problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
23. Sam wants to complete the diagram so that each of the nine circles contains one of the digits from 1 to 9 inclusive and each contains a different digit. Also, the digits in each of the three lines of four circles must have the same total. What is this total?
A 17
B 18
C 19
D 20
E 21 | 20 |
G1.3 If $1+3+3^{2}+3^{3}+\ldots+3^{8}=\frac{3^{c}-1}{2}$, find $c$. | 9 |
397. Find the conditions under which the squares of three consecutive terms of an arithmetic progression are three consecutive terms of a geometric progression. | =0or=(-2\\sqrt{2}) |
Among the real numbers $\frac{2}{3}$, $\sqrt{2}$, $\pi$, $\sqrt{9}$, $2.12112111211112\ldots$, which ones are irrational numbers?
A: $1$ number
B: $2$ numbers
C: $3$ numbers
D: $4$ numbers | C |
On the exam, there are three trigonometry problems, two algebra problems, and five geometry problems. Vanya solves trigonometry problems with a probability of
$p_{1}=0.2$, geometry problems with a probability of $p_{2}=0.4$, and algebra problems with a probability of $p_{3}=0.5$. To get a grade of three, Vanya needs ... | )\approx0.27;b)-) |
Let $\{a_n\}^{\infty}_0$ and $\{b_n\}^{\infty}_0$ be two sequences determined by the recursion formulas
\[a_{n+1} = a_n + b_n,\]
\[ b_{n+1} = 3a_n + b_n, n= 0, 1, 2, \cdots,\]
and the initial values $a_0 = b_0 = 1$. Prove that there exists a uniquely determined constant $c$ such that $n|ca_n-b_n| < 2$ for all nonnegati... | \sqrt{3} |
6. For any $\theta \in\left[0, \frac{\pi}{2}\right]$, such that
$$
|\sin \theta-p \cos \theta-q| \leqslant \frac{\sqrt{2}-1}{2}
$$
the real number pairs $(p, q)$ that always satisfy the condition are | \left(-1, \frac{1+\sqrt{2}}{2}\right) |
A tangent line is drawn from the left focus $F(-c,0)$ ($c > 0$) of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) to the circle $x^2 + y^2 = \frac{a^2}{4}$. Let the point of tangency be $E$. Extend $FE$ to intersect the right branch of the hyperbola at point $P$. If $\overrightarrow{OP} = 2\ov... | \frac{\sqrt{10}}{2} |
$\frac{2}{10}+\frac{4}{100}+\frac{6}{1000}=$
$\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$ | 0.246 |
10.5. Find all numbers $a$ such that for any natural $n$ the number $a n(n+2)(n+3)(n+4)$ is an integer.
(O. Podlipsky) | \frac{k}{6} |
In the arithmetic sequence $\{a_n\}$, $(a_2=2, a_3=4)$, find the value of $(a_{10}= )$. | 18 |
Problem 6.4. Given a quadrilateral $A B C D$, in which $A D \| B C$. The bisector of angle $A$ intersects side $C D$ at point $X$, and the extension of side $B C$ beyond point $C$ - at point $Y$. It turns out that $\angle A X C=90^{\circ}$. Find the length of segment $A B$, if it is known that $A D=19$ and $C Y=16$.
!... | 17.5 |
7.5. There are 200 matches. How many ways are there to form, using all the matches, a square and (separately) an equilateral triangle? (Different ways differ in the sizes of the square and the triangle). | 16 |
Calculate the value of $2\sin 60^{\circ}$ as ( ).
A: $\sqrt{3}$
B: $\frac{\sqrt{3}}{2}$
C: $1$
D: $\frac{1}{2}$ | A |
Shnol D.e.
In the morning, a dandelion blooms, it flowers yellow for three days, on the fourth morning it turns white, and by the evening of the fifth day, it withers. On Monday afternoon, there were 20 yellow and 14 white dandelions on the meadow, and on Wednesday - 15 yellow and 11 white. How many white dandelions w... | 6 |
If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?
