problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given the function $f(x) = \begin{cases} x^{a} & (x\geq 0) \\ |x-2| & (x<0) \end{cases}$, and $f(-2) = f(2)$, then $f(4) =$ （ ）
A: 2
B: 4
C: 8
D: 16 | D |
1. Given the set
$$
M=\{1,2, \cdots, 2020\},
$$
for any non-empty subset $A$ of $M, \lambda_{A}$ is the sum of the largest and smallest numbers in the set $A$. Then the arithmetic mean of all such $\lambda_{A}$ is $\qquad$ . | 2021 |
Let $a_{10} = 10$, and for each positive integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least positive $n > 10$ such that $a_n$ is a multiple of $99$. | 45 |
Given the circle $x^{2}+y^{2}+2x-2y+a=0$ intersects the line $x+y+2=0$ to form a chord of length $4$, the value of the real number $a$ is ( ).
A: $-2$
B: $-4$
C: $-6$
D: $-8$ | B |
Given the parabola $C: y^2 = kx$ ($k > 0$) with focus $F$, and a moving point $N$ on the parabola, the point $M(1, \sqrt{2})$ is not on the parabola.
(1) If $k=4$, find the minimum value of $|MN| + |NF|$;
(2) Let $p: 2k^2 - 11k + 5 < 0$, and $q$: the line segment $MF$ intersects the parabola $C$. If $p \land q$ is ... | \left(\frac{1}{2}, 2\right) |
According to the following conditions, the only triangle $\triangle ABC$ that can be constructed is ( )
A: $AB=3$, $BC=4$, $CA=8$
B: $AB=4$, $BC=3$, $\angle A=60^{\circ}$
C: $\angle A=60^{\circ}$, $\angle B=45^{\circ}$, $AB=4$
D: $\angle C=90^{\circ}$, $\angle B=30^{\circ}$, $\angle A=60^{\circ}$ | C |
7. (10 points) 30 tigers and 30 foxes are divided into 20 groups, with 3 animals in each group. Tigers always tell the truth, while foxes always lie. When asked if there are any foxes in the group, 39 out of the 60 animals answered “no”. How many groups consist of 3 tigers?
Groups. | 3 |
115. In how many non-empty subsets of the set $\{1,2,3, \ldots, 10\}$ will there be no two consecutive numbers? | 143 |
Given the parabola $y^{2}=2 p x$ and the point $P(3 p, 0)$. Determine the point $Q$ on the parabola such that the distance $P Q$ is minimal. | Q_{1}(2p,2p)Q_{2}(2p,-2p) |
4. An ant walks on the numeric keypad of a mobile phone, which consists of 10 buttons arranged as shown in the figure. The ant always moves from the button it is on to an adjacent button horizontally or vertically; it starts from the button 1 and walks around the keypad for a while, finally stopping on the button 0, wh... | 3 |
1. Given $\mathrm{i}$ is the imaginary unit. Then the complex number $\frac{1+2 \mathrm{i}}{\mathrm{i}-2}=$ ( ).
(A)i
(B) $-\mathrm{i}$
(C) $-\frac{4}{5}-\frac{3}{5} i$
(D) $-\frac{4}{5}+\frac{3}{5}$ i | B |
In the Mathematical Competition of HMS (Hellenic Mathematical Society) take part boys and girls who are divided into two groups : [i]Juniors[/i] and [i]seniors.[/i]The number of the boys taking part of this year competition is 55% of the number of all participants. The ratio of the number of... | \frac{11}{9} |
Given the sets $A={2,m}$ and $B={1,m^2}$, if $A\cup B={1,2,3,9}$, then the real number $m=$ . | 3 |
7.098. $\lg \sqrt{x-3}+\lg \sqrt{x+3}=2-0.5 \lg 625$. | 5 |
9. A team of 8 people participates in a sports tournament. The rules stipulate that there must always be 5 players on the field for each team and that, during each match (which lasts 60 minutes), all 8 members of each team must play the same number of minutes. For how many minutes will each player be on the field durin... | 37.5 |
\section*{Problem 1}
A book contains 30 stories. Each story has a different number of pages under 31. The first story starts on page 1 and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?
| 23 |
Example 6. Given a three-digit integer that is a multiple of 5, the sum of its digits is 20, and the sum of one digit and the hundreds digit is a multiple of 3, find this integer. | 785 |
Given $a=(\frac{1}{2})^{\frac{1}{3}}$, $b=\log_{\2}\frac{1}{3}$, and $c=\log_{\2}3$, determine the correct relationship among $a$, $b$, and $c$.
