problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
10.186. The lengths of the bases of an isosceles trapezoid are in the ratio 5:12, and the length of its height is 17 cm. Calculate the radius of the circle circumscribed around the trapezoid, given that its midline is equal to the height. | 13 |
6. Let $f(x)$ be a function defined on the set of real numbers $\mathbf{R}$, and satisfies the following relations:
$$
f(10+x)=f(10-x), f(20-x)=-f(20+x) \text {. }
$$
Then $f(x)$ is
(A) an even function, and also a periodic function
(B) an even function, but not a periodic function
(C) an odd function, and also a peri... | C |
Four, (50 points) On a roadside, there are $n$ parking spaces, and $n$ drivers each driving a car. Each driver parks their car in front of their favorite parking space. If that space is already occupied, they park in the nearest available space down the road. If that space and all the spaces below it are occupied, they... | (n+1)^{n-1} |
3. How to find the lightest and heaviest stones among $2 N$ stones, any two of which differ in weight, in $3 N-2$ weighings? All weighings are performed on a two-pan balance without weights. | 3N-2 |
A cuboid, if its height is reduced by 4 cm, becomes a cube, and its volume decreases by 64 cubic centimeters. The surface area of this cube is ____ square centimeters. | 96 |
138. Calculate the sum for any $\alpha$
$$
\sin ^{2} \alpha+\sin ^{2}\left(\alpha+1^{\circ}\right)+\sin ^{2}\left(\alpha+2^{\circ}\right)+\ldots+\sin ^{2}\left(\alpha+179^{\circ}\right)
$$ | 90 |
Given that A is a moving point on the ray $x+y=0$ (where $x \leq 0$), and B is a moving point on the positive half of the x-axis, if line AB is tangent to the circle $x^2+y^2=1$, the minimum value of $|AB|$ is _______ . | 2 + 2\sqrt{2} |
A fair six-sided die is rolled twice. Let $a$ be the number that comes up on the first roll, and $b$ be the number that comes up on the second roll. What is the probability that the function $y=ax^2-2bx+1$ is decreasing on the interval $(-\infty, 2]$? | \frac{1}{4} |
Given an ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with its foci on the $x$-axis, and the left vertex of the ellipse $E$ is $A$. A line with slope $k(k > 0)$ intersects the ellipse $E$ at points $A$ and $B$. Point $C$ is on the ellipse $E$, and $AB \perpendicular AC$. The line $AC$ intersects the $y$-axis at... | \sqrt[3]{2} < k < 2 |
[Law of Cosines] [Law of Sines]
Find the radius of the circumcircle of a triangle with sides 5 and 8 and an angle of $60^{\circ}$ between them. | \frac{7\sqrt{3}}{3} |
Develop $(a+b)^{5}$. | (+b)^{5}=^{5}+5^{4}b+10^{3}b^{2}+10^{2}b^{3}+5^{4}+b^{5} |
## Zadatak B-3.1.
Riješite nejednadžbu
$$
2 \log _{\frac{1}{5}}\left(49^{\sqrt{x^{2}-2}}-1\right)+\log _{5}\left(7^{\sqrt{4 x^{2}-8}}+\frac{1}{5}\right) \leqslant-1
$$
| x\in\langle-\infty,-\frac{3}{2}]\cup[\frac{3}{2},+\infty\rangle |
Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares. | 4 |
## Task B-3.5.
A regular right quadrilateral pyramid with apex $V$ and base $ABCD$ is inscribed in a sphere of radius $R$. Let $\cos \angle AVB = \frac{3}{4}$. What is the height of the pyramid expressed in terms of $R$? | \frac{3}{2}R |
Suppose there are $2017$ spies, each with $\frac{1}{2017}$th of a secret code. They communicate by telephone; when two of them talk, they share all information they know with each other. What is the minimum number of telephone calls that are needed for all 2017 people to know all parts of the code? | 4030 |
Example X (China Mathematical Olympiad 2000) Given that $a, b, c$ are the sides of $\triangle ABC$, $a \leqslant b \leqslant c$, and $R$ and $r$ are the radii of the circumcircle and incircle of $\triangle ABC$, respectively. Let $f=a+b-2 R-2 r$, try to determine the sign of $f$ using the size of angle $C$.
