problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
1. (17 points) Masha's tablet, which she needed for a presentation at school, was completely drained. Using additional equipment, the tablet can be fully charged in 2 hours and 40 minutes, without it in 8 hours. Masha first put the discharged tablet on regular charging, and when she found the equipment, she switched to... | 288 |
16th Australian 1995 Problem B4 Find all real-valued functions f on the positive reals such that f(1) = 1/2 and f(xy) = f(x) f(3/y) + f(y) f(3/x). | f(x)=\frac{1}{2}forallx |
Given $y=f(x)$ is an odd function, and $g(x)=f(x)+2$. If $g(1)=1$, then $g(-1)=$? | 3 |
12. Let $S$ be a set of points in the Cartesian coordinate system, satisfying that two of the quantities $3$, $x+2$, $y-4$ are equal and the third quantity is not greater than the other two. Which of the following statements about the point set $S$ is correct? ( ).
(A) a single point
(B) two intersecting lines
(C) thre... | E |
1. (5 points) Wang Yun calculated $325-\square \times 5$ by performing the subtraction first, resulting in 1500. What should the correct result be? $\qquad$ . | 200 |
3. Give all possible triples $(a, b, c)$ of positive integers with the following properties:
- $\operatorname{gcd}(a, b)=\operatorname{gcd}(a, c)=\operatorname{gcd}(b, c)=1$;
- $a$ is a divisor of $a+b+c$;
- $b$ is a divisor of $a+b+c$;
- $c$ is a divisor of $a+b+c$.
(For given numbers $x$ and $y$, $\operatorname{gcd... | (1,1,1),(1,1,2),(1,2,1),(2,1,1),(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1) |
Find all $ x,y,z\in\mathbb{N}_{0}$ such that $ 7^{x} \plus{} 1 \equal{} 3^{y} \plus{} 5^{z}$.
[i]Alternative formulation:[/i] Solve the equation $ 1\plus{}7^{x}\equal{}3^{y}\plus{}5^{z}$ in nonnegative integers $ x$, $ y$, $ z$. | (0, 0, 0) |
A triangle $A B C$ has side lengths $A B=50 \mathrm{~cm}, B C=20 \mathrm{~cm}$ and $A C=40 \mathrm{~cm}$. Let $M$ and $N$ be points on side $A B$ such that $C M$ is the angle bisector of $\angle A C B$ and $C N$ is the altitude to side $A B$. What is the measure, in centimeters, of $M N$? | \frac{11}{3} |
In the quadrilateral $A B C D$, we have: $A B=5, B C=17, C D=5, D A=9$, and the measure of the segment $D B$ is an integer. Determine $D B$.
 | 13 |
8. There are six measuring cups $A, B, C, D, E, F$, with volumes of $16 \mathrm{~mL}, 18 \mathrm{~mL}, 22 \mathrm{~mL}, 23 \mathrm{~mL}, 24 \mathrm{~mL}$, and $34 \mathrm{~mL}$. Some of the cups are filled with alcohol, some are filled with distilled water, and one cup is empty. The volume of alcohol is twice the volum... | D |
A total of 1000 students participating in a math competition are numbered as follows: 0001, 0002, 0003, …, 1000. A sample of size 50 is to be drawn for a systemic sampling. The sample will be divided into 50 parts. If the first part includes numbers 0001, 0002, 0003, …, 0020, and the randomly selected number from the f... | 0195 |
11. When seven dice are rolled simultaneously, the probability that the sum of the numbers on the seven faces is 10 is equal to the probability that the sum of the numbers on the seven faces is $a(a \neq 10)$. Then, $a=$ $\qquad$ | 39 |
8.2. Find the sum of all three-digit natural numbers that do not contain the digit 0 or the digit 5. | 284160 |
Let $\overrightarrow{a}$ represent moving 10km east, and $\overrightarrow{b}$ represent moving 10$\sqrt{3}$km north, then $\overrightarrow{a} - \overrightarrow{b}$ represents ( )
A: Moving 20km in a direction 30° south of west
B: Moving 20km in a direction 30° north of west
C: Moving 20km in a direction 30° south of e... | \text{C} |
A class has 50 students, and their scores in a math test $\xi$ follow a normal distribution $N(100, 10^2)$. It is known that $P(90 \leq \xi \leq 100) = 0.3$. Estimate the number of students scoring above 110. | 10 |
If set $P=\{x|3x-x^2\leq0\}$, and $Q=\{x||x|\leq2\}$, then $P\cap Q=$ （ ）
A: $[-2，2]$
B: $[-2，0]$
C: $[2，3]$
D: $[-2，3]$ | B |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{\sqrt{2}} \frac{x^{4} \cdot d x}{\left(4-x^{2}\right)^{3 / 2}}
$$ | 5-\frac{3\pi}{2} |
In the tangential trapezoid $ABCD$, the inscribed circle touches the base $AB$ at point $T$ and the parallel base $CD$ at point $U$. Let $M$ be the intersection of the non-parallel sides $AD$ and $BC$, and let $V$ be the intersection of the side $AB$ and the line $MU$. Show that $AT=VB$.
