problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
A line passing through point M(2, 1) intersects the X-axis and Y-axis at points P and Q, respectively, and $|MP| = |MQ|$. The equation of line L is ( )
A: $x-2y+3=0$
B: $2x-y-3=0$
C: $2x+y-5=0$
D: $x+2y-4=0$ | \text{D} |
If a real number $t$ satisfies $f(t) = -t$, then $t$ is called a sub-fixed point of the function $f(x)$. Suppose the function $f(x) = \ln x$ and its inverse function have all sub-fixed points summing up to $m$, then $m = \boxed{0}$. | 0 |
Given that the square root of $2a-1$ is $\pm 3$, and the arithmetic square root of $3a+b$ is $4$. Find the cube root of $5a+2b$. | 3 |
9.169. $\frac{1}{x^{2}-4}+\frac{4}{2 x^{2}+7 x+6} \leq \frac{1}{2 x+3}+\frac{4}{2 x^{3}+3 x^{2}-8 x-12}$. | x\in(-2;-\frac{3}{2})\cup[1;2)\cup[5;\infty) |
Given that $\{a_n\}$ is an arithmetic sequence, $S_n$ is the sum of its first $n$ terms, and $S_{11}= \frac {22\pi}{3}$, $\{b_n\}$ is a geometric sequence, and $b_{5}\cdot b_{7}= \frac {\pi^{2}}{4}$, then the value of $\tan (a_{6}+b_{6})$ is __( ).
A: $ \sqrt {3}$
B: $± \sqrt {3}$
C: $ \frac { \sqrt {3}}{3}$
D: $± ... | C |
How many five-digit natural numbers have the product of their digits equal to 2000? | 30 |
10. Find the result of the following binary division operations:
(i) $(1100011)_{2} \div(100001)_{2}=$ ?
(ii) $(110001)_{2} \div(111)_{2}=$ ? | (11)_{2} |
7.3. Katya wrote a four-digit number on the board that was divisible by each of its digits without a remainder (there were no zeros in the number). Then she erased the first and last digits, and the number 89 remained on the board. What could have been written on the board initially? (Find all options and show that the... | 4896 |
Steaks are mainly divided into filet steak, ribeye steak, sirloin steak, and T-bone steak. A beef purchaser randomly selects 100 boxes from a batch of steaks and obtains the following data based on the classification standard of steaks:
| Type of Steak | Filet Steak | Ribeye Steak | Sirloin Steak | T-bone Steak |
|---... | \frac{3}{5} |
Let $a,b,c$ be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: $ax^2+bx+c, bx^2+cx+a,$ and $cx^2+ax+b $. | 4 |
The value of $\cos 585^{\circ}$ is ( ).
A: $\dfrac{\sqrt{3}}{2}$
B: $-\dfrac{\sqrt{3}}{2}$
C: $\dfrac{\sqrt{2}}{2}$
D: $-\dfrac{\sqrt{2}}{2}$ | D |
Example 9 $\triangle A B C$ is inscribed in a unit circle, the angle bisectors of the three interior angles $A, B, C$ are extended to intersect this circle at points $A_{1}, B_{1}, C_{1}$, respectively. Find the value of $\frac{A A_{1} \cos \frac{A}{2}+B B_{1} \cos \frac{B}{2}+C C_{1} \cos \frac{C}{2}}{\sin A+\sin B+\s... | 2 |
Determine all distinct positive integers $x$ and $y$ such that
$$
\frac{1}{x}+\frac{1}{y}=\frac{2}{7}
$$ | \frac{1}{28}+\frac{1}{4}=\frac{2}{7} |
16. Ulysses, Kim, Mei and Tanika have their 12th, 14th, 15th and 15th birthdays today. In what year will their ages first total 100 ?
A 2023
B 2024
C 2025
D 2057
E 2113 | 2024 |
50.What is the 1997th digit in the decimal expansion of the fraction $\frac{1}{7}=0.142857 \ldots ?$ | 5 |
Which one of the following combinations of given parts does not determine the indicated triangle?
