problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
There are 5 different novels in one backpack, and 6 different textbooks of various subjects in another backpack. How many ways are there to take one book from each backpack? | 30 |
Given that $\{a_{n}\}$ is an arithmetic progression, $\{b_{n}\}$ is a geometric progression, and $a_{2}+a_{5}=a_{3}+9=8b_{1}=b_{4}=16$.
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$.
$(2)$ Arrange the terms of $\{a_{n}\}$ and $\{b_{n}\}$ in ascending order to form a new sequence $\{c_{n}\}$. Let t... | 2726 |
Given the quadratic function $y=x^{2}-2mx+5$, when $x \gt -1$, $y$ increases as $x$ increases. The range of real number $m$ is ( ).
A: $m \lt -1$
B: $m\geqslant -1$
C: $m \gt -1$
D: $m\leqslant -1$ | D |
Find the domain of the function $y=\frac{\sqrt{9-x^2}}{\log_2(x+1)}$.
A) $(-1,3)$
B) $(-1,3]$
C) $(-1,0) \cup (0,3)$
D) $(-1,0) \cup (0,3]$ | D |
The degree of each interior angle of a regular hexagon is ____ degrees. | 120^{\circ} |
(2012•Chongqing) For any real number $k$, the position relationship between the line $y=kx+1$ and the circle $x^2+y^2=2$ is always ( )
A: Separate
B: Tangent
C: Intersect but the line does not pass through the center of the circle
D: Intersect and the line passes through the center of the circle | \text{C} |
The sequence $\{a_n\}$ is an arithmetic sequence with a common difference of $d$. The function $f(x) = (x-a_1)(x-a_2)(x-a_3)(x-a_4)$. Then, the value of $f'(a_1) + f'(a_2) + f'(a_3) + f'(a_4) =$ ． | 0 |
In the geometric sequence $\{a_n\}$, the first term is 2, and the second term is 4. What is the third term?
A: 3
B: 5
C: 6
D: 8 | \text{D} |
Given a random variable $\xi$ follows a normal distribution $N(0,\sigma^{2})$, if $P(\xi > 3)=0.023$, then $P(-3\leqslant \xi\leqslant 3)=$ ()
A: $0.954$
B: $0.023$
C: $0.977$
D: $0.046$ | A |
22. One side of a rhombus is equal to the leg of an isosceles right triangle. If one angle of the rhombus is $60^{\circ}$, then the ratio of the area of the rhombus to the area of the isosceles right triangle is ( ).
(A) $\sqrt{3}: 2$
(B) $\sqrt{3}: 1$
(C) $1: \sqrt{3}$
(D) $\sqrt{3}: 4$ | B |
4-1. Katya attached a square with a perimeter of 40 cm to a square with a perimeter of 100 cm as shown in the figure. What is the perimeter of the resulting figure in centimeters?
 | 120 |
1. As shown in the figure, let quadrilateral $ABCD$ be inscribed in a circle, $M$ be the intersection of line $AB$ and $CD$, and $N$ be the intersection of line $BC$ and $AD$. If $BM=DN$, prove that $CM=CN$. | CM=CN |
B1. Find all real numbers $x$ for which $\left(x^{2}-7 x+11\right)^{x^{2}-13 x+42}=1$. | 2,3,4,5,6,7 |
Problem 1. Find all pairs $(x, y)$ of integers that satisfy the equation
$$
x^{2} y+y^{2}=x^{3} \text {. }
$$
(Daniel Paleka) | (0,0),(-4,-8) |
## Aufgabe 2
$$
\begin{array}{rr}
7-5-2 & 9+0-4 \\
10-3+3 & 2+2+2
\end{array}
$$
| 0,5,10,6 |
If $x\geqslant 0$, $y\geqslant 0$, and $x+2y=1$, then the minimum value of $2x+3y^{2}$ is ____. | 0.75 |
Two circles \\(C_{1}\\): \\((x+2)^{2}+(y-m)^{2}=9\\) and \\(C_{2}\\): \\((x-m)^{2}+(y+1)^{2}=4\\) are externally tangent. The value of \\(m\\) is ( ).