Let's proceed with the proof s... | AT=VB |
If the function $f(x)=m\sin \left(x+ \dfrac {\pi}{4}\right)- \sqrt {2}\sin x$ has both a maximum and a minimum value in the open interval $\left(0, \dfrac {7\pi}{6}\right)$, then the range of the positive real number $m$ is \_\_\_\_\_\_. | 2 < m < 3+ \sqrt {3} |
Determine the maximum possible real value of the number $ k$, such that
\[ (a \plus{} b \plus{} c)\left (\frac {1}{a \plus{} b} \plus{} \frac {1}{c \plus{} b} \plus{} \frac {1}{a \plus{} c} \minus{} k \right )\ge k\]
for all real numbers $ a,b,c\ge 0$ with $ a \plus{} b \plus{} c \equal{} ab \plus{} bc \plus{} ca$. | 1 |
Given the inequality $|2x-a| \leq 3$ has the solution set $[-1, 2]$.
(Ⅰ) Find the value of $a$;
(Ⅱ) If $|x-m| < a$, prove that $|x| < |m| + 1$. | a=1 |
Let the quadratic function be $y=(x+1)(ax+2a+2)$ ($a$ is a constant, $a\neq 0$).$(1)$ If $a=1$, find the coordinates of the vertex of the graph of this function.$(2)$ If the graph of this quadratic function passes through one of the points $\left(-1,1\right)$, $\left(-2,3\right)$, or $\left(0,-2\right)$, find the expre... | a < -\frac{2}{5} |
Given the function $f(x)=\frac{1}{2}\sin 2x-\cos^2(x+\frac{\pi}{4})$.
(1) Find the interval(s) where $f(x)$ is monotonically increasing for $x\in(0,\pi)$.
(2) In an acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $f(\frac{B}{2})=0$ and $b=1$, find t... | \frac{2+\sqrt{3}}{4} |
3. (16 points) In a garden plot, it was decided to create a rectangular flower bed. Due to a lack of space, the length of the flower bed was reduced by $10 \%$, and the width was reduced by $20 \%$. As a result, the perimeter of the flower bed decreased by $12 \%$. However, this was not enough, so it was decided to red... | 18 |
4. In a triangular pyramid, two of the three faces are isosceles right triangles, and the other is an equilateral triangle with a side length of 1. Then, the number of such triangular pyramids is $\qquad$. | 3 |
In a triangle, there are at least acute angles among the three interior angles, and at most acute angles among the three exterior angles. | 2, 1 |
At a school Science Fair, it was observed that half of the elementary school students and a quarter of the high school students present at this event bought one sticker each.
| SCIENCE FAIR | |
| :---: | :---: |
| Price of Stickers (unit) | |
| $\mathrm{R} \$ 0.30$ | elementary school students |
| $\mathrm{R} \$ 0.5... | 120,160 |
The sum of the first $n$ terms of an arithmetic sequence $\left\{a_n\right\}$ is denoted as $S_n$. Given that $S_{k-2}=-4$, $S_k=0$, and $S_{k+2}=8$ $(k > 2,k\in\mathbb{N}^*)$, find the value of $k$. | 6 |
Find all strictly increasing sequences of positive integers $a_1, a_2, \ldots$ with $a_1=1$, satisfying $$3(a_1+a_2+\ldots+a_n)=a_{n+1}+\ldots+a_{2n}$$ for all positive integers $n$. | a_n = 2n - 1 |
Example 2 Given that $a$ is a root of the equation $x^{2}+x-\frac{1}{4}=0$. Then the value of $\frac{a^{3}-1}{a^{5}+a^{4}-a^{3}-a^{2}}$ is $\qquad$ .