$\textbf{(A)}\ \text{base angle and vertex angle; isosceles triangle} \\ \textbf{(B)}\ \text{vertex angle and the base; isosceles triangle} \\ \textbf{(C)}\ \text{the radius of the circumscribed circle; equilateral triang... | A |
For any integer $r \geq 1$, determine the smallest integer $h(r) \geq 1$ such that for any partition of the set $\{1, 2, \cdots, h(r)\}$ into $r$ classes, there are integers $a \geq 0 \ ; 1 \leq x \leq y$, such that $a + x, a + y, a + x + y$ belong to the same class.
[i]Proposed by Romania[/i] | 2r |
The function $f(x)=ax^{3}-bx-\ln x (a>0, b\in R)$ has a critical point at $1$. Then the value of $\ln a-b+1$ is ()
A: always greater than $0$
B: always less than $0$
C: always equal to $0$
D: uncertain | \text{B} |
Two examinees, A and B, are filling out their college application preferences and can only choose one college out of the three options: College A, College B, or College C. Assuming that the choice of each college by each student is equally likely, the probability that at least one of College A or College B is chosen is... | \frac{8}{9} |
176. "Square" Dad. Having once entered into a legal marriage in California, my neighbor has now reached an age that is a perfect square. The product of the digits representing the father's age equals the age of his wife. The age of their daughter is equal to the sum of the digits in the father's age, and the age of the... | 49,36,13,9 |
9. Merryn wrote down the numbers $2,0,2,3$ and one further number. What was the median of her five numbers?
A 0
B 2
C 2.5
D 3
E more information required | 2 |
4. 15 bags of potatoes and 12 bags of flour weigh 1 ton 710 kg, with the weight of one bag of potatoes being 30 kg less than the weight of one bag of flour. How much does one bag of potatoes and one bag of flour weigh? | one\bag\of\flour\weighs\80\,\\one\bag\of\potatoes\weighs\50\ |
Task B-1.3. Two trains start simultaneously from their initial stations towards each other. The trains travel at constant speeds and pass each other at point $M$. One train travels twice as fast as the other. If the slower train starts 5 minutes later, it will pass the other train at a point which is $4 \mathrm{~km}$ f... | 72 |
A certain store purchased a household item at a cost of $8$ yuan per item. During the sales process, it was found that there is a linear relationship between the daily sales quantity $y$ (items) and the selling price per item $x$ (yuan) (where $8\leqslant x\leqslant 15$, and $x$ is an integer). Some corresponding value... | 13 |
Simplify:$ (1) 5x^{2}+x+3+4x-8x^{2}-2$;$(2)(5a^{2}+2a-1)-4(3-8a+2a^{2})$. | -3a^{2} + 34a - 13 |
8. The value of $100-98+96-94+\cdots+4-2$ is ( ).
(A) 20
(B) 40
(C) 50
(D) 80
(E) 100 | 50 |
15. Given the parabola $y=a x^{2}$ passes through the point $P(-1,1)$, a line $l$ with a positive slope is drawn through the point $Q\left(-\frac{1}{2}, 0\right)$ intersecting the parabola at points $M, N$ (point $M$ is between $Q$ and $N$). A line parallel to the $x$-axis is drawn through point $M$, intersecting $O P$... | S_{1}>3S_{2} |
18. Let $P(x)$ be a polynomial with integer coefficients such that $P(-4)=5$ and $P(5)=-4$. What is the maximum possible remainder when $P(0)$ is divided by 60 ? | 41 |
2. Divide a square into 25 smaller squares, where 24 of the smaller squares are unit squares, and the remaining piece can still be cut into some squares with a side length of 1. Try to find the area of the original square. ${ }^{\text {[2] }}$
(1997, Global City Mathematics Competition (Junior Group)) | 49 |
9.4 In the thirtieth kingdom, there are three types of coins in circulation: bronze rubles, silver coins worth 9 rubles, and gold coins worth 81 rubles. From the treasury, which contains an unlimited supply of each type of coin, a certain amount was issued with 23 coins, which is less than 700 rubles. Find this amount,... | 647 |
9.1. Several consecutive natural numbers are written on the board. Exactly $52 \%$ of them are even. How many even numbers are written on the board | 13 |
8. If a four-digit number $n$ contains at most two different digits among its four digits, then $n$ is called a "simple four-digit number" (such as 5555 and 3313). Then, the number of simple four-digit numbers is | 576 |
Anjanan A.
