problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
The range of the angle $\theta$ formed by two skew lines is ( )
A: $0^\circ < \theta < 90^\circ$
B: $0^\circ < \theta < 180^\circ$
C: $0^\circ < \theta \leq 90^\circ$
D: $0^\circ \leq \theta < 90^\circ$ | \text{C} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an angle of $120^{\circ}$ between them, and $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=3$. Find:
(Ⅰ) $\overrightarrow{a} \cdot \overrightarrow{b}$;
(Ⅱ) $|\overrightarrow{a} + 2\overrightarrow{b}|$. | 2\sqrt{7} |
Given that the even function $f(x)=x^{2}+(a-b)x+1$ is defined on the interval $[a-1,2a+4]$, determine the solution set for the inequality $f(x) > f(b)$.
A: $[1,2]$
B: $[-2,-1]$
C: $(1,2]$
D: $[-2,-1)∪(1,2]$ | D |
$7 \cdot 77$ Given five functions:
(1) $f(x)=0, x \in R$;
(2) $f(x)=x^{2},-3<x \leqslant 3$;
(3) $f(x)=\log _{2}|x|, \quad x \neq 0, \quad x \in R$;
(4) $f(x)=(1+x)^{2 n}-(1-x)^{2 n}, \quad x \in R, n \in N$;
(5) $f(x)=\sin (\sin x), x \in R$.
Among them, the number of odd functions is
(A) 1.
(B) 2.
(C) 3.
(D) 4.
(2nd... | C |
# Task 8.2 (7 points)
The shares of the company "Nu-i-Nu" increase in price by 10 percent every day. Businessman Borya bought shares of the company for 1000 rubles every day for three days in a row, and on the fourth day, he sold them all. How much money did he make from this operation? | 641 |
17. A five-digit number is divisible by 2025, and the quotient is exactly the sum of the digits of this five-digit number. There are $\qquad$ such five-digit numbers. | 2 |
The equations of the sides of a triangle are
$$
y=x+1, \quad y=-\frac{1}{2} x-2, \quad y=3 x-9
$$
Determine the equation of the circumscribed circle. | (x-2)^{2}+(y-2)^{2}=25 |
Given that the sum of all coefficients in the expansion of ($\frac{1}{x} + \sqrt{x})^n$ is 64, determine the constant term in its expansion. | 15 |
V OM - I - Problem 7
In the plane, a line $ p $ and points $ A $ and $ B $ are given. Find a point $ M $ on the line $ p $ such that the sum of the squares $ AM^2 + BM^2 $ is minimized. | M |
6. Let $M=\{1,2,3, \cdots, 1995\}, A$ be a subset of $M$ and satisfy the condition: when $x \in$ $A$, $15 x \notin A$, then the maximum number of elements in $A$ is $\qquad$ . | 1870 |
Given that \\(f(x)\\) is an odd function, and for any \\(x \in \mathbb{R}\\), it satisfies \\(f(2-x) = f(x)\\). When \\(0 < x \leqslant 1\\), \\(f(x) = \ln x + 2\\). Find the number of zeros of the function \\(y = f(x)\\) on the interval \\((-2, 4]\).
A: \\(7\\)
B: \\(8\\)
C: \\(9\\)
D: \\(10\\) | C |
Given the complex number $z= \frac{2-i}{1+i}$ (where $i$ is the imaginary unit), the real part of the complex number $z$ is ( )
A: $$- \frac{3}{2}$$
B: $$- \frac{1}{2}$$
C: $$\frac{1}{2}$$
D: $$\frac{3}{2}$$ | \text{C} |
4. If in the sequence $\left\{a_{n}\right\}$, $a_{n}=\frac{100^{n}}{n!}$, then
A. $\left\{a_{n}\right\}$ is an increasing sequence
B. $\left\{a_{n}\right\}$ is a decreasing sequence
C. It becomes decreasing from a certain term
D. It becomes increasing from a certain term | C |
For example, $3 z \in \mathbf{C}, z^{7}=1(z \neq 1), z$ has an argument of $\alpha$, find the value of $\cos \alpha+\cos 2 \alpha+\cos 4 \alpha$. | -\frac{1}{2} |
A class has 5 students participating in the duty roster from Monday to Friday, with one student arranged each day. Among them, student A can only be arranged on Monday or Tuesday, and student B cannot be arranged on Friday. The number of different duty arrangements for them is ( )
A: 288
B: 72
C: 42
D: 36 | D |
Define the function $f(x)$ on $\mathbb{R}$ such that $f(x+y) = f(x) + f(y) + 2xy$ for $x, y \in \mathbb{R}$, and $f(1) = 2$. Find the value of $f(-3)$. | 6 |
Given the complex number $z= \frac {1+i}{i}$ (where $i$ is the imaginary unit), find the imaginary part of the complex number $z$.