A: \\(2\\)
B: \\(-5\\)
C: \\(2\\) or \\(-5\\)
D: Uncertain | C |
What is the value of the product $\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right) \cdots\left(1-\frac{1}{225}\right)$ ?
(a) $\frac{10}{125}$
(b) $\frac{5}{9}$
(c) $\frac{3}{5}$
(d) $\frac{8}{15}$
(e) $\frac{1}{120}$ | \frac{8}{15} |
Given $2a+4b=3$, find the value of $4a+8b-2$. | 4 |
Example 1 Suppose $a^{2}+2 a-1=0, b^{4}-2 b^{2}-1=$ 0 , and $1-a b^{2} \neq 0$. Then the value of $\left(\frac{a b^{2}+b^{2}+1}{a}\right)^{1990}$ is (1990, Hefei Junior High School Mathematics Competition) | 1 |
At the tournament, teams Akouska, Bovenska, Colska, and Demecka met. Each team played against each other exactly once. The winning team received three points, the losing team received no points, and in the event of a draw, each of the drawing teams received one point. After all six matches were played, Akouska had 7 po... | 2 |
Rice grains were placed on the cells of a chessboard. The number of grains on any two adjacent cells differed by exactly
1. At the same time, there were three grains on one of the cells of the board, and 17 grains on another. A rooster pecked all the grains from one of the main diagonals of the board, and a hen - from... | 80 |
$36 \cdot 29$ There are 50 girls, whose skin is either white or light black, and their eyes are either blue or brown. If 14 have blue eyes and white skin, 31 have light black skin, and 18 have brown eyes, then the number of girls with brown eyes and light black skin is
(A) 5 .
(B) 7 .
(C) 9 .
(D) 11 .
(E) 13 .
(22nd Am... | 13 |
Problem 2. Given 10 equal segments in a plane. All intersection points are colored. Each colored point divides each of the given segments it lies on in the ratio $3: 4$. What is the maximum number of colored points? | 10 |
6. 23 Given the function $y=\frac{\sqrt{4-x^{2}}}{\lg (|x|+x)}$, then its domain is
(A) $[-2,0) \cup(0,2]$.
(B) $\left(0, \frac{1}{2}\right) \cup\left(\frac{1}{2}, 2\right]$.
(C) $\left[-2, \frac{1}{2}\right) \cup\left(\frac{1}{2}, 2\right]$.
(D) $(0,2]$.
(China Beijing High School Mathematics Competition, 1990) | (0,\frac{1}{2})\cup(\frac{1}{2},2] |
6. Let $a$, $b$, $c$ and $m_{a}$, $m_{b}$, $m_{c}$ represent the lengths of the three sides and the lengths of the medians to these sides of $\triangle ABC$, respectively. Given the following conclusions:
(1) $m_{a}+m_{b}+m_{c} \leq \frac{3}{4}(a+b+c)$;
(4) $m_{a}^{2}+m_{b}^{2}+m_{c}^{2}=\frac{3}{4}\left(a^{2}+b^{2}+c^... | C |
The function $f(x)=\sqrt{3}\sin 2x+2\cos^2 x$, if $f(x_{1})\cdot f(x_{2})=-3$, then the minimum value of $|x_{1}+x_{2}|$ is ( )
A: $\frac{π}{6}$
B: $\frac{π}{4}$
C: $\frac{π}{3}$
D: $\frac{{2π}}{3}$ | A |
1. Given $\triangle A B C$ is an acute triangle,
$$
\begin{array}{l}
f(x)=-x^{2}+2 x+m, \\
p=\sin A+\sin B+\sin C, \\
q=\cos A+\cos B+\cos C .
\end{array}
$$
Then ( ).