A) $a > b > c$
B) $c > a > b$
C) $a > c > b$
D) $c > b > a$ | \text{B} |
2.11 people jointly manage an arsenal, and some locks are added to the door, and each person is given several keys. To ensure that any 6 people together can open all the locks, but any 5 people together cannot open all the locks, how many locks should be installed on the door at least? How should the keys be distribute... | C_{11}^{5} |
It's pouring down rain, and the amount of rain hitting point $(x,y)$ is given by
$$f(x,y)=|x^3+2x^2y-5xy^2-6y^3|.$$
If you start at the origin $(0,0)$, find all the possibilities for $m$ such that $y=mx$ is a straight line along which you could walk without any rain falling on you. | -1, \frac{1}{2}, -\frac{1}{3} |
In the sequence $\{a_{n}\}$, where $a_{1}=4$, $a_{n} \gt 0$, and the sum of the first $n$ terms is $S_{n}$, if $a_{n}=\sqrt{{S}_{n}}+\sqrt{{S}_{n-1}}$, $(n\in N^{*}$, $n\geqslant 2)$. Find:$(1)$ the general formula for the sequence $\{a_{n}\}$;$(2)$ if the sum of the first $n$ terms of the sequence $\{\frac{1}{{a}_{n}{... | \frac{1}{20}\leqslant T_{n} < \frac{3}{20} |
The numbers 1447, 1005, and 1231 have something in common: each is a four-digit number beginning with 1 that has exactly two identical digits. How many such numbers are there? | 432 |
8. Let $F$ be the set of all sequences of the form $\left(A_{1}, A_{2}, \cdots, A_{n}\right)$, where $A_{i}$ is a subset of $B=\{1,2, \cdots, 10\}$, and let $|A|$ denote the number of elements in the set $A$. Then $\sum_{F}\left|\bigcup_{i=1}^{n} A_{i}\right|=$ $\qquad$ . | 10\left(2^{10 n}-2^{9 n}\right) |
Given sets $A = \{x \mid a-1 < x < a+1, x \in \mathbb{R}\}$ and $B = \{x \mid 1 < x < 5, x \in \mathbb{R}\}$:
1. If $a=1$, find $A \cap B$.
2. If $A \subseteq A \cap B$, find the range of values for $a$. | 2 \leq a \leq 4 |
Three circles, each of radius 3, are drawn with centers at $(14,92)$, $(17,76)$, and $(19,84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absol... | 24 |
10. Rectangle $ABCD$ with sides $AB=1$ and $AD=10$ serves as the base of pyramid $SABCD$, and the edge $SA=4$ is perpendicular to the base. Find a point $M$ on edge $AD$ such that triangle $SMC$ has the smallest perimeter. Find the area of this triangle.
# | 6 |
10. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a smooth function such that $f^{\prime}(x)=f(1-x)$ for all $x$ and $f(0)=1$. Find $f(1)$. | \1+\tan1 |
7. The sum of the reciprocals of all positive integers $n$ that make $\left[\frac{n^{2}}{5}\right]$ a prime number is . $\qquad$ | \frac{37}{60} |
Given that $α$ is an angle in the third quadrant, $f(α)= \dfrac {\sin (α- \dfrac {π}{2})\cos ( \dfrac {3π}{2}+α)\tan (π-α)}{\tan (-α-π)\sin (-α-π)}$.
(1) Simplify $f(α)$;
(2) If $\cos (α- \dfrac {3π}{2})= \dfrac {1}{5}$, find the value of $f(α)$. | \dfrac {2 \sqrt {6}}{5} |
The necessary and sufficient condition for the vectors $\overrightarrow{a}$ ($\overrightarrow{a} \neq \overrightarrow{0}$) and $\overrightarrow{b}$ to be collinear is: there exists a unique real number $\lambda$ such that ______. | \overrightarrow{b} = \lambda \overrightarrow{a} |
Given the universal set \\(U=\{0,1,2,3,4,5\}\\), and the sets \\(A=\{1,2,3,5\}\\) and \\(B=\{2,4\}\\), then \\((C_{U}A) \cup B =\) \()
A: \\(\{0,2,3,4\}\\)
B: \\(\{4\}\\)
C: \\(\{1,2,4\}\\)
D: \\(\{0,2,4\}\\) | D |
The sequence $\{a\_n\}$ is an arithmetic sequence with a non-zero common difference. Furthermore, $a\_1$, $a\_3$, and $a\_7$ are three consecutive terms in a geometric sequence $\{b\_n\}$. Find the common ratio of the geometric sequence $\{b\_n\}$.