$\textbf{(A) }$ $n$ and $m$ are even $\qquad\textbf{(B) }$ $n$ and $m$ are odd $\qquad\textbf{(C) }$ $n+m$ is even $\qquad\textbf{(D) }$ $n+m$ is odd $\qquad \textbf{(E) }$ none of these are impossible | D |
7. Encode the letters $a \sim z$ as $(\mathbf{a}=1, \mathbf{b}=2, \ldots, \mathbf{z}=26)$, so that each English word (all letters in the word are considered lowercase) can have the product $\mathbf{p}$ of the encodings of all its letters calculated. For example, for the word good, the corresponding $\mathbf{p}$ value i... | 106 |
Given an ellipse $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ and a circle $x^{2} + y^{2} = \left(\dfrac{b}{2} + c\right)^{2}$, where $c$ is the semi-focal distance of the ellipse, and they intersect at four distinct points, the range of the eccentricity $e$ of the ellipse is ( ).
A: $\left(\dfrac{\sq... | C |
18. In the calculation result of the multiplication $15 \times 16 \times 17 \times 18 \times 19 \times 20 \times 21 \times 22 \times 23 \times 24 \times 25$, how many consecutive 0s are there at the end? | 4 |
Nathan has discovered a new way to construct chocolate bars, but it’s expensive! He starts with a single $1\times1$ square of chocolate and then adds more rows and columns from there. If his current bar has dimensions $w\times h$ ($w$ columns and $h$ rows), then it costs $w^2$ dollars to add another row and $h^2$ dolla... | 5339 |
Given vectors $\overrightarrow{a}=(2,3)$ and $\overrightarrow{b}=(6,x)$, where $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, find the value of $x$. | -4 |
Three, (1) Prove: $\sqrt{a^{2}+\frac{1}{b^{2}}+\frac{a^{2}}{(a b+1)^{2}}}$ $=\left|a+\frac{1}{b}-\frac{a}{a b+1}\right|$
(2) Using or not using (1), calculate
$$
\sqrt{1+1990^{2}+\frac{1990^{2}}{1991^{2}}}-\frac{1}{1991}
$$ | 1990 |
In the sequence $\{a_{n}\}$, $a_{1}=-2$, $a_{n}a_{n+1}=a_{n}-1$. What is the value of $a_{2021}$?
A: $-2$
B: $\frac{1}{3}$
C: $\frac{1}{2}$
D: $\frac{3}{2}$ | D |
Given that the center of the ellipse $(C)$ is at the origin of the coordinate, the focus is on the $x$-axis, and the eccentricity is $\frac{1}{2}$, a line passes through the left focus $(F)$ of the ellipse at an angle of $60^{\circ}$ and is tangent to the circle $x^{2}+y^{2}= \frac{b^{2}}{a^{2}}$.
(I) Find the equation... | (\frac{2}{7},0) |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $S_3=9$ and $S_6=36$, then $a_7+a_8+a_9=$ ______. | 45 |
2. How many numbers from 1 to 100 are there, each of which is divisible by 3 but does not contain the digit 3 in its representation? | 26 |
10,11 | | |
Point $M$ is at distances of 5 and 4 from two parallel lines $m$ and $n$, and at a distance of 3 from the plane passing through these lines. Find the distance between the lines $m$ and $n$. | 4+\sqrt{7} |
A vocational test was conducted on 1000 students from a school. The table below presents the results by field of study and gender.