(1995, National Junior High School Mathematics League) | 20 |
\section*{Problem 4 - 181024}
For a triangle \(A B C\) with \(\angle C A B=\alpha=120^{\circ}\) and \(\angle B C A=\gamma=30^{\circ}\), the length \(r\) of the circumradius is known.
Calculate the perimeter and the area of this triangle! | (2+\sqrt{3})r\frac{r^2}{4}\sqrt |
Problem 2. Determine the perimeter of the scalene triangle $\mathrm{ABC}$ knowing that
$$
p^{2} \cdot \overrightarrow{G I}=(4 p-b-c) \cdot \overrightarrow{A I}+(4 p-c-a) \cdot \overrightarrow{B I}+(4 p-a-b) \cdot \overrightarrow{C I}
$$
where $\mathrm{G}$ is the centroid of $\triangle \mathrm{ABC}$, I is the incenter... | 12 |
A rectangular hyperbola with coordinate axes as its axes of symmetry passes through the point $(2, \sqrt{2})$. The equation of this hyperbola is __________. | x^2 - y^2 = 2 |
26. In the game "Sanguosha", there is 1 Lord, 3 Loyalists, 5 Rebels, and 1 Traitor, a total of 10 people participating in the game. Among them, the Lord and the Loyalists are on the same team, the Rebels are on the same team, and the Traitor is on their own team. Now, two brothers are participating in the same game of ... | \frac{16}{45} |
Example 2 In the sequence $\left\{a_{n}\right\}$,
$$
a_{1}=2, a_{n+1}=4 a_{n}-3 n+1\left(n \in \mathbf{N}_{+}\right) \text {. }
$$
Find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=4^{n-1}+n |
Seven dwarfs stood around the perimeter of their garden, one in each corner, and stretched a rope around the entire garden. Snow White came out from Smudla and walked along the rope. First, she walked four meters to the east, where she met Prof. From there, she continued two meters to the north until she reached Rejpal... | 22 |
During a secret meeting, 20 interns elect their favorite animator. Each votes for two animators. It is known that if you take any two interns, there is always at least one animator for whom they have both voted. What is the minimum number of votes received by the animator who wins the election? | 14 |
The following statements are given:
1. If a line in one plane is parallel to a line in another plane, then the two planes are parallel.
2. If two lines in one plane are each parallel to another plane, then the two planes are parallel.
3. Two different planes that are parallel to the same plane are parallel to each othe... | 3, 4 |
Given $\left(1+i\right)z=3-i$, where $i$ is the imaginary unit, then $|z|=\left(\ \ \right)$
A: $5$
B: $\sqrt{5}$
C: $2$
D: $\sqrt{2}$ | B |
A company has 160 employees, including 10 senior managers, 30 middle managers, and 120 staff members. If 32 people are to be selected for a medical examination using stratified sampling, then the number of middle managers to be selected should be ______. | 6 |
18. As shown in Figure 2, in a $5 \times 5$ grid, the numbers in each row and each column form an arithmetic sequence. Then the value of $x$ is ( ).