A: $i$
B: $-i$
C: $1$
D: $-1$ | \text{D} |
6. The sphere touches all the edges of the tetrahedron, two opposite edges of which are equal to $a$ and $b$, and all other edges are equal to each other. Find the radius of this sphere. | \frac{\sqrt{2ab}}{4} |
Let $f(x) = |2 - x^2|$, if $0 < a < b$, and $f(a) = f(b)$, then the range of values for $ab$ is ( )
A: $(0, 2)$
B: $(0, 2]$
C: $(0, 4]$
D: $(0, 4)$ | A |
Example 6 Given that the sum of several positive integers is 1976, find the maximum value of their product. | 2\times3^{658} |
9. Can swapping any two digits of 1234567891011 result in a perfect square? Please explain your reasoning. | No |
2. Find the real numbers $a, b, c>0$ if
$$
\lim _{n \rightarrow \infty}\left(\sqrt{a^{2} n^{2}+2014 n+1}-b n+c\right)=\sqrt{\frac{2 c}{a}} \cdot \sqrt{2014} \text { and } a+c=72 \text {. }
$$
Prof.Voiculeț Septimius, Videle | (=53,b=53,=19),(=19,b=19,=53) |
Among the following functions, which is both an odd function and monotonically increasing on \\((-\\infty ,0)\\)?
A: \\(y={{x}^{-1}}\\)
B: \\(y={{x}^{2}}\\)
C: \\(y={{x}^{\\frac{1}{3}}}\\)
D: \\(y={{x}^{-\\frac{1}{2}}}\\) | \text{C} |
Factorization: $m^{2}-9=\_\_\_\_\_\_.$ | (m+3)(m-3) |
Among \\(3.14\\), \\(\sqrt{4}\\), \\(\dfrac{22}{7}\\), \\(-\sqrt{3}\\), \\(2\pi\\), and \\(38\\), there are \\((\quad)\) irrational numbers.
A: \\(1\\) number
B: \\(2\\) numbers
C: \\(3\\) numbers
D: \\(4\\) numbers | \text{B} |
The coordinates of the vertex of the parabola $y=2x^2-4x-7$ are. | (1, -9) |
A2. Lucijana chose 3 different numbers from the set $\{1,2,3,4,5,6,7,8,9\}$. With these, she wrote the largest possible three-digit number and the smallest possible three-digit number. She added the obtained numbers and got the number 545. What is the sum of the 3 numbers that Lucijana chose?
(A) 6
(B) 7
(C) 9
(D) 11
(... | 7 |
Given that $\sin \alpha + \cos \alpha = \frac{1}{5}$, where $0 \leqslant \alpha \leqslant \pi$, the value of $\sqrt{2}\sin \left( 2\alpha -\frac{\pi }{4} \right)$ is ( )
A: $-\frac{31}{25}$
B: $-\frac{17}{25}$
C: $\pm \frac{31}{25}$
D: $\pm \frac{17}{25}$ | B |
4. (10 points) Each basketball game is divided into four quarters. In a certain game, Garcia took 20 shots in the first two quarters and made 12. In the third quarter, he took 10 shots, but his shooting percentage dropped to 50% of his overall shooting percentage from the first two quarters. In the last quarter, his sh... | 8 |
If one side of a square is increased by 3.5 times and the other side is decreased by 30 cm, resulting in a rectangle that has twice the area of the square, then the area of the square is ( ) square centimeters.
A: 2500
B: 4900
C: 22500
D: 44100 | B |
In the geometric sequence $\{a_n\}$, $a_2=9$, $a_5=243$. The sum of the first $4$ terms of $\{a_n\}$ is ( ).
