problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
7.2. A rectangle is cut into several rectangles, the perimeter of each of which is a number of meters divisible by 4. Is it true that the perimeter of the original rectangle is divisible by 4? | No |
G2.3 Let $a \oplus b=a b+10$. If $C=(1 \oplus 2) \oplus 3$, find the value of $C$. | 46 |
If the domain of the function $f(x)$ is $D$, and the set $M \subseteq D$, if there exists a nonzero real number $t$ such that for all $x \in M$, $x-t \in D$, and $f(x-t) < f(x)$, then $f(x)$ is called a $t\oplus$ function on $M$.
$(1)$ Given functions $g(x)=x$ and $h(x)=\frac{1}{x}$, determine whether $g(x)$ and $h(x... | n = 1 |
12・104 Find all positive integer solutions \(x, y, z\) to the equation \(3^{x}+4^{y}=5^{z}\).
(32nd International Mathematical Olympiad Preliminary Question, 1991) | 2 |
5. Twenty students participating in a math camp have decided to send messages to each other, with each of them sending messages to exactly ten of the remaining students. Determine the minimum possible number of mutual messages, i.e., find an example of a message-sending schedule in which the number of mutual messages i... | 10 |
In the arithmetic sequence $\{a_{n}\}$, $a_{3}+a_{9}=27-a_{6}$, and $S_{n}$ represents the sum of the first $n$ terms of the sequence $\{a_{_{n}}\}$. Determine the value of $S_{11}$. | 99 |
$\mathrm{Az}$ asztalon fekvő (az asztalt érintő) négy golyó mindegyike érinti a másik hármat. Hármuknak ugyanakkora $R$ sugara van. Mekkora a negyedik golyó sugara.
Four spheres lying on a table (touching the table) each touch the other three. Three of them have the same radius $R$. What is the radius of the fourth sp... | \frac{R}{3} |
9. Given: $c>0$. Let $p$ : the function $y=c^{\prime}$ is monotonically decreasing on $\mathbf{R}$, $q$ : the solution set of the inequality $x+|x-2 c|>1$ is R. If $p$ and $q$ are such that exactly one of them is true, find the range of values for $c$. | (0,\frac{1}{2}]\cup[1,+\infty) |
In the plane, fixed points $A$, $B$, $C$, $D$ satisfy $|\overrightarrow{DA}| = |\overrightarrow{DB}| = |\overrightarrow{DC}|$, $\overrightarrow{DA} \cdot \overrightarrow{DB} = \overrightarrow{DB} \cdot \overrightarrow{DC} = \overrightarrow{DC} \cdot \overrightarrow{DA} = -2$. For moving points $P$, $M$ satisfying $|\ov... | \frac{49}{4} |
Given that $0 < a < 2$, and a complex number $z$ has a real part equal to $a$ and an imaginary part equal to $1$, the range of values for $|z|$ is __________. | (1, \sqrt{5}) |
If $a_1 ,a_2 ,\ldots, a_n$ are complex numbers such that
$$ |a_1| =|a_2 | =\cdots = |a_n| =r \ne 0,$$
and if $T_s$ denotes the sum of all products of these $n$ numbers taken $s$ at a time, prove that
$$ \left| \frac{T_s }{T_{n-s}}\right| =r^{2s-n}$$
whenever the denominator of the left-hand side is different from $0$. | \left| \frac{T_s}{T_{n-s}} \right| = r^{2s-n} |
The distance between two parallel lines $l_1$: $x-2y+1=0$ and $l_2$: $x-2y-4=0$ is __________. | \sqrt{5} |
Three, (25 points) A chemical plant, starting from January this year, if it does not improve its production environment and continues to produce as it is, will earn 700,000 yuan per month. At the same time, it will receive penalties from the environmental protection department, with the first month's penalty being 30,0... | 9 |
15th Chinese 2000 Problem A2 The sequence a n is defined by a 1 = 0, a 2 = 1, a n = (n a n-1 + n(n-1) a n-2 + (-1) n-1 n)/2 + (-1) n . Find a n + 2 nC1 a n-1 + 3 nC2 a n-2 + ... + n nC(n-1) a 1 , where nCm is the binomial coefficient n!/(m! (n-m)! ). Solution | 2n!-(n+1) |
Given the function $y=f\left(x\right)$, where $f(x)=\frac{{4^x}+k}{{2^x}}$ ($k∈R$).