(A) 21
(B) 31
(C) 36
(D) 40
(E) 42 | 31 |
Given two fixed points $F_1(5,0)$, $F_2(-5,0)$, and a moving point $M$ satisfies $|MF_1|+|MF_2|=10$, then the trajectory of point $M$ is ( )
A: Ellipse
B: Straight line
C: Line segment
D: Ray | \text{C} |
Example. Find the flux of the vector field
$$
\vec{a}=x \vec{i}+y \vec{j}+z \vec{k}
$$
through the part of the surface
$$
x^{2}+y^{2}=1
$$
cut by the planes $z=0$ and $z=2$. (The normal is outward to the closed surface formed by these surfaces). | 4\pi |
8. In $\triangle A B C$, $D$ is the midpoint of side $B C$. If $\overrightarrow{A D} \cdot \overrightarrow{A C}=0$, then the minimum value of $\tan C-\cot A$ is $\qquad$ | \sqrt{2} |
For any positive integer $k$, let $g(k)$ denote the greatest odd factor of $k$. For example, $g(1)=1$, $g(2)=1$, $g(10)=5$. Let $S_{n}=g(1)+g(2)+g(3)+…+g(2^{n})$. Determine the correct conclusions from the following:
$①g(3)+g(4)=10$;
$②∀m∈\mathbb{N}^{*}$, we have $g(2m)=g(m)$;
$③S_{1}+S_{2}+S_{3}=30$;
$④S_{n}-S_{n-... | B |
In all two-digit numbers, how many numbers have a units digit that is greater than the tens digit? | 36 |
3. Given two geometric progressions $a_{n}$ and $b_{n}$. It is known that $a_{0}=4$ and the sum of the denominators of these two progressions is 5. Find what $b_{0}$ can be, if it is known that $a_{0}, b_{0}, a_{1}, b_{1}$ are consecutive terms of some arithmetic progression (in that order) | 8;-2 |
Which of the following options is a linear equation with one variable?
A: $11x-7$
B: $4a-1=8$
C: $6x+y=3$
D: $x^{3}-x=4x$ | B |
Example 4.20. Find the particular solution of the equation $y^{\prime \prime}-3 y^{\prime}+2 y=0$, satisfying the given initial conditions $y(0)=1$, $y^{\prime}(0)=-1$. | 3e^{x}-2e^{2x} |
Given vectors $a = (1, 1, x)$, $b = (1, 2, 1)$, and $c = (1, 1, 1)$, satisfying the condition $(c - a) \cdot (2b) = -2$, find the value of $x$. | 2 |
Given the complex number $z$ that satisfies the equation: $z(2-i)=3+i$ (where $i$ is the imaginary unit), find the modulus of $z$. | \sqrt{2} |
12.1.7 Let $l$ and $m$ be two skew lines, and on $l$ there are three points $A$, $B$, and $C$, such that $AB = BC$. Through $A$, $B$, and $C$, perpendiculars $AD$, $BE$, and $CF$ are drawn to $m$, with feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $AD = \sqrt{15}$, $BE = \frac{7}{2}$, and ... | \sqrt{6} |
1. Let $\lfloor x\rfloor$ denote the greatest integer not exceeding the real number $x$. Find all positive integers $M$ such that the sequence defined by
$$
\begin{array}{l}
a_{0}=\frac{2 M+1}{2}, \\
a_{k+1}=a_{k}\left\lfloor a_{k}\right\rfloor(k=0,1, \cdots)
\end{array}
$$
contains at least one integer term. | M\geqslant2 |
12.131 Given 16 identical marbles, we want to select $n$ marbles to form a spatial structure, such that each marble is in contact with exactly 3 other marbles. Find all possible values of $n$, and explain how the $n$ marbles should be arranged. | {4,6,8,10,12,14,16} |
Margie bought $3$ apples at a cost of $50$ cents per apple. She paid with a 5-dollar bill. How much change did Margie receive?
$\textbf{(A)}\ \textdollar 1.50 \qquad \textbf{(B)}\ \textdollar 2.00 \qquad \textbf{(C)}\ \textdollar 2.50 \qquad \textbf{(D)}\ \textdollar 3.00 \qquad \textbf{(E)}\ \textdollar 3.50$ | \textdollar3.50 |
There are 20 students, numbered from 1 to 20. Now, 4 students are to be selected for a survey using the systematic sampling method. The numbers of the students selected are ( )
A: 5, 10, 15, 20
B: 2, 6, 10, 14
C: 2, 4, 6, 8
D: 5, 8, 11, 14 | \text{A} |
2. In square $ABCD$ with side length $1$, there are points $P, Q$ on $AB, AD$ respectively. If the perimeter of $\triangle APQ$ is 2, find $\angle PCQ$.
| 45 |
I. Multiple Choice
This section consists of 6 questions, each offering four options: (A), (B), (C), and (D). Only one of these options is correct. Please write the letter representing the correct answer in the parentheses after the question.