A: $81$
B: $120$
C: $168$
D: $192$ | B |
2. In quadrilateral $A B C D$, it is known that $A B=B C, \angle A B C=$ $\angle A D C=90^{\circ}$. A perpendicular $B H$ is dropped from vertex $B$ to side $A D$. Find the area of quadrilateral $A B C D$, if it is known that $B H=h$. | ^2 |
A sequence $\{a_n\}$ with the sum of its first $n$ terms denoted by $S_n$ is called a harmonic series if $\frac{S_n}{S_{2n}}$ is a constant. If an arithmetic sequence with first term 1 and common difference $d$ (where $d \neq 0$) is a harmonic series, then the common difference $d$ equals _____________. | d = 2 |
Example 14 (7th Canadian Mathematical Olympiad) Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{2}, a_{1}+a_{2}+\cdots+$ $a_{n}=n^{2} a_{n}, n \geqslant 1$. Find the general term formula of $\left\{a_{n}\right\}$. | a_{n}=\frac{1}{n(n+1)} |
Example 2 As shown in Figure 2, in $\triangle A B C$, $A B=A C$, $\angle B A C=100^{\circ}$, point $D$ is on the extension of side $A B$, and $A D=B C$. Find the measure of $\angle B C D$. ${ }^{[2]}$ | 10 |
Given $U=\mathbb{R}$, $A=\{x|y=\ln (1-x)\}$, $B=\{x|x^{2}-x-2 < 0\}$, then $B\cap(\complement_{U}A)=$ ()
A: $\{x|x\geqslant 1\}$
B: $\{x|1\leqslant x < 2\}$
C: $\{x|0 < x\leqslant 2\}$
D: $\{x|x\leqslant 1\}$ | B |
Positive $x,y,z$ satisfy a system: $\begin{cases} x^2 + xy + y^2/3= 25 \\
y^2/ 3 + z^2 = 9 \\
z^2 + zx + x^2 = 16 \end{cases}$
Find the value of expression $xy + 2yz + 3zx$. | 24\sqrt{3} |
## Task 5
Marion missed the start of a running competition and started as the last runner. However, after overtaking Birgit, Karin, Carola, Gabi, and Katja, she still managed to finish third.
How many girls participated in the race? | 8 |
Given that the equation $p: x^2 + mx + 1 = 0$ has two distinct negative roots, and the equation $q: 4x^2 + 4(m - 2)x + 1 = 0$ ($m \in \mathbb{R}$) has no real roots, find the range of values for $m$ that makes both $p$ and $q$ true. | (2, 3) |
2. A rectangle with perimeter $30 \mathrm{~cm}$ is divided by two lines, forming a square of area $9 \mathrm{~cm}^{2}$, as shown in the figure.
What is the perimeter of the shaded rectangle?
A $14 \mathrm{~cm}$
B $16 \mathrm{~cm}$
C $18 \mathrm{~cm}$
D $21 \mathrm{~cm}$
E $24 \mathrm{~cm}$ | 18 |
2. Petya wrote down five consecutive terms of an arithmetic progression and encrypted them according to the principle: each digit was replaced with a letter, different digits with different letters, and vice versa. Here's what he got: A, BV, DS, EE, SD. What numbers did Petya write? | 6,15,24,33,42 |
Given the function $f(x) = A\sin(wx + \varphi)$ ($A > 0$, $w > 0$, $0 < \varphi < \frac{\pi}{2}$) has a minimum positive period of $\pi$, and on its graph, a lowest point is $M\left( \frac{2\pi}{3}, -2\right)$
(1) Find the analytical expression of $f(x)$.
(2) When $x \in \left[0, \frac{\pi}{2}\right]$, find the ran... | [-1, 2] |
$\begin{array}{l}\text { 1. If } n \text { satisfies }(n-2003)^{2}+(2004-n)^{2} \\ =1 \text {, then }(2004-n)(n-2003)=\end{array}$ | 0 |
Determine all functions $f:\mathbb R\to\mathbb R$ such that equality
$$f(x + y + yf(x)) = f(x) + f(y) + xf(y)$$
holds for all real numbers $x$, $y$.
Proposed by Athanasios Kontogeorgis | f(x) \equiv x |
Given vectors $\overrightarrow{AB}=({-3,2}),\overrightarrow{AC}=({1,-2})$, then $\overrightarrow{BC}=\underline{\hspace{1cm}}$
A: $(1,1)$
B: $(4,-4)$
C: $(2,0)$
D: $(-4,4)$ | B |
The concept of negative numbers first appeared in the book "Nine Chapters on the Mathematical Art". In this book, moving $2km$ east is denoted as "$+2km$". Moving $1km$ west should be denoted as ( ).
