{2}t\end{cases}$$ (t is the parameter), and the polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of curve C is ρ²(3+sin²θ)=12.
1. Find the general equation of line l and the rectangular coordinate equation of curve C.
2. If line l intersects curve C at points A and B, and point P is defined as (2,1), find the value of $$\frac {|PB|}{|PA|}+ \frac {|PA|}{|PB|}$$.
|
\frac{86}{7}
| 11.71875 |
10,802 |
During intervals, students played table tennis. Any two students played against each other no more than one game. At the end of the week, it turned out that Petya played half, Kolya played a third, and Vasya played a fifth of all the games played during the week. How many games could have been played during the week if it is known that Vasya did not play with either Petya or Kolya?
|
30
| 57.8125 |
10,803 |
Triangle $PQR$ has sides of length 7, 24, and 25 units, and triangle $XYZ$ has sides of length 9, 40, and 41 units. Calculate the ratio of the area of triangle $PQR$ to the area of triangle $XYZ$, and also find the ratio of their perimeters.
|
\frac{28}{45}
| 83.59375 |
10,804 |
Six male middle school students from a school participated in a pull-up physical fitness test and scored respectively: $8$, $5$, $2$, $5$, $6$, $4$. Calculate the variance of this dataset.
|
\frac{10}{3}
| 37.5 |
10,805 |
When dividing the numbers 312837 and 310650 by some three-digit natural number, the remainders are the same. Find this remainder.
|
96
| 44.53125 |
10,806 |
The center of one sphere is on the surface of another sphere with an equal radius. How does the volume of the intersection of the two spheres compare to the volume of one of the spheres?
|
\frac{5}{16}
| 69.53125 |
10,807 |
Given a circle with radius $R$ and a fixed point $A$ on the circumference, a point is randomly chosen on the circumference and connected to point $A$. The probability that the length of the chord formed is between $R$ and $\sqrt{3}R$ is ______.
|
\frac{1}{3}
| 81.25 |
10,808 |
Find all 10-digit numbers \( a_0 a_1 \ldots a_9 \) such that for each \( k \), \( a_k \) is the number of times that the digit \( k \) appears in the number.
|
6210001000
| 67.96875 |
10,809 |
Given that \( x, y, z \in \mathbf{R}_{+} \) and \( x^{2} + y^{2} + z^{2} = 1 \), find the value of \( z \) when \(\frac{(z+1)^{2}}{x y z} \) reaches its minimum.
|
\sqrt{2} - 1
| 20.3125 |
10,810 |
Seven couples are at a social gathering. If each person shakes hands exactly once with everyone else except their spouse and one other person they choose not to shake hands with, how many handshakes were exchanged?
|
77
| 57.8125 |
10,811 |
How many four-digit positive integers exist, all of whose digits are 0's, 2's, and/or 5's, and the number does not start with 0?
|
54
| 100 |
10,812 |
Given the real numbers \( x \) and \( y \) that satisfy \( xy + 6 = x + 9y \) and \( y \in (-\infty, 1) \), find the maximum value of \((x+3)(y+1)\).
|
27 - 12\sqrt{2}
| 31.25 |
10,813 |
Let \( a, b \) and \( c \) be positive integers such that \( a^{2} = 2b^{3} = 3c^{5} \). What is the minimum possible number of factors of \( abc \) (including 1 and \( abc \))?
|
77
| 13.28125 |
10,814 |
Eight balls are randomly and independently painted either black or white with equal probability. What is the probability that each ball is painted a color such that it is different from the color of at least half of the other 7 balls?
A) $\frac{21}{128}$
B) $\frac{28}{128}$
C) $\frac{35}{128}$
D) $\frac{42}{128}$
|
\frac{35}{128}
| 58.59375 |
10,815 |
Let \( a \in \mathbf{R}_{+} \). If the function
\[
f(x)=\frac{a}{x-1}+\frac{1}{x-2}+\frac{1}{x-6} \quad (3 < x < 5)
\]
achieves its maximum value at \( x=4 \), find the value of \( a \).