1. For each natural number $n$, the parabola $y=\left(n^{2}+n\right) x^{2}-(2 ... | \frac{1992}{1993} |
【Question 4】On a farm, the grass grows evenly every day. The farm can supply 10 sheep for 20 days, or 14 sheep for 12 days. Then, the new grass that grows on the farm every day is enough for 2 sheep to eat $\qquad$ days. | 2 |
In China's Beidou satellite navigation system, the altitude of a medium-high orbit satellite is approximately $21500000$ meters. Express the number $21500000$ in scientific notation as ( ).
A: $2.15\times 10^{7}$
B: $0.215\times 10^{8}$
C: $2.15\times 10^{6}$
D: $21.5\times 10^{6}$ | A |
3. Given $\alpha, \beta \in\left(\frac{3 \pi}{4}, \pi\right)$,
$$
\cos (\alpha+\beta)=\frac{4}{5}, \sin \left(\alpha-\frac{\pi}{4}\right)=\frac{12}{13} \text {. }
$$
Then $\cos \left(\beta+\frac{\pi}{4}\right)=$ $\qquad$ | -\frac{56}{65} |
Which of the following equations is correct?
A: $0-\left(-3\right)=3$
B: $(-14)-5=-9$
C: $(-3)-\left(-3\right)=-6$
D: $|5-3|=-\left(5-3\right)$ | A |
Given condition A: The function $f(x)$ satisfies $f(x) + f(-x) = 0$, and condition B: The function $f(x)$ is an odd function, then condition A is ( ) to condition B.
A: A sufficient but not necessary condition
B: A necessary but not sufficient condition
C: A necessary and sufficient condition
D: Neither a suffi... | C |
If $1$ and $9$ are two terms in the geometric sequence $\{a_{n}\}$ of the first $6$ terms, then the minimum value of $a_{5}$ is ( )
A: $-81$
B: $-27$
C: $-9$
D: $-3$ | B |
Four. (25 points) As shown in Figure 2, in the right trapezoid $ABCD$, $\angle ABC = \angle BAD = 90^{\circ}$, $AB = 16$. The diagonals $AC$ and $BD$ intersect at point $E$. A line $EF \perp AB$ is drawn through $E$ at point $F$, and $O$ is the midpoint of side $AB$, with $FE + EO = 8$. Find the value of $AD + BC$. | 16 |
Let the universal set $U=\{1,2,3,4,5,6,7\}$, $\complement_U(A\cup B)=\{1,3\}$, and $A\cap(\complement_UB)=\{2,5\}$. Then, the set $B$ equals to ( )
A: $\{4,6,7\}$
B: $\{1,2,3,5\}$
C: $\{2,4,6\}$
D: $\{1,3,5\}$ | \text{A} |
The original proposition is "In $\triangle ABC$, if $\sin A > \sin B$, then $A > B$". Among its converse, inverse, and contrapositive propositions, the number of true propositions is $\boxed{\phantom{5}}$.
A: $0$
B: $1$
C: $2$
D: $3$ | D |
6. Let $[x]$ denote the greatest integer not exceeding the real number $x$. If
$$
A=\left[\frac{7}{8}\right]+\left[\frac{7^{2}}{8}\right]+\cdots+\left[\frac{7^{2019}}{8}\right]+\left[\frac{7^{2020}}{8}\right],
$$
then the remainder when $A$ is divided by 50 is | 40 |
5. Find the largest real number $C$, such that for any positive integer $n$ and $0=x_{0}<x_{1}<\cdots<x_{n}=1$, the following inequality holds:
$$
\sum_{i=1}^{n} x_{i}^{2}(x_{i}-x_{i-1})>C .
$$ | \frac{1}{3} |
4. If the solution set of the inequality $k x^{2}-2|x-1|+$ $6 k<0$ with respect to $x$ is an empty set, then the range of values for $k$ is $\qquad$. | k \geqslant \frac{1+\sqrt{7}}{6} |
Given the complex number $z=(m^2+m-2)+(m^2+4m-5)i$ is a pure imaginary number, then $m=$ （ ）
A: -2
B: 1
C: -2 or 1
D: -5 | A |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $\sin \frac{C}{2} = \frac{2\sqrt{2}}{3}$.