A. $\sqrt{2}$
B. $4$
C. $2$
D. $\frac{1}{2}$ | C |
In triangle $ABC$, let the sides opposite angles $A$, $B$, and $C$ be $a$, $b$, and $c$, respectively. Given vectors $\overrightarrow{m} = (2\sin{B}, -\sqrt{3})$ and $\overrightarrow{n} = (\cos{2B}, 2\cos^2{\frac{B}{2}} - 1)$, and knowing that $\overrightarrow{m} \parallel \overrightarrow{n}$,
(Ⅰ) Find the measure of t... | \sqrt{3} |
4.5 If $a \geqslant 2, \quad b>2$, then which of the following expressions is correct?
(A) $a b < a+b$.
(B) $a b \leqslant a+b$, equality can be achieved.
(C) $a b > a+b$.
(D) $a b \geqslant a+b$, equality can be achieved.
(1st "Five Sheep Cup" Junior High School Mathematics Competition, 1989) | C |
. Let $p$ be a prime number.
a) Show that if $a \equiv b(\bmod p)$, then $a^{p} \equiv b^{p}\left(\bmod p^{2}\right)$.
b) How many integers $n \in\left\{1,2, \ldots, p^{2}\right\}$ are there for which the equation $x^{p} \equiv n\left(\bmod p^{2}\right)$ has at least one solution? | p |
A [i]substring [/i] of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ that have the property that any substring of $p$ is also prime.
| 576 |
Consider a sequence $\{a_n\}$, where $a_1 = \frac{1}{2}$, $a_{n+1} = \frac{a_n^2}{a_n^2 - a_n + 1}$ ($n\in \mathbb{N}^*$).
(Ⅰ) Prove that $a_{n+1} < a_n$;
(Ⅱ) Let $S_n$ denote the sum of the first n terms of the sequence $\{a_n\}$. Prove that $S_n < 1$. | S_n < 1 |
Arrange 5 products A, B, C, D, E in a row on a shelf. The number of arrangements where A and B are not next to each other is ( )
A: 18
B: 24
C: 36
D: 72 | D |
A school needs to arrange 6 intern teachers into three classes, A, B, and C, with 2 teachers per class. It is known that Teacher A cannot be arranged into Class A, and Teachers B and C cannot be arranged into the same class. How many possible arrangements are there?
A: 24
B: 36
C: 48
D: 72 | C |
For an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is denoted as $S_n$. Given $a_3 = 3$ and $S_4 = 10$, find the value of $\sum\limits_{k=1}^{n} \frac{1}{S_k}$. | \sum\limits_{k=1}^{n} \frac{1}{S_k} = \frac{2n}{n+1} |
4. (20 points) Before the geometry lesson, the teacher wrote on the board the values of all the angles (in degrees) of a certain convex polygon. However, the duty students erased one of the written numbers. When the lesson began, it turned out that the sum of the remaining numbers was 1703. What number did the duty stu... | 97 |
11. Vera bought 6 fewer notebooks than Misha and Vasya together, and Vasya bought 10 fewer notebooks than Vera and Misha together. How many notebooks did Misha buy? | 8 |
Given $-\frac{π}{2} \lt x \lt 0$, $\sin x+\cos x=\frac{1}{5}$. Find:
$(1)$ $\sin x\cos x$;
$(2)$ the value of $\sin x-\cos x$. | -\frac{7}{5} |
Given that $(1+\sin t)(1+\cos t)=5/4$ and \[ (1-\sin t)(1-\cos t)=\frac mn-\sqrt{k}, \] where $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$ | 27 |
34. (1984 American Invitational Mathematics Examination) A gardener is to plant three maple trees, four oaks, and five birch trees in a row. He randomly determines the order of these trees, with all different arrangements being equally likely. Express the probability that no two birch trees are adjacent as $\frac{m}{n}... | 106 |
Given that the random variable $X$ follows $N(1, \sigma ^{2})$, if $P(X \geq 0) = 0.8$, then $P(1 \leq X < 2) = \_\_\_\_\_\_$. | 0.3 |
In $\triangle ABC$, the ratio of $\sin A : \sin B : \sin C = 3 : 2 : 4$. Find $\cos C$. | \cos C = -\frac{1}{4} |
Given the function f(x) = |x - 2| + |x - 1|.
(1) Find the solution set for the inequality f(x) ≤ 7.