(A) $f(p)>f(q)$
(B) $f(p)=f(q)$
(C) $f(p)<f(q)$
(D) Any of the above three cases is possible | C |
Given the function $f\left(x\right)=-x^{3}+3x+a$, $a\in R$, if there exist three distinct real numbers $m$, $n$, $p$ such that $f\left(m\right)=f\left(n\right)=f\left(p\right)=2022$, then the range of real number $a$ is ______. | (2020, 2024) |
Question 175, Given point $E(m, n)$ is a fixed point inside the parabola $y^{2}=2 p x(p>0)$, two lines with slopes $k_{1}$ and $k_{2}$ are drawn through $E$, intersecting the parabola at points $A, B$ and $C, D$ respectively, and $M, N$ are the midpoints of segments $A B$ and $C D$ respectively.
(1) When $\mathrm{n}=0$... | p^2 |
Among the following expressions, which one is calculated correctly?
A: $2^{-3}=6$
B: $a^{3}b\cdot (a^{-1}b)^{-2}=\frac{a}{b}$
C: $(-\frac{1}{2})^{-1}=2$
D: $(\pi -3.14)^{0}=1$ | \text{D} |
Bogganov I.I.
Given an infinite supply of white, blue, and red cubes. Any \$N\$ of them are arranged in a circle. A robot, starting at any point on the circle, moves clockwise and, until only one cube remains, repeatedly performs the following operation: it destroys the two nearest cubes in front of it and places a ne... | 2^k |
Given the proposition $p$: "For all $x$ in $[1, 2]$, $x^2 - a \geq 0$", and the proposition $q$: "There exists an $x$ in $\mathbb{R}$ such that $x^2 + 2ax + 2 - a = 0$". If the proposition "$\neg p \land q$" is true, then the range of values for the real number $a$ is ____. | a > 1 |
II. (This question is worth 25 points) Arrange all positive integers that are coprime with 105 in ascending order. Try to find the 1000th term of this sequence. | 2186 |
Given the hyperbola $x^2-y^2=4$ and the line $l:y=k(x-1)$, discuss the range of the real number $k$ under the following conditions:
$(1)$ The line $L$ and the hyperbola have two common points.
$(2)$ The line $L$ and the hyperbola have exactly one common point.
$(3)$ The line $L$ and the hyperbola have no common poin... | \left(-\infty, -\frac{2\sqrt{3}}{3}\right) \cup \left(\frac{2\sqrt{3}}{3}, +\infty\right) |
For the hyperbola $$\frac {x^{2}}{m}- \frac {y^{2}}{6}=1$$, if one of its asymptotes is the line $y=x$, find the value of the real number $m$. | 6 |
Problem 8-3. $\quad$ In a right triangle $A B C$, angle $B$ is a right angle. On the leg $A B$, a point $M$ is chosen such that $A M=B C$, and on the leg $B C$, a point $N$ is chosen such that $C N=M B$. Find the acute angle between the lines $A N$ and $C M$.
Answer: $45^{\circ}$.
 and B($x_2$, $y_2$) on the graph. When $x_1 0$
B: $m > \frac{1}{2}$
C: $m < 0$
D: $m < \frac{1}{2}$ | D |
15. The Grand Old Duke of York had 10000 men. He lost $10 \%$ of them on the way to the top of the hill, and he lost $15 \%$ of the rest as he marched them back down the hill. What percentage of the 10000 men were still there when they reached the bottom of the hill?
A $76 \frac{1}{2} \%$
B $75 \%$
C $73 \frac{1}{2} \%... | 76\frac{1}{2} |
5. Given that the lengths of the three sides of a triangle are all integers, the difference between the lengths of two of its sides is 5. If the perimeter of this triangle is odd, then the minimum value of the third side is ( ).