All possible non-empty subsets are taken from the set of numbers $1,2,3, \ldots, n$. For each subset, the reciprocal of the product of all its numbers is taken. Find the sum of all such reciprocal values. | n |
$(2008\cdot $Shunyi District Second Mock Exam) If the value of the fraction $\dfrac{x+1}{x-1}$ is $0$, then the value of $x$ is ( ).
A: $-1$
B: $1$
C: $0$
D: $\pm 1$ | \text{A} |
18.13 In trapezoid $A B C D$, $A D / / B C, \angle B=30^{\circ}, \angle C=60^{\circ}, E$, $M$, $F$, $N$ are the midpoints of $A B$, $B C$, $C D$, $D A$ respectively. Given that $B C=7, M N=3$. Then the value of $E F$ is
(A) 4.
(B) $4 \frac{1}{2}$.
(C) 5.
(D) 6.
(China Junior High School Mathematics League, 1997) | 4 |
$$\cos^{2} \frac {\pi}{12}-\sin^{2} \frac {\pi}{12} = \_\_\_\_\_\_.$$ | \frac {\sqrt {3}}{2} |
Given the function $f(x)= \begin{cases} \log_{2}x-1, & x > 0 \\ |2x-6|, & x\leqslant 0 \end{cases}$, then $f(f(-1)-1)=$ ()
A: $2\log_{2}3-2$
B: $\log_{2}7-1$
C: $2$
D: $\log_{2}6$ | B |
4. In a pond, there are 25 hungry pikes. For one pike to be satiated, it needs to eat three other pikes. What is the maximum number of pikes that can remain in the pond, with all of them being full? | 6 |
Example 1 Let $x, y, z \in \mathbf{R}^{+}, \sqrt{x^{2}+y^{2}}+z=1$, try to find the maximum value of $x y+2 x z$.
| \frac{\sqrt{3}}{3} |
The domain of the function $f(x) = x^2 - 4x - 6$ is $[0, m]$, and its range is $[-10, -6]$. The range of $m$ is ( )
A: $[0, 4]$
B: $[2, 4]$
C: $[2, 6]$
D: $[4, 6]$ | B |
Given an arithmetic sequence $\{a_{n}\}$ with a non-zero common difference, the sum of its first $n$ terms is $S_{n}$, and $a_{1} \lt 0$, $S_{7}=S_{13}$. Find the value of $n$ when $S_{n}$ reaches its minimum. | 10 |
10. (20 points) Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
\begin{array}{l}
a_{1}=a_{2}=1, \\
a_{n+1}=\frac{n^{2} a_{n}^{2}+5}{\left(n^{2}-1\right) a_{n-1}}(n>1) .
\end{array}
$$
Question: Does there exist real numbers $x, y (x \neq 0)$, such that $\left\{\frac{b_{n+2}+b_{n}}{b_{n+1}}\right\}$ (where, $b_... | 1,0 |
3. Given a cyclic quadrilateral $A B C D$ with perpendicular diagonals. On the circle circumscribed around it, a point $E$ is marked, diametrically opposite to $D$, and the segments $A B$ and $D E$ do not intersect. Find the ratio of the areas of triangle $B C D$ and quadrilateral $A B E D$. | 1 |
7. On the number line, points with coordinates $0,1,2,3,5,8,2016$ are marked. Consider the set of lengths of segments with endpoints at these points. How many elements does it contain? ANSWER: 14. | 14 |
Problem 3. Determine the numbers of the form $\overline{a b c d}$ that simultaneously satisfy the following conditions: a) the sum of the squares of the digits is divisible by 4;
b) the remainder of the division of the number $\overline{a b c d}$ by $c$ is 7. | 1393,3193,3391,1195,1591,5191,7999,9799,9997 |
Find the value of $\tan \left( -\frac{7\pi }{6} \right)$. | -\frac{\sqrt{3}}{3} |
The arithmetic sequence $\{a_n\}$ satisfies $a_n > 0$, and $a_4^2 + a_7^2 + 2a_4a_7 = 9$. The sum of the first $10$ terms is ( ).
