problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
For two lines $m$, $n$ in space and a plane $\alpha$, which of the following propositions is true? ( )
A: If $m \parallel \alpha$, $n \parallel \alpha$, then $m \parallel n$
B: If $m \parallel \alpha$, $n \subset \alpha$, then $m \parallel n$
C: If $m \parallel \alpha$, $n \perp \alpha$, then $m \parallel n$
D: If $m ... | D |
8. Let the rational number $r=\frac{p}{q} \in(0,1)$, where $p, q$ are coprime positive integers, and $pq$ divides 3600. The number of such rational numbers $r$ is $\qquad$ . | 112 |
On a trip, a car traveled $80$ miles in an hour and a half, then was stopped in traffic for $30$ minutes, then traveled $100$ miles during the next $2$ hours. What was the car's average speed in miles per hour for the $4$-hour trip?
$\text{(A)}\ 45 \qquad \text{(B)}\ 50 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 75 \qqu... | 45 |
Use the Fundamental Theorem of Calculus or the geometric meaning of definite integrals to calculate the following integrals:
(1) $$\int_{0}^{1}(x^{2}-x)dx$$
(2) $$\int_{1}^{3}|x-2|dx$$
(3) $$\int_{0}^{1} \sqrt{1-x^{2}}dx$$. | \frac{\pi}{4} |
2. Factorize:
$$
a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b)-a^{3}-b^{3}-c^{3}-2 a b c .
$$ | (a+b-c)(b+c-a)(c+a-b) |
If $\frac{1}{3}{a^2}{b^m}$ and $-0.5a^{n}b^{4}$ are like terms, then $m=$______, $n=______. | 4; 2 |
For each prime $p$, let $\mathbb S_p = \{1, 2, \dots, p-1\}$. Find all primes $p$ for which there exists a function $f\colon \mathbb S_p \to \mathbb S_p$ such that
\[ n \cdot f(n) \cdot f(f(n)) - 1 \; \text{is a multiple of} \; p \]
for all $n \in \mathbb S_p$.
[i]Andrew Wen[/i] | 2 |
Let $\overrightarrow{a}$ be a non-zero vector, and $\lambda$ be a non-zero real number. Which of the following statements is correct? ( )
A: $\overrightarrow{a}$ and $-\lambda\overrightarrow{a}$ have opposite directions.
B: $|-\lambda\overrightarrow{a}| \geq |\overrightarrow{a}|$
C: $\overrightarrow{a}$ and $\lambda^2... | C |
What is the value of the expression $\sqrt{16\sqrt{8\sqrt{4}}}$?
$\textbf{(A) }4\qquad\textbf{(B) }4\sqrt{2}\qquad\textbf{(C) }8\qquad\textbf{(D) }8\sqrt{2}\qquad\textbf{(E) }16$ | 8 |
8. If $a$, $b$, $c$ are distinct integers, then
$$
3 a^{2}+2 b^{2}+4 c^{2}-a b-3 b c-5 c a
$$
the minimum value is $\qquad$ | 6 |
3. Let $f(x)=\cos (\omega x)$ have the smallest positive period of 6, then the value of $f(1)+f(2)+\cdots+f(2018)$ is $($ )
A: 0
B: 1
C: $\frac{1}{2}$
D: $\frac{\sqrt{3}}{2}$ | 0 |
5. How many five-digit numbers of the form $\overline{a b 16 c}$ are divisible by 16? $(a, b, c-$ arbitrary digits, not necessarily different).
ANSWER: 90. | 90 |
11. As shown in the figure, the ellipse $C_{1}: \frac{x^{2}}{4}+y^{2}=1$, the parabola $C_{2}: x^{2}=2 p y(p>0)$, let $C_{1}, C_{2}$ intersect at points $A, B$, and $O$ be the origin.