(3) If the minimum value of the function g(x) = x^2 - 2x + |a^2 - 3| is not less than the minimum value of f(x), find the range of values for a. | a \in (-\infty, -\sqrt{5}] \cup [-1, 1] \cup [\sqrt{5}, \infty) |
90. On a plane, there are 4 points $A_{1}, A_{2}, A_{3}, A_{4}$, the distance between any two of which is not less than 1. What is the maximum possible number of line segments $A_{i} A_{j}$ of length 1 connecting these points pairwise? | 5 |
"x^2 - 2x < 0" is a ( ) condition for "0 < x < 4".
A: Necessary and sufficient
B: Sufficient but not necessary
C: Necessary but not sufficient
D: Neither sufficient nor necessary | \text{B} |
Given that $i$ is the imaginary unit, find the product of $(1-2i)(2+i)=$ \_\_\_\_\_\_. | 4-3i |
Among the following angles, the one whose terminal side is the same as $330^{\circ}$ is ( )
A: $30^{\circ}$
B: $-30^{\circ}$
C: $630^{\circ}$
D: $-630^{\circ}$ | B |
## Task Condition
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{1+2+\ldots+n}{n-n^{2}+3}$ | -\frac{1}{2} |
The sum of six consecutive positive integers is 2013. What is the largest of these six integers?
$\textbf{(A)}\ 335 \qquad \textbf{(B)}\ 338 \qquad \textbf{(C)}\ 340 \qquad \textbf{(D)}\ 345 \qquad \textbf{(E)}\ 350$ | 338 |
Suppose $a$ is a real number such that the equation \[a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)}\]
has more than one solution in the interval $(0, \pi)$. The set of all such $a$ that can be written
in the form \[(p,q) \cup (q,r),\]
where $p, q,$ and $r$ are real numbers with $p < q< r$. What is $p+q+r$?
$\textbf{(A) } {-}... | -4 |
Example 4. Team A and Team B each send out 7 players to participate in a Go competition according to a pre-arranged order. Both sides start with Player 1 competing, the loser is eliminated, and the winner then competes with the next (Player 2) of the losing side, $\cdots \cdots \cdots$, until one side is completely eli... | 2 C_{13}^{6} |
List the set $A = \{ x | x^2 - 3x - 4 < 0, x \in \mathbb{Z} \}$ using the enumeration method. | A = \{0, 1, 2, 3\} |
06.3. A sequence of positive integers $\left\{a_{n}\right\}$ is given by
$$
a_{0}=m \quad \text { and } \quad a_{n+1}=a_{n}^{5}+487
$$
for all $n \geq 0$. Determine all values of $m$ for which the sequence contains as many square numbers as possible.
| 9 |
Calculate $\left(2+1\right)(2^{2}+1)(2^{4}+1)(2^{8}+1)\ldots (2^{64}+1)$, and the unit digit of the result is ( ).
A: $6$
B: $5$
C: $8$
D: $7$ | B |
Suppose $P(x)$ is a monic polynomial of degree $2023$ such that $P(k) = k^{2023}P(1-\frac{1}{k})$ for every positive integer $1 \leq k \leq 2023$. Then $P(-1) = \frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Compute the unique integer $0 \leq n < 2027$ such that $bn-a$ is divisible by the prime $2027$. | 0 |
Given that $x_{1}$ and $x_{2}$ are two real roots of the equation $2x^{2}+kx-2=0$, and $(x_{1}-2)(x_{2}-2)=10$, find the value of $k$. | 7 |
Example 7 As shown in Figure 1.5.7, given that the area of quadrilateral $A B C D$ is equal to $1 ; H, G$ and $E, F$ are the $n$-division points (closer to the endpoints) of $A D$ and $B C$ respectively. Prove: The area of quadrilateral $E F G H$ is $\frac{n-2}{n}$. (A generalization of Example 4 in Chapter 4 of this b... | \frac{n-2}{n} |
[ The ratio in which the bisector divides the side. [The ratio of linear elements of similar triangles] A point on the hypotenuse, equidistant from both legs, divides the hypotenuse into segments of length 30 and 40. Find the legs of the triangle.