| When\C\in[\frac{\pi}{3},\frac{\pi}{2}),\f>0;\when\C=\frac{\pi}{2},\f=0;\when\C\in(\frac{\pi}{2},\pi),\f<0 |
The little sprite wrote down consecutive natural numbers starting from 1 in a row:
From the 30th digit, 2021 (the four digits marked in red) first appears. The little sprite continues to write, and from the $\qquad$th digit, 2022 first appears. | 496 |
Problem 8. For what values of the parameter $a$ does the equation
$$
\log _{2}\left(2 x^{2}-x-2 a-4 a^{2}\right)+3 \log _{1 / 8}\left(x^{2}-a x-2 a^{2}\right)=0
$$
have two distinct roots, the sum of whose squares lies in the interval $(4 ; 8)$? | (\frac{3}{5};1) |
In space, the number of correct propositions is ( )
① Two lines parallel to the same line are parallel.
② Two lines perpendicular to the same line are parallel.
③ Two lines parallel to the same plane are parallel.
④ Two lines perpendicular to the same plane are parallel.
A: 1
B: 2
C: 3
D: 4 | \text{B} |
If the decimal number 26 is equal to the base-$k$ number 32, then $k$ equals \_\_\_\_\_\_. | 8 |
(14) There are 20 cards, each with a number from $1,2,3, \cdots, 20$, placed in a box. Four people draw one card each without replacement. The two people who draw the smaller numbers are in one group, and the two people who draw the larger numbers are in another group. If two of them draw 5 and 14, the probability that... | \frac{7}{51} |
Given $\overrightarrow{m}=(\sqrt{3}\sin \omega x,\cos \omega x)$, $\overrightarrow{n}=(\cos \omega x,-\cos \omega x)$ ($\omega > 0$, $x\in\mathbb{R}$), and $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}- \frac{1}{2}$, and the distance between two adjacent axes of symmetry on the graph of $f(x)$ is $\frac{\pi}{2}$.
$... | \frac{3\sqrt{3}}{4} |
You have 54 poker cards. You throw away the 1st card, put the 2nd card at the end, throw away the 3rd card, put the 4th card at the end, and so on. When you are left with only one card, what is the position of this card in the original sequence of cards? | 44 |
607. The sum of a two-digit number and its reverse is a perfect square. Find all such numbers.
(Definition of the number reversed to the given one, see § 12, p. 12.4.) | 29,38,47,56,65,74,83,92 |
Let $a$ and $b$ be non-zero vectors. Among the following inequalities, the one that does not always hold is
A: $|a+b| \leq |a| + |b|$
B: $|a| - |b| \leq |a+b|$
C: $|a| - |b| \leq |a| + |b|$
D: $|a| \leq |a+b|$ | \text{D} |
Given vectors $p = (a_n, 2^n)$ and $q = (2^{n+1}, -a_{n+1})$, where $n \in \mathbb{N}^*$, $p$ is perpendicular to $q$, and $a_1 = 1$.
(1) Find the general formula for the sequence $\{a_n\}$;
(2) If the sequence $\{b_n\}$ satisfies $b_n = \log_2 a_n + 1$, find the sum of the first $n$ terms of the sequence $\{a_n \cd... | 1 + (n - 1)2^n |
12. In the tetrahedron $ABCD$, $AD=2\sqrt{3}$, $\angle BAC=60^{\circ}$, $\angle BAD=\angle CAD=45^{\circ}$; if a sphere passing through $D$ is tangent to the plane $ABC$ and internally tangent to the circumscribed sphere of the tetrahedron with a radius of 1, then the radius of the circumscribed sphere of the tetrahedr... | 3 |
Pegs are put in a board $1$ unit apart both horizontally and vertically. A rubber band is stretched over $4$ pegs as shown in the figure, forming a [quadrilateral](https://artofproblemsolving.com/wiki/index.php/Quadrilateral). Its [area](https://artofproblemsolving.com/wiki/index.php/Area) in square units is
[asy] int ... | 6 |
3.220
$$
\frac{1+\cos (2 \alpha-2 \pi)+\cos (4 \alpha+2 \pi)-\cos (6 \alpha-\pi)}{\cos (2 \pi-2 \alpha)+2 \cos ^{2}(2 \alpha+\pi)-1}=2 \cos 2 \alpha
$$ | 2\cos2\alpha |
Consider the function $f(x)=e^x+2x-a$ (where $a \in \mathbb{R}$, and $e$ is the base of natural logarithms). If there exists a point $(x_0, y_0)$ on the curve $y=\sin x$ such that $f(f(y_0))=y_0$, find the range of value for $a$. | [-1+e^{-1}, e+1] |
In the "Nine Chapters on the Mathematical Art," there is the following problem: "Now there is a woman skilled in weaving, who doubles her output daily. In five days, she weaves five meters. How much does she weave each day?" The length of cloth woven by the woman on the second day is ( ).
