problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given $a=(\cos\alpha, \sin\alpha)$, $b=(\cos\beta, \sin\beta)$, then ( )
A: $a \perp b$
B: $a \parallel b$
C: $(a+b) \perp (a-b)$
D: The angle between $a$ and $b$ is $\alpha+\beta$ | C |
If vector $\vec{a}=(1,2)$, $\vec{b}=(-3,4)$, then $(\vec{a} \cdot \vec{b})(\vec{a} + \vec{b})$ equals to ( )
A: $20$
B: $(-10,30)$
C: $54$
D: $(-8,24)$ | \text{B} |
1. Let $f(x)=\frac{x}{\sqrt{1+x^{2}}}, f_{1}(x)=f(x), f_{2}(x)=f\left(f_{1}(x)\right), \cdots, f_{n}(x)=f\left(f_{n-1}(x)\right)$, then $f_{n}(x)=$ | f_{n}(x)=\frac{x}{\sqrt{1+nx^{2}}} |
Distribute 8 students from four classes (2 students per class) to four different districts for a social survey, with 2 students in each district. The number of assignment schemes where exactly 2 districts are assigned 2 students from the same class is ( )
A: 48
B: 72
C: 144
D: 288 | D |
If a random variable $X$ follows the normal distribution $X \sim N(\mu, \sigma^2)$, and $P(X > 5) = P(X < -1) = 0.2$, then find $P(2 < X < 5)$. | 0.3 |
Given the line $x+2y-4=0$ intersects the parametric equations $\begin{cases} \left.\begin{matrix}x=2-3\cos\theta \\ y=1+3\sin\theta\end{matrix}\right\} \end{cases}$ ($\theta$ is the parameter) at points A and B, find the length of $|AB|$. | 6 |
Given the circle $C_{1}:x^{2}+y^{2}+4x-4y-3=0$, and a moving point $P$ on the circle $C_{2}:x^{2}+y^{2}-4x-12=0$, the maximum area of $\triangle PC_{1}C_{2}$ is ( )
A: $2\sqrt{5}$
B: $4\sqrt{5}$
C: $8\sqrt{5}$
D: $20$ | B |
4. Fold a triangle with side lengths of $10$, $12$, and $14$ along its three midlines to form a tetrahedron. Then the diameter of the circumscribed sphere of the tetrahedron is $\qquad$ . | \sqrt{55} |
7.290. $\left\{\begin{array}{l}y^{x}=1.5+y^{-x} \\ y^{2.5+x}=64(y>0)\end{array}\right.$ | (\frac{1}{2};4) |
Given $f(x)= \sqrt {3}\cos 2x-\sin 2x$, if $y=f(x-m)(m > 0)$ is an odd function, then the minimum value of $m$ is ( ).
A: $ \dfrac {\pi}{6}$
B: $ \dfrac {5\pi}{6}$
C: $ \dfrac {\pi}{3}$
D: $ \dfrac {2\pi}{3}$ | C |
Given the function $f(x)=4\cos x \sin\left(x+ \frac{\pi}{6}\right)$.
$(1)$ Find the axis of symmetry for $f(x)$;
$(2)$ Find the maximum and minimum values of $f(x)$ in the interval $\left[- \frac{\pi}{6}, \frac{\pi}{4}\right]$. | 0 |
How many subsets are there of a set of cardinality $n$? | 2^{n} |
Calculate:
(1) $$\frac {lg2+lg5-lg8}{lg50-lg40}$$
(2) $$2^{2+log_{ \sqrt {2}} \frac {1}{4}}$$． | \frac {1}{4} |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $a\cos C + \sqrt{3}a\sin C = b + c$.
$(1)$ Find the value of angle $A$.