(A) 8
(B) 7
(C) 6
(D) 4 | C |
Find at least one non-zero polynomial $P(x, y, z)$ such that $P(a, b, c)=0$ for every three real numbers that satisfy $\sqrt[3]{a}+\sqrt[3]{b}=$ $\sqrt[3]{c}$
Remark. Polynomial in three variables refers to any expression built from $x, y, z$ and numerlas using only addition, subtraction, and multiplication. Parenthes... | 27xyz-(z-x-y)^{3} |
For which natural numbers $a$ and $b$ do the conditions $90<a+b<100$ and $0.9<\frac{a}{b}<0.91$ hold? | (46,51)(47,52) |
(1 + 1/x^2)(1 + x)^6, the coefficient of x^2 in the expansion is:
A: 15
B: 20
C: 30
D: 35 | C |
5. Let the increasing function $f(x)=k x-2$. It is stipulated: given an initial value $a$ of a real number $x$, assign $x_{1}=f(a)$, if $x_{1} \leqslant$ 2012, then assign $x_{2}=f\left(x_{1}\right)$, and so on; if $x_{n-1} \leqslant 2012$, then assign $x_{n}=f\left(x_{n-1}\right)$, otherwise stop the assignment; if $x... | \left(\frac{1007}{1006},+\infty\right) |
\section*{Problem 3 - 101233}
Let \(f\) be the function defined for all real numbers \(x\) by \(f(x)=\frac{1-x^{2}}{x^{6}+4}\).
It is to be decided whether among all function values \(f(x)\) there is a greatest and a smallest value. These values are to be determined if they exist. | -\frac{1}{12} |
Example 4.23. Find the general solution of the equation
$$
y^{\prime \prime}-4 y^{\prime}+5 y=2 \cos x+6 \sin x
$$ | e^{2x}(C_{1}\cosx+C_{2}\sinx)+\cosx+\frac{1}{2}\sinx |
4. Calculate $\sqrt{6+\sqrt{27}}-\sqrt{6-\sqrt{27}}$. | \sqrt{6} |
In the plane rectangular coordinate system $xOy$, the point $A(3,a)$ is symmetric with respect to the $x$-axis to point $B(b,4)$. Find the value of $a+b$. | -1 |
Squirrels prepare supplies of hazelnuts, beech nuts, and spruce cones for the winter. Zrzečka, Pizizubka, and Křivoouško have the same number of supplies. Zrzečka has twice as many hazelnuts as Pizizubka. Křivoouško has 20 more hazelnuts than Pizizubka. All three have the same number of beech nuts, which is 48 pieces. ... | \begin{pmatrix}\text{Zrzečka:}&48 |
G1.2 Let $p$ and $q$ be prime numbers that are the two distinct roots of the equation $x^{2}-13 x+R=0$, where $R$ is a real number. If $b=p^{2}+q^{2}$, find the value of $b$. | 125 |
4. For which $p$ do the quadratic equations
$$
9 x^{2}-3(p+6) x+6 p+5=0 \text{ and } 6 x^{2}-3(p+4) x+6 p+14=0
$$
have a common root? Find this root. | p=-\frac{32}{9},-1orp=\frac{32}{3},3 |
Given $m>0$ and $|x+1|+|2x-1|\geq m$ always holds, $a, b, c \in \mathbb{R}$ satisfy $a^2+2b^2+3c^2=m$. Find the minimum value of $a+2b+3c$. | -3 |
18.100 Construct a trapezoid with the segments $a=5, b=10, c=15, d=20$. The number of such trapezoids (not congruent)
(A) can be at least three.
(B) can be exactly two.
(C) can be only one.
(D) cannot be made at all.
（1st "Five Sheep Cup" Junior High School Mathematics Competition, 1989） | C |
If the solutions of the equations $2x+1=3$ and $2-\frac{a-x}{3}=1$ with respect to $x$ are the same, then the value of $a$ is ____. | 4 |
一、(50 points) As shown in Figure 3, in the isosceles right triangle $\triangle A B C$, $D$ is the midpoint of the hypotenuse $A B$. A line $C C^{\prime} / / A B$ is drawn through the right-angle vertex $C$. Points $E$, $F$, and $M$ are on sides $A C$, $B C$, and $E F$ respectively, such that $\frac{C E}{A E}=$ $\frac{B... | M D = M H |
Problem 6. Calculate $2 \operatorname{arctg} 3+\arcsin \frac{3}{5}$. | \pi |
Given that positive numbers $x$ and $y$ satisfy $x^{2}y\left(4x+3y\right)=3$, then the minimum value of $2x+3y$ is ______. | 2\sqrt{3} |
If $x=3$, which of the following expressions is an even number?
