problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
4. Given prime numbers $p$ and $q$, such that $p^{3}-q^{5}=(p+q)^{2}$. Then $\frac{8\left(p^{2013}-p^{2010} q^{5}\right)}{p^{2011}-p^{2009} q^{2}}=$ $\qquad$. | 140 |
：Given $M$ as the midpoint of the height $S O$ of a regular tetrahedron $S-A B C$ with edge length 1. A plane is drawn through $M$ parallel to the lateral edge $S A$ and the base triangle $B C$. The area of the section formed by this plane cutting through the tetrahedron is $\qquad$
$\qquad$ . | \frac{2}{9} |
5. Find all solutions of the equation $(x+1)(x+2)(x+3)(x+4)=120$ in the set of complex
(8) numbers. | x_{1}=-6,x_{2}=1,x_{3}=\frac{-5-i\sqrt{39}}{2},x_{4}=\frac{-5+i\sqrt{39}}{2} |
Given that $y = f(x)$ is an even function, among the points $(-a, f(a))$, $(-a, -f(-a))$, $(-a, -f(a))$, and $(a, -f(-a))$, the point that definitely lies on the graph of $y = f(x)$ is __________. | (-a, f(a)) |
$(1)$ Calculate: $[(x^{2}+y^{2})-(x-y)^{2}+2y(x-y)]\div 4y$.
$(2)$ Solve the equation: $\frac{x}{x-2}-\frac{3}{x}=1$. | x=6 |
A spherical iron ball is placed into a cylindrical water bucket with a diameter of $32$ cm. After the ball is completely submerged, the water level rises by $9$ cm. What is the radius of the iron ball? | 12\,\text{cm} |
196. For what values of $x$ will the numbers $\frac{2 x+1}{3}, \frac{3 x+1}{4}$, and $\frac{4 x+1}{5}$ be integers simultaneously? | 60t+1 |
Given the complex number $z = (m - 2) + (m + 1)i$ (where $i$ is the imaginary unit) and $m \in \mathbb{R}$, find the value of $|z|$. | \sqrt{2m^2 - 2m + 5} |
Example 5 Given $P(x)=x^{5}+a_{1} x^{4}+a_{2} x^{3}+$ $a_{3} x^{2}+a_{4} x+a_{5}$, and when $k=1,2,3,4$, $P(k)$ $=k \times 1$ 997. Then $P(10)-P(-5)=$ $\qquad$ | 75315 |
1. Find any four different natural numbers that have the following property: if the product of any two of them is added to the product of the other two numbers, the result is a prime number. | 1,2,3,5 |
Two regular hexagons of side length $2$ are laid on top of each other such that they share the same center point and one hexagon is rotated $30^\circ$ about the center from the other. Compute the area of the union of the two hexagons. | 48\sqrt{3} - 72 |
A rational number $a$ equals its reciprocal. Then $a^{2014}$ is ( )
A: The smallest positive integer
B: The smallest non-negative number
C: The integer with the smallest absolute value
D: The largest negative number | \text{A} |
Example 5 Let real numbers $a, b, c, d, e \geqslant -1$, and
$$
\begin{aligned}
a+b+
c+d+e & =5 . \text{ Find } \\
S & =(a+b)(b+c)(c+d)(d+e)(e+a)
\end{aligned}
$$
the minimum and maximum values of $S$. ${ }^{[2]}$
(34th China Mathematical Olympiad) | -512288 |
Given that the function $f(x)$ is an odd function defined on the domain $\mathbb{R}$, and is monotonically increasing on the interval $[0, +\infty)$, if $$\frac {|f(\ln x)-f(\ln \frac {1}{x})|}{2} < f(1),$$ then the range of $x$ is ( )