|
-\frac{9}{2}
| 0 |
10,816 |
Dragoons take up \(1 \times 1\) squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the \(x\)-coordinates of their centers and the difference in the \(y\)-coordinates of their centers are both at most 6, regardless of any dragoons in between. For example, a dragoon centered at \((4,5)\) can fire at a dragoon centered at the origin, but a dragoon centered at \((7,0)\) cannot. A dragoon cannot fire at itself. What is the maximum number of dragoons that can fire at a single dragoon simultaneously?
|
168
| 94.53125 |
10,817 |
In a regular hexagon divided into 6 regions, plant ornamental plants such that the same type of plant is planted within one region, and different types of plants are planted in adjacent regions. There are 4 different types of plants available. How many planting schemes are possible?
|
732
| 79.6875 |
10,818 |
Find the smallest constant $N$, such that for any triangle with sides $a, b,$ and $c$, and perimeter $p = a + b + c$, the inequality holds:
\[
\frac{a^2 + b^2 + k}{c^2} > N
\]
where $k$ is a constant.
|
\frac{1}{2}
| 37.5 |
10,819 |
Nicholas is counting the sheep in a flock as they cross a road. The sheep begin to cross the road at 2:00 p.m. and cross at a constant rate of three sheep per minute. After counting 42 sheep, Nicholas falls asleep. He wakes up an hour and a half later, at which point exactly half of the total flock has crossed the road since 2:00 p.m. How many sheep are there in the entire flock?
|
624
| 85.9375 |
10,820 |
Consider the infinite series: $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \cdots$. Let $T$ be the limiting sum of this series. Find $T$.
**A)** $\frac{3}{26}$
**B)** $\frac{15}{26}$
**C)** $\frac{27}{26}$
**D)** $\frac{1}{26}$
**E)** $\frac{40}{26}$
|
\frac{15}{26}
| 47.65625 |
10,821 |
Compute the sum $\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor$.
|
75
| 100 |
10,822 |
Given $m \gt 1$, $n \gt 0$, and $m^{2}-3m+n=0$, find the minimum value of $\frac{4}{m-1}+\frac{m}{n}$.
|
\frac{9}{2}
| 62.5 |
10,823 |
Evaluate the integral $\int_{0}^{\frac{\pi}{2}} \sin^{2} \frac{x}{2} dx =$ \_\_\_\_\_\_.
|
\frac{\pi}{4} - \frac{1}{2}
| 55.46875 |
10,824 |
Given the line $y=-x+1$ and the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, which intersect at points $A$ and $B$.
(1) If the eccentricity of the ellipse is $\frac{\sqrt{3}}{3}$ and the focal length is $2$, find the length of the line segment $AB$.
(2) If vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are perpendicular to each other (where $O$ is the origin), find the maximum length of the major axis of the ellipse when its eccentricity $e \in [\frac{1}{2}, \frac{\sqrt{2}}{2}]$.
|
\sqrt{6}
| 7.8125 |
10,825 |
Given that point $P$ lies on the line $3x+4y+8=0$, and $PA$ and $PB$ are the two tangents drawn from $P$ to the circle $x^{2}+y^{2}-2x-2y+1=0$. Let $A$ and $B$ be the points of tangency, and $C$ be the center of the circle. Find the minimum possible area of the quadrilateral $PACB$.
|
2\sqrt{2}
| 58.59375 |
10,826 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $m=(\sin \frac{A}{2},\cos \frac{A}{2})$ and $n=(\cos \frac{A}{2},-\cos \frac{A}{2})$, and $2m\cdot n+|m|=\frac{ \sqrt{2}}{2}$, find $\angle A=$____.
|
\frac{5\pi }{12}
| 78.125 |
10,827 |
Given \( \theta_{1}, \theta_{2}, \theta_{3}, \theta_{4} \in \mathbf{R}^{+} \) and \( \theta_{1} + \theta_{2} + \theta_{3} + \theta_{4} = \pi \), find the minimum value of \( \left(2 \sin^{2} \theta_{1} + \frac{1}{\sin^{2} \theta_{1}}\right)\left(2 \sin^{2} \theta_{2} + \frac{1}{\sin^{2} \theta_{2}}\right)\left(2 \sin^{2} \theta_{3} + \frac{1}{\sin^{2} \theta_{3}}\right)\left(2 \sin^{2} \theta_{4} + \frac{1}{\sin^{2} \theta_{1}}\right) \).