(A) $9 x$
(B) $x^{3}$
(C) $2\left(x^{2}+9\right)$
(D) $2 x^{2}+9$
(E) $3 x^{2}$ | C |
## Problem Statement
Find the angle between the planes:
$$
\begin{aligned}
& 2 x+2 y+z+9=0 \\
& x-y+3 z-1=0
\end{aligned}
$$ | \arccos\frac{1}{\sqrt{11}}\approx7227^{\}6^{\\} |
$\mathrm{P}$ is a point inside the equilateral triangle $\mathrm{ABC}$ such that $\mathrm{PA}=5, \mathrm{~PB}=7, \mathrm{PC}=8$. Find $\mathrm{AB}$. | \sqrt{129} |
11. Given the quadratic function $f(x)=x^{2}+a x+b(a, b \in \mathbf{R})$ has two distinct zeros, if $f\left(x^{2}+2 x-1\right)$ $=0$ has four distinct roots $x_{1}<x_{2}<x_{3}<x_{4}$, and $x_{1}, x_{2}, x_{3}, x_{4}$ form an arithmetic sequence, find the range of $a-b$. | (-\infty,\frac{25}{9}] |
Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that
$a^4 - 2019a = b^4 - 2019b = c$.
Prove that $- \sqrt{c} < ab < 0$. | -\sqrt{c} < ab < 0 |
Calculate lg $$\frac {5}{2}$$+21g2-（ $$\frac {1}{2}$$）-1+$${8}^{ \frac{2}{3}}$$\=\_\_\_\_\_\_． | 3 |
8.4. Four boys and three girls went to the forest to pick mushrooms. Each found several mushrooms, in total they collected 70. No two girls collected the same amount, and any three boys together brought no fewer than 43 mushrooms. The number of mushrooms collected by any two children differed by no more than 5 times. M... | 5 |
If $n$ is a positive integer such that $2n$ has $28$ positive divisors and $3n$ has $30$ positive divisors, then how many positive divisors does $6n$ have?
$\text{(A)}\ 32\qquad\text{(B)}\ 34\qquad\text{(C)}\ 35\qquad\text{(D)}\ 36\qquad\text{(E)}\ 38$ | 35 |
Given the function $f(x) = |e^{x} + \frac{a}{e^{x}}|$, $(a \in \mathbb{R})$ is monotonically increasing in the interval $[0, 1]$, the range of the real number $a$ is _______ . | a \in [-1, 1] |
Xavier, Yolanda, and Zixuan have a total of $\$ 50$. The ratio of the amount Xavier has to the total amount Yolanda and Zixuan have is $3: 2$. Yolanda has $\$ 4$ more than Zixuan. How much does Zixuan have?
(A) $\$ 16$
(B) $\$ 8$
(C) $\$ 14$
(D) $\$ 13$
(E) $\$ 30$ | 8 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. Given $a-b=2$, $c=4$, and $\sin A=2\sin B$.
$(1)$ Find the values of $a$, $b$, and $\cos B$;
$(2)$ Find the value of $\sin \left(2B-\frac{\pi}{6}\right)$. | \frac{21\sqrt{5}-17}{64} |
Given the universal set $U=\mathbb{R}$, set $A=\{x\mid y=\frac{1}{\sqrt{a-x}}\}$, and set $B=\{x\mid x^2-x-6=0\}$.
1. If $a=-1$, find $A\cap B$.
2. If $(\neg_{U}A)\cap B=\varnothing$, find the range of values for the real number $a$. | a\in(3,+\infty) |
3.297. $\sin ^{2}(2 \alpha-\beta)-\sin ^{2} 2 \alpha-\sin ^{2} \beta$. | -2\sin2\alpha\sin\beta\cos(2\alpha-\beta) |
There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that
($1$) for each $k,$ the mathematician who was seated in seat $k$ befor... | 332 |
Let the universal set be $\mathbb{R}$, $A=\{x | a \leq x < 4\}$, $B=\{x | 3x - 7 \geq 8 - 2x\}$
(1) If $a=2$, find $A \cap B$, $(\complement_{\mathbb{R}}A) \cup (\complement_{\mathbb{R}}B)$;
(2) If $A \cap B = A$, find the range of values for $a$. | a \geq 3 |
Given sets $A=\{x|x^{2}-3x-10 \lt 0\}$ and $B=\{x|2^{x} \lt 2\}$, what is $A\cap B$?