$(2)$ If triangle $\triangle ABC$ is an acute triangle and $b = 2$, find the range of possible values ... | \left(\frac{\sqrt{3}}{2}, 2\sqrt{3}\right) |
The total cost of 4 one-litre cartons of milk is $\$ 4.88$. The cost of 1 one-litre carton of milk is
(A) $\$ 0.88$
(B) $\$ 1.44$
(C) $\$ 1.88$
(D) $\$ 4.22$
(E) $\$ 1.22$ | 1.22 |
I4.1 Solve the equation $2^{a+1}+2^{a}+2^{a-1}=112$ in $a$.
I4.2 If $a$ is one root of the equation $x^{2}-b x+35=0$, find $b$.
I4.3 If $\sin \theta=\frac{-b}{15}$, where $180^{\circ}<\theta<270^{\circ}$, and $\tan \theta=\frac{c}{3}$, find $c$.
I4.4 The probability of getting a sum of $c$ in throwing two dice is $\fra... | =5,b=12,=4,=12 |
Given the parabola $y^2 = 8x$ with directrix $l$, and a point Q on the circle $C: x^2 + y^2 + 2x - 8y + 13 = 0$, let the distance from any point P on the parabola to the directrix $l$ be $d$. Find the minimum value of $d + |PQ|$. | 3 |
Given $1+x+x^2+x^3+x^4=0$, then the polynomial $1+x+x^2+x^3+\ldots+x^{1989}=$ （ ）
A: 1
B: 1+x
C: 0
D: 1989 | C |
II. (40 points) Given that there exist integers $x_{1}, x_{2}, \cdots, x_{n}$ $\left(n \in \mathbf{Z}_{+}\right)$, such that
$$
(4 n+1)\left(\sum_{i=1}^{n} x_{i}^{2}-1\right)=4\left(\sum_{i=1}^{n} x_{i}\right)^{2} \text {. }
$$
Find all possible values of $n$ and the corresponding integer solutions $\left(x_{1}, x_{2}... | 2or6 |
Given propositions p: $1 \in \{x | x^2 < a\}$; q: $2 \in \{x | x^2 < a\}$
(1) If the proposition "p ∨ q" is true, find the range of values for the real number $a$.
(2) If the proposition "p ∧ q" is true, find the range of values for the real number $a$. | a > 4 |
8. The numbers $1059, 1417, 2312$ leave the same remainder when divided by $d$, $d>1$, then the integer $d=$
保留了源文本的换行和格式。 | 179 |
Example 6. Find the sum $S_{n}=1^{4}+2^{4}+\cdots+n^{4}$. | S_{n} = \frac{1}{5} n^{5} + \frac{1}{2} n^{4} + \frac{1}{3} n^{3} - \frac{1}{30} n |
7.131. $\left\{\begin{array}{l}\log _{4} x+\log _{4} y=1+\log _{4} 9, \\ x+y-20=0 .\end{array}\right.$ | (2;18),(18;2) |
The area of the enclosed shape formed by the parabola $y=x^{2}-4x+3$ and the $x$-axis is ( ).
A: $\dfrac {2}{3}$
B: $\dfrac {4}{3}$
C: $2$
D: $\dfrac {8}{3}$ | B |
8. The system of equations $\left\{\begin{array}{l}x^{3}+y^{3}=26 \\ x y(x+y)=-6\end{array}\right.$ has the real solutions as | {\begin{pmatrix}-1\\3\end{pmatrix}. |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$:
$$
f\left(x^{2022}+y\right)=f\left(x^{1747}+2 y\right)+f\left(x^{42}\right)
$$ | 0 |
Let $a, b$ be two known constants, and $a>b$. Points $P^{3}, Q$ are on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. If the line connecting point $A(-a, 0)$ and $Q$ is parallel to the line $O P$, and intersects the $y$-axis at point $R$, then $\frac{|A Q| \cdot |A R|}{|O P|^{2}}=$ $\qquad$ (where $O$ is the ... | 2 |
If the equation $x^{2}+y^{2}-2ax+4ay+6a^{2}-a=0$ represents a circle whose center is in the fourth quadrant, then the range of the real number $a$ is \_\_\_\_\_\_. | 0 < a < 1 |
On a plate, there are a total of 58 slices of beigli, a mixture of walnut and poppy seed. Three walnut ones can be chosen from them in the same number of ways as two poppy seed ones and one walnut one. How many poppy seed beigli are on the plate? | 21 |
Given the function $y=\log(x^2-2kx+k)$ has a range of $\mathbb{R}$, then the range of the real number $k$ is ( )
A: $(0,1)$
B: $(-\infty,0]\cup[1,+\infty)$
C: $[0,1)$
D: $k=0$ or $k\geq1$ | \text{B} |
A line with a slope of $1$ passes through the focus of the parabola $y^{2}=4x$ and intersects the parabola at points $A$ and $B$. Find the length of the line segment $|AB|=$ ( ).