| | Exatas | Humanas | Biológicas |
| :---: | :---: | :---: | :---: |
| Masculino | 232 | 116 | 207 |
| Feminino | 112 | 153 | 180 |
If a student is chosen at random, determine the proba... | 0.344,0.209,0.465 |
6. For any positive integers $x, y, z$, if $x \mid y^{3}$, $y \mid z^{3}$, $z \mid x^{3}$, and $x y z \mid (x+y+z)^{n}$, find the smallest positive integer $n$ that satisfies the condition. | 13 |
1. (7 points) The first term of the sequence is 934. Each subsequent term is equal to the sum of the digits of the previous term, multiplied by 13. Find the $2019-$th term of the sequence. | 130 |
3.356. $\sin 20^{\circ} \cos 50^{\circ} \sin 60^{\circ} \cos 10^{\circ}$. | \frac{3}{16} |
Let $ABC$ be a triangle with $BC = 5$, $CA = 3$, and $AB = 4$. Variable points $P, Q$ are on segments $AB$, $AC$, respectively such that the area of $APQ$ is half of the area of $ABC$. Let $x$ and $y$ be the lengths of perpendiculars drawn from the midpoint of $PQ$ to sides $AB$ and $AC$, respectively. Find the range o... | [6, 6.5] |
2. Calculate: $\frac{2 \times 4.6 \times 9+4 \times 9.2 \times 18}{1 \times 2.3 \times 4.5+3 \times 6.9 \times 13.5}=$ $\qquad$ | \frac{18}{7} |
Example 11 (2003 Tianjin Competition Question) Let the quadratic function $f(x)=a x^{2}+b x+c$ (where $a, b, c$ are integers), and 4 students calculate the function values. Student A gets: $f(7)=-1$; Student B gets: $f(1)=3$; Student C gets: $f(4)=-4$; Student D gets: $f(2)=4$. Among them, only 1 student made a calcula... | B |
5. In $\triangle A B C$, $A B=3, A C=4, B C=5, I$ is the incenter of $\triangle A B C$, and $P$ is a point within $\triangle I B C$ (including the boundary). If $\overrightarrow{A P}=\lambda \overrightarrow{A B}+\mu \overrightarrow{A C}\left(\lambda, \mu \in \mathbf{R}\right)$, then the minimum value of $\lambda+\mu$ i... | \frac{7}{12} |
In the figure, $O$ is the center of the semicircle with diameter $P Q, R$ is a point on the semicircle, and $R M$ is perpendicular to $P Q$. If the measure of arc $\widehat{P R}$ is twice the measure of arc $\overparen{R Q}$, what is the ratio between $P M$ and $M Q$?
$
A: $-\frac{3}{2}$
B: $3$
C: $-1$
D: Does not exist | C |
Example 8 (1990 Japanese Spring Selection Problem) Let $x, y, z > 0$, and $x+y+z=1$, find the minimum value of $\frac{1}{x}+\frac{4}{y}+\frac{9}{z}$. | 36 |
Given a point $P$ on the sine curve $y=\sin x$, the tangent line at point $P$ is line $l$. Then, the range of the inclination angle of line $l$ is $(\quad)$
A: $\left[0, \dfrac {\pi}{4}\right]\cup\left[ \dfrac {3\pi}{4},\pi\right)$
B: $\left[0,\pi\right)$
C: $\left[ \dfrac {\pi}{4}, \dfrac {3\pi}{4}\right]$
D: ... | A |
Given $y=\log_{a}(2-ax)$ ($a > 0$ and $a \neq 1$) is a decreasing function on the interval $[0,1]$, the range of $a$ is ( ).
A: $(0,1)$
B: $(1,2)$
C: $(0,2)$
D: $[2,+\infty)$ | B |
2. Twenty numbers are arranged in a sequence. It is known that each number in the sequence, except the first one, is twice as large as the number immediately preceding it. How many times smaller is the sum of the first ten numbers in the sequence compared to the sum of the last ten numbers in the sequence? | 1024 |
Given that $x$ and $y$ satisfy $|x+5|+\left(y-4\right)^{2}=0$, find the value of $\left(x+y\right)^{99}$. | -1 |
3. What is the maximum number of pawns that can be placed on a chessboard (no more than one pawn per square), if: 1) a pawn cannot be placed on the $e4$ square; 2) no two pawns can stand on squares that are symmetric with respect to the $e4$ square?
Answer: 39 pawns. | 39 |
Given $f(x) = ax^2 + bx + c$ ($ac \neq 0$), and $g(x) = cx^2 + bx + a$
① If $f(x)$ has no real roots, then $g(x) > 0$ for all $x \in \mathbb{R}$.