(1) Find the value of $\sin(A+B)$;
(2) If $a+b=2\sqrt{2}$, find the maximum value of the area $S$ of $\triangle ABC$. | \frac{4\sqrt{2}}{9} |
Given a line $a$ contained in plane $\alpha$, consider the following three propositions:
① If plane $\alpha$ is parallel to plane $\beta$, then line $a$ is parallel to plane $\beta$;
② If line $a$ is parallel to plane $\beta$, then plane $\alpha$ is parallel to plane $\beta$;
③ If line $a$ is not parallel to plan... | D |
Which of the following surveys is suitable for a census?
A: Understanding the current situation of the height of ninth-grade students nationwide
B: Understanding the average lifespan of a batch of light bulbs
C: Understanding the global male-to-female ratio
D: Conducting medical examinations on passengers in the sa... | D |
Among the following sets of conditions, the one that cannot determine the congruence of \\( \triangle ABC \\) and \\( \triangle DEF \\) is \\( ( \quad ) \\)
A: \\( AB=DE \\), \\( \angle B=\angle E \\), \\( \angle C=\angle F \\)
B: \\( AC=DF \\), \\( BC=DE \\), \\( \angle C=\angle D \\)
C: \\( AB=EF \\), \\( \angle A=\a... | D |
1. Let $x, y, z \in R$ and $x+2 y+3 z=1, y z+z x+x y=-1$, then the range of $x+y+z$ is $\qquad$ | [\frac{3-3\sqrt{3}}{4},\frac{3+3\sqrt{3}}{4}] |
Given in $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, with $a= \sqrt {2}$, $A=45°$, and $B=60°$, find the area $S_{\triangle ABC}$. | \frac {3+ \sqrt {3}}{4} |
Given that $a > b > 0$, determine the relationship between $\frac{a}{b}$ and $\frac{a+1}{b+1}$. | \frac{a}{b} > \frac{a+1}{b+1} |
A natural number $N$ is called bombastic if it does not contain any zero in its representation and if no smaller natural number has the same product of digits as the number $N$.
Karel first became interested in bombastic prime numbers and claimed that there are not many of them. List all two-digit bombastic prime numb... | 8 |
4. The positive integers $x, y, z$ satisfy
$$
x+2 y=z, \quad x^{2}-4 y^{2}+z^{2}=310 .
$$
Find all possible values of the product $xyz$. | 11935,2015 |
$17 \cdot 63$ In $\triangle A B C$, $A B=8, B C=7, C A=6$. Also, as shown in the figure, extend side $B C$ to point $P$ so that $\triangle P A B$ is similar to $\triangle P C A$. The length of $P C$ is
(A) 7 .
(B) 8 .
(C) 9 .
(D) 10 .
(E) 11 .
(37th American High School Mathematics Examination, 1986) | 9 |
Each cell of an $100\times 100$ board is divided into two triangles by drawing some diagonal. What is the smallest number of colors in which it is always possible to paint these triangles so that any two triangles having a common side or vertex have different colors? | 8 |
Let $P$ be a polynomial with integer coefficients, and let $n$ be the cardinality of the set
$$
\{k \in \mathbb{Z}: k P(k)=2020\}
$$
What are the possible values of $n$? | 0,1,2,3,4,5,6 |
32. Express the length of a) the median and b)* the bisector of a triangle in terms of the lengths of its sides. | m_{}^{2}=\frac{b^{2}+^{2}}{2}-\frac{^{2}}{4},\quadn^{2}=\frac{(b+)^{2}-^{2}}{(b+)^{2}} |
A positive integer is called a perfect power if it can be written in the form $a^{b}$, where $a$ and $b$ are positive integers with $b \geq 2$. For example, 32 and 125 are perfect powers because $32=2^{5}$ and $125=5^{3}$.