|
81
| 78.125 |
10,828 |
Given a regular pyramid V-ABCD with a base edge length of 4 and a lateral edge length of $\sqrt{13}$, its surface area is ______.
|
40
| 97.65625 |
10,829 |
a) What is the maximum number of squares on an $8 \times 8$ board that can be painted black such that in each corner of three squares, there is at least one unpainted square?
b) What is the minimum number of squares on an $8 \times 8$ board that need to be painted black such that in each corner of three squares, there is at least one black square?
|
32
| 52.34375 |
10,830 |
Minimize \(\boldsymbol{F}=\boldsymbol{x}_{2}-\boldsymbol{x}_{1}\) for non-negative \(x_{1}\) and \(x_{2}\), subject to the system of constraints:
$$
\left\{\begin{aligned}
-2 x_{1}+x_{2}+x_{3} &=2 \\
x_{1}-2 x_{2}+x_{4} &=2 \\
x_{1}+x_{2}+x_{5} &=5
\end{aligned}\right.
$$
|
-3
| 23.4375 |
10,831 |
A bus arrives randomly sometime between 2:00 and 3:00 and waits for 15 minutes before leaving. If Carla arrives randomly between 2:00 and 3:00, what is the probability that the bus will still be there when Carla arrives?
|
\frac{7}{32}
| 3.125 |
10,832 |
Determine the number of ways to arrange the letters of the word MAMMAAD.
|
140
| 0.78125 |
10,833 |
Given that four people A, B, C, D are randomly selected for a volunteer activity, find the probability that A is selected and B is not.
|
\frac{1}{3}
| 25 |
10,834 |
In an international mathematics conference in 2024, a puzzle competition involves finding distinct positive integers $A$, $B$, and $C$ such that the product $A\cdot B\cdot C = 2401$. Determine the largest possible value of the sum $A+B+C$.
|
351
| 82.8125 |
10,835 |
Construct the cross-section of a triangular pyramid \( A B C D \) with a plane passing through the midpoints \( M \) and \( N \) of edges \( A C \) and \( B D \) and the point \( K \) on edge \( C D \), for which \( C K: K D = 1: 2 \). In what ratio does this plane divide edge \( A B \)?
|
1:2
| 36.71875 |
10,836 |
Grain warehouses A and B each originally stored a certain number of full bags of grain. If 90 bags are transferred from warehouse A to warehouse B, the number of bags in warehouse B will be twice the number in warehouse A. If an unspecified number of bags are transferred from warehouse B to warehouse A, the number of bags in warehouse A will be six times the number in warehouse B. What is the minimum number of bags originally stored in warehouse A?
|
153
| 12.5 |
10,837 |
Let \( x \in \left(0, \frac{\pi}{2}\right) \). Find the minimum value of the function \( y = \frac{1}{\sin^2 x} + \frac{12\sqrt{3}}{\cos x} \).
|
28
| 46.09375 |
10,838 |
This year is 2017, and the sum of the digits of the year is 10. Find the sum of all the years in this century whose digits sum to 10.
|
18396
| 2.34375 |
10,839 |
Given the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a＞0，b＞0)$$, the sum of the two line segments that are perpendicular to the two asymptotes and pass through one of its foci is $a$. Find the eccentricity of the hyperbola.
|
\frac{\sqrt{5}}{2}
| 86.71875 |
10,840 |
Perform the calculations.
36×17+129
320×(300-294)
25×5×4
18.45-25.6-24.4.
|
-31.55
| 85.9375 |
10,841 |
Find the largest natural number \( n \) with the following property: for any odd prime number \( p \) less than \( n \), the difference \( n - p \) is also a prime number.
|
10
| 95.3125 |
10,842 |
Given that $\sin\alpha = \frac{3}{5}$, and $\alpha \in \left(\frac{\pi}{2}, \pi \right)$.