A: $-9$
B: $15$
C: $-15$
D: $\pm15$ | B |
8. If $\forall x \in(-\infty,-1)$, the inequality $\left(m-m^{2}\right) 4^{x}+2^{x}+1>0$ always holds, then the range of real number $m$ is $\qquad$ | -2\leqslant\leqslant3 |
Given an even function $f(x) = a^x + b \cdot a^{-x}$ defined on $\mathbb{R}$, where $a > 0$, $a \neq 1$, and $b \in \mathbb{R}$.
(1) Find the value of the real number $b$;
(2) Determine and prove the monotonicity of $f(x)$;
(3) If $f((\log_{2}x)^{2}-\log_{2}x+1) \geq f(m+\log_{\frac{1}{2}}x^{2})$ holds for any $x ... | m \in \left[\frac{5}{4}, 3\right] |
Simplify first, then evaluate: $3(a^{2}-ab+7)-2(3ab-a^{2}+1)+3$, where $a=2$ and $b=\frac{1}{3}$. | 36 |
1.5. The base of the triangle is $30 \mathrm{~cm}$, and the lateral sides are 26 and $28 \mathrm{~cm}$. The height is divided in the ratio $2: 3$ (measured from the vertex), and a line parallel to the base is drawn through the point of division. Determine the area of the resulting trapezoid. | 282.24\mathrm{~}^{2} |
Given the universal set $U=\{-4,-1,0,1,2,4\}$, $M=\{x\in \mathbb{Z} | 0 \leqslant x < 3\}$, $N=\{x|x^{2}-x-2=0\}$,
$(1)$ Find sets $M$ and $N$;
$(2)$ If the set $\{m^{2}, m-2\} = \overline{U}(M \cup N)$, find the value of the real number $m$. | m=-2 |
6. If a sequence of numbers $a_{1}, a_{2}, \cdots$ satisfies for any positive integer $n$ that
$$
a_{n}=\frac{n^{2}+n-2-\sqrt{2}}{n^{2}-2},
$$
then the value of $a_{1} a_{2} \cdots a_{2016}$ is | 2016\sqrt{2}-2015 |
1. [ $x$ ] represents the greatest integer not greater than $x$, then the real solution $x$ of the equation $\frac{1}{2} \times\left[x^{2}+x\right]=19 x+99$ is $\qquad$. | x=-\frac{181}{38} \text{ or } \frac{1587}{38} |
Given $\sin (\pi+\alpha)= \frac {3}{5}$, and $\alpha$ is an angle in the fourth quadrant, then the value of $\cos (\alpha-2\pi)$ is ( )
A: $ \frac {4}{5}$
B: $- \frac {4}{5}$
C: $± \frac {4}{5}$
D: $ \frac {3}{5}$ | A |
What is the value of the [expression](https://artofproblemsolving.com/wiki/index.php/Expression)
\[\frac{1}{\log_2 100!} + \frac{1}{\log_3 100!} + \frac{1}{\log_4 100!} + \cdots + \frac{1}{\log_{100} 100!}?\]
$\mathrm{(A)}\ 0.01 \qquad\mathrm{(B)}\ 0.1 \qquad\mathrm{(C)}\ 1 \qquad\mathrm{(D)}\ 2 \qquad\mathrm{(E)}\ 10... | 1 |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=\dfrac{1}{2}t \\ y=1-\dfrac{\sqrt{3}}{2}t \end{cases}$ (where $t$ is the parameter). In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of circle $C$ ... | 2 |
Calculate: $4a^{2}(3a-1)=\_\_\_\_\_\_$. | 12a^{3}-4a^{2} |
Given the circle $x^2+y^2+2x-2y+2a=0$ intersects the line $x+y+2=0$ and the length of the chord is 4, then the value of the real number $a$ is ( )
A: $-1$
B: $-2$
C: $-3$
D: $-4$ | B |
Solve the following system of equations:
$$
\begin{aligned}
& \lg \left(\frac{x^{2}}{y^{3}}\right)=1 \\
& \lg \left(x^{2} y^{3}\right)=7
\end{aligned}
$$ | \100,\quad10 |
Find the smallest positive integer that, when multiplied by 1999, the last four digits of the resulting number are 2001. | 5999 |