A: $8$
B: $6$
C: $12$
D: $7 \sqrt {3}$ | A |
5.8. $x^{4}+x^{3}+x^{2}+x+12$. | (x^{2}-2x+3)(x^{2}+3x+4) |
\section*{Problem 5 - 331035}
Determine all pairs \((m ; n)\) of positive integers \(m\) and \(n\) for which
\[
\frac{m^{2}}{m+1}+\frac{n^{2}}{n+1}
\]
is an integer. | (1;1) |
If the sequence $\{a_n\}$ satisfies $(2n+3)a_{n+1}-(2n+5)a_n=(2n+3)(2n+5)\lg (1+ \frac{1}{n} )$ and $a_1=5$, then the 2017th term of the sequence $\left\{ \frac{a_n}{2n+3} \right\}$ is ( )
A: $\lg 2017$
B: $\lg 2016$
C: $1+\lg 2016$
D: $1+\lg 2017$ | D |
8. As shown in Figure 4, in parallelogram $A B C D$, $A B=2, B C=$ $2 \sqrt{3}, A C=4$, a line $E F \perp A C$ is drawn through the midpoint $O$ of $A C$, intersecting $A D$ at $E$ and $B C$ at $F$. Then $E F=$ $\qquad$ | \frac{4 \sqrt{3}}{3} |
Problem 9.5. Point $M$ is the midpoint of side $B C$ of triangle $A B C$, where $A B=17$, $A C=30, B C=19$. A circle is constructed with side $A B$ as its diameter. An arbitrary point $X$ is chosen on this circle. What is the minimum value that the length of segment $M X$ can take?
 $f(x + y) + f(x - y) = 2f(x)f(y)$ for all $x, y \in \mathbb R$, and
(b) $\lim_{x\to \infty} f(x) = 0$. | f(x) = 0 |
Example: $x, y$ are prime numbers, solve the equation $x^{y}-y^{x}=x y^{2}-19$.
| 2,7;2,3 |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to $\angle A$, $\angle B$, $\angle C$ respectively. If $\cos 2B + \cos B + \cos (A-C) = 1$ and $b = \sqrt{7}$, find the minimum value of $a^2 + c^2$. | 14 |
$2 \cdot 11$ The equation $x=k^{2}(x-1)(x-2)$ has real roots, the value of $k$ is
(A) does not exist.
(B) $-21$ or $k<-2$.
(E) all real numbers.
(13th American High School Mathematics Examination, 1962) | E |
Given that one focus of the ellipse $\frac{x^{2}}{2m^{2}} + \frac{y^{2}}{m+1} = 1$, with foci on the $x$-axis, lies on the line $\sqrt{2}x - y + 2 = 0$, find the eccentricity of the ellipse. | \frac{2}{3} |
Problem 1. The sum of the first four terms of an arithmetic progression, as well as the sum of the first nine terms, are natural numbers. In addition, its first term $b_{1}$ satisfies the inequality $b_{1} \leqslant \frac{3}{4}$. What is the greatest value that $b_{1}$ can take? | \frac{11}{15} |
1. How many times is the value of the expression
$$
2 \cdot \frac{7 \frac{2}{3}:\left(2 \frac{2}{3}+0.5: 1 \frac{1}{2}\right)}{\left(2 \frac{1}{3}-1 \frac{1}{9} \cdot 0.75\right): 7 \frac{1}{3}}
$$
smaller than the number 2024? | \frac{2024}{81} |
Simplify first, then evaluate: $({\frac{2}{{m-n}}-\frac{1}{{m+n}}})÷\frac{{mn+3{n^2}}}{{{m^3}-m{n^2}}}$, where $m=-1$ and $n=\frac{1}{2}$. | -2 |
Which of the following is the condition for "$\sin \theta +\tan \theta\ \ \gt 0$" when $\theta$ is an angle in the first or third quadrant?