A: $-2km$
B: $-1km$
C: $1km$
D: $+2km$ | B |
Given \\(U=\{1,2,3,4,5\}\\), \\(P=\{2,4\}\\), \\(Q=\{1,3,4,6\}\\), then \\(({C}_{U}P)∩Q= \\) ( )
A: \\(\{ 1,4 \}\\)
B: \\(\{ 1,3 \}\\)
C: \\(\{ 3,6 \}\\)
D: \\(\{ 3,4 \}\\) | \text{B} |
$(1)$ Simplify the expression using factorization: $99^{2}+202\times 99+101^{2}$;$(2)$ Simplify: $(\frac{1}{x-1}-2)÷\frac{2x-3}{{x}^{2}-1}$. | -x-1 |
23. Given 5 numbers: $a_{1}=1, a_{2}=-1, a_{3}=-1, a_{4}=1, a_{5}=-1$. The following numbers are determined as follows: $a_{6}=a_{1} \cdot a_{2}, a_{7}=a_{2} \cdot a_{3}, a_{8}=a_{3} \cdot a_{4}$ and so on. What is $a_{1988}$? | -1 |
9. Given that a line passing through the focus $F$ of the parabola $y^{2}=4 x$ intersects the parabola at points $M$ and $N$, and $E(m, 0)$ is a point on the $x$-axis. The extensions of $M E$ and $N E$ intersect the parabola at points $P$ and $Q$ respectively. If the slopes $k_{1}$ and $k_{2}$ of $M N$ and $P Q$ satisf... | 3 |
6. Cesare owns a large number of toy soldiers, between 2000 and 2500. When he tries to arrange them in rows of 2, 3, 4, 5, 6, 7, there is always one left over. If he wants to arrange all of them in rows of the same length (greater than 1), what is the minimum length of such a row?
(A) 19
(B) 23
(C) 11
(D) 17
(E) 13 | 11 |
If the directrix of the parabola $y^2 = ax$ is the line $x = -1$, then the coordinates of its focus are ( )
A: $(1, 0)$
B: $(2, 0)$
C: $(3, 0)$
D: $(-1, 0)$ | \text{A} |
11. Find the real number $m$, such that the inequality $\frac{x^{2}-8 x+20}{m x^{2}+2(m+1) x+9 m+4}<0$ has the solution set as the set of all real numbers $\mathbf{R}$ | <-\frac{1}{2} |
Bakayev E.V.
In a circle, there are boys and girls (both are present), a total of 20 children. It is known that for each boy, the neighbor in the clockwise direction is a child in a blue T-shirt, and for each girl, the neighbor in the counterclockwise direction is a child in a red T-shirt. Can the number of boys in th... | 10 |
89. If prime numbers $p, q$ satisfy $p q+p^{2}+q^{2}=199(p<q)$, find the value of the algebraic expression $\frac{2}{p(p+1)}+\frac{2}{(p+1)(p+2)}+\cdots+\frac{2}{(q-1) q}$. | \frac{11}{13} |
1. (17 points) Find the area of the triangle cut off by the line $y=2x+2$ from the figure defined by the inequality $|x-2|+|y-3| \leq 3$. | 3 |
49.* A two-digit number, when added to the number written with the same digits but in reverse order, gives a perfect square. Find all such numbers. | 29,38,47,56,65,74,83,92 |
GS. 2 Given that $x_{0}$ satisfies the equation $x^{2}+x+2=0$. If $b=x_{0}^{4}+2 x_{0}^{3}+3 x_{0}^{2}+2 x_{0}+1$, find the value of $b$. | 1 |
1. [2] Let $S=\{1,2,3,4,5,6,7,8,9,10\}$. How many (potentially empty) subsets $T$ of $S$ are there such that, for all $x$, if $x$ is in $T$ and $2 x$ is in $S$ then $2 x$ is also in $T$ ? | 180 |
Given $a \gt 0$, $b \gt 0$, and $\frac{1}{a}+\frac{4}{b}=2$, then for $a+b\geqslant m$ to always hold, find the range of values for $m$ is ( ).