A: $(0, \frac {1}{e})$
B: $(0, e)$
C: $(\frac {1}{e}, e)$
D: $(e, +\infty)$ | C |
30. Calculate the value of the expression $\log ^{2} 5+\log 2 \cdot \log 50$.
(8) | 1 |
Given that \\(a\\) is a real number, if the set \\(\{x|ax=1\}\\) is a subset of any set, then the value of \\(a\\) is \_\_\_\_\_\_. | 0 |
If $\alpha$ is an angle in the fourth quadrant, then $180^\circ-\alpha$ is ( )
A: An angle in the first quadrant
B: An angle in the second quadrant
C: An angle in the third quadrant
D: An angle in the fourth quadrant | \text{C} |
The derivative of the function $f(x) = (x + 2a)(x - a)^2$ is ( )
A: $2(x^2 - a^2)$
B: $2(x^2 + a^2)$
C: $3(x^2 - a^2)$
D: $3(x^2 + a^2)$ | C |
Calculate the value of the function $f(x) = 3x^3 + 2x^2 + x + 1$ at $x = 2$ using the Horner's method. | 35 |
In how many ways can 25 forints be paid using 1, 2, 5, 10, and 20 forint coins? | 68 |
To make the fraction $\frac{3}{{x-2}}$ meaningful, the value of $x$ should satisfy ______. | x \neq 2 |
A person rolls a die twice, obtaining the numbers $m$ and $n$, which are used as the coefficients of a quadratic equation $x^2 + mx + n = 0$. The probability that the equation has real roots is ______. | \dfrac{19}{36} |
17-130 Condition A: The sum of the areas of two triangles and two corresponding sides are equal. Condition B: The two triangles are congruent. Then
(A) A is a sufficient and necessary condition for B.
(B) A is a necessary condition for B.
(C) A is a sufficient condition for B.
(D) A is neither a necessary condition nor... | B |
4. Given the polynomial $f(x)=x^{12}-x^{6}+1$ divided by $x^{2}+1$ has a quotient $q(x)$ and a remainder $r(x)=a x+b$, where $a$ and $b$ are real numbers. Then the value of $b$ is ( ).
(A) 0
(B) 1
(C) 2
(D) 3 | 3 |
The constant term in the expansion of $$(x+ \frac {1}{ \sqrt {x}}-2)^{5}$$ is \_\_\_\_\_\_. | 88 |
Rotate the curve $x+y^{2}=1$ counterclockwise around the origin by $45^{\circ}$. Determine the equation of the resulting curve $C$. | x^{2} + y^{2} - 2xy + \sqrt{2}x + \sqrt{2}y - 2 = 0 |
(21) (12 points) Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ passes through the point $(0,1)$, and the eccentricity is $\frac{\sqrt{3}}{2}$.
(1) Find the equation of the ellipse $C$;
(2) Suppose the line $l: x=m y+1$ intersects the ellipse $C$ at points $A$ and $B$, and the point $A$ is symm... | (4,0) |
73 Given the equation of a circle as $x^{2}+y^{2}=4$. Try to find two points $A(s, t), B(m, n)$ on the coordinate plane, such that the following two conditions are satisfied:
(1) The ratio of the distance from any point on the circle to point $A$ and to point $B$ is a constant $k$;
(2) $s>m, t>n$, and $m, n$ are both n... | (2,2),(1,1) |
33.24 If $[a]$ represents the integer part of the real number $a$, then $\left[\frac{1}{\sqrt{16-6 \sqrt{7}}}\right]$ equals
(A) 1.
(B) 2.
(C) 3.
(D) 4.
(Shaanxi Province, China Junior High School Mathematics Competition, 1997) | 2 |
1) Federico has a collection of toy soldiers; he knows he has a bit less than 100, but certainly at least 30. He arranges them in rows of 7 and has one soldier left over; then he arranges them in rows of 10 and this time has 2 soldiers left over. How many soldiers does he have in total $\begin{array}{lllll}\text { (A) ... | 92 |
Ex. 129. In an integer-sided triangle, two sides are equal to 10. Find the third side, given that the radius of the inscribed circle is an integer. | 12 |
4. For any $x, y \in \mathbf{R}, y \neq 0$, define the remainder
$$
\operatorname{rem}(x, y)=x-y\left[\frac{x}{y}\right],
$$
where $[a]$ denotes the greatest integer not exceeding the real number $a$. Then the value of $\operatorname{rem}\left(\frac{3}{8},-\frac{2}{5}\right)$ is $(\quad)$.