$(1)$ Is there a real number $k$ such that the function $y=f\left(x\right)$ is an odd function? If it exists, please provide a proof.
$(2)$ When $k=1$, if the inequality $f\left(x\right)\geqslant a$ holds for all $x$, find the range... | (-\infty, 2] |
The degree measures of the angles in a [convex](https://artofproblemsolving.com/wiki/index.php/Convex_polygon) 18-sided polygon form an increasing [arithmetic sequence](https://artofproblemsolving.com/wiki/index.php/Arithmetic_sequence) with integer values. Find the degree measure of the smallest [angle](https://artofp... | 143 |
A bus travels from the starting station to the terminal station (including both the starting and terminal stations) passing through a total of 8 stations. It is known that 100 people boarded the bus at the first 6 stations, and a total of 80 people got off the bus at all stations except the terminal station. How many p... | 20 |
Four vehicles were traveling on the country road at constant speeds: a car, a motorcycle, a truck, and a bicycle. The car caught up with the truck at 12:00, met the cyclist at 14:00, and the motorcyclist at 16:00. The motorcyclist met the truck at 17:00 and caught up with the cyclist at 18:00.
At what time did the cyc... | 15\frac{1}{3} |
9.2. Given two coprime natural numbers $p$ and $q$, differing by more than one. a) Prove that there exists a natural number $n$ such that the numbers $p+n$ and $q+n$ will not be coprime. b) Find the smallest such $n$ for $p=2, q=2023$. | 41 |
Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of $6$ hops, and end up back on the ground?
(For example, one sequence of hops is up-up-down-down-up-down.)
[2024-AMC8-q13.png](https://artofproblemsolving.com/wiki/index.php/File:2024-A... | 5 |
8.2. On weekdays (from Monday to Friday), Petya worked out in the gym five times. It is known that in total he spent 135 minutes in the gym, and the time spent in the gym on any two different days differed by at least 7 minutes. What is the maximum duration that the shortest workout could have been? | 13 |
Iodine-131 ($^{131}_{53}\text{I}$) can be used to treat hyperthyroidism. Which of the following statements about $^{131}_{53}\text{I}$ is correct? ( )
A: The number of neutrons is 131
B: The number of protons is 131
C: The number of electrons is 53
D: The atomic mass is 53 | C |
Example 13. Let $\alpha, \beta$ be acute angles, and $3 \sin \alpha=2 \sin \beta$, $3 \cos \alpha+2 \cos \beta=3$. Find the degree measure of $\beta + \frac{\alpha}{2}$. | 90^{\circ} |
How many integers can be expressed as a sum of three distinct numbers chosen from the set $\{4,7,10,13, \ldots, 46\}$ ?
(A) 45
(B) 37
(C) 36
(D) 43
(E) 42 | 37 |
4. In a decimal positive integer, if it contains an even number (including zero) of the digit 8, it is called a "good number"; otherwise, it is called a "non-good number". Then, the number of all "good numbers" with length (number of digits) not exceeding $n$ ($n$ is a positive integer) is $\qquad$ | \frac{1}{2}(8^{n}+10^{n})-1 |
7.2. Agent 007 wants to encrypt his number using two natural numbers $m$ and $n$ such that $0.07=\frac{1}{m}+\frac{1}{n} \cdot$ Can he do this? | 0.07=\frac{1}{20}+\frac{1}{50} |
Katie and Allie are playing a game. Katie rolls two fair six-sided dice and Allie flips two fair two-sided coins. Katie’s score is equal to the sum of the numbers on the top of the dice. Allie’s score is the product of the values of two coins, where heads is worth $4$ and tails is worth $2.$ What is the probability Kat... | \frac{25}{72} |
There are four people in a room.
For every two people, there is a $50 \%$ chance that they are friends.