A: $ \dfrac {40}{31}$
B: $ \... | C |
7. (10 points) As shown in the figure, a square is divided into 4 identical rectangles, and the perimeter of each rectangle is 20 cm. Then the area of this square is $\qquad$ 64 square cm. | 64 |
# Task № 1.3
## Condition:
Five friends - Sasha, Yulia, Rita, Alina, and Natasha - meet in the park every day after buying ice cream from the little shop around the corner. One day, the girls had a conversation.
Sasha: There were five people in front of me.
Alina: There was no one after me.
Rita: I was the first i... | 3 |
2. 7511 sets $M_{1}, M_{2}, \cdots, M_{11}$. Each set has 5 elements, and the intersection of any two sets is non-empty. Find the minimum possible value of the maximum number of sets that have a common element. | 4 |
2. (10 points) Choose the correct statements, assuming that $A$ is the area of a square with a diagonal of $2 \sqrt{2}$, $B$ is the area of a rectangle whose center and two adjacent vertices have coordinates $(2,2)$, $(4,1)$, and $(4,3)$ respectively, and $C$ is the area of the triangle formed by the coordinate axes an... | )) |
The coordinates of the point on the parabola $y=x^2$ that is closest to the line $2x-y-4=0$ are ( )
A: (1,1)
B: $\left( \frac {1}{2}, \frac {1}{4}\right)$
C: $\left( \frac {1}{3}, \frac {1}{9}\right)$
D: (2,4) | A |
Let $f(x) = \begin{cases} \sin \frac {\pi x}{6} & \text{if } x \leq 0 \\ 1-2x & \text{if } x > 0 \end{cases}$. Then, $f(f(3)) =$ （ ）
A: 1
B: $-1$
C: $-\frac {1}{2}$
D: $\frac {1}{2}$ | C |
Observe the following equations: $4-1=3$, $9-4=5$, $16-9=7$, $25-16=9$, $36-25=11$…, these equations reflect a certain relationship between natural numbers. Let $n$ represent a natural number, express this pattern with an equation involving $n$. | (n+1)^2 - n^2 = 2n + 1 |
Alex, Pei-Hsin, and Edward got together before the contest to send a mailing to all the invited schools. Pei-Hsin usually just stuffs the envelopes, but if Alex leaves the room she has to lick them as well and has a $25 \%$ chance of dying from an allergic reaction before he gets back. Licking the glue makes Edward a b... | \frac{1}{2} |
If $ab\neq 0$, and $3b=2a$, then $\frac{{2a+b}}{b}=$____. | 4 |
[ Decimal fractions]
How are the decimal representations of the numbers $\left\{\frac{p}{q}\right\}$ and $\left\{\frac{10^{k} p}{q}\right\}$ related?