A: $(-\infty ,9)$
B: $(0,1]$
C: $({-∞，\frac{9}{2}}]$
D: $(-\infty ,8]$ | C |
20. $63 x_{1}+110 x_{2}=6893$ Does it have a positive solution? | x_{1}=71, x_{2}=22 |
7. In the expansion of $(2+\sqrt{x})^{2 n+1}$, the sum of the coefficients of the terms where the power of $x$ is an integer is $\qquad$ . | \frac{3^{2 n+1}+1}{2} |
Problem 3. In a certain company, there are 100 shareholders, and any 66 of them collectively own at least $50\%$ of the company's shares. What is the largest percentage of all shares that one shareholder can own? (The percentage of shares in the company owned by a shareholder can be non-integer.) | 25 |
Among the following shapes, the one that is both axially symmetric and centrally symmetric is ( ):
A: Parallelogram
B: Circle
C: Equilateral triangle
D: Regular pentagon | \text{B} |
A finite set $M$ of real numbers has the following properties: $M$ has at least $4$ elements, and there exists a bijective function $f:M\to M$, different from the identity, such that $ab\leq f(a)f(b)$ for all $a\neq b\in M.$ Prove that the sum of the elements of $M$ is $0.$ | 0 |
Black and white checkers are placed on an $8 \times 8$ chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones? | 48 |
11.4. How many Pythagorean triangles exist with one of the legs equal to 2013? (A Pythagorean triangle is a right triangle with integer sides. Equal triangles are counted as one.). | 13 |
[b]5.[/b] Find the continuous solutions of the functional equation $f(xyz)= f(x)+f(y)+f(z)$ in the following cases:
(a) $x,y,z$ are arbitrary non-zero real numbers;
(b) $a<x,y,z<b (1<a^{3}<b)$.
[b](R. 13)[/b] | f(x) = \lambda \ln x |
Given the function $f(x)=ax(x+1)-\ln x$.
(1) When $a=1$, find the equation of the tangent line to $f(x)$ at the point $(1,f(1))$;
(2) If the function $g(x)=f(x)+\ln x-ax^{2}+e^{x}$, find the extreme values of $g(x)$ when $a < -1$. | a(\ln (-a)-1) |
1. The number of integer solutions to the equation $\left(x^{2}-x-1\right)^{x+2}=1$ is ( ).
(A) 5
(B) 4
(C) 3
(D) 2 | B |
Given the line $l: kx - y + 1 + 2k = 0$ ($k \in \mathbb{R}$).
(1) Prove that line $l$ passes through a fixed point.
(2) If line $l$ intersects the negative half of the x-axis at point A and the positive half of the y-axis at point B, with O being the origin, and let the area of $\triangle AOB$ be $S$, find the minimu... | x - 2y + 4 = 0 |
## SUBJECT 3.
$O, A, B, C$ are collinear points in this order, $O A=2^{x} \mathrm{~cm}, O B=2^{x+1} \mathrm{~cm}, O C=2^{x+2} \mathrm{~cm}, x \in \mathbf{N}^{*}$.
a) Show that $B C=O A+A B$;
b) If $M$ is the midpoint of segment $[O A]$ and $N$ is the midpoint of segment $[B C]$, and $M N=20 \mathrm{~cm}$, find the l... | AC=24\mathrm{~} |
Proposition 8 Let $P$ be any point inside (including the boundary) $\triangle ABC$, draw perpendiculars from $P$ to the three sides $BC$, $CA$, $AB$, with the feet of the perpendiculars being $A'$, $B'$, $C'$, respectively. Connect $A'$, $B'$, $C'$ to form $\triangle A'B'C'$. If the areas of $\triangle ABC$ and $\trian... | \Delta' \leqslant \frac{1}{4} \Delta |
7. Variant 1.
In the "Triangle" cinema, the seats are arranged in a triangular shape: in the first row, there is one seat with number 1, in the second row - seats with numbers 2 and 3, in the third row - 4, 5, 6, and so on (the figure shows an example of such a triangular hall with 45 seats). The best seat in the cine... | 1035 |
Determine all triples $(x, y, z)$ of positive real numbers which satisfies the following system of equations
\[2x^3=2y(x^2+1)-(z^2+1), \] \[ 2y^4=3z(y^2+1)-2(x^2+1), \] \[ 2z^5=4x(z^2+1)-3(y^2+1).\] | (1, 1, 1) |
5、 $A, B$ are distinct points on the parabola $y=3-x^{2}$ that are symmetric with respect to the line $x+y=0$, then $|A B|$ equals $($ )
A. 3
B. 4
C. $3 \sqrt{2}$
D. $4 \sqrt{2}$ | 3\sqrt{2} |
Let $n$ be the largest integer for which $14 n$ has exactly 100 digits. Counting from right to left, what is the 68 th digit of $n$ ?