(1) If the circumcenter of $\triangle A B O$ lies on the ellipse, find the value of the real number $p$;
(2) If the circumcircle of $\tr... | \frac{7-\sqrt{13}}{6} |
4. Alberto, Barbara, Carlo, and Daria are participating in a game. At the beginning, each is assigned a number through a draw: Alberto gets $2^{101}+2^{121}+2^{180}$, Barbara gets $2^{100}+2^{202}+2^{400}$, Carlo gets $2^{101}+2^{109}$, and Daria gets $2^{100}+2^{108}$. Then various rounds take place: in each round, ea... | B |
1. (2003 Beijing Competition Question) The three sides of a triangle form a geometric sequence. Then, the range of the common ratio $q$ is | \frac{\sqrt{5}-1}{2}<\frac{\sqrt{5}+1}{2} |
Let $f(x)=\left\{\begin{array}{l}{\frac{2}{3}x-1 (x\geq 0)}\\{\frac{1}{x} \;\;\;\; (x<0)}\end{array}\right.$. If $f(a)=a$, then the value of the real number $a$ is ____. | -1 |
1. Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation:
$$
a !+b !=2^{n}
$$
Notation: $k !=1 \times 2 \times \cdots \times k$, for example: $1 !=1$, and $4 !=1 \times 2 \times 3 \times 4=24$.
| (1,1,1),(2,2,2),(2,3,3),(3,2,3) |
12. In a regular hexagon divided into six areas for观赏 plants, 加园: $\mathrm{i}$, it is required that the same type of plant be in the same area, and adjacent areas must have different plants. There are 4 different types of plants available, resulting in $\qquad$ planting schemes.
Note: The term "观赏 plants" is directly ... | 732 |
$f(x)$ is an even function on $R$. When $x \geqslant 0$, $f(x) = x^{3} + \ln(x + 1)$. Then, when $x < 0$, $f(x) = ($ $)$.
A: $-x^{3} - \ln(x - 1)$
B: $x^{3} + \ln(x - 1)$
C: $x^{3} - \ln(1 - x)$
D: $-x^{3} + \ln(1 - x)$ | D |
11.5. A square plot of 14 by 14 cells needs to be paved with rectangular tiles of size $1 \times 4$. The tiles can only be laid along the grid (not diagonally), and the tiles cannot be broken. What is the maximum number of tiles required? Will there be any uncovered area left? | 48 |
Which of the following correspondences $f$ from set $A$ to set $B$ is a mapping?
A. $A=\mathbb{R}, B=\{x | x \text{ is a positive real number}\}, f$: The absolute value of numbers in $A$
B. $A=\{0, 1\}, B=\{-1, 0, 1\}, f$: The square root of numbers in $A$
C. $A=\mathbb{Z}, B=\mathbb{Q}, f$: The reciprocal of numbers ... | D |
2. A square is inscribed in a circle, as shown in the figure. If the area of the circle is $16 \pi \mathrm{cm}^{2}$ and the area of the square is $S \mathrm{~cm}^{2}$, what is the value of $\mathrm{S}$ ? | 32 |
11. (10 points) Above are the building blocks with one star, two stars, and three stars as shown in figures $(a)$, $(b)$, and $(c)$. There are 5 blocks with one star, 2 blocks with two stars, and 1 block with three stars. If several of these blocks are used to form a strip of five stars, how many different ways can thi... | 13 |
A circle with equation $x^2+y^2=1$ intersects the x-axis at two points A and B. Any point P on the curve C satisfies $|PA|+|PB|=2\sqrt{2}$, with O being the origin of the coordinate system.
(I) Find the equation of the curve C;
(II) If a tangent line of the circle intersects the curve C at two points M and N, find the ... | \frac{\sqrt{2}}{2} |
The graphs $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have y-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer x-intercepts. Find $h$. | 36 |
Exercise 2. We have a deck containing 52 cards. Each card has a value among «1, 2, $3,4,5,6,7,8,9,10$, jack, queen, king» and a suit among «hearts, diamonds, spades, clubs», such that for each value and each suit, the deck contains a unique card with this value and this suit. A hand of 5 cards is a selection of 5 cards... | 624 |
Jonas wrote a sequence with the positive multiples of 13 in ascending order.