(1) Find the value of $\tan\left(\alpha+\frac{\pi}{4}\right)$;
(2) If $\beta \in (0, \frac{\pi}{2})$, and $\cos(\alpha-\beta) = \frac{1}{3}$, find the value of $\cos\beta$.
|
\frac{6\sqrt{2} - 4}{15}
| 0 |
10,843 |
Assume that savings banks offer the same interest rate as the inflation rate for a year to deposit holders. The government takes away $20 \%$ of the interest as tax. By what percentage does the real value of government interest tax revenue decrease if the inflation rate drops from $25 \%$ to $16 \%$, with the real value of the deposit remaining unchanged?
|
31
| 5.46875 |
10,844 |
Let $\triangle ABC$ be an equilateral triangle with side length $s$. Point $P$ is an internal point of the triangle such that $AP=1$, $BP=\sqrt{3}$, and $CP=2$. Find the value of $s$.
|
$\sqrt{7}$
| 0 |
10,845 |
Calculate: $1.23 \times 67 + 8.2 \times 12.3 - 90 \times 0.123$
|
172.2
| 85.9375 |
10,846 |
Three boys \( B_{1}, B_{2}, B_{3} \) and three girls \( G_{1}, G_{2}, G_{3} \) are to be seated in a row according to the following rules:
1) A boy will not sit next to another boy and a girl will not sit next to another girl,
2) Boy \( B_{1} \) must sit next to girl \( G_{1} \).
If \( s \) is the number of different such seating arrangements, find the value of \( s \).
|
40
| 17.96875 |
10,847 |
Let \( ABC \) be a triangle in which \( \angle ABC = 60^\circ \). Let \( I \) and \( O \) be the incentre and circumcentre of \( ABC \), respectively. Let \( M \) be the midpoint of the arc \( BC \) of the circumcircle of \( ABC \), which does not contain the point \( A \). Determine \( \angle BAC \) given that \( MB = OI \).
|
30
| 12.5 |
10,848 |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}(1-\ln (1+\sqrt[3]{x}))^{\frac{x}{\sin ^{4} \sqrt[3]{x}}}
$$
|
e^{-1}
| 32.8125 |
10,849 |
Let \( n \) be a fixed integer with \( n \geq 2 \). Determine the minimal constant \( c \) such that the inequality
\[ \sum_{1 \leq i < j \leq n} x_i x_j (x_i^2 + x_j^2) \leq c \left( \sum_{i=1}^{n} x_i \right)^4 \]
holds for all non-negative real numbers \( x_1, x_2, \ldots, x_n \). Additionally, determine the necessary and sufficient conditions for which equality holds.
|
\frac{1}{8}
| 68.75 |
10,850 |
Given the function $f(x)= \frac{x^2}{1+x^2}$.
(1) Calculate the values of $f(2)+f(\frac{1}{2})$, $f(3)+f(\frac{1}{3})$, and $f(4)+f(\frac{1}{4})$, respectively, and conjecture a general conclusion (proof not required);
(2) Compute the value of $2f(2)+2f(3)+\ldots+2f(2017)+f(\frac{1}{2})+f(\frac{1}{3})+\ldots+f(\frac{1}{2017})+\frac{1}{2^2}f(2)+\frac{1}{3^2}f(3)+\ldots+\frac{1}{2017^2}f(2017)$.
|
4032
| 46.09375 |
10,851 |
Consider all four-digit numbers (including leading zeros) from $0000$ to $9999$. A number is considered balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. Calculate the total number of such balanced four-digit numbers.
|
670
| 61.71875 |
10,852 |
In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. Given that $\frac{{c\sin C}}{{\sin A}} - c = \frac{{b\sin B}}{{\sin A}} - a$ and $b = 2$, find:
$(1)$ The measure of angle $B$;
$(2)$ If $a = \frac{{2\sqrt{6}}}{3}$, find the area of triangle $\triangle ABC$.
|
1 + \frac{\sqrt{3}}{3}
| 6.25 |
10,853 |
Given positive real numbers \(a, b, c\) satisfy \(2(a+b)=ab\) and \(a+b+c=abc\), find the maximum value of \(c\).
|
\frac{8}{15}
| 41.40625 |
10,854 |
Evaluate the volume of solid $T$ defined by the inequalities $|x| + |y| \leq 2$, $|x| + |z| \leq 2$, and $|y| + |z| \leq 2$.
|
\frac{32}{3}
| 60.15625 |
10,855 |
Let $m$ be the smallest integer whose cube root is of the form $n+s$, where $n$ is a positive integer and $s$ is a positive real number less than $1/500$. Find $n$.
