problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Simplify first, then evaluate: $\frac{1}{2}x-2(x-\frac{1}{3}y^2)+(-\frac{3}{2}x+\frac{1}{3}y^2)$, where $x=-2$ and $y=2$. | 10 |
Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that $S$ is a positive integer, compute the number of positive di... | 1054 |
The sum of the interior angles of a dodecagon is ( )
A: 180°
B: 360°
C: 1800°
D: Cannot be calculated | C |
What is the ones digit of\[222,222-22,222-2,222-222-22-2?\]$\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$ | B |
Square pyramid $ABCDE$ has base $ABCD$, which measures $3$ cm on a side, and altitude $AE$ perpendicular to the base, which measures $6$ cm. Point $P$ lies on $BE$, one third of the way from $B$ to $E$; point $Q$ lies on $DE$, one third of the way from $D$ to $E$; and point $R$ lies on $CE$, two thirds of the way from ... | 2\sqrt{2} |
1. How many times in a day does the angle between the hour and minute hands equal exactly $19^{\circ}$? | 44 |
2. (6 points) The 50th term of the arithmetic sequence $3, 5, 7, 9 \cdots$ is $\qquad$ | 101 |
Given $-1 \lt a \lt 0$, arrange the four numbers $\frac{1}{a}$, $a^{2}$, $a$, $|a|$ in ascending order, separated by "$ \lt $". | \frac{1}{a} < a < a^{2} < |a| |
Given a system of linear equations in two variables $x$ and $y$: $\left\{\begin{array}{c}x-y=3-a\\ x+2y=5a\end{array}\right.$ ($a$ is a real number).$(1)$ If the solutions of the system always satisfy $y=a+1$, find the value of $a$;$(2)$ It is known that the solutions of the system are also solutions of the equation $b... | -22 |
7.020. $\left(1+\frac{1}{2 x}\right) \log 3+\log 2=\log \left(27-3^{1 / x}\right)$. | \frac{1}{2} |
78. Arrange the polynomial $f(x)=x^{4}+2 x^{3}-3 x^{2}-4 x+1$ in powers of $x+1$. | f(x)=(x+1)^{4}-2(x+1)^{3}-3(x+1)^{2}+4(x+1)+1 |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(y f(x+y)+f(x))=4 x+2 y f(x+y)
$$
for all $x, y \in \mathbb{R}$.
Origin. Netherlands (Birgit van Dalen). | f(x)=2x |
Two straight lines intersect at a right angle, and the centers of two circles move toward the intersection point along each line. The radius of the first circle is $981 \mathrm{~m}$, and the speed of its center is $7 \mathrm{~m}$ per second. The distance of the center from the intersection point at the start is $2442 \... | 111\, |
Given the function $f(x) = |\log_{3}x|$, if there exist two different real numbers $a$ and $b$ such that $f(a) = f(b)$, then $ab = \_\_\_\_\_\_$. | 1 |
[ Properties and characteristics of an isosceles triangle.] [Sum of the angles of a triangle. Theorem of the exterior angle.]
Triangle $A B C$ is isosceles $(A B=B C)$. Segment $A M$ divides it into two isosceles triangles with bases $A B$ and $M C$. Find angle $B$. | 36 |
The line $l: x-2y+5=0$ intersects the circle $C: x^2+y^2=9$ at points A and B. If point D is another point on circle C different from A and B, then the maximum area of $\triangle ABD$ is \_\_\_\_\_. | 6 + 2\sqrt{5} |
10. Given that $n$ is a natural number, the polynomial $P(x)=\sum_{h=0}^{n} \mathrm{C}_{n}^{h} x^{n-h} \cdot(x-1)^{h}$ can be expanded into an ascending power series of $x$ as $a_{0}+a_{1} x+a_{2} x^{2}+$ $\cdots+a_{n} x^{n}$. Then $\left|a_{0}\right|+\left|a_{1}\right|+\left|a_{2}\right|+\cdots+\left|a_{n}\right|=$ $\... | 3^{n} |
Calculate:$(1)(\sqrt{16})^{2}-\sqrt{25}+\sqrt{(-2)^{2}}$;$(2)\sqrt{\frac{1}{2}}\times \sqrt{48}\div \sqrt{\frac{1}{8}}$. | 8\sqrt{3} |
Given that the arc length of a sector is $6cm$, and its area is $18cm^{2}$, what is the measure of the sector's central angle in radians? | 1 |
Task B-1.2. If $\left[\left(\frac{a}{b}-\frac{b}{a}\right):\left((a+b)\left(\frac{1}{b}-\frac{1}{a}\right)\right)-\frac{a}{b}\right]:\left(\frac{a}{b}+1\right)=\frac{1}{2014}$, what is $\frac{a}{b}$? | \frac{2013}{2015} |