(A) 1
(B) 2
(C) 4
(D) 5
(E) 8 | 1 |
Three candles can burn for $30$, $40$, and $50$ minutes respectively (but they are not lit at the same time). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. Then, the time when exactly two candles are burning simultaneously is $\qquad$ minute... | 35 |
2. The number of non-negative integer solutions to the equation $2 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}=3$ is. $\qquad$ groups. | 174 |
Example 10. $M$ is any point on the side $AB$ of $\triangle ABC$. $r_{1}, r_{2}$, $r$ are the radii of the incircles of $\triangle AMC, \triangle BMC, \triangle ABC$, and $q_{1}, q_{2}, q$ are the radii of the excircles of the above triangles that lie inside $\angle ACB$. Prove: $\frac{r_{1}}{q_{1}} \cdot \frac{r_{2}}{... | \frac{r_{1}}{q_{1}} \cdot \frac{r_{2}}{q_{2}}=\frac{r}{q} |
There are 5 boys and 4 girls. How many ways can 3 of them be selected and arranged in a row? (Express the result with a number). | 504 |
Given the function $f(x) = |a-3x| - |2+x|$.
$(1)$ If $a=2$, solve the inequality $f(x) \leqslant 3$;
$(2)$ If there exists a real number $a$ such that the inequality $f(x) \geqslant 1-a+2|2+x|$ holds, find the range of values for the real number $a$. | \left[-\frac{5}{2}, +\infty \right) |
7. Farmer John is grazing his cows at the origin. There is a river that runs east to west 50 feet north of the origin. The barn is 100 feet to the south and 80 feet to the east of the origin. Farmer John leads his cows to the river to take a swim, then the cows leave the river from the same place they entered and Farme... | 40\sqrt{29} |
2. [7] Let $n$ be a positive integer. Alice writes $n$ real numbers $a_{1}, a_{2}, \ldots, a_{n}$ in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors $\left(a_{n}\right.$ is not a neighbor of $a_{1}$, nor vice versa). A number changes sign if it chang... | n-1 |
3. Among the 8 vertices, the midpoints of the 12 edges, the centers of the 6 faces, and the center of the cube, a total of 27 points, the number of groups of three collinear points is
A. 57
B. 49
C. 43
D. 37 | 49 |
1. (12 points) Consider the quadratic functions $\mathrm{y}=x^{2}+p x+q$, for which $-2 p+q=2023$. Show that the parabolas, which are the graphs of these functions, intersect at one point. | (-2;2027) |
The sequence $\{a_{n}\}$ satisfies $a_{1}+\frac{1}{2}{a_2}+\frac{1}{3}{a_3}+…+\frac{1}{n}{a_n}={a_{n+1}}-1$, $n\in N^{*}$, and $a_{1}=1$.$(1)$ Find the general formula for the sequence $\{a_{n}\}$;$(2)$ Let $S_{n}=a_{1}\cdot a_{n}+a_{2}\cdot a_{n-1}+a_{3}\cdot a_{n-2}+\ldots +a_{n}\cdot a_{1}$, $b_{n}=\frac{1}{{3{S_n}}... | 1012 |
Example 5 (2005 National High School Mathematics League Test Question) Define the function $f(k)=\left\{\begin{array}{l}0, k \text { is a perfect square } \\ {\left[\frac{1}{\{\sqrt{k}\}}\right], k \text { is not a perfect square, }}\end{array}\right.$ Find $\sum_{k=1}^{240} f(k)$. | 768 |
Suppose m and n are positive odd integers. Which of the following must also be an odd integer?