A: $(-2,1)$
B: $(-5,1)$
C: $\varnothing$
D: $\{0\}$ | \text{A} |
Given that one focus of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ is $F(1,0)$, and the two trisection points $M$, $N$ of the minor axis and $F$ form an equilateral triangle, find the equation of the ellipse. | \frac{x^{2}}{4} + \frac{y^{2}}{3} = 1 |
Given a hyperbola with its center at the origin, and its foci $F_1$ and $F_2$ are on the coordinate axes, the eccentricity is $\sqrt{2}$, and it passes through the point $(4, -\sqrt{10})$. Point $M(3, m)$ is on the hyperbola.
(1) Find the equation of the hyperbola;
(2) Find the area of $\triangle F_1MF_2$. | 6 |
Let $z=\frac{1}{1+i} +i$ (where $i$ is the imaginary unit), then $|z|=$
A: $\frac{1}{2}$
B: $\frac{\sqrt{2}}{2}$
C: $\frac{\sqrt{3}}{2}$
D: $2$ | B |
The function $f(x)=\begin{cases}10^{1-x}+1, & x\leqslant 0 \\ \lg (x+2), & x > 0\end{cases}$ If $f(a)=1$, then $f(8-a)=$ ( )
A: $4$
B: $6$
C: $8$
D: $11$ | \text{D} |
Given lines $m$, $l$ and planes $\alpha$, $\beta$, $\gamma$ satisfying $\beta \cap \gamma = l$, $l \parallel \alpha$, $m \subset \alpha$, $m \perp \gamma$, which of the following statements is necessarily true?
A: $\alpha \perp \gamma$ and $l \perp m$
B: $\alpha \perp \gamma$ and $m \parallel \beta$
C: $m \parallel \be... | \text{A} |
Convert 1920° to radians.
A: $$\frac {16}{3}$$
B: $$\frac {32}{3}$$
C: $$\frac {16}{3}$$π
D: $$\frac {32}{3}$$π | \text{D} |
Four. (20 points) Given the ellipse $C: \frac{x^{2}}{4}+y^{2}=1$ and a fixed point $P(t, 0)(t>0)$, a line $l$ with a slope of $\frac{1}{2}$ passes through point $P$ and intersects the ellipse $C$ at two distinct points $A$ and $B$. For any point $M$ on the ellipse, there exists $\theta \in[0,2 \pi]$, such that $O M=\co... | 2 |
Selected Optional 4-4: Coordinate Systems and Parametric Equations
In the polar coordinate system, the ray $l$: $θ = \frac{π}{6}$ intersects with the circle $C$: $ρ = 2$ at point $A$. The equation of the ellipse $Γ$ is: $ρ^2 = \frac{3}{1 + 2\sin^2 θ}$. Establish a rectangular coordinate system $xOy$ with the pole as t... | \sqrt{13} + 5 |
A circle and a square overlap such that the overlapping area is $50\%$ of the area of the circle, and is $25\%$ of the area of the square, as shown in the figure. Find the ratio of the area of the square outside the circle to the area of the whole figure.