## $1326395265 \ldots$
a) What is the $2,019^{\text{th}}$ digit of Jonas's sequence?
b) Will the number 2,019 appear in this sequence? | 8 |
3. For any function $y=f(x)$, in the same Cartesian coordinate system, the graphs of the functions $y=f(x-1)$ and $y=f(-x+1)$ are always
A. symmetric with respect to the $x$-axis
B. symmetric with respect to the line $x=1$
C. symmetric with respect to the line $x=-1$
D. symmetric with respect to the $y$-axis | B |
9. The multiple-digit number $\underbrace{202120212021 \cdots 2021}_{n \text { 2021s }} 21$ can be divided by 21, the smallest $n$ is $\qquad$ . $(n>0)$ | 3 |
2. Solve the inequality $\log _{x}\left(25-40 x+16 x^{2}\right)<0$. | x\in(0;1)\cup(1;5/4)\cup(5/4;3/2) |
4. Let $\alpha, \beta, \gamma$ be the three interior angles of an obtuse triangle, then the relationship between $\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}}$ and $\frac{27}{\pi^{2}}$ is
A. $\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}}>\frac{27}{\pi^{2}}$
B. $\frac{1}{\alpha^{2}}+\fra... | A |
2. How many solutions does the equation $\left[2 x-x^{2}\right]+2\{\cos 2 \pi x\}=0$ have? Indicate the smallest and the largest of them. Here $[a]$ is the integer part of the number $a$ - the greatest integer not exceeding $a$, $\{a\}=a-[a]$ is the fractional part of the number $a$. | x_{\}=-\frac{1}{3},x_{\max}=\frac{7}{3} |
## Problema I - 6
Se considera la inecuación
$$
|x-1|<a x
$$
donde $a$ es un parámetro real.
a) Discutir la inecuación según los valores de $a$.
b) Caracterizar los valores de $a$ para los cuales la inecuación tiene exactamente DOS soluciones enteras. | \frac{1}{2}<\leq\frac{2}{3} |
The monotonic increasing interval of the function $f(x)=3+x\ln x$ is $(\quad)$
A: $(0, \frac {1}{e})$
B: $(e,+\infty)$
C: $(\frac {1}{e},+\infty)$
D: $(\frac {1}{e},e)$ | C |
Let $k \geqslant 0$ be given. Determine the minimum of the expression $A=\frac{x}{1+y^{2}}+\frac{y}{1+x^{2}}+\frac{z}{1+t^{2}}+$ $\frac{t}{1+z^{2}}$ under the condition $x+y+z+t=k$ and $x, y, z, t$ positive. | \frac{4k}{4+k^{2}} |
13. The product of the two roots of $\sqrt{2014} x^{\log _{2014} x}=x^{2014}$ is an integer. Find its units digit. | 6 |
A prism has $9$ faces, then it has ______ vertices. | 14 |
1. [3] Suppose that $x$ and $y$ are positive reals such that
$$
x-y^{2}=3, \quad x^{2}+y^{4}=13
$$
Find $x$. | \frac{3+\sqrt{17}}{2} |
400. Milk Delivery. One morning, a milkman was transporting two 80-liter barrels of milk to his shop when he encountered two women who begged him to sell them 2 liters of milk each right away. Mrs. Green had a 5-liter jug, and Mrs. Brown had a 4-liter jug, while the milkman had nothing to measure the milk with.
How di... | 80&76&2&2 |
If $\sqrt{x-2}$ is meaningful, then the range of values for $x$ is ______. | x \geqslant 2 |
50. How many divisors does the integer
$$
N=a^{\alpha} \cdot b^{\beta} \cdot c^{\gamma} \ldots l^{\lambda}
$$
have, where $a, b, c, \ldots l$ are prime numbers and $\alpha, \beta, \gamma, \ldots \lambda$ are any integers? | (\alpha+1)(\beta+1)(\gamma+1)\cdots(\lambda+1) |
Example 6.23. The probability of placing an order in a tailor shop is $p<1.0$. A persistent customer visits all the tailor shops in the city until he succeeds. What is the probability that he will succeed no earlier than on the third attempt, if on average the number of attempts is 5. | 0.64 |
4. Given that $H$ is the orthocenter of acute $\triangle A B C$, the rays $A H, B H, C H$ intersect $B C, C A, A B$ at points $D, E, F$ respectively. Let $B C=a, C A=b, A B=c$. If line segment $E F$ bisects the area of $\triangle A B C$, then $D E^{2}+E F^{2}+F D^{2}=$ ( ).