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither sufficient nor necessary condition | C |
17・142 As shown in the figure, given $\angle C=90^{\circ}, A D=D B, D E \perp A B, A B=20$ and $A C=12$, then the area of quadrilateral $A D E C$ is
(A) 75.
(B) $58 \frac{1}{2}$.
(C) 48.
(D) $37 \frac{1}{2}$.
(E) Other. | 58\frac{1}{2} |
218. Someone bought 30 birds for 30 coins, among these birds, for every 3 sparrows, 1 coin was paid, for every 2 doves, also 1 coin, and finally, for each pigeon - 2 coins. How many birds of each kind were there?
## Problem of John of Palermo. | 9,10,11 |
8. On a circular track that is 660 m long, a relay race is being held, with each stage being $150 \mathrm{~m}$. The start and finish are in the same place. What is the minimum number of stages that can be in this relay? | 22 |
Let $N$ is the set of all positive integers. Determine all mappings $f: N-\{1\} \to N$ such that for every $n \ne m$ the following equation is true $$f(n)f(m)=f\left((nm)^{2021}\right)$$
| f(x) = 1 \quad \forall x \geq 2 |
Problem 10.8. For what least natural $a$ does the numerical interval ( $a, 3 a$ ) contain exactly 50 perfect squares? | 4486 |
8.1. (13 points) Find the smallest natural number such that after multiplying it by 9, the result is a number written with the same digits but in some different order. | 1089 |
Given the sets $A=\left\{x \mid 3 < 3^{x} < 9 \right\}$, $B=\left\{x \mid 1 < \log_{2}x < 4 \right\}$, $C=(a,a+3)$.
$(1)$ Find $(C_{R}A) \cap B$;
$(2)$ If $A \subseteq C$, find the range of the real number $a$. | -1 \leqslant a \leqslant 1 |
The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$. | 937 |
Given the vertices of a triangle A(-5, 0), B(3, -3), C(0, 2), find:
(1) The equation of the line on which side AC lies
(2) The equation of the line on which the median to side BC lies. | x+13y+5=0 |
Find the number of $n$-digit numbers composed of $1, 2, 3$ where each of $1, 2, 3$ appears at least once. | 3^{n}-3\cdot2^{n}+3 |
10 Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=\frac{1}{4}, a_{2}=\frac{3}{4}, a_{n+1}=2 a_{n}-a_{n-1}(n \geqslant$
2), the sequence $\left\{b_{n}\right\}$ satisfies: $b_{1} \neq \frac{1}{4}, 3 b_{n}-b_{n-1}=n(n \geqslant 2)$, and the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$ is ... | S_{n}=\frac{1}{4}n^{2}-(\frac{1}{3})^{n}+1 |
9.1. Petya wrote 10 integers on the board (not necessarily distinct).
Then he calculated the pairwise products (that is, he multiplied each of the written numbers by each other). Among them, there were exactly 15 negative products. How many zeros were written on the board? | 2 |
Example 9 In $\triangle A B C$, $\angle A>\angle B>\angle C$, $\angle A \neq 90^{\circ}$, draw a line that divides $\triangle A B C$ into two parts, and makes one of the parts similar to $\triangle A B C$. There are ( ) such non-parallel lines.