1.23. The sides of a triangle are 13, 14, and 15 cm. Find the ratio of the areas of the circumscribed and inscribed circles of the triangle. | (\frac{65}{32})^2 |
Let $ABC$ be a triangle such that $BA > BC$. Let $M$ be the midpoint of $AC$, the internal bisector of angle $\widehat{B}$ intersects $AC$ at $I$. The line parallel to $BC$ passing through $I$ intersects $AM$ at $E$. The line parallel to $BA$ passing through $M$ intersects $AI$ at $D$. What is the measure of angle $\wi... | 90 |
4) A family consisting of two parents and two young children wants to cross a river. Their boat can carry at most two young children or one adult. Counting both the crossings in one direction and those in the other, what is the minimum number of crossings the boat must make? (Obviously, the boat cannot cross the river ... | 9 |
G2.2 In Figure $1, \triangle A B C$ is a right-angled triangle, $A B=3 \mathrm{~cm}$, $A C=4 \mathrm{~cm}$ and $B C=5 \mathrm{~cm}$. If $B C D E$ is a square and the area of $\triangle A B E$ is $b \mathrm{~cm}^{2}$, find the value of $b$. | \frac{9}{2} |
If there exists an angle $\alpha \in \left[0,2\pi \right)$ such that there is an angle $\beta \in \left[0,2\pi \right)$ satisfying $\cos \left(\alpha +\beta \right)=\cos \alpha +\cos \beta $, then $\beta$ is called the "complementary angle" of $\alpha$. There are two propositions as follows, and the correct judgment is... | B |
Let be two real numbers $ a,b, $ that satisfy $ 3^a+13^b=17^a $ and $ 5^a+7^b=11^b. $
Show that $ a<b. $ | a < b |
How many zeros are there at the end of the number $9^{2007}+1$? | 1 |
10. Which of these integers cannot be expressed as the difference of two squares?
A 5
B 7
C 8
D 9
E 10 | E |
Problem 12. (6 points)
Victor received a large sum of money as a birthday gift in the amount of 45 thousand rubles. The young man decided to save this part of his savings in dollars on a currency deposit. The term of the deposit agreement was 2 years, with an interest rate of 4.7% per annum, compounded quarterly. On t... | 873 |
## Task 2 - 201232
Let $f$ be the function defined by
$$
f(x)=x^{4}-(x+1)^{4}-(x+2)^{4}+(x+3)^{4}
$$
where the domain of $f$ is
a) the set of all integers,
b) the set of all real numbers.
Investigate, for both case a) and case b), whether the function $f$ attains a minimum value, and determine, if applicable, thi... | 10 |
Given the functions $f(x)=x+a$ and $g(x)=x+ \frac {4}{x}$, if for every $x_{1} \in [1,3]$, there exists $x_{2} \in [1,4]$ such that $f(x_{1}) \geqslant g(x_{2})$, then the range of values for the real number $a$ is ( ).
A: $a \geqslant 1$
B: $a \geqslant 2$
C: $a \geqslant 3$
D: $a \geqslant 4$ | C |
Three pumpkins are weighed two at a time in all possible ways. The weights of the pairs of pumpkins are $12 \mathrm{~kg}, 13 \mathrm{~kg}$ and $15 \mathrm{~kg}$. How much does the lightest pumpkin weigh?
(A) $4 \mathrm{~kg}$
(B) $5 \mathrm{~kg}$
(C) $6 \mathrm{~kg}$
(D) $7 \mathrm{~kg}$
(E) $8 \mathrm{~kg}$ | 5\mathrm{~} |
Let $n\ge 4$ be a positive integer and let $M$ be a set of $n$ points in the plane, where no three points are collinear and not all of the $n$ points being concyclic. Find all real functions $f:M\to\mathbb{R}$ such that for any circle $\mathcal{C}$ containing at least three points from $M$, the following equality holds... | f(P) = 0 |
## Task 5 - 280835
Let $ABC$ be a triangle, $\alpha$ be the measure of the angle $\angle BAC$, and $\beta$ be the measure of the angle $\angle ABC$. The incircle of the triangle touches the side $AB$ at $D$, the side $BC$ at $E$, and the side $AC$ at $F$.