(A) $a^{2}+b^{2}+c^{2}$
(B) $a^{2}$
(C) $b^{2... | B |
24. C4 (BUL) Let \( T \) be the set of ordered triples \((x, y, z)\), where \( x, y, z \) are integers with \( 0 \leq x, y, z \leq 9 \). Players \( A \) and \( B \) play the following guessing game: Player \( A \) chooses a triple \((x, y, z)\) in \( T \), and Player \( B \) has to discover \( A \)'s triple in as few m... | 3 |
1. Solve the equation: $2^{x^{5}} 4^{x^{4}}+256^{4}=3 \cdot 16^{x^{3}}$. | x=2 |
Example 4. Find the volume of the ellipsoid enclosed by the ellipsoid surface $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$. | \frac{4}{3} \pi a b c |
Lao Li is a well-known expert in fish farming in the local area. He is preparing to contract a fish farm and sign a contract. After calculation and research, it is predicted that the growth rate of fish weight in the first year is $200\%$, and the weight growth rate in the following years is half of the previous year's... | 7 \text{ years} |
Example 4 (1992 "Friendship Cup" International Mathematics Invitational for Grade 8) Find three real numbers $x, y, z$, such that they simultaneously satisfy the following equations: $2 x+3 y+z=13,4 x^{2}+9 y^{2}+z^{2}-2 x+15 y+$ $3 z=82$. | x=3, y=1, z=4 |
Which of the following propositions is true?
A: If $a+b \lt 0$, then $ab \lt 0$
B: Corresponding angles are equal
C: The sum of interior angles of a triangle is $180^{\circ}$
D: Equal angles are vertical angles | C |
Problem C/1
You are traveling in a foreign country whose currency consists of five different-looking kinds of coins. You have several of each coin in your pocket. You remember that the coins are worth 1, 2, 5, 10, and 20 florins, but you have no idea which coin is which and you don't speak the local language. You find ... | 4 |
Let $n > 1$ be an integer and $X = \{1, 2, \cdots , n^2 \}$. If there exist $x, y$ such that $x^2\mid y$ in all subsets of $X$ with $k$ elements, find the least possible value of $k$. | n^2 - n + 1 |
When two numbers are added, the result is -26 . If one of the numbers is 11 , what is the other number?
(A) -37
(B) 37
(C) -15
(D) 15
(E) -48 | -37 |
If \\((x-1)(x-2) < 2\\), then the range of values for \\((x+1)(x-3)\\) is ( )
A: \\((0,3)\\)
B: \\(\left[-4,-3 \right)\\)
C: \\(\left[-4,0 \right)\\)
D: \\(\left( -3,4 \right]\\) | \text{C} |
Problem 9.2. At a party, 24 people gathered. A guest is considered an introvert if they have no more than three acquaintances among the other guests. It turned out that each guest has at least three acquaintances who are introverts. How many introverts could there have been at the party? Provide all possible answers an... | 24 |
2. In a square table $11 \times 11$, we have written the natural numbers $1,2, \ldots, 121$ sequentially from left to right and from top to bottom. Using a square tile $4 \times 4$, we covered exactly 16 cells in all possible ways. How many times was the sum of the 16 covered numbers a perfect square? | 5 |
3. A circle $\omega$ is circumscribed around triangle $A B C$. Circle $\omega_{1}$ touches the line $A B$ at point $A$ and passes through point $C$, while circle $\omega_{2}$ touches the line $A C$ at point $A$ and passes through point $B$. A tangent is drawn from point $A$ to circle $\omega$, which intersects circle $... | \frac{1}{2} |
Let $ n $ $(n\geq2)$ be an integer. Find the greatest possible value of the expression $$E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2}$$ if the positive real numbers $a_1,a_2,\ldots,a_n$ satisfy $a_1+a_2+\ldots+a_n=\frac{n}{2}.$ What are the values of $a_1,a_2,\ldots,a_n$ when the greatest value ... | \frac{2n}{5} |
Given that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, and a circle with diameter $F_{1}F_{2}$ intersects the right branch of the hyperbola at point $P$. The line $PF_{1}$ intersects the hyperbola at point $Q$, and $|PQ| = 2|QF_{1}|$. ... | A |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 8} \frac{\sqrt{9+2 x}-5}{\sqrt[3]{x}-2}$ | \frac{12}{5} |
Which of the following statements is correct? ( )
A: A quadrilateral with a pair of adjacent equal sides is a rhombus.