[img]https://cdn.artofproblemsolving.com/attachments/e/2/c209a95... | \frac{3}{5} |
Example 5 Given the set $M=\left\{x \mid x^{2}-2 x+m=0, x \in \mathbf{R}\right\}$ is non-empty, then the sum of all elements in set $M$ is $\qquad$ . | 2 |
Let $n$ be a positive integer not less than 2. Prove that: $$\frac {4}{7} < 1- \frac {1}{2} + \frac {1}{3} - \frac {1}{4} + \ldots + \frac {1}{2n-1} - \frac {1}{2n} < \frac { \sqrt {2}}{2}.$$ | \frac {4}{7} < 1- \frac {1}{2} + \frac {1}{3} - \frac {1}{4} + \ldots + \frac {1}{2n-1} - \frac {1}{2n} < \frac { \sqrt {2}}{2} |
Given vectors $\overrightarrow {a}$=(1,−3) and $\overrightarrow {b}$=(m,6), find the value of m such that $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$. | m = -2 |
3. What is the greatest value that the sum $S_{n}$ of the first $n$ terms of an arithmetic progression can take, given that the sum $S_{3}=327$ and the sum $S_{57}=57$? | 1653 |
(2) Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=2,(n+1) a_{n+1}=a_{n}+n\left(n \in \mathbf{N}^{*}\right)$, then $a_{n}=$ $\qquad$. | a_{n}=\frac{1}{n!}+1 |
5. Lei Lei went to buy instant noodles, handing over a 100 yuan note to the boss. After giving her the change, the boss said to her: “You only gave me 1 note, but I gave you 16 notes, plus instant noodles worth 5 yuan, you really got a great deal!” If the change the boss gave Lei Lei was either 10 yuan or 5 yuan notes,... | 3 |
"$x < 0$" is a condition for "$\ln(x+1) < 0$" to be \\((\\) \\()\\).
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Both sufficient and necessary
D: Neither sufficient nor necessary | B |
Example 1 Let $O(0,0), A(1,0), B(0,1)$, and $P$ be a moving point on the line segment $AB$, $\overrightarrow{A P}=\lambda \overrightarrow{A B}$. If
$$
\overrightarrow{O P} \cdot \overrightarrow{A B} \geqslant \overrightarrow{P A} \cdot \overrightarrow{P B} \text {, }
$$
then the range of the real number $\lambda$ is $... | \left[1-\frac{\sqrt{2}}{2}, 1\right] |
4. Find all integer pairs $(x, y)$ such that
$$
1+2^{x}+2^{2 x+1}=y^{2} .
$$ | (0,2),(0,-2),(4,23),(4,-23) |
What is the maximum number of edges we can select from a regular dodecahedron so that no two of them have a common point? | 10 |
11.126. The side of the base of a regular hexagonal pyramid is equal to $a$. All diagonal sections of it are equal. Find the volume and lateral surface area of the pyramid. | \frac{3^{3}}{4};\frac{3^{2}\sqrt{6}}{2} |
The constant term in the expansion of the binomial $(x+ \frac {1}{x^{2}})^{6}$ is ( )
A: 64
B: 30
C: 15
D: 1 | C |
6. A person shoots once, the probability of hitting the target is 0.9. He shoots 4 times and the probability of hitting exactly 2 times is | 0.0486 |
Given $$f(x)= \begin{cases} -\cos\pi x & (x>0) \\ f(x+1)+1 & (x\leq 0)\end{cases}$$, then the value of $$f\left( \frac {4}{3}\right)+f\left(- \frac {4}{3}\right)$$ is equal to \_\_\_\_\_\_\_\_\_\_\_\_. | 3 |
4.1. All natural numbers from 1 to 2017 inclusive were written in a row. How many times was the digit 7 written? | 602 |
[ Properties of bisectors, concurrency ] [ The ratio in which the bisector divides the side $]$ [ Properties of bisectors, concurrency ]
In triangle $A B C$, points $M$ and $N$ are marked on sides $A B$ and $B C$ respectively, such that $B M=B N$. A line is drawn through point $M$ perpendicular to $B C$, and a line th... | 5 |
Problem 7.2. (15 points) Given an isosceles triangle $A B C(A B=B C)$. On the sides $A B, B C, A C$ points $K, L, M$ are marked respectively such that $\angle A K M=90^{\circ}, \angle B L K=90^{\circ}$ and $K M=K L$. What is the measure of angle $C M L ?$ | 90 |
13. Given $x=2 \sqrt{2}+1$. Then the fraction
$$
\frac{x^{2}-2 x-9}{x^{3}-11 x-15}=
$$
$\qquad$ | 2 |
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