Determine the measure of the angle $\angle FDE$ in terms of $\... | \frac{\alpha+\beta}{2} |
1. Two parabolas param 1 and param 2 touch at a point lying on the $O x$ axis. Through point $D$, the second intersection point of the first parabola with the $O x$ axis, a vertical line is drawn, intersecting the second parabola at point $A$ and the common tangent to the parabolas at point $B$. Find the ratio $D A: D ... | 4 |
4.047. Find three numbers that form a geometric progression, given that their product is 64, and their arithmetic mean is $14 / 3$. | 2,4,88,4,2 |
Use the Euclidean algorithm to find the greatest common divisor (GCD) of 108 and 45, and then verify it with the method of consecutive subtraction. | \text{GCD}(108, 45) = 9 |
Given that the function $f(x)$ is an odd function defined on the set of real numbers R. When $x \in [0, 1]$, $f(x) = \log_2(x+1)$. Find the value of $f(1- \sqrt{2})$. | -\frac{1}{2} |
Given that $x_1$ and $x_2$ are two distinct real roots of the quadratic equation $f(x)$, and $x_3$ and $x_4$ are two distinct real roots of the quadratic equation $g(x)=0$. If $g(x_1)g(x_2)<0$, then ( )
A: $x_1$ and $x_2$ are between $x_3$ and $x_4$
B: $x_3$ and $x_4$ are between $x_1$ and $x_2$
C: $x_1$ and $x_2$ are ... | D |
Let proposition p be: $\exists x \in \mathbb{N}, x^3 < 3^x$, then $\neg p$ is ( )
A: $\forall x \in \mathbb{N}, x^3 < 3^x$
B: $\exists x \in \mathbb{N}, x^3 \geq 3^x$
C: $\forall x \in \mathbb{N}, x^3 \geq 3^x$
D: $\exists x \in \mathbb{N}, x^3 \neq 3^x$ | \text{C} |
7.1. Write the number 1.11 ten times and the number 1.01 eleven times. Cross out one or several numbers so that the sum of the remaining numbers is equal to 20.19. | 20.19 |
4. If for the real numbers $x$ and $y$ the equality $(x+5)^{2}+(y-12)^{2}=14^{2}$ holds, find the smallest value of the expression $x^{2}+y^{2}$ and determine for which values of $x$ and $y$ this value is achieved. | 1 |
The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37$? | 217 |
Given the hyperbola $E$: $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a＞0,b＞0)$ with its right focus at $F$, $O$ as the origin, and $Q$ a point on the right branch of the hyperbola $E$, such that $2＜\frac{\overrightarrow{OF}⋅\overrightarrow{OQ}}{|\overrightarrow{OQ}|}≤4$. What is the eccentricity of the hyperbola... | A |
In the expansion of $(a-3b^2-c)^6$, the coefficient of the term $a^3b^2c^2$ is ______. | -180 |
7. (10 points) 2016 students line up in a row, and from left to right, they report numbers $1, 2, \cdots, n$ $(n \geqslant 2)$. If the 2016th student reports the number $n$, then all students who report $n$ in this round will receive a New Year's gift. No matter what value $n$ takes, there will be
students who will not... | 576 |
2. A parallelogram has 3 of its vertices at $(1,2),(3,8)$, and $(4,1)$. Compute the sum of the possible $x$-coordinates for the 4 th vertex. | 8 |
Given the general term formula of the sequence \\(\\{a\_{n}\\}\\) as \\(a\_{n}= \\dfrac {n-2017.5}{n-2016.5}\\), determine the option that correctly describes the minimum and maximum terms of the sequence.
A: The minimum term is \\(-1\\), and the maximum term is \\(3\\)
B: The minimum term is \\(-1\\), and there is no... | A |
11. Through the vertices $A, B, D$ of parallelogram $A B C D$, draw a circle intersecting line $A C$ at point $E$. Prove: $A B^{2}+A D^{2}=A C \cdot A E$. | AB^{2}+AD^{2}=AC\cdotAE |
6. (15 points) From a homogeneous straight rod, a piece of length $s=80 \mathrm{~cm}$ was cut. By how much did the center of gravity of the rod move as a result? | 40\mathrm{~} |
Find $x$ so that the arithmetic mean of $x, 3x, 1000$, and $3000$ is $2018$. | 1018 |
There are 1955 points. What is the maximum number of triples that can be chosen from them so that any two triples have exactly one point in common?