# | 4256 |
Given the plane vectors $a=(2m+1,3)$ and $b=(2,m)$, and $a\parallel b$, then the real number $m=$ ( )
A: $2$ or $-\dfrac{3}{2}$
B: $\dfrac{3}{2}$
C: $-2$ or $\dfrac{3}{2}$
D: $-\dfrac{2}{7}$ | C |
4. Through a random point inside a given triangle, three lines parallel to the sides of the triangle pass. These lines divide the area of the triangle into six parts, three of which are triangles with areas $s_{1}, s_{2}, s_{3}$. Determine the area of the given triangle. | (\sqrt{s_{1}}+\sqrt{s_{2}}+\sqrt{s_{3}})^{2} |
To prove "In the angles of a triangle, there is at least one angle that is greater than or equal to 60°" using the method of contradiction, one should first assume ( )
A: Every angle of the triangle is less than 60°
B: There is an angle in the triangle that is greater than 60°
C: Every angle of the triangle is greater... | A |
Given sets $A=\{x|x^{2}-3x \lt 0\}$ and $B=\{x|y=\ln (x-2)\}$, then $A\cap B=$
A: $(2,+\infty)$
B: $(2,3)$
C: $(3,+\infty)$
D: $(-\infty ,2)$ | \text{B} |
5. There are two alloys. The first alloy contains $p \%$ impurities, and the second - respectively $q \%$ impurities. Determine the proportion in which these alloys should be combined to obtain a new alloy containing $r \%$ impurities. In your answer, indicate the ratio of the mass of the first alloy to the mass of the... | 1.17 |
A circle $k$ has its center at $O$, and points $A, B, C, D$ are consecutive points on the circumference of $k$, such that $\angle AOB = \angle BOC = \angle COD = \alpha < 60^{\circ}$. The projection of $D$ onto the diameter $AO$ is $E$, and the point $F$ is the closer trisection point of segment $DE$ to $E$. Finally, t... | 60 |
A school has six classes in one day: Chinese, Mathematics, Foreign Language, History, Politics, and Physical Education. Among all possible arrangements, what is the probability that Mathematics is not the last class, and Physical Education is not the first class? | \frac{7}{10} |
Pick out three numbers from $0,1,\cdots,9$, their sum is an even number and not less than $10$. We have________different ways to pick numbers. | 51 |
1. Given $x>0, y>0, x^{2}+y^{2}=1$. Find the minimum value of $\frac{x^{6}+y^{6}}{x y}$. | \frac{1}{2} |
The equation of the tangent line to the curve $y = x^2 + \frac{1}{x}$ at the point $(1,2)$ is __________. | x - y + 1 = 0 |
Determine all integers $n \geq 1$ for which there exists a pair $(a, b)$ of positive integers with the following properties:
i) No third power of a prime divides $a^{2}+b+3$.
ii) It holds that $\frac{a b+3 b+8}{a^{2}+b+3}=n$.
Answer: The only integer with these properties is $n=2$. | 2 |
Suppose $ABCD$ is a trapezoid with $AB\parallel CD$ and $AB\perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB=10$ and $CD=15$, compute the maximum possible value of $XY$. | 6 |
Given that the center of the hyperbola $C$ is at the origin, the left focus is at $(-2\sqrt{5}$,$0)$, and the eccentricity is $\sqrt{5}$.
$(1)$ Find the equation of $C$;
$(2)$ Let the left and right vertices of $C$ be $A_{1}$ and $ $A_{2}$ respectively. A line passing through the point $\left(-4,0\right)$ intersect... | x = -1 |
3. Three positive integers $a$, $b$, and $c$ satisfy the conditions:
(1) $a<b<c<30$;
(2) For some positive integer base, the logarithms of $a(2 b-a)$ and $c^{2}+$ $60 b-11 a$ are 9 and 11, respectively.
Then the value of $a-2 b+c$ is $\qquad$. | -4 |
Problem 8.8.1. Different positive numbers $a, b, c$ are such that
$$
\left\{\begin{array}{l}
a^{2}+b c=115 \\
b^{2}+a c=127 \\
c^{2}+a b=115
\end{array}\right.
$$
Find $a+b+c$. | 22 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. It is given that $3a\cos C=2c\cos A$ and $\tan C= \frac {1}{2}$.
(Ⅰ) Find $B$;
(Ⅱ) If $b=5$, find the area of $\triangle ABC$. | S_{\triangle ABC} = \frac {5}{2} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. The value of $a\cos C + c\cos A$ is ( )
A: $b$
B: $c$
C: $2\cos B$
D: $2\sin B$ | A |
Let $a$ and $b$ be such that $a+b=13$ and $a b=41$. Calculate $a^{3}+b^{3}$. | 598 |
1.35 A four-digit number is composed of four consecutive digits in sequence. If the first two digits from the left are swapped, the resulting number is a perfect square. Find the four-digit number.