The increasing sequence
$$
2,3,5,6,7,10, \ldots
$$
consists of all positive integers which are... | 21 |
There are $N$ students in a class. Each possible nonempty group of students selected a positive integer. All of these integers are distinct and add up to 2014. Compute the greatest possible value of $N$. | 5 |
Given positive real numbers $x$ and $y$ such that $x^{2}+y^{2}=1$, if $f(x,y)= \frac {x^{3}+y^{3}}{(x+y)^{3}}$, find the range of $f(x,y)$. | [\frac {1}{4},1) |
7. Two people are walking towards each other from points $A$ and $B$ with speeds $v_{1}=6 \mathrm{~m} / \mathrm{s}$ and $v_{2}=4 \mathrm{~m} / \mathrm{s}$. At the moment of their meeting, one of them turned around and walked in the opposite direction, while the other continued in the same direction. The person who turn... | 20 |
When the number of sides of a polygon increases by $1$, the sum of its interior angles will ( )
A: increase by $180^{\circ}$
B: increase by $360^{\circ}$
C: remain the same
D: cannot be determined | \text{A} |
Given the function $f(x)=x^{2}-3x$. If for any $x_{1}$, $x_{2}$ in the interval $[-3,2]$, we have $|f(x_{1})-f(x_{2})| \leqslant m$, then the minimum value of the real number $m$ is _______. | \frac{81}{4} |
Given the space vectors $\overrightarrow{a}=(0,1,1)$ and $\overrightarrow{b}=(1,0,1)$, the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$ is ( )
A: $60^{\circ}$
B: $120^{\circ}$
C: $30^{\circ}$
D: $150^{\circ}$ | A |
Given the sample data $x_{1}$, $x_{2}$, ..., $x_{20}$ with mean and variance of $1$ and $8$ respectively, if $y_{i}=2x_{i}+3$ $(i=1,2,…,20)$, then the mean and variance of $y_{1}$, $y_{2}$, ..., $y_{20}$ are respectively ( ).
A: $5$, $32$
B: $5$, $19$
C: $1$, $32$
D: $4$, $35$ | A |
Example 7 Let $x_{1}, x_{2}, \cdots, x_{7}$ all be integers, and
$$
\begin{array}{l}
x_{1}+4 x_{2}+9 x_{3}+16 x_{4}+25 x_{5}+36 x_{6} \\
+49 x_{7}=1, \\
4 x_{1}+9 x_{2}+16 x_{3}+25 x_{4}+36 x_{5}+49 x_{6} \\
+64 x_{7}=12, \\
9 x_{1}+16 x_{2}+25 x_{3}+36 x_{4}+49 x_{5}+64 x_{6} \\
+81 x_{7}=123 .
\end{array}
$$
Find $1... | 334 |
1. Choose 5 numbers (repetition allowed) from $\{1,2, \cdots, 100\}$. Then the expected number of composite numbers taken is | \frac{37}{10} |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $1 \leq a_5 \leq 4$ and $2 \leq a_6 \leq 3$, then the range of values for $S_6$ is. | [-12, 42] |
The value range of the function $g(x)=x^{2}-2x$ $(x\in[0,3])$ is ( )
A: $[0,3]$
B: $[-1,3]$
C: $[-1,0]$
D: $[1,3]$ | \text{B} |
Suppose that $ABC$ and $PQR$ are triangles such that $A,P$ are the midpoints of $QR,BC$ respectively, and $QR,BC$ are the bisectors of $\angle BAC,\angle QPR$. Prove that $AB+AC=PQ+PR$. | AB + AC = PQ + PR |
Select 4 students from 10 students, including A and B, to participate in a charity event, with the requirement that at least one of A and B must participate. How many different selection methods are there? | 140 |
4. For what values of the parameter $a$ does the system of equations
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+z^{2}+2 y=0 \\
x+a y+a z-a=0
\end{array}\right.
$$
have a unique solution? | \\frac{\sqrt{2}}{2} |
Given vectors $\vec{a}$ and $\vec{b}$ satisfy $|\vec{a}|=1$, $|\vec{a}|=1$, and $\vec{a} \cdot \vec{b}=-1$, then $\vec{a} \cdot (2\vec{a}-\vec{b})=$ ()
A: $4$
B: $3$
C: $2$
D: $0$ | B |
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