# | 977 |
$30 \cdot 29$ Among the following numbers, the one that definitely cannot be the square of some natural number (where $n$ is a natural number) is
(A) $3 n^{2}-3 n+3$.
(B) $4 n^{2}+4 n+4$.
(C) $5 n^{2}-5 n-5$.
(D) $7 n^{2}-7 n+7$.
(E) $11 n^{2}+11 n-11$.
(Tianjin Junior High School Mathematics Competition; China Junior ... | B |
For a given value of $x$, when using the Horner's method to evaluate the polynomial $6x^3+5x^2+4x+3$, the polynomial should first be transformed into: | x(x(6x+5)+4)+3 |
You know that the binary function $\diamond$ takes in two non-negative integers and has the following properties:
\begin{align*}0\diamond a&=1\\ a\diamond a&=0\end{align*}
$\text{If } a<b, \text{ then } a\diamond b\&=(b-a)[(a-1)\diamond (b-1)].$
Find a general formula for $x\diamond y$, assuming that $y\gex>0$. | (y - x)^x |
11.36 Solve graphically the equation
$$
|x-1|+2x-5=0
$$ | 2 |
5. (7 points) There is a sequence of numbers: $1,1,2,3,5,8,13,21, \cdots$, starting from the third number, each number is the sum of the two preceding ones. How many of the first 2007 numbers are even? | 669 |
If set $M=\{x\,|\,|x|<1\}$, and $N=\{x\,|\,x^2-x<0\}$, then $M\cap N=$ （ ）
A: $\{x\,|\,-1<x<1\}$
B: $\{x\,|\,0<x<1\}$
C: $\{x\,|\,-1<x<0\}$
D: $\{-1,0,1\}$ | B |
2. Find all three-digit numbers that are equal to the third power of the sum of their digits. | 512 |
2. The average of 30 odd prime numbers is closest to ( ).
(A) 12
(B) 13
(C) 14
(D) 15 | C |
6. In the chicken and rabbit cage, there are a total of 40 heads, the number of rabbit feet is 8 less than 10 times the number of chicken feet, so the number of rabbits is $\qquad$. | 33 |
11. (20 points) Let $[x]$ denote the greatest integer not exceeding the real number $x$. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=1, a_{n+1}=1+\frac{1}{a_{n}}+\ln a_{n} .
$$
Let $S_{n}=\left[a_{1}\right]+\left[a_{2}\right]+\cdots+\left[a_{n}\right]$. Find $S_{n}$. | 2n-1 |
The equation $2x^3+5x-2=0$ has a unique real root $r$, and there exists a unique strictly increasing sequence of positive integers $a_n$, such that $$\frac {2}{5}= \lim_{n\rightarrow \infty }(r^{a_{1}}+r^{a_{2}}+r^{a_{3}}+\cdots +r^{a_{n}})$$ holds. Then $a_n=\_\_\_\_\_\_\_\_\_\_\_\_$．($n\in\mathbb{N}^*$) | 3n-2 |
In the arithmetic sequence $\{a_n\}$, if $a_2 + 4a_7 + a_{12} = 96$, then $2a_3 + a_{15}$ equals to what? | 48 |
A light bulb factory randomly sampled $100$ light bulbs to measure their lifespans. The lifespans of these light bulbs are shown in the table below:
| Lifespan $x/h$ | $60\leqslant x \lt 100$ | $100\leqslant x \lt 140$ | $140\leqslant x \lt 180$ |
|----------------|------------------------|-------------------------|--... | 124 |
Solve the following quadratic equations:
(1) $3x^2 = 6x$
(2) $x^2 - 6x + 5 = 0$． | x_1 = 5, x_2 = 1 |
7. In how many different ways can 7 different prizes be awarded to 5 students in such a way that each student has at least one prize? | 16800 |
Given the function $f(x)=|a-4x|+|2a+x|$.
(Ⅰ) If $a=1$, solve the inequality $f(x) \geqslant 3$.