Two people are connected if:
- they are friends, or
- a third person is friends with both of them, or
- they have different friends who are friends of each other.
What is the probability that every pair of people... | \frac{19}{32} |
Given the hyperbola $C$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ has an asymptote equation $y = \dfrac{\sqrt{5}}{2}x$, and it shares a common focus with the ellipse $\dfrac{x^2}{12} + \dfrac{y^2}{3} = 1$, then the equation of $C$ is ( )
A: $\dfrac{x^2}{8} - \dfrac{y^2}{10} = 1$
B: $\dfrac{x^2}{4}... | \text{B} |
Given that variables $x$ and $y$ are linearly related, and the sample means obtained from observational data are $\overset{.}{x}=2$ and $\overset{.}{y}=3$, the linear regression equation obtained from this observational data cannot be
A: $y=3x-3$
B: $y=2x+1$
C: $y=x+1$
D: $y=0.5x+2$ | \text{B} |
Find all triples of continuous functions $f, g, h$ from $\mathbb{R}$ to $\mathbb{R}$ such that $f(x+y)=$ $g(x)+h(y)$ for all real numbers $x$ and $y$. | f(x)=++b,(x)=+,(x)=+b |
(For liberal arts) There are two identical, uniformly made tetrahedral toys, with the numbers 1, 2, 3, and 5 written on their faces. When these two toys are thrown at the same time, let $m$ be the sum of the numbers on the faces that are facing down.
(1) Find the probability that "$m$ is not less than 6";
(2) Are the p... | \frac{1}{2} |
Example 4.7. Find the general integral of the equation
$$
\left(x^{2}+y^{2}\right) d x-x y d y=0
$$ | Ce^{\frac{y^{2}}{2x^{2}}} |
9. (12 points) For the sequence of positive numbers $\left\{a_{n}\right\}$, the sum of the first $n$ terms is $b_{n}$, and the product of the first $n$ terms of the sequence $\left\{b_{n}\right\}$ is $c_{n}$, and it is given that $b_{n}+2 c_{n}=1$ $\left(n \in \mathbf{Z}_{+}\right)$. Find the number in the sequence $\l... | 2024 \frac{3}{4} |
[Limitedness, monotonicity]
For what value of $K$ is the quantity $A_{\mathrm{k}}=\frac{19^{k}+66^{k}}{k!}$ maximal? | 65 |
4. Let real numbers $a, b, c$ satisfy $\left\{\begin{array}{l}a^{2}-b c-8 a+7=0 \\ b^{2}+c^{2}+b c-6 a+6=0,\end{array} \quad a\right.$, the range of values for $a$ is | 1\leqslant\leqslant9 |
Consider $n$ points in the plane in general position: no three of them are ever collinear. Moreover, assume that each of these $n$ points is the center of a circle passing through at least $k$ of the remaining $n-1$ points. Prove that $k<1 / 2+\sqrt{2 n}$. | k<\frac{1}{2}+\sqrt{2n} |
Let O be the origin of coordinates, F be the focus of the parabola $y^2 = 2px$ ($p > 0$), and A be a point on the parabola. The angle between $\overrightarrow{FA}$ and the positive direction of the x-axis is 60°. Find the magnitude of $\overrightarrow{OA}$. | \frac{\sqrt{21}}{2}p |
## Problem Statement
Calculate the lengths of the arcs of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=3(2 \cos t-\cos 2 t) \\
y=3(2 \sin t-\sin 2 t)
\end{array}\right. \\
& 0 \leq t \leq 2 \pi
\end{aligned}
$$ | 48 |
4B. On the sides of a square with side length 2, isosceles trapezoids are constructed outward, such that the vertices of all trapezoids are simultaneously vertices of a regular dodecagon.