39. Junior high school student Xiao Peng was surprised to find that: his age this year (2018) is exactly the sum of the four digits of his birth year plus 10. He told this discovery to his math teacher, Mr. Wang, who thought for a moment and said: My age this year is also exactly the sum of the four digits of my birth ... | 18 |
The graph of the function $y=\sin x- \sqrt {3}\cos x$ can be obtained by shifting the graph of the function $y=2\sin x$ to the right by at least \_\_\_\_\_\_ units. | \frac {\pi}{3} |
In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units?
[asy] size(4cm); filldraw(scale(2)*unitcircle,gray,black); filldraw(shift(-1,0)*unitcircl... | 1 |
A class has 55 students, and all students are randomly numbered from 1 to 55. If a systematic sampling method is used to draw a sample of size 5, and it is known that students with numbers 6, a, 28, b, and 50 are in the sample, then the sum of a and b is $\boxed{56}$. | 56 |
1. The sum of 86 consecutive integers is 2021. By how much is the arithmetic mean of all positive numbers among them greater than the arithmetic mean of all negative numbers? | 43.5 |
Given points M(-3, 3), N(-5, -1), then $\overrightarrow {MN}$ equals ( )
A: (-2, -4)
B: (-4, -2)
C: (2, 4)
D: (4, 2) | A |
16.15. Given $n$ integers $a_{1}=1, a_{2}, a_{3}, \ldots, a_{n}$, and $a_{i} \leqslant a_{i+1} \leqslant 2 a_{i}(i=1,2, \ldots, n-1)$, and the sum of all numbers is even. Can these numbers be divided into two groups such that the sums of the numbers in these groups are equal? | S_{1}=S_{2} |
9.1. In the village, seven people live. Some of them are liars (always lie), and the rest are knights (always tell the truth). Each of them said about each of the others whether they are a knight or a liar. Out of the 42 answers received, 24 were “He is a liar.” What is the smallest number of knights that can live in t... | 3 |
Which of the following calculations is correct?
A: $(2a^{2})^{3}=8a^{6}$
B: $a^{3}+a^{2}=2a^{5}$
C: $2a^{2}\cdot 3a^{3}=6a^{6}$
D: $(2a+1)^{2}=4a^{2}+1$ | \text{A} |
If the function $f(x) = x^2 - 2x + 3$, then $f'(1) = \_\_\_\_\_\_\_\_\_$. | 0 |
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $2$, $a_n$, $S_n$ form an arithmetic sequence.
(1) Prove that the sequence $\{a_n\}$ is a geometric sequence;
(2) If $b_n = a_n + \log_{2} \frac{1}{a_n}$, and $T_n = b_1 + b_2 + \ldots + b_n$, find the smallest positive integer $n$ such that ... | 10 |
Given that $\sin(30^\circ + \alpha) = \frac{3}{5}$ and $60^\circ < \alpha < 150^\circ$, find the value of $\cos \alpha$. | \frac{3 - 4\sqrt{3}}{10} |
1A. Dimitar arranged the digits $1,2,3,4,5,6,7,8,9$ in a circle in some way. Any three consecutive digits, in the clockwise direction, form a three-digit number. Dimitar added all such numbers. What sum did Dimitar get? | 4995 |
## Problem Statement
Find the point of intersection of the line and the plane.