B: A parallelogram with one right angle is a rectangle.
C: A parallelogram with perpendicular diagonals is a square.
D: A quadrilateral with one pair of parallel sides is a parallelogram. | B |
7. In $\triangle A B C$, the maximum value of $\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$ is | \frac{1}{8} |
140. An example of multiplication. One morning at breakfast, the Crackhams were discussing high matters, when suddenly George asked his sister Dora to quickly multiply
$$
1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 0
$$
How long will it take the reader to find this product? | 0 |
11. Xiao Ming brought 24 handmade souvenirs to sell at the London Olympics. In the morning, each souvenir was sold for 7 pounds, and the number of souvenirs sold was less than half of the total. In the afternoon, he discounted the price of each souvenir, and the discounted price was still an integer. In the afternoon, ... | 8 |
In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$- and $y$- axes, and call a set of points $nice$ if no two of them have the same $x$- or $y$- coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert, Ernie then cho... | 2015 |
Suppose that points $A$ and $B$ lie on circle $\Omega$, and suppose that points $C$ and $D$ are the trisection points of major arc $AB$, with $C$ closer to $B$ than $A$. Let $E$ be the intersection of line $AB$ with the line tangent to $\Omega$ at $C$. Suppose that $DC = 8$ and $DB = 11$. If $DE = a\sqrt{b}$ for intege... | 37 |
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $a_1=1$, $a_{n+1}=\frac{n+2}{n}S_n$ ($n\in\mathbb{N}^*$), prove that the sequence $\left\{\frac{S_n}{n}\right\}$ is a geometric sequence. | q=2 |
B3. In the fog, I see a quarter of my herd of camels. The number of camels that have gone to the hillside is twice the square root of the total number of my herd. In addition, three times five camels are resting by the river. How many camels are there in my herd? | 36 |
20.110 Three circles with centers at $A, B, C$ have radii $r$, where $1<r<2$. The distance between each pair of centers is 2. If $B$ is a point of intersection of circles $A$ and $C$ and lies outside circle $B$, and $C'$ is a point of intersection of circles $A$ and $B$ and lies outside circle $C$, then the length of $... | 1+\sqrt{3(r^2-1)} |
The domain of the function $y=x^{-3}$ and $y=\log_{3}x$ are respectively $P$ and $Q$, then ( )
A: $P \subset Q$
B: $P \cap Q = P$
C: $P \cup \complement_{\mathbb{R}}Q = P$
D: $Q \cap \complement_{\mathbb{R}}P = \emptyset$ | D |
2. The sequence $1,1,2,2,2,2,2, \cdots, k, k, \cdots, k, \cdots$, where each positive integer $k$ appears $3 k-1$ times. Then the 2004th term of this sequence is $\qquad$ | 37 |
If both $x$ and $y$ in the fraction $\frac{2xy}{{x}^{2}+\square}$ are increased by $3$ times, and the value of the fraction remains unchanged, then $\square$ can be ( )
A: $2$
B: $y$
C: $y^{2}$
D: $3y$ | C |
5.75 Find all values of $a$ such that the polynomial
$$x^{3}-6 x^{2}+a x+a$$
has roots $x_{1}, x_{2}, x_{3}$ satisfying
$$\left(x_{1}-3\right)^{3}+\left(x_{2}-3\right)^{3}+\left(x_{3}-3\right)^{3}=0$$ | -9 |
1. Seeing a fox several meters away, the dog chased after it along a straight dirt road. The dog's jump is $23 \%$ longer than the fox's jump. There is a time interval during which both the fox and the dog make a whole number of jumps. Each time it turns out that the dog manages to make $t \%$ fewer jumps than the fox,... | 19 |
2. Let $\left\{a_{n}\right\}_{n \geq 1}$ be an arithmetic sequence and $\left\{g_{n}\right\}_{n \geq 1}$ be a geometric sequence such that the first four terms of $\left\{\bar{a}_{n}+g_{n}\right\}$ are $0,0,1$, and 0 , in that order. What is the 10 th term of $\left\{a_{n}+g_{n}\right\}$ ? | -54 |
Example 7 Given that $n$ is a positive integer, and $n^{2}-71$ is divisible by $7 n+55$. Try to find the value of $n$.