What is the perimeter of the dodecagon? | 12(\sqrt{3}-1) |
6.20 The sum of an infinite geometric progression with a common ratio $|q|<1$ is 4, and the sum of the cubes of its terms is 192. Find the first term and the common ratio of the progression. | 6-0.5 |
In the geometric sequence $\{a_n\}$, if $a_n + a_{n+1} = 3 \cdot \left(\frac{1}{2}\right)^n$, then $a_5$ = （ ）
A: $\frac{1}{4}$
B: $\frac{1}{8}$
C: $\frac{1}{16}$
D: $\frac{1}{32}$ | C |
251. Find all real solutions of the system
$$
\left\{\begin{array}{c}
x^{3}+y^{3}=1 \\
x^{4}+y^{4}=1
\end{array}\right.
$$ | 1,00,1 |
Show that $r = 2$ is the largest real number $r$ which satisfies the following condition:
If a sequence $a_1$, $a_2$, $\ldots$ of positive integers fulfills the inequalities
\[a_n \leq a_{n+2} \leq\sqrt{a_n^2+ra_{n+1}}\]
for every positive integer $n$, then there exists a positive integer $M$ such that $a_{n+2} = a_n$... | r = 2 |
1. Through four points not lying on the same straight line, the maximum number of planes that can be determined is _, and the minimum number of planes that can be determined is _. | 4, 1 |
11. (GDR 1) Given real numbers $x_{1}, x_{2}, \ldots, x_{n}(n \geq 2)$, with $x_{i} \geq 1 / n$ $(i=1,2, \ldots, n)$ and with $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1$, find whether the product $P=x_{1} x_{2} x_{3} \cdots x_{n}$ has a greatest and/or least value and if so, give these values. | P_{\min }=\sqrt{n^{2}-n+1} / n^{n}, P_{\max }=1 / n^{n / 2} |
1. Given the function $S=|x-2|+|x-4|$.
(1) Find the minimum value of $S$;
(2) If for any real numbers $x, y$ the inequality
$$
S \geqslant m\left(-y^{2}+2 y\right)
$$
holds, find the maximum value of the real number $m$. | 2 |
13. (6 points) Both two-digit numbers $\overline{a b}$ and $\overline{\mathrm{ba}}$ are prime numbers, then $\overline{a b}$ has $\qquad$.
保留源文本的换行和格式，直接输出翻译结果如下：
13. (6 points) Both two-digit numbers $\overline{a b}$ and $\overline{\mathrm{ba}}$ are prime numbers, then $\overline{a b}$ has $\qquad$.
| 9 |
4. The height to the base of an isosceles triangle has a length of $\sqrt{2+\sqrt{2}}$. If the measure of the angle opposite the base is $45^{\circ}$, what is the length of the height to the leg of the triangle? | \sqrt{2} |
2.160. $\frac{\left(a^{2} b \sqrt{b}-6 a^{5 / 3} b^{5 / 4}+12 a b \sqrt[3]{a}-8 a b^{3 / 4}\right)^{2 / 3}}{a b \sqrt[3]{a}-4 a b^{3 / 4}+4 a^{2 / 3} \sqrt{b}}$. | 1 |
1067. Kolya had to square a certain natural number for his homework. Instead, he mistakenly doubled the number and got a two-digit number, written with the same digits as the square of the number, but in reverse order. What should the correct answer be? | 81 |
6. Given that the line $l$ passes through the left focus $F$ of the ellipse $C: \frac{x^{2}}{2}+y^{2}=1$ and intersects the ellipse $C$ at points $A$ and $B$, and $O$ is the origin. If $O A \perp O B$, then the distance from point $O$ to the line $A B$ is $\qquad$ | \frac{\sqrt{6}}{3} |
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $20$ cents. If she had one more quarter, the average value would be $21$ cents. How many dimes does she have in her purse?
$\text {(A)}\ 0 \qquad \text {(B)}\ 1 \qquad \text {(C)}\ 2 \qquad \text {(D)}\ 3\qquad \text {(E)}\ 4$ | 0 |
In the sequence $1,3,2, \ldots$ each term after the first two is equal to the preceding term subtracted from the term that precedes it, that is: if $n>2$ then $a_{n}=a_{n-1}-a_{n-2}$. What is the sum of the first 100 terms of this sequence? | 5 |
1. (3 points) $20.16 \times 32 + 2.016 \times 680=$ | 2016 |
Given $p: (x-1)(x-2) < 0$, $q: x-a < 0$, if $p$ is a sufficient condition for $q$, find the range of values for $a$. | a \geq 2 |
Given an ellipse C with its center of symmetry at the origin O, and its foci on the x-axis, with the left and right foci being $F_1$ and $F_2$ respectively, and the distance between $F_1$ and $F_2$ is 2. Point P (1, $\frac{3}{2}$) lies on this ellipse.