(A) 2
(B) 3
(C) 6
(D) 12 | C |
1. Find all complex numbers $z$ for which the equality
$$
z^{2}=\bar{z}
$$
holds. | z_{1}=0,z_{2}=1,z_{3}=-\frac{1}{2}+i\frac{\sqrt{3}}{2},z_{4}=-\frac{1}{2}-i\frac{\sqrt{3}}{2} |
In the independence test, the method that can roughly judge whether two categorical variables are related is ( )
A: Residuals
B: Contour bar chart
C: The idea of hypothesis testing
D: None of the above | \text{B} |
Three, 1. Sketch the graph of the function $\mathrm{y}=2 \sin \left(\frac{\mathrm{x}}{2}-\frac{\pi}{6}\right)+1$, and answer the following questions:
(1) Domain is $\qquad$ , Range is $\qquad$ .
(2) Period is $\qquad$ , Frequency is $\qquad$ . $\mathrm{Y}_{\text {max is }}$ , $\mathbf{Y}_{\text {min is }}$ $\qquad$ | y_{\text{max}}=3, y_{\text{min}}=-1 |
3$\cdot$34 A person is sailing a sailboat downstream along a river and encounters a speedboat coming upstream. He asks the speedboat driver: "Is there a ship coming behind you?" The speedboat driver replies: "I passed a ship half an hour ago." The sailboat continues to sail for another half hour and meets a ship coming... | 5 |
Given a natural number $k$, find the smallest natural number $C$ such that
$$\frac C{n+k+1}\binom{2n}{n+k}$$is an integer for every integer $n\ge k$. | 2k+1 |
If the lines \\(l_1: x-2y+1=0\\) and \\(l_2: 2x+ay-2=0\\) are parallel, then the distance between \\(l_1\\) and \\(l_2\\) is __________. | \frac{2\sqrt{5}}{5} |
19) How many equilateral triangles are present in this figure?
(A) 16
(B) 20
(C) 25
(D) 26
(E) 27
 | 27 |
Which of the following options represent quantities with opposite meanings?
A: Winning $1$ game and losing $2$ ten thousand dollars
B: Traveling $5km$ east and $10km$ north
C: Transporting $6kg$ of apples and selling $5kg$ of apples
D: Water level rising by $0.6$ meters and water level dropping by $1$ meter | D |
If $\log _{3 n} 675 \sqrt{3}=\log _{n} 75$, determine the value of $n^{5}$. | 5625 |
3-4. How many natural ${ }^{2}$ numbers less than a thousand are there that are not divisible by 5 or 7? | 686 |
Ex. 169. On one side of an angle with vertex at point $M$, equal to $30^{\circ}$, points $A$ and $B$ are marked such that $M A=1, M B=3$. A point $C$ moves along the other side of the angle. Find $M C$, if it is known that
a) the angle $B C A$ is the largest;
b) the sum of the squares of the distances from $C$ to poi... | MC=\sqrt{3} |
8. A point $A$ on the $x$-axis is equidistant from the fixed points $(-3,2)$ and $(4,-5)$. Then the coordinates of $A$ are | (2,0) |
6. On the number line, points with coordinates $0,1,2,3,5,8,2016$ are marked.
Consider the set of lengths of segments with endpoints at these points. How many elements does it contain?