(A) $-\frac{3}{8}$
(B) $-\fr... | -\frac{1}{40} |
Define the "harmonic mean" of $n$ positive numbers $a_1, a_2, \cdots, a_n$ as $\dfrac{n}{a_1 + a_2 + \cdots + a_n}$. If it is known that the harmonic mean of the first $n$ terms of the sequence $\{a_n\}$ is $\dfrac{1}{2n+1}$, and $b_n = \dfrac{a_n + 1}{4}$, then the value of $\dfrac{1}{b_1b_2} + \dfrac{1}{b_2b_3} + \cd... | \text{A} |
If $a^{2}+3a-4=0$, then $2a^{2}+6a-3=$
A: $5$
B: $1$
C: $-1$
D: $0$ | A |
Example 5 (2003 Belarusian Mathematical Olympiad) Given a cyclic quadrilateral $A B C D$ satisfying $A B=B C=A D$ $+C D, \angle B A D=\alpha, A C=d$. Find the area of $\triangle A B C$. | \frac{1}{2}^{2}\sin\alpha |
The number of zeros of the function $f(x)=2^x + x^3 - 2$ in the interval (0, 1) is __________. | 1 |
5. Let $a$ and $n$ be natural numbers, and it is known that $a^{n}-2014$ is a number with $2014$ digits. Find the smallest natural number $k$ such that $a$ cannot be a $k$-digit number. | 49 |
5.3. Find the sum of the digits in the decimal representation of the integer part of the number $\sqrt{\underbrace{44.44}_{2017} \underbrace{22 \ldots 22}_{2018}} 5$. | 12107 |
If $\frac{m}{n}=\frac{4}{3}$ and $\frac{r}{t}=\frac{9}{14}$, the value of $\frac{3mr-nt}{4nt-7mr}$ is:
$\textbf{(A)}\ -5\frac{1}{2}\qquad\textbf{(B)}\ -\frac{11}{14}\qquad\textbf{(C)}\ -1\frac{1}{4}\qquad\textbf{(D)}\ \frac{11}{14}\qquad\textbf{(E)}\ -\frac{2}{3}$ | \frac{-11}{14} |
Find all functions $f: R \rightarrow R$, such that for any real numbers $x, y$ the following is satisfied:
$$
f(x y)=f(x) f(y)-x-y .
$$ | f(x)=x+1 |
## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{x^{3}+2 x^{2}+3}{(x-1)(x-2)(x-3)} d x
$$ | x+3\cdot\ln|x-1|-19\cdot\ln|x-2|+24\cdot\ln|x-3|+C |
Find the equation of the line that passes through the intersection point of the lines $2x-y=0$ and $x+y-6=0$, and is perpendicular to the line $2x+y-1=0$. The options are:
A: $x-2y+6=0$
B: $x-2y-6=0$
C: $x+2y-10=0$
D: $x+2y-8=0$ | \text{A} |
Example 3.18. Calculate the volume of the body formed by rotating the figure bounded by one half-wave of the sine curve $y=\sin x$ around the $O x$ axis. | \frac{\pi^{2}}{2} |
Points A, B, and C on the number line correspond to the numbers 0, -1, and x, respectively. If the distance between C and A is greater than the distance between C and B, then the range of values for x is . | x < -\frac{1}{2} |
4. In a convex pentagon $A B C D E$,
$$
A B=B C=C D=D E=E A,
$$
and $\angle C A D=\angle B A C+\angle E A D$. Then the degree measure of $\angle B A E$ is ( ).