② If $f(x)$ has exactly one real root, then $g(x)$ must have two real roots.
③ If the equation $f(x) = 0$ has two distinct real roots, then the equation $g(x) = 0$ cann... | 1 |
1. (2 points) Does there exist a four-digit natural number with the sum of its digits being 14, which is divisible by $14 ?$ | 6314 |
Find the area enclosed by the parabola $y=x^{2}$ and the line $2x+y-3=0$. | \frac{32}{3} |
## Task 4 - 050714
Rolf presents his friend with the following problem:
In a chess tournament, everyone played exactly one game against each other. In total, 28 games were played.
How many participants were there in this tournament? | 8 |
Problem 10.7. Two circles have a common chord $AB$. Through point $B$, a line is drawn that intersects the circles at points $C$ and $D$, such that $B$ lies between $C$ and $D$. The tangents to the circles, drawn through points $C$ and $D$, intersect at a point $E$. Compare $AD \cdot AC$ with $AB \cdot AE$. | AD\cdotAC=AB\cdotAE |
Problem 3. Six numbers are written in a row. It is known that among them there is a one and any three adjacent numbers have the same arithmetic mean. Find the maximum value of the geometric mean of any three adjacent numbers in this row, if the arithmetic mean of all 6 numbers is A. | \sqrt[3]{(3A-1)^{2}/4} |
In a class of 50 students, it is decided to use systematic sampling to select 5 students from these 50. The students are randomly assigned numbers from 1 to 50 and grouped, with the first group being numbers 1 to 10, the second group 11 to 20, and so on, up to the fifth group which is 41 to 50. If the student with numb... | A |
Given that $0 < x < 1$, find the minimum value of $\frac{1}{x} + \frac{4}{1 - x}$. | 9 |
Among the following propositions, the correct one(s) is(are):
① If the proposition "p or q" is true, then both propositions "p" and "q" are true propositions.
② The negation of the proposition "∃x₀ ∈ R, x₀² + 1 > 3x₀" is "∀x ∈ R, x² + 1 ≤ 3x".
③ The statement "The smallest positive period of the function f(x) = co... | A |
Problem 3. For natural numbers $x$ and $y$ and an odd integer $z$, it is known that $x! + y! = 8z + 2017$. Find all possible such triples of numbers $(x, y, z)$. (Recall that $1! = 1, 2! = 1 \cdot 2, n! = 1 \cdot 2 \cdot \ldots \cdot n$) Answer: $(1; 4; -249), (4; 1; -249), (1; 5; -237), (5; 1; -237)$. | (1;4;-249),(4;1;-249),(1;5;-237),(5;1;-237) |
Given that $m > 0$, $n > 0$, and $2m + n = 4$, find the minimum value of $\frac{1}{m} + \frac{2}{n}$. | 2 |
Given $f(x) = xe^x$, find $f'(1)$. | 2e |
Students A, B, and C are running for the president of the student council at their school. During the election, 1500 valid votes were received, and the results of the first 1000 votes were as follows: A received 350 votes, B received 370 votes, and C received 280 votes. To ensure that A wins the election with the most ... | 261 |
1. The number of five-digit numbers where the sum of any two adjacent digits is divisible by 3 is $\qquad$ (answer with a number) . | 1254 |
2. If the three points
$$
\begin{array}{l}
(1, a, b) \\
(a, 2, b) \\
(a, b, 3)
\end{array}
$$
are collinear (in 3-space), what is the value of $a+b$ ? | 4 |
Let the function $f(x)=\begin{cases} (1-2m)x-3m, & x < 1 \\ \log_{m}x, & x\geqslant 1 \end{cases}$ where $m\in \left[\frac{1}{5},\frac{1}{2}\right)$, if $a=f\left(-\frac{3}{2}\right)$, $b=f(1)$, $c=f(2)$, then ( )
A: $a < c < b$
B: $a < b < c$
C: $b < a < c$
D: $c < b < a$ | A |
4.1. Solve the equation $\left(x^{2}-2 x+4\right)^{x^{2}-2 x+3}=625$. In the answer, specify the sum of the squares of all its roots. If there are no roots, put 0. | 6 |
# Problem 5. (3 points)
In how many ways can natural numbers from 1 to 14 (each used exactly once) be arranged in a $2 \times 7$ table so that the sum of the numbers in each of the seven columns is odd?