|
13
| 26.5625 |
10,856 |
Given that point \( P \) moves on the circle \( C: x^{2}+(y+2)^{2}=\frac{1}{4} \), and point \( Q \) moves on the curve \( y=a x^{2} \) (where \( a > 0 \) and \( -1 \leq x \leq 2 \)), if the maximum value of \(|PQ|\) is \(\frac{9}{2}\), then find \( a \).
|
\frac{\sqrt{3} - 1}{2}
| 2.34375 |
10,857 |
Given that $a, b \in R^{+}$, and $a + b = 1$, find the maximum value of $- \frac{1}{2a} - \frac{2}{b}$.
|
-\frac{9}{2}
| 67.1875 |
10,858 |
Find the minimum sample size for which the precision of the estimate of the population mean $a$ based on the sample mean with a confidence level of 0.975 is $\delta=0.3$, given that the standard deviation $\sigma=1.2$ of the normally distributed population is known.
|
62
| 78.90625 |
10,859 |
A box contains 5 white balls and 6 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate in color starting with a black ball?
|
\frac{1}{462}
| 54.6875 |
10,860 |
The polynomial \( p(x) = x^2 - 3x + 1 \) has zeros \( r \) and \( s \). A quadratic polynomial \( q(x) \) has a leading coefficient of 1 and zeros \( r^3 \) and \( s^3 \). Find \( q(1) \).
|
-16
| 89.84375 |
10,861 |
Find all three-digit numbers \( \overline{\mathrm{MGU}} \) consisting of distinct digits \( M, \Gamma, \) and \( U \) for which the equality \( \overline{\mathrm{MGU}} = (M + \Gamma + U) \times (M + \Gamma + U - 2) \) holds.
|
195
| 91.40625 |
10,862 |
10 people attend a meeting. Everyone at the meeting exchanges business cards with everyone else. How many exchanges of business cards occur?
|
45
| 71.875 |
10,863 |
Given the sequence $503, 1509, 3015, 6021, \dots$, determine how many of the first $1500$ numbers in this sequence are divisible by $503$.
|
1500
| 75 |
10,864 |
We consider a white \( 5 \times 5 \) square consisting of 25 unit squares. How many different ways are there to paint one or more of the unit squares black so that the resulting black area forms a rectangle?
|
225
| 78.90625 |
10,865 |
The total corn yield in centners, harvested from a certain field area, is expressed as a four-digit number composed of the digits 0, 2, 3, and 5. When the average yield per hectare was calculated, it was found to be the same number of centners as the number of hectares of the field area. Determine the total corn yield.
|
3025
| 28.125 |
10,866 |
Cara is sitting at a circular table with her seven friends. Determine the different possible pairs of people Cara could be sitting between on this table.
|
21
| 75 |
10,867 |
Find the smallest four-digit number whose product of all digits equals 512.
|
1888
| 76.5625 |
10,868 |
The graph of the function \( y = x^2 + ax + b \) is drawn on a board. Julia drew two lines parallel to the \( O x \) axis on the same diagram. The first line intersects the graph at points \( A \) and \( B \), and the second line intersects the graph at points \( C \) and \( D \). Find the distance between the lines, given that \( A B = 5 \) and \( C D = 11 \).
|
24
| 66.40625 |
10,869 |
Through the end of a chord that divides the circle in the ratio 3:5, a tangent is drawn. Find the acute angle between the chord and the tangent.
|
67.5
| 41.40625 |
10,870 |
In the following diagram, \(\angle ACB = 90^\circ\), \(DE \perp BC\), \(BE = AC\), \(BD = \frac{1}{2} \mathrm{~cm}\), and \(DE + BC = 1 \mathrm{~cm}\). Suppose \(\angle ABC = x^\circ\). Find the value of \(x\).
|
30
| 27.34375 |
10,871 |
Given the shadow length $l$ is equal to the product of the table height $h$ and the tangent value of the solar zenith angle $\theta$, and $\tan(\alpha-\beta)=\frac{1}{3}$, if the shadow length in the first measurement is three times the table height, determine the shadow length in the second measurement as a multiple of the table height.
|
\frac{4}{3}
| 47.65625 |
10,872 |
Given the function \( f(x) = \frac{1}{\sqrt[3]{1 - x^3}} \). Find \( f(f(f( \ldots f(19)) \ldots )) \), calculated 95 times.