ANSWER: 14. | 14 |
Given that in triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Angle $B$ is obtuse. Let the area of $\triangle ABC$ be $S$. If $4bS=a(b^{2}+c^{2}-a^{2})$, then the maximum value of $\sin A + \sin C$ is ____. | \frac{9}{8} |
When Paulo turned 15, he invited 43 friends to a party. The cake was in the shape of a regular 15-sided polygon and had 15 candles on it. The candles were placed in such a way that no three candles were in a straight line. Paulo divided the cake into triangular pieces where each cut connected two candles or connected a... | 43 |
7. Three people are playing a ball-passing game, starting from $A$, and the ball can be passed from $A$ to $B, C$, from $B$ to $A, C$, and from $C$ to $A, B$ with equal probability. Then the probability that the ball returns to $A$ after 2021 passes is $\qquad$ . | \frac{1}{3}(1-\frac{1}{2^{2020}}) |
Five balls, numbered 1 to 5 , are placed in order on a table. A sequence of steps is performed on the balls. In step 1, the rightmost ball is picked up and put in the middle of the four remaining balls. (The remaining balls are shifted to make room for the inserted ball.) Then in step 2, the leftmost ball is picked up ... | 2025 |
If $\cos 60^{\circ}=\cos 45^{\circ} \cos \theta$ with $0^{\circ} \leq \theta \leq 90^{\circ}$, then $\theta$ equals
(A) $0^{\circ}$
(B) $15^{\circ}$
(C) $30^{\circ}$
(D) $45^{\circ}$
(E) $60^{\circ}$ | 45 |
Let $P_{1}=\{x|x^{2}+ax+1 \gt 0\}$, $P_{2}=\{x|x^{2}+ax+2 \gt 0\}$ ($a\in R$), which of the following statements is correct?
A: For any $a$, $P_{1}$ is a subset of $P_{2}$.
B: For any $a$, $P_{1}$ is not a subset of $P_{2}$.
C: There exists $a$ such that $P_{1}$ is not a subset of $P_{2}$.
D: There exists $a$ such ... | A |
Let's find such an $n$ and such numbers $a_{i}, b_{i}, c_{i},(i=1,2, \ldots, n)$, for which the points $(x, y)$ satisfying the condition
$$
\sum_{i=1}^{n}\left|a_{i} x+b_{i} y+c_{i}\right|=10
$$
when plotted on the coordinate system, form the vertices of a regular octagon! | 4 |
The minimum value of the function $f(x)=(\frac{1}{2})^{x}$ in the interval $[-2,-1]$ is $\_\_\_\_\_\_\_.$ | 2 |
Daniel can hack a finite cylindrical log into $3$ pieces in $6$ minutes. How long would it take him to cut it into $9$ pieces, assuming each cut takes Daniel the same amount of time?
[i]2015 CCA Math Bonanza Lightning Round #1.3[/i] | 24 \text{ minutes} |
Given that the domain of the function $f(x)$ is $[-2, 2]$, and $f(x)$ is an increasing function on $[-2, 2]$, if $f(1-m) < f(m)$, then the range of the real number $m$ is ( )
A: $( \frac {1}{2}, +\infty)$
B: $(-\infty, \frac {1}{2})$
C: $( \frac {1}{2}, 2]$
D: $[-2, \frac {1}{2})$ | C |
12. (9) On a plane, there are 9 points arranged in a $3 \times 3$ grid, as shown in the figure.
a) Lines are drawn through all possible pairs of points. How many different lines are obtaine... | 20 |
Example 1.29. In the equations of the line $\frac{x}{2}=\frac{y}{-3}=\frac{z}{n}$, determine the parameter $n$ so that this line intersects with the line $\frac{x+1}{3}=\frac{y+5}{2}=\frac{z}{1}$, and find the point of their intersection. | M(2,-3,1) |
10. (10 points) The teacher used the ten digits from 0 to 9 to form five two-digit numbers, with each digit used exactly once; then gave these five two-digit numbers to five clever and honest students $A, B, C, D, E$, each student could only see their own two-digit number, and the following conversation took place:
$A$... | 180 |
7. The axial section $SAB$ of the circular cone is an equilateral triangle with a side length of 2, $O$ is the center of the base, $M$ is the midpoint of $SO$, and the moving point $P$ is within the base of the cone (including the circumference). If $AM \perp MP$, then the length of the trajectory formed by point $P$ i... | \frac{\sqrt{7}}{2} |
Given that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and $a_{1}=1$, $a_{n+1}=2S_{n}$ for $n\geqslant 1$, then $a_{2022}=\left(\ \ \right)$
A: $2\times 3^{2021}$
B: $3^{2021}$
C: $3^{2020}$
D: $2\times 3^{2020}$ | D |
## Task 3 - 100933
If $x$ is a real number, let $[x]$ denote the greatest integer not greater than $\mathrm{x}$. (For example, $[3.7]=3,[-3.7]=-4,[4]=4$.)