Translating the text into English while preserving the original formatting and line breaks, the result is as follows:
Example 7 Given that $n$ is a positive integer, and $n^{2}-71$ is divisible by $7... | 57 |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{\sin (\pi(x+2))}$ | \frac{1}{2\pi} |
Given vectors $\overrightarrow{a}=(3,-2)$, $\overrightarrow{b}=(-1,4)$, $\overrightarrow{c}=(1,m)$ ($m\in \mathbb{R}$). If $\overrightarrow{c}\perp (\overrightarrow{a}+\overrightarrow{b})$, then $m=\_\_\_\_\_\_$. | -1 |
# Task 7. (14 points)
In a company, there are 168 employees. Among any four people, at least one can be chosen who is acquainted with the other three. What is the minimum possible number of people who are acquainted with everyone? # | 165 |
4B. Given $y=x^{2}-2(k-1) x+k^{2}$, where $k$ is a real number, is a set of parabolas (each number $k$ defines one parabola). Determine the geometric place of the points which are the vertices of the given parabolas. | 2x+1 |
9.087. $\left|3-\log _{2} x\right|<2$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
9.087. $\left|3-\log _{2} x\right|<2$. | x\in(2;32) |
Example 2 Let $P_{i}(i=1,2, \cdots, n)$ be $n$ points on an ellipse, $F$ be the left focus of the ellipse, and the line segments $F P_{1}, F P_{2}$, $\cdots, F P_{n}$ divide the angle at $F$ into $n$ equal parts. Prove:
$$
\sum_{i=1}^{n} \frac{1}{\left|F P_{i}\right|}=\frac{n}{e p} .
$$ | \frac{n}{e p} |
Calculate: $(a^{-2})^{3} = \_\_\_\_\_\_$. | a^{-6} |
The opposite of $-6$ is ____,The absolute value of $-8$ is ____,The reciprocal of $-3$ is ____. | -\frac{1}{3} |
Which of the following descriptions is not an algorithm for solving a problem?
A: To go from Zhongshan to Beijing, first take a bus, then take a train.
B: The steps to solve a linear equation are to eliminate the denominator, remove the brackets, transpose terms, combine like terms, and make the coefficient 1.
C: The e... | C |
Diagonals of a quadrilateral $ABCD$ are equal and meet at point $O$. The perpendicular bisectors to segments $AB$ and $CD$ meet at point $P$, and the perpendicular bisectors to $BC$ and $AD$ meet at point $Q$. Find angle $\angle POQ$.
by A.Zaslavsky | 90^\circ |
A positive integer $n$ is called "strong" if there exists a positive integer $x$ such that $x^{nx} + 1$ is divisible by $2^n$.
a. Prove that $2013$ is strong.