Answer: $2^{7} \cdot(7!)^{2}$
# | 2^{7}\cdot(7!)^{2} |
$$
\begin{aligned}
f(x)= & |a \sin x+b \cos x-1|+ \\
& |b \sin x-a \cos x| \quad(a, b \in \mathbf{R})
\end{aligned}
$$
If the maximum value of the function is 11, then $a^{2}+b^{2}=$ $\qquad$ . | 50 |
Given a positive integer $m$. Let
$$A_l = (4l+1)(4l+2)...(4(5^m+1)l)$$
for any positive integer $l$. Prove that there exist infinite number of positive integer $l$ which
$$5^{5^ml}\mid A_l\text{ and } 5^{5^ml+1}\nmid A_l$$
and find the minimum value of $l$ satisfying the above condition. | l = \frac{5^{m+1} - 1}{4} |
[ Arithmetic. Mental calculation, etc.]
Gena went to the shooting range with his dad. They agreed that Gena would make 5 shots and for each hit, he would get the right to make 2 more shots. In total, Gena made 17 shots. How many times did he hit the target?
# | 6 |
Given two sets $\{a^2, 0, -1\} = \{a, b, 0\}$, find the value of $a^{2014} + b^{2014}$. | 2 |
3. Given that $x$, $y$, and $z$ are positive numbers, and satisfy:
$$
\left\{\begin{array}{l}
x+y+x y=8, \\
y+z+y z=15, \\
z+x+z x=35 .
\end{array}\right.
$$
Then $x+y+z+x y=$ $\qquad$ | 15 |
Example 3. Compute the integral $\int_{C} e^{\bar{z}} d z$, where $C$ is the line segment $y=-x$, connecting the points $z_{1}=0$ and $z_{2}=\pi-i \pi$. | (e^{\pi}+1)i |
Given that $\cos \alpha = -\frac{4}{5}$ and $\frac{\pi}{2} < \alpha < \pi$, find the value of $\tan \alpha$. | -\frac{3}{4} |
Given $0＜θ＜\frac{π}{2}$, if ${(sinθ+cosθ)}^{2}+\sqrt{3}cos2θ=3$, then $\tan \theta =\left(\ \ \right)$
A: $2\sqrt{2}-\sqrt{7}$
B: $\sqrt{3}-\sqrt{2}$
C: $2-\sqrt{3}$
D: $3-2\sqrt{2}$ | C |
The function $f(x) = x^2 + bx + c$ satisfies $f(2+t) = f(2-t)$ for any real number $t$. Then, the relationship among $f(1)$, $f(2)$, and $f(4)$ is ( )
A: $f(1) < f(2) < f(4)$
B: $f(2) < f(1) < f(4)$
C: $f(4) < f(2) < f(1)$
D: $f(4) < f(1) < f(2)$ | \text{B} |
Let $\alpha \in (0, \pi)$. If $\cos(\pi - \alpha) = \frac{1}{3}$, then $\tan(\alpha + \pi) = \_\_\_\_\_\_$. | -2\sqrt{2} |
Example 3 Find the minimum value of the function $y=\sin x+\frac{2}{\sin x}(0<x<\pi)$.