保留源文本的换行和格式，直接输出翻译结果。 | 3456 |
Four. (50 points) The International Mathematical Olympiad Chief Examination Committee has $n$ countries participating, with each country being represented by a team leader and a deputy team leader. Before the meeting, the participants shake hands with each other, but the team leader does not shake hands with their own ... | m=0, n=50 |
Find $\frac{area(CDF)}{area(CEF)}$ in the figure.
[asy]
/* File unicodetex not found. */
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(5.75cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(... | \frac{14}{15} |
A $6 \times 6$ board is given such that each unit square is either red or green. It is known that there are no $4$ adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A $2 \times 2$ subsquare of the board is [i]chesslike[/i] if it has one red and one green diagonal. Find the maximal pos... | 25 |
Let's write down the consecutive terms of the sequence of natural numbers next to each other. Determine the 10000th digit of this number sequence. | 7 |
18 Let $a, b, c, d, m, n \in \mathbf{N}, a^{2}+b^{2}+c^{2}+d^{2}=1989, a+b+c+d=m^{2}$, and the largest of $a, b, c, d$ is $n^{2}$, determine and prove the values of $m, n$.
Determine and prove the values of $m, n$. | =9,n=6 |
How many distinct four letter arrangements can be formed by rearranging the letters found in the word [b]FLUFFY[/b]? For example, FLYF and ULFY are two possible arrangements. | 72 |
For which values of $p$ and $q$ integers strictly positive does $pq$ divide $3(p-1)(q-1) ?$ | (6,5),(4,9),(2,3) |
$1 \cdot 19$ For a given $n \in N$, find the number of all different natural number triples whose sum is $6 n$.
The above text has been translated into English, preserving the original text's line breaks and format. | 3n^2 |
6. Two acute angles $\alpha$ and $\beta$ satisfy the condition $\operatorname{Sin}^{2} \alpha+\operatorname{Sin}^{2} \beta=\operatorname{Sin}(\alpha+\beta)$. Find the sum of the angles $\alpha+\beta$ in degrees. | 90 |
69*. Determine the five last digits of the number
$$
\left.N=9^{\left(9^{(} \cdot 9^{\left(9^{9}\right)}\right)} \ldots\right),
$$
written using 1001 nines in a similar manner to the number in problem 68). | 45289 |
Given the sets $A={a+3, \log_{2}(a+1)}$ and $B={1, b}$, if $A=B$, find the value of $b$. | 4 |
Given a function $f(x)$ with domain $\mathbb{R}$ that satisfies $f(x) + f(x+2) = 2x^2 + 3$, and $f(x+1) - f(x-1) = 2x - 1$. If $f(t-1)$, $1$, and $f(t)$ form an arithmetic sequence, then the value of $t$ is. | 1 |
Given $a > 0$ and $a \neq 1$, $b > 0$ and $b \neq 1$, if for any values of $a$ and $b$ within the given range, the function $y=x+\log_{a}(x-2)$ and the function $y=b^{x-c}+2$ always pass through the same fixed point, then the real number $c=$ ______. | 3 |
To maximize the result of the expression $\sqrt{8}\square \sqrt{2}$, the operator to be filled in the square is ( ).
A: $+$
B: $-$
C: $\times $
D: $\div $ | \text{A} |
2. What is the sum of all of the prime numbers that are greater than 10 and less than 20? (A prime number is a positive integer greater than 1 whose only positive divisors are 1 and itself. For example. 7 is a prime number.) | 60 |
Let the complex number $z = m^2 - 2m - 3 + (m^2 + 3m + 2)i$. Determine the values of the real number $m$ for which:
(1) $z$ is a real number;
(2) $z$ is a pure imaginary number;
(3) The point corresponding to $z$ is located in the second quadrant of the complex plane. | -1 < m < 3 |
14. A teacher assigned the task of drawing a triangle $A B C$ at will on the notebook, drawing the three altitudes $A A^{\prime}, B B^{\prime}, C C^{\prime}$ and measuring their respective lengths. Five students found the measurements reported in the answers below. Which of them has definitely made a mistake?
(A) $\ov... | E |
12. Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\frac{2}{3}$, $A_{1}, A_{2}$ are the left and right vertices of the ellipse $C$, $B$ is the upper vertex of the ellipse, $F_{1}$ is the left focus of the ellipse $C$, and the area of $\triangle A_{1} F_{1} B$ is $\frac{... | \frac{1}{2} |
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