$$
\begin{aligned}
& \frac{x-1}{-1}=\frac{y+5}{4}=\frac{z-1}{2} \\
& x-3 y+7 z-24=0
\end{aligned}
$$ | (0,-1,3) |
If the one-variable linear equation $\frac{ax-1}{2}=7$ has a positive integer solution for $x$, and the system of inequalities about $x$ $\left\{\begin{array}{l}{2x-a\geq 0}\\{\frac{x-2}{2}<\frac{x+3}{3}}\end{array}\right.$ has at least $4$ integer solutions, then the number of integers $a$ that satisfy all conditions ... | B |
$O$ double of a number divided by 5 leaves a remainder of 1. What is the remainder of the division of this number by 5?
Translating the text as requested, while preserving the original line breaks and format. | 3 |
53. The natural numbers $16520, 14903, 14177$ give the same remainder when divided by $m$. The largest value of $m$ is | 33 |
6. Let the inradius of a regular tetrahedron be 1. Then the minimum value of its volume is $\qquad$ .
| 8 \sqrt{3} |
934. Find the derivative of the function $u=\sqrt{x^{2}+y^{2}}$ at the point $A(3 ; 4):$
1) in the direction of the bisector of the first coordinate angle
2) in the direction of the radius vector of the point $A$;
3) in the direction of the vector $\bar{q}\{4 ;-3\}$. | \frac{7\sqrt{2}}{10} |
Given $\log_{2}3=a$, $\log_{2}5=b$, then $\log_{2} \frac{9}{5}$ equals ( )
A: $a^{2}-b$
B: $2a-b$
C: $\frac{a^{2}}{b}$
D: $\frac{2a}{b}$ | B |
Given $a=\log_{3}9$, $b=\left(\frac{1}{3}\right)^{-\sqrt{3}}$, $c=2^{-\frac{1}{2}}$, then ()
A: $c \lt a \lt b$
B: $a \lt c \lt b$
C: $b \lt c \lt a$
D: $c \lt b \lt a$ | A |
In the complex plane, the point corresponding to the complex number $\frac {3i}{1-i}$ is located in ( )
A: The first quadrant
B: The second quadrant
C: The third quadrant
D: The fourth quadrant | \text{B} |
6. Given $\sqrt{2009}=\sqrt{x}+\sqrt{y}$, and $0<x<y$. Find all integer pairs $(x, y)$ that satisfy the equation. | (x, y) = (41, 1476), (164, 1025), (369, 656) |
Vanya believes that fractions are "reduced" by crossing out identical digits in the numerator and denominator. Seryozha noticed that sometimes Vanya gets correct equalities, for example, $49 / 98 = 4 / 8$. Find all proper fractions with numerators and denominators consisting of two non-zero digits that can be "reduced"... | 26/65,16/64,19/95,49/98 |
Given that $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, $a_1=2$, and $4S_n=a_n \cdot a_{n+1}$, for the sequence $\{b_n\}$, $b_1= \frac{1}{4}$, and $b_{n+1}= \frac{nb_{n}}{(n+1)-b_{n}}$, where $n\in\mathbb{N}^*$.
(1) Find the general formula for the sequence $\{a_n\}$.
(2) Let $c_n= \frac{a_{n}... | T_n=2- \frac{2+n}{2^{n}} |
Given that $i$ is the imaginary unit, and the complex number $z$ satisfies $z(1-i)=1+i$, then $z^{2017}=$ $(\quad)$
A: $1$
B: $-1$
C: $-i$
D: $i$ | D |
I3.2 If the sum of the coefficients in the expansion of $(x+y)^{a}$ is $b$, find $b$. | 16 |
4. On the coordinate plane, an isosceles triangle $A B C$ was drawn: $A B=2016, B C=$ $A C=1533$, with vertices $A$ and $B$ lying on nodes on the same horizontal line. Determine how many nodes lie within the triangle $A B C$ (including nodes lying on the sides). A node is a point on the coordinate plane where both coor... | 1165270 |
If $3 \times 3 \times 5 \times 5 \times 7 \times 9=3 \times 3 \times 7 \times n \times n$, what is a possible value of $n$ ?