untranslated part:
```
(0<x<\pi)
```
The untranslated part is kept as it is commonly acceptable to leave mathematical expressions and conditions in their original form. However, if you prefer the entire text to be translated, it ... | 3 |
The maximum distance from point P on the ellipse $7x^2+4y^2=28$ to the line $3x-2y-16=0$ is
A: $\sqrt {13}$
B: $\frac {16}{13}\sqrt {13}$
C: $\frac {24}{13}\sqrt {13}$
D: $\frac {28}{13}\sqrt {13}$ | C |
A school has 1600, 1200, and 800 students in grades 10, 11, and 12, respectively. To understand the dental health status of high school students, stratified sampling is conducted according to the number of students in each grade. If 20 students are sampled from grade 12, the total number of students that need to be sam... | 70 |
In groups A and B, there are three students each, and their scores in a test are as follows: Group A: 88, 89, 90; Group B: 87, 88, 92. If one student is randomly selected from each group, the probability that the absolute difference in their scores does not exceed 3 is ____. | \dfrac{8}{9} |
Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with one focus at $F(-1,0)$, and the product of the distances from point $F$ to the left and right vertices of $C$ is $5$.
$(1)$ Find the standard equation of the ellipse $C$;
$(2)$ Two lines $l_{1}$ and $l_{2}$ are drawn through point $F$ with a product... | (-\frac{6}{11}, 0) |
Given sets $A=\{x\left|\right.-1\leqslant x \lt 2\}$ and $B=\{-1,0,1,2\}$, then $A\cap B=\left(\ \ \right)$
A: $\{-1,0,1,2\}$
B: $\{-1,0,1\}$
C: $\{0,1,2\}$
D: $\{x\left|\right.-1\leqslant x \lt 2\}$ | B |
6. As shown in Figure 2, given that $N$ is a point on the semicircle with diameter $AB$, $NM \perp AB$ at point $M$, semicircles $\odot O_{1}$ and $\odot O_{2}$ have diameters $AM$ and $BM$ respectively, $\odot O_{3}$ and $\odot O_{4}$ are both internally tangent to the semicircle $\odot O$, also tangent to $MN$, and e... | D |
2. Calculate $A=\sqrt{\underbrace{111 \ldots 1}_{2 n}+\underbrace{111 \ldots 1}_{n+1}+\underbrace{666 \ldots 6}_{n}+8}$. | \underbrace{33\ldots33}_{n}6 |
If the graph of the function $y=mx^2-4x+1$ has common points with the x-axis, then the range of $m$ is. | m \leq 4 |
1. Find all the solutions of each of the following diophantine equations
a) $x^{2}+3 y^{2}=4$
b) $x^{2}+5 y^{2}=7$
c) $2 x^{2}+7 y^{2}=30$. | a) x= \pm 2, y=0 ; x= \pm 1, y= \pm 1 \quad b) \text{no solution} \quad c) x= \pm 1, y= \pm 2 |
## Task B-4.2.
Determine $n \in \mathbb{N}$ such that the polynomial $P(x)=\left(2 x^{2}+1\right)^{n+1}-(3 x+15)^{2 n}$ is divisible by the polynomial $Q(x)=x+2$. In this case, determine all the non-real roots of the polynomial $P$. | x_{1,2}=\frac{-3\i\sqrt{119}}{4} |
3.1 The tractor driver plowed three plots of land. The area of the first is $\frac{2}{5}$ of the total area of all three plots, and the area of the second is to the area of the third as $\frac{3}{2}: \frac{4}{3}$. How many hectares were there in all three plots if the third plot was 16 hectares smaller than the first? | 136 |
What is the value of the product $\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right) \ldots\left(1-\frac{1}{225}\right) ?$ | \frac{8}{15} |
2. Given $\theta=\arctan \frac{5}{12}$, then the principal value of the argument of the complex number $z=\frac{\cos 2 \theta+i \sin 2 \theta}{239+i}$ is $\qquad$ | \frac{\pi}{4} |
Given an ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, there is a point $P$ on the ellipse such that the distance to the two foci of the ellipse, $F_{1}$ and $F_{2}$, satisfies $|PF_{1}|+|PF_{2}|=10$, and the eccentricity $e= \dfrac {4}{5}$.
$(1)$ Find the standard equation of the ellipse.