$\textbf{(A)}\ m+3n\qquad\textbf{(B)}\ 3m-n\qquad\textbf{(C)}\ 3m^2 + 3n^2\qquad\textbf{(D)}\ (nm + 3)^2\qquad\textbf{(E)}\ 3mn$ | 3mn |
3. The original equation is equivalent to
$$
\begin{aligned}
\left(\frac{1-\cos 2 x}{2}\right)^{5}+\left(\frac{1+\cos 2 x}{2}\right)^{5}=\frac{29}{16} \cos ^{4} 2 x & \Longleftrightarrow \\
& \Longleftrightarrow \frac{2+20 \cos ^{2} 2 x+10 \cos ^{4} 2 x}{32}=\frac{29}{16} \cos ^{4} 2 x
\end{aligned}
$$
From which $24... | \frac{\pi}{8}+\frac{\pik}{4} |
Given $|\vec{a}|=1$, $|\vec{b}|=2$, if $(\vec{a}+\vec{b}) \perp \vec{a}$, then the angle between $\vec{a}$ and $\vec{b}$ is $(\ )$
A: $30^{\circ}$
B: $60^{\circ}$
C: $120^{\circ}$
D: $150^{\circ}$ | C |
4. Given real numbers $a, b$ satisfy $\arcsin \left(1+a^{2}\right)-\arcsin (b-1)^{2} \geqslant \frac{\pi}{2}$. Then $\arccos \left(a^{2}-b^{2}\right)=$ $\qquad$ . | \pi |
18. (6 points) Two ribbons, A and B, were originally the same length. If ribbon A is cut by 18 meters and ribbon B is cut by 12 meters, the length of ribbon B is then twice the length of ribbon A. What was the original length of ribbons A and B? | 24 |
Find the number of permutations of $1, 2, 3, 4, 5, 6$ such that for each $k$ with $1$ $\leq$ $k$ $\leq$ $5$, at least one of the first $k$ terms of the permutation is greater than $k$. | 461 |
The line $l$ intersects the circle $x^2+y^2+2x-4y+a=0$ ($a<3$) at points $A$ and $B$. If the midpoint of chord $AB$ is $C(-2, 3)$, then the equation of line $l$ is ( )
A: $x-y+5=0$
B: $x+y-1=0$
C: $x-y-5=0$
D: $x+y-3=0$ | \text{A} |
Given that the sum of the first n terms of the sequence $\{a_n\}$ is $S_n = k + 3^n$, if $\{a_n\}$ is a geometric sequence, then the value of $k$ is ( )
A: -1
B: 0
C: 1
D: None of the above are correct | A |
7. Given that the parabola $C$ has the center of the ellipse $E$ as its focus, the parabola $C$ passes through the two foci of the ellipse $E$, and intersects the ellipse $E$ at exactly three points. Then the eccentricity of the ellipse $E$ is | \frac{2 \sqrt{5}}{5} |
## Task 3B - 301233B
Let $D_{1}, \ldots, D_{n}$ be cans, for whose sizes (diameters) $d_{1}, \ldots, d_{n}$ in a suitable unit of measurement
$$
d_{1}=2, \quad d_{2}=3, \quad \ldots, \quad d_{n}=n+1
$$
holds. Furthermore, let $G_{1}, \ldots, G_{n}$ be objects, for whose sizes $g_{1}, \ldots, g_{n}$
$$
g_{1}=1, \qua... | 2^{n-1} |
II. (50 points)
Given that $x, y, z$ are all positive real numbers, find the minimum value of $\iota=\frac{(x+y-z)^{2}}{(x+y)^{2}+z^{2}}+\frac{(z+x-y)^{2}}{(z+x)^{2}+y^{2}}+\frac{(y+z-x)^{2}}{(y+z)^{2}+x^{2}}$. | \frac{3}{5} |
14. Given an arithmetic sequence $\left\{a_{n}\right\}$ satisfies: $a_{3}=7, a_{5}+a_{7}=26$, and the sum of the first $n$ terms of $\left\{a_{n}\right\}$ is $S_{n}$.
(1) Find $a_{n}$ and $S_{n}$;
(2) Let $b_{n}=\frac{1}{a_{n}^{2}-1}\left(n \in \mathbf{N}_{+}\right)$, find the sum of the first $n$ terms of the sequence... | a_{n}=2n+1,S_{n}=n^2+2n,T_{n}=\frac{n}{4(n+1)} |
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