|
\sqrt[3]{1 - \frac{1}{19^3}}
| 0 |
10,873 |
Given the sample data set $3$, $3$, $4$, $4$, $5$, $6$, $6$, $7$, $7$, calculate the standard deviation of the data set.
|
\frac{2\sqrt{5}}{3}
| 25 |
10,874 |
A notebook with 75 pages numbered from 1 to 75 is renumbered in reverse, from 75 to 1. Determine how many pages have the same units digit in both the old and new numbering systems.
|
15
| 78.125 |
10,875 |
Solve the inequalities:
(1) $(2x - 4)(x - 5) < 0$;
(2) $3x^{2} + 5x + 1 > 0$;
(3) $-x^{2} + x < 2$;
(4) $7x^{2} + 5x + 1 \leq 0$;
(5) $4x \geq 4x^{2} + 1$.
|
\left\{\frac{1}{2}\right\}
| 78.125 |
10,876 |
Two circles are inscribed in an angle of 60 degrees and they touch each other. The radius of the smaller circle is 24. What is the radius of the larger circle?
|
72
| 39.84375 |
10,877 |
Suppose $d$ and $e$ are digits. For how many pairs of $(d, e)$ is $2.0d06e > 2.006$?
|
99
| 10.15625 |
10,878 |
What is the least positive integer that is divisible by the next three distinct primes larger than 5?
|
1001
| 81.25 |
10,879 |
Let the operation $x*y$ be defined as $x*y = (x+1)(y+1)$. The operation $x^{*2}$ is defined as $x^{*2} = x*x$. Calculate the value of the polynomial $3*(x^{*2}) - 2*x + 1$ when $x=2$.
|
32
| 65.625 |
10,880 |
Suppose that the angles of triangle $PQR$ satisfy
\[\cos 3P + \cos 3Q + \cos 3R = 1.\]Two sides of the triangle have lengths 12 and 15. Find the maximum length of the third side.
|
27
| 19.53125 |
10,881 |
Let $x$, $y$, $z$ be the three sides of a triangle, and let $\xi$, $\eta$, $\zeta$ be the angles opposite them, respectively. If $x^2 + y^2 = 2023z^2$, find the value of
\[\frac{\cot \zeta}{\cot \xi + \cot \eta}.\]
|
1011
| 67.96875 |
10,882 |
Let's call a number "remarkable" if it has exactly 4 different natural divisors, among which there are two such that neither is a multiple of the other. How many "remarkable" two-digit numbers exist?
|
30
| 28.125 |
10,883 |
In the Cartesian coordinate system $(xOy)$, the parametric equations of curve $C_{1}$ are given by $\begin{cases}x=2t-1 \\ y=-4t-2\end{cases}$ $(t$ is the parameter$)$, and in the polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar equation of curve $C_{2}$ is $\rho= \frac{2}{1-\cos \theta}$.
(1) Write the Cartesian equation of curve $C_{2}$;
(2) Let $M_{1}$ be a point on curve $C_{1}$, and $M_{2}$ be a point on curve $C_{2}$. Find the minimum value of $|M_{1}M_{2}|$.
|
\frac{3 \sqrt{5}}{10}
| 4.6875 |
10,884 |
A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 63. If $m$ is also divisible by 11, what is the greatest possible value of $m$?
|
9702
| 0 |
10,885 |
Given that the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n=3n^{2}+8n$, and $\{b\_n\}$ is an arithmetic sequence with $a\_n=b\_n+b_{n+1}$:
1. Find the general term formula for the sequence $\{b\_n\}$.
2. Find the maximum value of $c\_n=\frac{3a\_n}{b\_n-11}$ and specify which term it corresponds to.
|
\frac{87}{2}
| 5.46875 |
10,886 |
In the Cartesian coordinate plane $xOy$, an ellipse $(E)$ has its center at the origin, passes through the point $A(0,1)$, and its left and right foci are $F_{1}$ and $F_{2}$, respectively, with $\overrightarrow{AF_{1}} \cdot \overrightarrow{AF_{2}} = 0$.