(Ⅰ) Find the equation of ellipse C and its eccentricity $e$;
(Ⅱ... | (x-1)^2 + y^2 = 2 |
8. William has four cards with different integers written on them. Three of these integers are 2,3 and 4 . He puts one card in each cell of the $2 \times 2$ grid shown.
The sum of the two integers in the second row is 6 . The sum of the two integers in the second column is 10 . Which number is on the card he places in ... | 3 |
Example 4 Try to find the minimum value of the function $f(x, y)=6\left(x^{2}+y^{2}\right)(x+y)-4\left(x^{2}+x y+y^{2}\right)-3(x+y)+5$ in the region $D=\{(x, y) \mid x>0, y>0\}$. | 2 |
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$:
$(i)$ $f(x)=-f(-x);$
$(ii)$ $f(x+1)=f(x)+1;$
$(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$
| f(x) = x |
If $x^2+x-1=0$, then $x^3+2x^2+2007=$ ? | 2008 |
The angle bisectors of angles $A$ and $B$ of trapezoid $A B C D$ ($B C \| A D$) intersect at point $O$. Find the sides $A B$ and $B C$, if $\angle A$
$=2 \arccos \sqrt{\frac{5}{6}}, O C=\sqrt{7}, O D=3 \sqrt{15}, A D=5 B C$ | 2\sqrt{3},\frac{5\sqrt{3}}{3} |
5.2. $x^{4}+4$. | (x^{2}-2x+2)(x^{2}+2x+2) |
The remainder can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by \[\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor\]where $\left \lfloor \tfrac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\tfrac{x}{y}$. What is the value of $\text{rem} (\tfrac{3}{8}, -\tfrac{2... | -\frac{1}{40} |
3.15. The height of an isosceles trapezoid is $h$, and the angle between its diagonals, opposite the lateral side, is $\alpha$. Find the midline of the trapezoid. | \operatorname{ctg}\frac{\alpha}{2} |
Solve the system
$x^{2}+y^{2}=1$
$4 x y\left(2 y^{2}-1\right)=1$ | (\\frac{\sqrt{2-\sqrt{2}}}{2},\\frac{\sqrt{2+\sqrt{2}}}{2}),(\\frac{\sqrt{2+\sqrt{2}}}{2},\\frac{\sqrt{2-\sqrt{2}}}{2}) |
9. There are 50 consecutive even numbers: $2,4,6,8, \cdots \cdots, 98,100$. From these 50 numbers, the minimum number of numbers to be taken such that their sum is 2022 (each number can be taken at most once) is | 28 |
Given $z= \frac {a-i}{1-i}$, where $a>0$, and the imaginary part of the complex number $\omega=z(z+i)$ minus its real part equals $\frac {3}{2}$, find the real number $a$. | a=2 |
I3.3 If $0<x<1$, find the value of $c=\left(\frac{\sqrt{1+x}}{\sqrt{1+x}-\sqrt{1-x}}+\frac{1-x}{\sqrt{1-x^{2}}+x-1}\right) \times\left(\sqrt{\frac{1}{x^{2}-b^{2}}-1}-\frac{1}{x-b}\right)$. | -1 |
A boat takes 4 hours to travel downstream from dock A to dock B, and 5 hours to travel upstream from dock B to dock A. Given that the speed of the current is 3 kilometers per hour, the average speed of the boat in still water is ( )
A: 6 kilometers per hour
B: 9 kilometers per hour
C: 27 kilometers per hour
D: 54 kilo... | C |
5. From the set $\{1,2,3, \cdots, 99,100\}$, arbitrarily select $a$, and then from the same set arbitrarily select $b$, the probability that the last digit of $3^{a}+7^{b}$ is 8 equals $\qquad$. | \frac{3}{16} |