## Answer
If $\left\{\frac{p}{q}\right\}=0, a_{1} a_{2} \ldots a_{k} a_{k+1} \cdots, \quad$ then $\left\{\frac{10^{k} p}{q}\right\}=0, a_{k+1} \cdots$
Find the possib... | 11,33,99 |
For any rational numbers $x$, $y$, define the operation as follows: $x \triangle y = ax + by + cxy$, where $a$, $b$, $c$ are given numbers, and the right side of the equation involves the usual addition and multiplication of numbers. For example, when $a=1$, $b=2$, $c=3$, $1 \triangle 3 = 1 \times 1 + 2 \times 3 + 3 \t... | 4 |
12. A. As shown in Figure 5, in rectangle $A B C D$, $A B=a$, $B C=b$, and $a<b$. A line through the center $O$ of rectangle $A B C D$ intersects segments $B C$ and $D A$ at points $E$ and $F$, respectively. The quadrilateral $E C D F$ is folded along $E F$ to the plane of quadrilateral $B E F A$, such that point $C$ c... | \frac{11}{8} |
352. Concerning graphics. For what value of $a$ does the graph of the function $a^{x}$ touch the graph of the function $\log _{a} x ?$ | e^{\frac{1}{e}} |
8. Bear Big and Bear Small invited 4 friends to watch a movie together, but there were no 6 consecutive seats available. They bought 2 adjacent seats in the fourth row and 4 consecutive seats in the fifth row. Bear Small sits next to Bear Big. There are $\qquad$ ways to arrange the seats. | 192 |
In downtown Gaussville, there are three buildings with different heights: The Euclid (E), The Newton (N) and The Galileo (G). Only one of the statements below is true.
1. The Newton is not the shortest.
2. The Euclid is the tallest.
3. The Galileo is not the tallest.
Ordered from shortest to tallest in height, the ... | E,N,G |
A [sphere](https://artofproblemsolving.com/wiki/index.php/Sphere) is inscribed in the [tetrahedron](https://artofproblemsolving.com/wiki/index.php/Tetrahedron) whose vertices are $A = (6,0,0), B = (0,4,0), C = (0,0,2),$ and $D = (0,0,0).$ The [radius](https://artofproblemsolving.com/wiki/index.php/Radius) of the spher... | 5 |
Given that the solution set of the quadratic inequality $ax^2+2x+c\leq0$ is $\{x | x= -\frac{1}{a}\}$, and $a>c$, the maximum value of $\frac{a-c}{a^2+c^2}$ is __________. | \frac{\sqrt{2}}{4} |
The sum of the coefficients of all terms except the constant term in the expansion of $\left(\frac{2}{\sqrt{x}} - x\right)^9$ is ______. | 5377 |
6.189. $\left\{\begin{array}{l}(x-y)\left(x^{2}+y^{2}\right)=5, \\ (x+y)\left(x^{2}-y^{2}\right)=9 .\end{array}\right.$ | (2;1),(-1;-2) |
The polar coordinate system and the rectangular coordinate system $(xOy)$ have the same unit of length. The origin $(O)$ is the pole, and the positive semi-axis of the $x$-axis is the polar axis $(.)$. The polar coordinate equation of the curve $C_{1}$ is $ρ-2\cos θ=0$, and the parametric equation of the curve $C_{1}$ ... | \dfrac {1- \sqrt {5}}{2} < m < \dfrac {1+ \sqrt {5}}{2} |
9. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=1, a_{2}=2, a_{n+2}-3 a_{n+1}+2 a_{n}=1$. Let $T_{n}$ be the sum of the first $n$ terms of the sequence $\left\{\frac{8}{a_{n}}\right\}$, then $\left[T_{2023}\right]=$ $\qquad$ . (The symbol $[x]$ denotes the greatest integer less than or equal to $x$) | 14 |
## Task Condition
Write the canonical equations of the line.
$x+5 y-z-5=0$
$2 x-5 y+2 z+5=0$ | \frac{x}{5}=\frac{y-1}{-4}=\frac{z}{-15} |
In the Cartesian coordinate system $xoy$, the parametric equation of curve $C_{1}$ is $\begin{cases} x= \sqrt {3}\cos \alpha \\ y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter), with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar... | \left( \frac {3}{2}, \frac {1}{2}\right) |
Simplify: $-a-a=$____. | -2a |
Given a triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their corresponding sides $a$, $b$, $c$. It is known that $a^{2}=b^{2}+c^{2}-bc$.