(A) $30^{\circ}$
(B) $45^{\circ}$
(C) $60^{\circ}$
(D) $75^{\circ}$ | C |
8. Let $0 \leqslant x \leqslant 2 \pi$. Then the range of $x$ that satisfies the inequality
$$
\sin \left(x-\frac{\pi}{6}\right)>\cos x
$$
is | \frac{\pi}{3}<x<\frac{4 \pi}{3} |
3. Find all pairs of prime numbers $(p, q)$ such that $p q \mid\left(p^{p}+q^{q}+1\right)$.
(2007, Korean Mathematical Olympiad) | (2,5),(5,2) |
Given $\overrightarrow{m}=(2\sin x,\sin x-\cos x)$, $\overrightarrow{n}=( \sqrt {3}\cos x,\sin x+\cos x)$, let $f(x)= \overrightarrow{m} \cdot \overrightarrow{n}$.
(1) Find the set of all possible values of $x$ when the function $f(x)$ reaches its maximum value.
(2) In triangle $ABC$, if $f(C)=2$, $c= \sqrt {3}$, find ... | \frac {3\sqrt {3}}{4} |
Given the function $f(x)= \frac {1}{3}x^{3}+x$, the solution set of the inequality $f(2-x^{2})+f(2x+1) > 0$ is $(\quad)$
A: $(-\infty,- \sqrt {2}-1)\cup( \sqrt {2}-1,+\infty)$
B: $(- \sqrt {2}-1, \sqrt {2}-1)$
C: $(-\infty,-1)\cup(3,+\infty)$
D: $(-1,3)$ | \text{D} |
16. Let the function $f(x)=\ln x+a\left(\frac{1}{x}-1\right), a \in \mathbf{R}$, and the minimum value of $f(x)$ is 0.
(1) Find the value of $a$;
(2) Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n+1}=f\left(a_{n}\right)+2\left(n \in \mathbf{N}_{+}\right)$, let $S_{n}=\left[a_{1}\right]+\left[a_{2}\r... | 2n-1 |
72. In how many ways can 10 athletes be divided into two teams of 5 people each, if two athletes wish to play on different teams? | 70 |
How many integers between $2$ and $100$ inclusive [i]cannot[/i] be written as $m \cdot n,$ where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$? Note that there are $25$ primes less than $100.$ | 35 |
In a bag, there are 4 distinct red balls and 6 distinct white balls.
(1) How many ways can you draw 4 balls from the bag such that the number of red balls is not less than the number of white balls?
(2) If drawing a red ball scores 2 points, and a white ball scores 1 point, how many ways can you draw 5 balls from t... | C_4^2C_6^3 + C_4^3C_6^2 + C_4^4C_6^1 = 6 \times 20 + 4 \times 15 + 1 \times 6 = 120 + 60 + 6 = 186 |
Two rectangles $P Q U V$ and $W S T V$ overlap as shown. What is the area of $P Q R S T V$ ?
(A) 35
(B) 24
(C) 25
(D) 17
(E) 23
 | 23 |
+ Find all functions $f: \mathbf{Q}^{+} \rightarrow \mathbf{Q}^{+}$, such that
$$
f(x)+f(y)+2 x y f(x y)=\frac{f(x y)}{f(x+y)},
$$
where $\mathbf{Q}^{+}$ denotes the set of positive rational numbers. (Li Shenghong) | f(x)=\frac{1}{x^{2}} |
Given a set $A=\{a_1, a_2, a_3, \ldots, a_n\}$, denote the number of distinct values in the sums $a_i + a_j$ ($1 \leq i < j \leq n$) as $M(A)$. For example, when $A=\{1, 2, 3, 4\}$, since $1+2=3$, $1+3=4$, $1+4=2+3=5$, $2+4=6$, $3+4=7$, we have $M(A)=5$. If $A=\{1, 2, 3, \ldots, n\}$, then $M(A)=$ . | 2n-3 |
13. A. Real numbers $a, b$ make the system of equations about $x, y$
$$
\left\{\begin{array}{l}
x y - x^{2} = 1, \\
x y^{2} + a x^{2} + b x + a = 0
\end{array}\right.
$$
have real solutions $(x, y)$.