3. In a certain football league, a double round-robin system (i.e., two teams play each other twice) is used, with $m$ teams participating. At the end of the competition, a total of $9 n^{2}+6 n+32$ matches were played, where $n$ is an integer. Then $m=$ $\qquad$ | 8or32 |
Two chords AB and CD of circle O intersect at point P. Given that AP = 2 cm, BP = 6 cm, and the ratio CP : PD = 1 : 3, then DP = _________. | 6 |
In an $n \times n$ table, we write down the numbers $1,2, \ldots, n^{2}$ in sequence:
$$
\begin{array}{cccc}
1 & 2 & \ldots n \\
n+1 & n+2 & \ldots 2 n \\
\vdots & \vdots & \vdots \\
n^{2}-n+1 & n^{2}-n+2 & \ldots n^{2}
\end{array}
$$
From each row, we select one number such that no two are in the same column. What a... | \frac{n^3+n}{2} |
4. Find the number of pairs of integers $(x ; y)$ that satisfy the equation $x^{2}+x y=30000000$. | 256 |
14. One of the longest-living animals in the world is the Arctic quahog, and generally, Arctic quahogs can live for hundreds of years. Now there is a large Arctic quahog, 70 years old, and 4 smaller Arctic quahogs whose ages are 3 years, 4 years, 5 years, and 6 years, respectively. In $\qquad$ years, the sum of the age... | 18 |
If two functions have the same corresponding relationship and range, but different domains, then these two functions are called homologous functions. Then, the number of functions that are homologous to the function $y=x^2$, where $x \in \{-1, 0, 1, 2\}$, is ( )
A: 5
B: 6
C: 7
D: 8 | \text{D} |
Three. (25 points) Given $m, n, p (m<n)$ are positive integers. Let $A(-m, 0), B(n, 0), C(0, p)$, and $O$ be the origin. If $\angle ACB=90^{\circ}$, and
$$
OA^2 + OB^2 + OC^2 = 3(OA + OB + OC),
$$
(1) Prove: $m+n=p+3$;
(2) Find the analytical expression of the quadratic function whose graph passes through points $A, B,... | y=-\frac{1}{2} x^{2}+\frac{3}{2} x+2 |
Of the following statements, the one that is incorrect is:
$\textbf{(A)}\ \text{Doubling the base of a given rectangle doubles the area.}$
$\textbf{(B)}\ \text{Doubling the altitude of a triangle doubles the area.}$
$\textbf{(C)}\ \text{Doubling the radius of a given circle doubles the area.}$
$\textbf{(D)}\ \text{Doub... | \textbf{(C)} |
"x≠3" is a ( ) condition for "|x-3|>0".
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Necessary and sufficient
D: Neither sufficient nor necessary | C |
If $\left(1+i\right)\cdot \left(z+2\right)=2$, then $z+\overline{z}=$____. | -2 |
## Task 10/83
Given is a straight circular frustum of a cone, whose generatrices are inclined at $60^{\circ}$ to the base, and for whose base and top areas the relationship $A_{G}=4 A_{D}$ holds. A rubber band is stretched taut between a point $P$ on the edge of the base and a point $Q$ on the edge of the top.
How lo... | 2r_{G}\sqrt{3} |
2. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ for which $f(x+y)=f(x-y)+2 f(y) \cos x$ holds for all real numbers $x$ and $y$. | f(x)=\cdot\sinx |
9. (3 points) On a 440-meter circular track, two people, A and B, start from the starting line at the same time. A runs 5 meters per second, and B runs 6 meters per second.
(1) If they run in opposite directions, they will meet in $\qquad$ seconds.