(1) Find the measure of angle $A$;
(2) If $a=2 \sqrt {3}$, $b=2$, find $\cos C$. | \cos C = \cos 90^{\circ} = 0 |
Point $A(\cos 2023^{\circ}$,$\tan 8)$ is located in which quadrant in the Cartesian coordinate system?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | C |
8.6. Let the function $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a x+b, a, b \in \mathbb{R}^{*}$. If $m, n$ are distinct natural numbers such that
$$
\begin{gathered}
n>2, m>2, f(1)+f(2)+\cdots+f(m)=-n \text { and } f(1)+f(2)+\cdots+f(n)=-m \text {, calculate } \\
\frac{f(1)+f(2)+\cdots+f(m+n)}{m+n}
\end{gathered}
$$ | 1 |
2. If $x, y \in \mathbf{R}$, and $3 x^{2}+2 y^{2}=6 x$, then the range of the bivariate function $f(x, y)=2 x^{2}+3 y^{2}-4 x-6 y$ is ( ).
(A) $\left[\frac{5}{2}-3 \sqrt{6}, \frac{5}{2}+3 \sqrt{6}\right]$
(B) $[5,10]$
(C) $[2 \sqrt{6}-1,2 \sqrt{6}+1]$
(D) $[7-2 \sqrt{6}, 7+2 \sqrt{6}]$ | A |
Given three people, A, B, and C, they stand in a row in any random order. What is the probability that A stands in front of B and C does not stand in front of A? | \frac{1}{3} |
## Problem Statement
Find the derivative $y_{x}^{\prime}$
$$
\left\{\begin{array}{l}
x=\frac{t^{2} \ln t}{1-t^{2}}+\ln \sqrt{1-t^{2}} \\
y=\frac{t}{\sqrt{1-t^{2}}} \arcsin t+\ln \sqrt{1-t^{2}}
\end{array}\right.
$$ | \frac{\arcsin\cdot\sqrt{1-^{2}}}{2\cdot\ln} |
## Task Condition
Find the derivative.
$y=x^{e^{\operatorname{arctan} x}}$ | y^{\}=x^{e^{\operatorname{arctg}x}}\cdote^{\operatorname{arctg}x}(\frac{\lnx}{1+x^{2}}+\frac{1}{x}) |
A set containing three real numbers can be represented as $\\{a,1, \dfrac {b}{a}\\}$, and also as $\\{a+b,0,a^{2}\\}$. Determine the value of $a^{2016}+b^{2016}$.
A: $0$
B: $1$
C: $-1$
D: $±1$ | \text{B} |
Factorize: $2m^{2}-18=\_\_\_\_\_\_.$ | 2\left(m+3\right)\left(m-3\right) |
Given that $f(x)$ is an even function, and for $x > 0$, $f(x) = x^2 - 2x$, find the expression for $f(x)$ when $x < 0$. | x^2 + 2x |
4. A rectangle $A B C D$ has been cut out of paper. A point $E$ is chosen on side $\overline{A B}$ such that $|A E|=8$ and $|B E|=17$. A point $F$ is chosen on side $\overline{C D}$ such that $|C F|=3$. After folding the paper along the line $\overline{E F}$, vertex $B$ coincides with a point $S$ on side $\overline{A D... | \frac{70}{3} |
## Task 1 - 320831
If $a, b, c$ are the hundreds, tens, and units digits of a three-digit natural number, then this number is briefly denoted by $\overline{a b c}$. Similarly, a two-digit number with tens and units digits $b$ and $c$ is denoted by $\overline{b c}$.
Determine all $a, b, c$ for which $\overline{a b c}$... | 625=25^{2} |
## Task A-4.5.
Prove that $\sum_{k=0}^{1005}(-3)^{k}\binom{2010}{2 k}=2^{2010}$. | 2^{2010} |
Find all natural values of $k$ for which the system
$$\begin{cases}x_1+x_2+\ldots+x_k=9\\\frac1{x_1}+\frac1{x_2}+\ldots+\frac1{x_k}=1\end{cases}$$
has solutions in positive numbers. Find these solutions.