(1) Prove: $|y| \geqslant 2$;
(2) Find the minimum value of $a^{2} + b^{2}$. | \frac{16}{5} |
$2 \cdot 23$ When solving a certain quadratic equation, student A made a mistake in the constant term of the equation, and thus obtained the two roots of the equation as 8 and 2. Student B made a mistake in the coefficient of the linear term, and thus obtained the two roots of the equation as -9 and -1. Then the correc... | x^{2}-10x+9=0 |
1. Find the range of the function $f(x)=2 x+3+\sqrt{-2 x^{2}+12 x-14}$. | [9-2\sqrt{2},9+2\sqrt{3}] |
The reciprocal of $-3$ is
A: $-\frac{1}{3}$
B: $\frac{1}{3}$
C: $-3$
D: $3$ | \text{A} |
20. Pauli and Bor are playing the following game. There is a pile of 99! molecules. In one move, a player is allowed to take no more than 1% of the remaining molecules. The player who cannot make a move loses. They take turns, with Pauli starting. Who among them can win, regardless of how the opponent plays? | Pauli |
Given an arithmetic sequence \(\{a_n\}\) where \(a_2 + a_6 = 14\), let \(S_n\) be the sum of the first \(n\) terms, and \(S_5 = 25\).
1. Find the general formula for \(\{a_n\}\).
2. Let \(b_n=\frac{2}{{a_n} \cdot {a_{n+1}}}\), calculate the sum \(T_n\) of the first \(n\) terms of the sequence \(\{b_n\}\). | T_n = \frac{2n}{2n + 1} |
(1) Given $\tan\alpha=3$, calculate the value of $$\frac {4\sin\alpha-2\cos\alpha}{5\cos\alpha +3\sin\alpha }.$$
(2) Given $\tan\theta=- \frac {3}{4}$, find the value of $2+\sin\theta\cos\theta-\cos^2\theta$. | \frac {22}{25} |
1. Given $x=\frac{3-\sqrt{5}}{2}$. Then $x^{3}-3 x^{2}+3 x+$ $\frac{6}{x^{2}+1}=$ . $\qquad$ | 6 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, with $B= \frac {\pi}{3}$, $\cos A= \frac {4}{5}$, and $b= \sqrt {3}$.
$(1)$ Find the value of $\sin C$;
$(2)$ Find the area of $\triangle ABC$. | \frac {9 \sqrt {3}+36}{50} |
If we set $\cos 80^\circ = m$, then $\tan (-440^\circ) =$ （ ）
A: $$\frac { \sqrt {1-m^{2}}}{|m|}$$
B: $$\frac { \sqrt {1-m^{2}}}{-m}$$
C: $$\frac { \sqrt {1+m^{2}}}{m}$$
D: $$\frac { \sqrt {1-m^{2}}}{m}$$ | \text{B} |
The number of subsets of the set $\{x | x^2 - 1 = 0\}$ is ( )
A: 1
B: 2
C: 3
D: 4 | D |
3. Let $E$ be a point on the side $AB$ of square $ABCD$ such that $|AE|=3|EB|$, and let $F$ be a point on the side $DA$ such that $|AF|=5|FD|$. Denote the intersection of segments $DE$ and $FC$ by $K$, the intersection of $DE$ and $BF$ by $L$, and the intersection of $FB$ and $EC$ by $M$. Let $p_{1}$ be the sum of the ... | 1 |
There are 6 students standing in a row. How many different arrangements are there if students A and B must stand together? | 240 |
Solve the equation $x^{3}+p x+q=0$ if the product of two roots is 1. | =\frac{1}{2}(\sqrt{q^{2}-4}),\quadb=\frac{1}{2}(\sqrt{q^{2}-4}),\quad=-q |
Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if nece... | \frac{1}{2013} |
G3.4 Given that $f(x)=\left(x^{4}+2 x^{3}+4 x-5\right)^{2004}+2004$. If $f(\sqrt{3}-1)=d$, find the value of $d$. | 2005 |
The real part of the complex number $z=(1+i)(1+2i)$ ($i$ is the imaginary unit) is. | -1 |