b. If $m$ is strong, determine the smallest $y$ (in terms of $m$) such that $y^{my} + 1$ is divisible by $2^m$. | 2^m - 1 |
$3+$ |
| :---: | :---: |
In the expression $1 * 2 * 3 * . . . * 9$, the asterisks are replaced with minus or plus.
a) Can the result be 0?
b) Can the result be 1?
c) What numbers can result? | )Itcannot;b)itcan;)alloddfrom-45to45 |
4. One of the lateral sides of the trapezoid is perpendicular to the bases and equals $2 R$. A circle is constructed on this side as a diameter, which divides the other lateral side into three segments. The ratio of the lengths of these segments is 12:15:5 (counting from the lower base). Find the area of the trapezoid.... | \frac{7R^{2}}{\sqrt{15}} |
15. (1999 American High School Mathematics Examination) Given six points on a circle, any combination of four chords can form a convex quadrilateral. The probability that they form a convex quadrilateral is ( ).
A. $\frac{1}{15}$
B. $\frac{1}{91}$
C. $\frac{1}{273}$
D. $\frac{1}{455}$
E. $\frac{1}{1365}$ | \frac{1}{91} |
If {**a**, **b**, **c**} is a basis in space, then among the following options, the set of vectors that can form a basis is
A: **a**, **a**＋**b**, **a**－**b**
B: **b**, **a**＋**b**, **a**－**b**
C: **c**, **a**＋**b**, **a**－**b**
D: **a**＋**b**, **a**－**b**, **a**＋2**b** | C |
Question 2. Given non-negative real numbers $a_{1} \geq a_{2} \geq \ldots \geq a_{n}, b_{1} \leq b_{2} \leq \ldots \leq b_{n}$, satisfying
$$
a_{1} a_{n}+a_{2} a_{n-1}+\ldots+a_{n} a_{1}=b_{1} b_{n}+b_{2} b_{n-1}+\ldots+b_{n} b_{1}=1 \text {, }
$$
Find
$$
S=\sum_{1 \leq i<j \leq n} a_{i} b_{j}
$$
the minimum value. | \frac{n-1}{2} |
3 [ Shortest path on the surface ]
A sphere of radius 2 is intersected by a plane at a distance of 1 from the center. Find the length of the shortest path on the surface of the sphere between the two most distant points of the intersection.
# | \frac{4\pi}{3} |
There are 4 different Chinese books and 3 different math books. If 2 books are randomly selected from these, find the probability of the following events:
(1) Event A: Both selected books are math books;
(2) Event B: One selected book is a Chinese book, and the other is a math book. | \frac{4}{7} |
In the complex plane, the point corresponding to the complex number $\dfrac {1+i}{1-i}+i^{2012}$ is located in which quadrant?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | \text{A} |
Which of the following is closest to one million $\left(10^{6}\right)$ seconds?
(A) 1 day
(B) 10 days
(C) 100 days
(D) 1 year
(E) 10 years | 10 |
The lines $l\_1$: $x+my-2=0$ and $l\_2$: $2x+(1-m)y+2=0$ are parallel, find the value of $m$ ___. | \frac{1}{3} |
187. Elementary arithmetic. Here is one question similar to those that were so popular in Venice (and not only there) in the mid-16th century. Their appearance was largely due to Niccolò Fontana, better known as Tartaglia (the stammerer).
If a quarter of twenty were four, then what would a third of ten be? | 2\frac{2}{3} |
Given $a+3b=0$, find the value of $a^{3}+3a^{2}b-2a-6b-5$. | -5 |
Let $f(n) = 2 + 2^4 + 2^7 + 2^{10} + \ldots + 2^{3n+1}$, where $n$ is a natural number. Find the value of $f(n)$. | \frac{2}{7}(8^{n+1}-1) |
Find the solution set of the inequality:
$$
\left|1+x+ \frac {x^{2}}{2}\right| < 1
$$
The solution set is:
A: $$\{x|-1 < x < 0\}$$
B: $$\{x|- \frac {3}{2} < x < 0\}$$
C: $$\{x|- \frac {5}{4} < x < 0\}$$
D: $$\{x|-2 < x < 0\}$$ | D |
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