(Ⅱ) Prove that: $f(x)+f\left(-\dfrac{1}{x}\right) \geqslant 10$. | f(x)+f\left(-\dfrac{1}{x}\right) \geqslant 10 |
7. In triangle $A B C$, angles $A$ and $B$ are $45^{\circ}$ and $30^{\circ}$ respectively, and $C M$ is a median. The incircles of triangles $A C M$ and $B C M$ touch segment $C M$ at points $D$ and $E$. Find the area of triangle $A B C$ if the length of segment $D E$ is $4(\sqrt{2}-1)$. | 16(\sqrt{3}+1) |
Example 6. Find the integral $\int \frac{\ln x}{x} d x$. | \frac{1}{2}\ln^{2}x+C |
Given that the right focus of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ is $F$, and point $A$ is on the asymptote of the hyperbola. If $\Delta OAF$ is an equilateral triangle with side length $2$ ($O$ is the origin), find the equation of the hyperbola. | x^2 - \frac{y^2}{3} = 1 |
Let $x \in \mathbb{Z}$. Set $A$ consists of odd numbers, and set $B$ consists of even numbers. If proposition $p$: $\forall x \in A, 2x \in B$, then ( )
A: $\neg p$: $\exists x \in A, 2x \in B$
B: $\neg p$: $\exists x \notin A, 2x \in B$
C: $\neg p$: $\exists x \in A, 2x \notin B$
D: $\neg p$: $\forall x \notin A, ... | \text{C} |
3. In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Curious Glafira found out that a herd of 17 cows completely drank the pond dry in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it dry in 30 days. How many days would it take for one cow... | 75 |
Among the following functions, the one that satisfies $f(x+y)=f(x)f(y)$ and is monotonically increasing within its domain is ( )
A: $f(x)=x^{3}$
B: $f(x)=3^{x}$
C: $f(x)=\lg x$
D: $f(x)=\tan x$ | B |
3. At a round table, several liars and knights were sitting. The first one said: "Not counting me, there is one more liar than knights here.". The second one said: "Not counting me, there are two more liars than knights here.", and so on up to the last one. How many people could be sitting at the table? | 2or3 |
2. Let the sequence $\left\{8 \times\left(-\frac{1}{3}\right)^{n-1}\right\}$ have the sum of its first $n$ terms as $S_{n}$. Then the smallest integer $n$ that satisfies the inequality
$$
\left|S_{n}-6\right|<\frac{1}{125}
$$
is | 7 |
1. In the following table, two of the logarithmic values are incorrect. Please correct them.
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline$x$ & 0.021 & 0.27 & 1.5 & 2.8 & 3 & 5 \\
\hline $\lg x$ & $2 a+b+c-3$ & $6 a-3 b-2$ & $3 a-b+c$ & $1-2 a+2 b-c$ & $2 a-b$ & $a+c$ \\
\hline$x$ & 6 & 7 & 8 & 9 & 14 & \\
\hline $\lg x$ & $... | \lg1.5=3-b+-1,\lg7=2b+ |
Convert the polar equation \\(\rho^{2}\cos \theta-\rho=0\\) into a Cartesian coordinate equation. The options are:
A: \\(x^{2}+y^{2}=0\\) or \\(y=1\\)
B: \\(x=1\\)
C: \\(x^{2}+y^{2}=0\\) or \\(x=1\\)
D: \\(y=1\\) | \text{C} |
In the geometric sequence $\{a_{n}\}$, if $a_{3}a_{5}a_{7}a_{9}a_{11}=243$, then the value of $\frac {a_{9}^{2}}{a_{11}}$ is ($\quad$). | 3 |
7. Which three-digit number increases 26 times when the digit 3 is added to its left? | 120 |
Problem 2. Solve the equation $[x] \cdot\{x\}=x-1$, where $[x]$ represents the integer part, and $\{x\}$ is the fractional part of the real number $x$. | x\in[1,2) |
Given the polar coordinate equation of a circle: $({{\rho }^{2}}-4\sqrt{2}\rho \cos (\theta -\dfrac{\pi }{4})+6=0)$, let $P(x,y)$ be a moving point on the circle. Find the maximum value of $x+2y$. | 6+ \sqrt{10} |
11. (10 points) If the equation with respect to $x$
$$
\frac{2 k}{x-1}-\frac{x}{x^{2}-x}=\frac{k x+1}{x}
$$
has only one solution, find the value of $k$. | k=0 \text{ or } \frac{1}{2} |
During a math competition organized in a certain city, the scores of all participating students approximately follow a normal distribution $N(60, 100)$. It is known that there are 13 students who scored 90 or above.
(1) Calculate the total number of students who participated in the competition.