1. Given the function $S=|x-2|+|x-4|$.
(1) Find the minimum value of $S$;
(2) If for any real numbers $x, y$ the inequality
$$
S \geqslant m\left(-y^{2}+2 y\right)
$$
holds, find the maximum value of the real number $m$. | 2 |
Given that the center of the ellipse $(C)$ is at the origin $(O)$, its foci are on the $x$-axis, its eccentricity is $\frac{1}{2}$, and the distance from the left focus to the left vertex is $1$.
$(1)$ Find the standard equation of the ellipse $(C)$;
$(2)$ A line passing through the point $M(1,1)$ intersects the elli... | 3x+4y-7=0 |
1. Given a natural number $n \geq 3$. For an $n$-element set of real numbers $S$, let $A(S)$ denote the set of all strictly increasing three-term arithmetic progressions composed of elements of $S$. How many elements can the set $A(S)$ have at most? | [\frac{n-1}{2}][\frac{n}{2}] |
Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$ for $n=1,2,3,\cdots$, and $a_{2}=3$, $S_{5}=25$.$(1)$ Find the arithmetic mean of $a_{2}$ and $a_{4}$.$(2)$ For a geometric sequence $\{b_{n}\}$ with the first term as $1$ and common ratio $q$, choose one of the following th... | m=\frac{3^{k-1}+1}{2} |
If \$\lim_{\Delta x \to 0} \frac{f(x_0 + 3\Delta x)-f(x_0)}{\Delta x} = 1\$, then \$f'(x_0)\$ is equal to \_\_\_\_\_\_. | \frac{1}{3} |
Given the graphs of $y=\sin x$ and $y=\cos x$, the three consecutive intersection points $A$, $B$, and $C$ form the triangle $\triangle ABC$. The area of $\triangle ABC$ equals \_\_\_\_\_\_. | \sqrt{2}\pi |
51 The equations of the asymptotes of the hyperbola are $y= \pm 2(x-1)$, and one of the foci has coordinates $(1+2 \sqrt{5}, 0)$. Then the equation of the hyperbola is . $\qquad$ | \frac{(x-1)^{2}}{4}-\frac{y^{2}}{16}=1 |
Given the function $f(x)= \frac{1}{4^{x}} + \frac{a}{2^{x}} +1$, where $a$ is a real constant.
(1) If the minimum value of $f(x)$ is $-7$ when $x\in[-1,0]$, find the value of $a$;
(2) If the inequality $|f(x)|\leqslant 3$ always holds when $x\in[0,+\infty)$, find the range of values for $a$. | [-5,1] |
Divide a set of data into 6 groups, where the sum of the frequencies of the first 3 groups is 0.65, and the sum of the frequencies of the last two groups is 0.32. Then, the frequency of the fourth group is ( )
A: 0.02
B: 0.01
C: 0.03
D: 0.04 | C |
Example 1.7. Find $\int \frac{6 d x}{x(x-1)(x-2)(x-3)}$. | -\ln|x|+3\ln|x-1|-3\ln|x-2|+\ln|x-3|+C |
The local minimum value of the function $f(x)=x^{3}-4x^{2}+4x$ is __________. | 0 |
Given the function $f\left(x\right)=x^{3}+ax^{2}+x+1$ achieves an extremum at $x=-1$. Find:$(1)$ The equation of the tangent line to $f\left(x\right)$ at $\left(0,f\left(0\right)\right)$;$(2)$ The maximum and minimum values of $f\left(x\right)$ on the interval $\left[-2,0\right]$. | -1 |
4.2. 12 * Given that $x, y, z$ are positive numbers, and satisfy $x y z(x+y+z)=1$. Find the minimum value of the expression $(x+y)(x+z)$.