[i]I. Dimovski[/i] | \left(x_1, x_2, x_3\right) = \left(3, 3, 3\right) |
1. Given the sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms $S_{n}=n^{2}-1\left(n \in \mathbf{N}_{+}\right)$, then $a_{1}+a_{3}+a_{5}+a_{7}+a_{9}=$
A. 40
B. 44
C. 45
D. 49 | 44 |
385. $(\tan \alpha+\cot \alpha)^{2}-(\tan \alpha-\cot \alpha)^{2}$. | 4 |
The following are four propositions:
① There is exactly one plane perpendicular to a given line through a point;
② If the distances from two points outside a plane to the plane are equal, then the line passing through these two points must be parallel to the plane;
③ The projections of two intersecting lines onto... | \text{D} |
2B. Determine the real number $x$ such that the sequence $\sqrt{x-7}, \sqrt[4]{2 x+19}$, $\sqrt{x+3}$ is a geometric progression. | 10 |
(a) Determine the positive integer $x$ for which $\frac{1}{4}-\frac{1}{x}=\frac{1}{6}$.
(b) Determine all pairs of positive integers $(a, b)$ for which $a b-b+a-1=4$.
(c) Determine the number of pairs of positive integers $(y, z)$ for which $\frac{1}{y}-\frac{1}{z}=\frac{1}{12}$.
(d) Prove that, for every prime numb... | (3,1),(2,3) |
Outstanding graduates from three universities, $A$, $B$, and $C$, two from each, are now arranged to go to three middle schools to carry out promotional activities. The requirement is that each school should have two graduates from different universities. The number of different arrangement plans is ( ).
A: $24$
B: $... | C |
(1) If $f(x)=x^{2}-2x-4\ln x$, then the decreasing interval of $f(x)$ is $\boxed{\left(0,2\right)}$.
(2) Let $z_{1},z_{2}$ be complex numbers, and consider the following four statements:
(i) If $|z_{1}-z_{2}|=0$, then $\overline{z_{1}}=\overline{z_{2}}$;
(ii) If $z_{1}=\overline{z_{2}}$, then $\overline{z_{1}}=z_{2... | (0,2e] |
Suppose two circles $\Omega_1$ and $\Omega_2$ with centers $O_1$ and $O_2$ have radii $3$ and $4$, respectively. Suppose that points $A$ and $B$ lie on circles $\Omega_1$ and $\Omega_2$, respectively, such that segments $AB$ and $O_1O_2$ intersect and that $AB$ is tangent to $\Omega_1$ and $\Omega_2$. If $O_1O_2=25$, f... | 84 |
Problem 3. If we divide the numbers 701 and 592 by the same natural number, we get remainders of 8 and 7, respectively. By which number did we divide the given numbers? | 9 |
# Task No. 8.1
## Condition:
Given triangle $\mathrm{ABC}$, where $2 \mathrm{BC}=\mathrm{AC}$ and angle $\mathrm{C}=74^{\circ}$. On ray $\mathrm{BC}$, segment $\mathrm{CD}=\mathrm{CB}$ is laid out. Then, from point $\mathrm{D}$, a perpendicular is drawn to the line containing the median of triangle $\mathrm{ABC}$, dr... | 37 |
If the equation $x^{a-1} - 5 = 3$ is a linear equation in terms of $x$, then $a=$ __ . | 2 |
The standard deviation of the data 5, 7, 7, 8, 10, 11 is | 2 |
The absolute value of the rational number $-3$ is:
A: $3$
B: $-3$
C: $\pm 3$
D: $\sqrt{3}$ | A |
A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement,
with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents?
$\text{(A)} \ \frac{37}{924} \qquad \text{(B)} \ \frac{91}{924} \qquad \text{(C)}... | \frac{127}{924} |
1. Let $p, q \in \mathbf{R}_{+}$, and satisfy
$$
\begin{array}{l}
\log _{9} p=\log _{12} q=\log _{16}(p+q) . \\
\text { Then } \frac{q}{p}=\text {. }
\end{array}
$$ | \frac{1+\sqrt{5}}{2} |
$[$ Mathematical logic (miscellaneous).]
Authors: Folklore, Folklore
In a school of witchcraft, there are 13 students. Before the clairvoyance exam, the teacher seated them at a round table and asked them to guess who would receive the clairvoyant diploma. They modestly kept quiet about themselves and their two neigh... | 2 |
$14.28 \star \star$ What is the largest even number that cannot be written as the sum of two odd composite numbers? | 38 |
Find the domain of the function \(f(x)=\ln (4-x)\).