(2) If it is planned to ... | 80 |
To investigate the linear correlation between two variables $x$ and $y$, two students, A and B, independently conducted 10 and 15 experiments, respectively. They used the linear regression method to obtain the regression lines $l_1$ and $l_2$. It is known that in their experiments, the average observed values of variab... | A |
Given a sequence $\{a_n\}$ that satisfies the recurrence $a_n = \frac{a_{n-1}}{a_{n-2}}$ for $n \in \mathbb{N}^{*}$ and $n \geq 3$, with $a_1=2$ and $a_2=\frac{1}{3}$, find the value of $a_{2016}$. | 6 |
Let $I$ be the set of points $(x,y)$ in the Cartesian plane such that $$x>\left(\frac{y^4}{9}+2015\right)^{1/4}$$ Let $f(r)$ denote the area of the intersection of $I$ and the disk $x^2+y^2\le r^2$ of radius $r>0$ centered at the origin $(0,0)$. Determine the minimum possible real number $L$ such that $f(r)<Lr^2$ for a... | \frac{\pi}{3} |
The line $y=e^x+b$ (where $e$ is the base of the natural logarithm) has at most one common point with the two functions $f(x)=e^x$ and $g(x)=\ln(x)$. The range of the real number $b$ is ______. | [-2,0] |
3. Find natural numbers $p$ and $q$ such that the roots of the trinomials $x^{2}-p x+q$ and $x^{2}-q x+p$ are natural numbers. | p=6,q=5;p=5,q=6p=q=4 |
4. Given $\frac{y+z-x}{x+y+z}=\frac{z+x-y}{y+z-x}=\frac{x+y-z}{z+x-y}$ $=p$. Then $p^{3}+p^{2}+p=$ $\qquad$ | 1 |
In May 2022, the Development and Reform Commission of Sichuan Province issued the first batch of central budget investment plans for the Affordable Housing Project in 2022. Luzhou City received 75,500,000 yuan in central budget support. Express 75,500,000 in scientific notation as ( ).
A: $7.55\times 10^{6}$
B: $75.5... | C |
A semicircle with diameter length $16$ contains a circle radius $3$ tangent both to the inside of the semicircle and its diameter as shown. A second larger circle is tangent to the inside of the semicircle, the outside of the circle, and the diameter of the semicircle. The diameter of the second circle can be written a... | 361 |
B2. Find all natural numbers $n$ and prime numbers $p$ for which $\sqrt[3]{n}+\frac{p}{\sqrt[3]{n}}$ is a square of a natural number. | n=1,p=3n=27,p=3 |
If a cylinder with equal diameter and height has a lateral surface area of $\pi$, then the volume of the cylinder is ______. | \frac{\pi}{4} |
Let $(1+x)^8 = a + a_1x + \ldots + a_8x^8$, then the number of odd numbers among $a, a_1, \ldots, a_8$ is ( )
A: 2
B: 3
C: 4
D: 5 | \text{A} |
2. Let $a, b \in \mathbb{R}^{+}, i^{2}=-1$, and there exists $z \in \mathbb{C}$, such that
$$
\left\{\begin{array}{l}
z+\bar{z}|z|=a+b i, \\
|z| \leqslant 1 .
\end{array}\right.
$$
Then the maximum value of $a b$ is | \frac{1}{8} |
10. As shown in Figure 5, in trapezoid $A B C D$, $A D / / B C, C E$ is the bisector of $\angle B C D$, and $C E \perp A B$, with $E$ being the foot of the perpendicular, $B E=2 A E$. If the area of quadrilateral $A E C D$ is 1, then the area of trapezoid $A B C D$ is $\qquad$ . | 2 \frac{1}{7} |
Given the function $f(x)=(x-a)^{2}+(e^{x}-a)^{2}$ where $a\in \mathbb{R}$, if there exists $x_{0} \in \mathbb{R}$ such that $f(x_{0}) \leq \dfrac{1}{2}$, then the value of the real number $a$ is (_).
A: $\dfrac{1}{3}$
B: $\dfrac{\sqrt{2}}{2}$
C: $\dfrac{\sqrt{2}}{4}$
D: $\dfrac{1}{2}$ | D |
12.280. The base of the pyramid is a rectangle $A B C D$ ($A B \| C D$). The lateral edge $O A$ is perpendicular to the base. The edges $O B$ and $O C$ form angles with the base, respectively equal to $\alpha$ and $\beta$. Find the angle between the edge $O D$ and the base. | \operatorname{arctg}\frac{\sin\alpha\sin\beta}{\sqrt{\sin(\alpha-\beta)\sin(\alpha+\beta)}} |
3. (17th Japan Mathematical Olympiad) Find the tens digit of $11^{12^{13}}$ (where $11^{12^{3}}$ represents 11 to the power of $12^{13}$) | 2 |
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