保留源文本的换行和格式，翻译结果如下：
4.2. 12 * Given that $x, y, z$ are positive numbers, and satisfy $x y z(x+y+z)=1$. Find the minimum value of the expression $(x+y)(x+z)$. | 2 |
9. (16 points) Let the function $f(x)$ satisfy $f\left(2^{x}\right)=x^{2}-2 a x+a^{2}-1$, and $f(x)$ has a range of $[-1,0]$ on $\left[2^{a-1}, 2^{a^{2}-2 a+2}\right]$, find the range of values for $a$.
| [\frac{3-\sqrt{5}}{2},1]\cup[2,\frac{3+\sqrt{5}}{2}] |
77. $\int \frac{\sin 2 x}{\cos x} d x$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
77. $\int \frac{\sin 2 x}{\cos x} d x$. | -2\cosx+C |
5. Put 1 cent, 2 cent, 5 cent, and 10 cent coins into 19 boxes, so that each box contains coins, and the amount of money in any two boxes is different. How many coins are needed at least? At this time, what is the minimum total amount of money in all the boxes? | 41 |
G3.4 Given that $5^{\log 30} \times\left(\frac{1}{3}\right)^{\log 0.5}=d$, find the value of $d$. | 15 |
A fair coin is flipped nine times. Which is more likely, having exactly four heads or having exactly five heads? | \frac{126}{512} |
Given a cube $A B C D A 1 B 1 C 1 D 1$ with edge $a$. a) Prove that $A A 1$ and $B C$ are skew lines; b) construct their common perpendicular; c) find the distance between these lines. | a |
2. Given that the two real roots of the equation $a x^{2}+b x+c=0$ are $a$ and $c (a c \neq 0)$. Then the root situation of the equation $9 c x^{2}+3 b x+a=0$ is ( ).
(A) One root must be $\frac{1}{3}$
(B) One root must be $\frac{1}{9}$
(C) The two roots are $\frac{1}{3},-\frac{1}{3}$
(D) One root must be $\frac{1}{3}$... | D |
Let's assume that the birth of a girl and a boy is equally likely. It is known that a certain family has two children.
a) What is the probability that one is a boy and one is a girl?
b) It is additionally known that one of the children is a boy. What is the probability now that there is one boy and one girl in the fa... | \frac{1}{2};\frac{2}{3};\frac{14}{27} |
Given an arithmetic sequence $\{a_n\}$, if $3(a_2+a_6) + 2(a_5+a_{10}+a_{15}) = 24$, what is the sum of the first 13 terms of this sequence? | 26 |
## Problem Statement
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(2 ; -1 ; -2) \)
\( A_{2}(1 ; 2 ; 1) \)
\( A_{3}(5 ; 0 ; -6) \)
\( A_{4}(-10 ; 9 ; -7) \) | 46\frac{2}{3},4\sqrt{14} |
7. $[\mathbf{3}]$ Express -2013 in base -4 . | 200203_{-4} |
Calculate: $(-1) \times (-4) + 2^{2} \div (7-5)$. | 6 |
[Reverse Process]
A peasant, buying goods, paid the first merchant half of his money and another 1 ruble; then he paid the second merchant half of the remaining money and another 2 rubles, and finally, he paid the third merchant half of what was left and another 1 ruble. After this, the peasant had no money left. How ... | 18 |
The Rhind Papyrus is one of the oldest known mathematical writings in the world. It contains the following problem: Divide 10 pounds of bread among 5 people in such a way that the amounts they receive form an arithmetic sequence and the sum of the larger three portions is $\frac{1}{7}$ of the sum of the smaller two por... | \frac{1}{6} |
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