A: \((-\infty,4]\)
B: \((-\infty,4)\)
C: \((0,4]\)
D: \((0,4)\) | B |
$x^{2}\cdot x^{3}=\left(\ \ \right)$
A: $x^{5}$
B: $x^{6}$
C: $x^{8}$
D: $x^{9}$ | A |
Given point $P(\sin \alpha - \cos \alpha, \tan \alpha)$ is in the first quadrant, then the range of angle $\alpha$ within $[0,2\pi]$ is ( )
A: $\left( \left. \frac{\pi}{4}, \frac{\pi}{2} \right. \right)$
B: $\left( \left. \pi, \frac{5}{4}\pi \right. \right)$
C: $\left( \left. \frac{3\pi}{4}, \frac{5}{4}\pi \right. \... | D |
A finite sequence of integers $ a_0, a_1, \ldots, a_n$ is called quadratic if for each $ i$ in the set $ \{1,2 \ldots, n\}$ we have the equality $ |a_i \minus{} a_{i\minus{}1}| \equal{} i^2.$
a.) Prove that any two integers $ b$ and $ c,$ there exists a natural number $ n$ and a quadratic sequence with $ a_0 \equal{... | n = 19 |
Square $A B C D$ has sides of length 2. The midpoint of $B C$ is $E$. Point $F$ is on $A E$ so that $D F$ is perpendicular to $A E$. Determine the length of $D F$.
 | \frac{4}{\sqrt{5}} |
3. As shown in Figure 1, in the regular triangular prism $A B C-A_{1} B_{1} C_{1}$, $D$ and $E$ are points on the side edges $B B_{1}$ and $C C_{1}$, respectively, with $E C=B C=2 B D$. Then the size of the dihedral angle formed by the section $A D E$ and the base $A B C$ is | 45^{\circ} |
3. As shown in the figure, the side lengths of the two squares $\mathrm{ABCD}$ and $\mathrm{DEFG}$ are both integer centimeters, point $\mathrm{E}$ lies on line segment $\mathrm{CD}$, and $\mathrm{CE}<\mathrm{DE}$, line segment $\mathrm{CF}=5$ centimeters, then the area of pentagon $\mathrm{ABCFG}$ is $\qquad$ square c... | 71 |
16.2.25 ** Arrange natural numbers in sequence:
$$
123456789101112131415161718192021 \cdots \cdots
$$
If the $10^{n}$-th digit is in an $m$-digit number, then define $f(n)=m$. For example, the $10^{1}$-th digit 1 is in the two-digit number 10, and the $10^{2}$-th digit 5 is in the two-digit number 55, so $f(1)=2, f(2)... | 1988 |
Which of the following calculations is correct?
A: $a^{3}+a^{3}=a^{6}$
B: $a^{3}\cdot a^{3}=a^{6}$
C: $(a^{2})^{3}=a^{5}$
D: $(ab)^{3}=ab^{3}$ | B |
140. Find the maximum value of the function
$$
y(x)=3 \sin x+4 \cos x
$$ | 5 |
The graph of the function $y=a^x+1$ passes through a fixed point. | (0, 2) |
Given the set $U={1,2,3,4,5}$, select 4 numbers from set $U$ to form a four-digit number with no repeating digits. This four-digit number should be greater than $2345$ and less than $4351$. The number of four-digit numbers that meet these conditions is \_\_\_\_\_\_. | 54 |
Given $f(x) = \begin{cases} \sqrt{1-x^{2}}, & x\in [-1,1] \\ x^{2}-1, & x\in (1,2] \end{cases}$, then $\int_{-1}^{2}f(x)dx=$\_\_\_\_\_\_. | \frac{\pi}{2} + \frac{4}{3} |
4. Let the $n$ roots of the equation $x^{n}=1$ (where $n$ is an odd number) be $1, x_{1}, x_{2}, \cdots, x_{n-1}$. Then $\sum_{i=1}^{n-1} \frac{1}{1+x_{i}}=$ $\qquad$ . | \frac{n-1}{2} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.