(A) 15
(B) 25
(C) 45
(D) 35
(E) 5 | 15 |
Regardless of the value of $a$, the line $ax + (2-a)y + 1 = 0$ always passes through a fixed point, which is ( )
A: $(0, 0)$
B: $(0, 1)$
C: $\left( \frac {1}{2}, - \frac {1}{2}\right)$
D: $\left(- \frac {1}{2}, - \frac {1}{2}\right)$ | D |
Example 6 Given real numbers $x, y, z$ satisfy
$$
\left\{\begin{array}{l}
x^{4}+y^{2}+4=5 y z, \\
y^{4}+z^{2}+4=5 z x, \\
z^{4}+x^{2}+4=5 x y .
\end{array}\right.
$$
Find the value of $x+y+z$. | x+y+z=3\sqrt{2} \text{ or } -3\sqrt{2} |
Given that $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, with $a_1=1$, $a_2=2$, $a_3=3$, and the sequence $\{a_n+a_{n+1}+a_{n+2}\}$ forms an arithmetic sequence with a common difference of $2$, then $S_{24}=$ .
A: $110$
B: $216$
C: $214$
D: $218$ | B |
At the elementary school U Tří lip, where Lukáš also attends, they are holding a knowledge competition with pre-arranged tasks. Each correctly solved task is scored as many points as its order. Each unsolved or partially solved task is not scored at all. Lukáš correctly solved the first 12 tasks. If he had correctly so... | 71 |
In the sequence $\{a_n\}$, $a_1=1$, and $a_n$, $2^n$, $a_{n+1}$ form a geometric sequence.
(1) Find $a_2$, $a_3$, $a_4$;
(2) Find the sum of the first $n$ terms, $S_n$, of the sequence $\{a_{2n}\}$. | S_n=\frac{4}{3}(4^n-1) |
Martina came up with a method for generating a numerical sequence. She started with the number 52. From it, she derived the next term in the sequence as follows: $2^{2}+2 \cdot 5=4+10=14$. Then she continued in the same way and from the number 14, she got $4^{2}+2 \cdot 1=16+2=18$. Each time, she would take a number, d... | 18 |
Given that the terminal side of angle $\theta$ passes through the point $(2,3)$, then $\tan\left(\frac{7\pi}{4}+\theta\right)$ equals to ( )
A: $-\frac{1}{5}$
B: $\frac{1}{5}$
C: $-5$
D: $5$ | B |
Problem 11.5. The tangent lines to the circumcircle $k$ of an isosceles $\triangle A B C$, $A C=B C$, at the points $B$ and $C$ meet at point $X$. If $A X$ meets $k$ at point $Y$, find the ratio $\frac{A Y}{B Y}$.
Emil Kolev | 2 |
Given that the two foci of an ellipse are $F_1(-\sqrt{3},0)$ and $F_2(\sqrt{3},0)$, with eccentricity $e= \frac{\sqrt{3}}{2}$.
$(1)$ Find the equation of this ellipse.
$(2)$ Suppose the line $y= \frac{x}{2}+m$ intersects the ellipse at points $P$ and $Q$, and $|PQ|$ equals the length of the minor axis of the ellipse.... | \pm \frac{\sqrt{30}}{5} |
(4) Given the ellipse $C: \frac{x^{2}}{2}+y^{2}=1$ with two foci $F_{1}$ and $F_{2}$, point $P\left(x_{0}, y_{0}\right)$ satisfies $0<\frac{x_{0}^{2}}{2}+y_{0}^{2} \leqslant 1$, then the range of $\left|P F_{1}\right|+\left|P F_{2}\right|$ is | [2,2\sqrt{2}] |
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