(2) If they run in the same direction, B will catch up to A in $\qquad$... | 40,440 |
The relationship between point $A(1,2)$ and circle $C$: $(x+1)^{2}+(y-2)^{2}=1$ is ( )
A: Inside the circle
B: Outside the circle
C: On the circle
D: Cannot be determined | B |
Problem 6. (30 points) Ten balls of the same radius are arranged in the form of a triangular pyramid such that each ball touches at least three others. Find the radius of the sphere in which the pyramid of balls is inscribed, if the radius of the sphere inscribed in the center of the pyramid of balls, touching six iden... | \sqrt{6}+1 |
There are 3 numbers, one is the smallest odd prime number, one is the largest prime number less than 50, and one is the smallest prime number greater than 60. The sum of these 3 numbers is ( )
A: 101
B: 110
C: 111
D: 113 | C |
7. Let $a, b, c$ be three distinct real numbers such that the equations $x^{2}+a x+1=0$ and $x^{2}+b x+c=0$ have a common real root, and the equations $x^{2}+x+a=0$ and $x^{2}+c x+b=0$ also have a common real root, then the value of $a+b+c$ is $\qquad$. | -3 |
Given a regression equation $$\hat y = 2 + 3x$$, when the variable $x$ increases by one unit, then ( )
A: $y$ increases by an average of 2 units
B: $y$ increases by an average of 3 units
C: $y$ decreases by an average of 2 units
D: $y$ decreases by an average of 3 units | B |
The contrapositive of the proposition "If at least one of the integers $a$ and $b$ is even, then $ab$ is even" is ( )
A: If at most one of the integers $a$ and $b$ is even, then $ab$ is even
B: If neither of the integers $a$ and $b$ is even, then $ab$ is not even
C: If $ab$ is not even, then neither of the integer... | C |
A good approximation of $\pi$ is $3.14.$ Find the least positive integer $d$ such that if the area of a circle with diameter $d$ is calculated using the approximation $3.14,$ the error will exceed $1.$ | 51 |
9. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=3$, and for any positive integers $m, n$, it holds that $a_{m+n}=a_{m}+a_{n}+2 m n$.
(1) Find the general term formula for the sequence $\left\{a_{n}\right\}$;
(2) If a real number $c$ makes $\sum_{i=1}^{k} \frac{1}{a_{i}}<c$ hold for all positive integers $... | [\frac{3}{4},+\infty) |
1. Find the range of real number $a$ such that the inequality
$$
\sin 2 \theta-(2 \sqrt{2}+\sqrt{2} a) \cdot \sin \left(\theta+\frac{\pi}{4}\right)-\frac{2 \sqrt{2}}{\cos \left(\theta-\frac{\pi}{4}\right)}>-3-2 a \text {, }
$$
holds for all $\theta \in\left[0, \frac{\pi}{2}\right]$. | >3 |
Xiao Li and Xiao Zhang both solved the system of equations $\left\{\begin{array}{l}{ax+by=2}\\{mx-7y=-8}\end{array}\right.$. Xiao Li solved it correctly and obtained $\left\{\begin{array}{l}{x=-2}\\{y=3}\end{array}\right.$, while Xiao Zhang copied the value of $m$ incorrectly and obtained $\left\{\begin{array}{l}{x=-2}... | -1 |
4. (5 points) As shown in the figure, a grasshopper stands at position 1. On the 1st jump, it moves 1 step and stands at position 2; on the 2nd jump, it moves 2 steps and stands at position 4; on the 3rd jump, it moves 3 steps and stands at position 1... On the $n$-th jump, it moves $n$ steps. When the grasshopper jump... | 5 |
Given that the equation $\log_{2}x + x - m = 0$ has a real root in the interval $(1, 2)$, find the range of the real number $m$. | (1, 3) |
The graph of the function $f(x)=a^{x-1}+4$ always passes through a fixed point $P$. The coordinates of point $P$ are \_\_\_\_\_\_. | (1,5) |
The distance from a point $P$ on the hyperbola $4x^{2}-y^{2}+64=0$ to one of its foci is equal to $1$. What is the distance from point $P$ to the other focus? | 17 |
# 10.5. (7 points)
At a joint conference of the party of liars and the party of truth-tellers, 32 people were elected to the presidium and seated in four rows of eight. During the break, each member of the presidium claimed that among their neighbors there are representatives of both parties. It is known that liars al... | 8 |
5. Find all pairs of positive integers $(a, b)$ such that
$$
a b+2=a^{3}+2 b .
$$ | (1,1),(3,25),(4,31),(5,41),(8,85) |
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