Determine all real numbers $\mathrm{x}$ for which:
$$
\left[\frac{10+3 x}{6}\right]=\frac{5 x+3}{7}
$$ | \frac{11}{5},\frac{18}{5},5 |
A new type of car costs \\(10\\) ten thousand yuan to purchase, with annual insurance, road maintenance, and fuel costs approximately \\(0.9\\) ten thousand yuan. The maintenance cost for the first year is \\(0.2\\) ten thousand yuan, increasing by \\(0.2\\) ten thousand yuan each subsequent year. What is the number of... | 10 |
Given that the eccentricity of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ is $\frac{\sqrt{3}}{2}$, and the point $P(2, \sqrt{3})$ is on the ellipse.
(1) Find the equation of the ellipse $C$;
(2) Suppose the left and right vertices of the ellipse are $A$ and $B$, respectively. A moving straight... | 8 |
3. There are 12 students in the class. Each of them was asked how many friends they have in the class. Each number \( n \) was either mentioned exactly \( n \) times or not at all. What is the maximum value that the sum of all the numbers mentioned can take? | 90 |
Find the range of values for $k$ such that the line $y = k(x-1)$ intersects with the line segment with endpoints $A(3,2)$ and $B(2,3)$.
A: $[1,\sqrt{3}]$
B: $[1,\frac{3}{2}]$
C: $[1,3]$
D: $[\frac{\sqrt{2}}{2},3]$ | C |
## Task 3 - 100613
Let $a, b, c, d, e, f, g, h$ be pairwise distinct single-digit natural numbers not equal to zero, and let it be given that:
$$
a+b=10, \quad c+d+e=16, \quad f+g+h=14
$$
Which single-digit natural number (not equal to zero) was not used in these three equations?
Provide a possible solution for $a,... | 5 |
346. Four villages are located at the vertices of a square with a side length of 2 km. The villages are connected by roads in such a way that it is possible to travel from any village to any other. Can the total length of the roads be less than 5.5 km? | 2+2\sqrt{3}<5.5 |
6. In tetrahedron $ABCD$, the opposite edges are equal, $E, F$ are the midpoints of $AB, CD$ respectively, and $EF \perp AB, EF \perp CD, EF=6, AC=10, BC=6\sqrt{2}$. Then the distance between the skew lines $AD$ and $BC$ is $\qquad$. | 8 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that ${b}^{2}+{c}^{2}={a}^{2}+bc$.
(I) Find the measure of angle $A$;
(II) If $\sin B= \frac{ \sqrt{3}}{3 } $, $b=2$, find the area of $\triangle ABC$. | \frac {3 \sqrt {2}+ \sqrt {3}}{2} |
A rope with a length of 4 meters is cut into 5 equal segments. What is the length of each segment in meters, and what fraction does each segment represent of the total rope length? Additionally, what fraction do 3 of these segments represent of the total rope length? | \frac{3}{5} |
If $ab > 0$, and points $A(a, 0)$, $B(0, b)$, $C(-2, -2)$ are collinear, find the minimum value of $ab$. | 16 |
## Problem Statement
Calculate the definite integral:
$$
\int_{1 / 24}^{1 / 3} \frac{5 \sqrt{x+1}}{(x+1)^{2} \sqrt{x}} d x
$$ | 3 |
Three numbers \(a=\log_{2}0.4\), \(b=0.4^{2}\), \(c=2^{0.4}\) are in the order of \((\quad)\).
A: \(b < a < c\)
B: \(a < c < b\)
C: \(a < b < c\)
D: \(b < c < a\) | C |
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