$(... | 3 \sqrt {3} |
7. Xiaoming mistakenly calculated addition as subtraction, getting a result of 86, which is 186 less than the correct answer. The larger of the original addends is $(\quad)$. | 179 |
6. A line $l$ is drawn through point $P(2,1)$, intersecting the two coordinate axes at points $A$ and $B$. When the area $S$ of $\triangle A O B$ takes values in $(0,+\infty)$, the number of lines $l$ that can be drawn is ( ) lines.
(A) 2
(B) 3
(C) 2 or 3
(D) 2 or 3 or 4 | D |
A deck of $ 2n\plus{}1$ cards consists of a joker and, for each number between 1 and $ n$ inclusive, two cards marked with that number. The $ 2n\plus{}1$ cards are placed in a row, with the joker in the middle. For each $ k$ with $ 1 \leq k \leq n,$ the two cards numbered $ k$ have exactly $ k\minus{}1$ cards between t... | n = 3, 4, 7, 8 |
9.1. The numbers $x=999$ and $x=1001$ satisfy the inequality $x^{2}+p x+q<0$. Prove that the discriminant of the quadratic trinomial $x^{2}+p x+q$ is greater than 4. | D>4 |
If the proposition $p: \exists x \gt 0$, $x^{2}-2x+1 \lt 0$, then ￢$p$ is ( ).
A: $\exists x \gt 0$, $x^{2}-2x+1\geqslant 0$
B: $\forall x \gt 0$, $x^{2}-2x+1\geqslant 0$
C: $\exists x\leqslant 0$, $x^{2}-2x+1 \gt 0$
D: $\forall x\leqslant 0$, $x^{2}-2x+1 \gt 0$ | B |
8-4. There are 50 parking spaces on a car park, numbered from 1 to 50. Currently, all parking spaces are empty. Two cars, a black one and a pink one, have arrived at the parking lot. How many ways are there to park these cars such that there is at least one empty parking space between them?
If the black and pink cars ... | 2352 |
Given the function $f(x)=(x^{2}-2x)\sin(x-1)+x+1$ has its maximum value $M$ and minimum value $m$ on the interval $[-1,3]$, then $M+m=\boxed{(\ )}$
A: $4$
B: $2$
C: $1$
D: $0$ | A |
$[\underline{\text { equations in integers }}]$
Solve the equation $\underbrace{\sqrt{n+\sqrt{n+\ldots \sqrt{n}}}}_{1964 \text { times }}=m$ in integers
# | 0 |
1. Let $x$ and $y$ be real numbers, and satisfy
$$
\left\{\begin{array}{l}
(x-1)^{5}+1999(x-1)=2519, \\
(y-1)^{5}+1999(y-1)=-2519 .
\end{array}\right.
$$
Then the value of $x+y$ is $\qquad$ | 2 |
4・76 Let $n \geqslant 4, \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} ; \beta_{1}, \beta_{2}, \cdots, \beta_{n}$ be two sets of real numbers, satisfying
$$
\sum_{j=1}^{n} \alpha_{j}^{2}<1, \sum_{j=1}^{n} \beta_{j}^{2}<1 .
$$
Let
$$
\begin{array}{l}
A^{2}=1-\sum_{j=1}^{n} \alpha_{j}^{2}, B^{2}=1-\sum_{j=1}^{n} \beta_{j}^... | \lambda=0 |
3. Given a triangle $A B C$, where $\angle B A C=60^{\circ}$. Point $S$ is the midpoint of the angle bisector $A D$. It is known that $\angle S B A=30^{\circ}$. Find DC/BS. | 2 |
Given $f_1(x) = (x^2 + 2x + 1)e^x$, $f_2(x) = [f_1(x)]'$, $f_3(x) = [f_2(x)]'$, ..., $f_{n+1}(x) = [f_n(x)]'$, where $n \in \mathbb{N}^*$. Let $f_n(x) = (a_nx^2 + b_nx + c_n)e^x$, find the value of $b_{2015}$.
A: 4034
B: 4032
C: 4030
D: 4028 | C |
841*. A natural number is equal to the cube of the number of its thousands. Find all such numbers. | 32768 |
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