(I) Find the equation of the ellipse $(E)$;
(II) A line $l$ passes through the point $(-\sqrt{3}, 0)$ and intersects the ellipse $(E)$ at exactly one point $P$. It also tangents the circle $(O): x^2 + y^2 = r^2 (r > 0)$ at point $Q$. Find the value of $r$ and the area of $\triangle OPQ$.
|
\frac{1}{4}
| 14.0625 |
10,887 |
Find the sum of all four-digit numbers in which the digits $0, 3, 6, 9$ are absent.
|
6479352
| 27.34375 |
10,888 |
$\Phi$ is the union of all triangles that are symmetric of the triangle $ABC$ wrt a point $O$ , as point $O$ moves along the triangle's sides. If the area of the triangle is $E$ , find the area of $\Phi$ .
|
2E
| 19.53125 |
10,889 |
Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ written in its lowest terms be $\frac{p}{q}$ . Find the value of $q - p$ .
|
83
| 58.59375 |
10,890 |
Starting with a list of three numbers, the “*Make-My-Day*” procedure creates a new list by replacing each number by the sum of the other two. For example, from $\{1, 3, 8\}$ “*Make-My-Day*” gives $\{11, 9, 4\}$ and a new “*MakeMy-Day*” leads to $\{13, 15, 20\}$ . If we begin with $\{20, 1, 8\}$ , what is the maximum difference between two numbers on the list after $2018$ consecutive “*Make-My-Day*”s?
|
19
| 62.5 |
10,891 |
It takes Mina 90 seconds to walk down an escalator when it is not operating, and 30 seconds to walk down when it is operating. Additionally, it takes her 40 seconds to walk up another escalator when it is not operating, and only 15 seconds to walk up when it is operating. Calculate the time it takes Mina to ride down the first operating escalator and then ride up the second operating escalator when she just stands on them.
|
69
| 57.03125 |
10,892 |
Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$ , the decimal representation of the number $c^n+2014$ has digits all less than $5$ .
*Proposed by Evan Chen*
|
10
| 56.25 |
10,893 |
Given three distinct points $A$, $B$, $C$ on a straight line, and $\overrightarrow{OB}=a_{5} \overrightarrow{OA}+a_{2012} \overrightarrow{OC}$, find the sum of the first 2016 terms of the arithmetic sequence $\{a_{n}\}$.
|
1008
| 89.0625 |
10,894 |
Select 5 different letters from the word "equation" to arrange in a row, including the condition that the letters "qu" are together and in the same order.
|
480
| 21.875 |
10,895 |
Calculate the result of the expression:
\[ 2013 \times \frac{5.7 \times 4.2 + \frac{21}{5} \times 4.3}{\frac{14}{73} \times 15 + \frac{5}{73} \times 177 + 656} \]
|
126
| 92.96875 |
10,896 |
In the final stage of a professional bowling competition, the top five players compete as follows:
- The fifth place player competes against the fourth place player.
- The loser of the match receives the 5th place award.
- The winner then competes against the third place player.
- The loser of this match receives the 4th place award.
- The winner competes against the second place player.
- The loser receives the 3rd place award.
- The winner competes against the first place player.
- The loser of this final match receives the 2nd place award, and the winner receives the 1st place award.
How many different possible sequences of award distribution are there?
|
16
| 96.875 |
10,897 |
Four friends rent a cottage for a total of £300 for the weekend. The first friend pays half of the sum of the amounts paid by the other three friends. The second friend pays one third of the sum of the amounts paid by the other three friends. The third friend pays one quarter of the sum of the amounts paid by the other three friends. How much money does the fourth friend pay?
|
65
| 82.03125 |
10,898 |
Another trapezoid \(ABCD\) has \(AD\) parallel to \(BC\). \(AC\) and \(BD\) intersect at \(P\). If \(\frac{[ADP]}{[BCP]} = \frac{1}{2}\), find \(\frac{[ADP]}{[ABCD]}\). (Here, the notation \(\left[P_1 \cdots P_n\right]\) denotes the area of the polygon \(P_1 \cdots P_n\)).
|
3 - 2\sqrt{2}
| 14.84375 |
10,899 |
Nyusha has 2022 coins, and Barash has 2023. Nyusha and Barash toss all their coins simultaneously and count how many heads each gets. The one who gets more heads wins, and in case of a tie, Nyusha wins. What is the probability that Nyusha wins?
|
0.5
| 49.21875 |
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