task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | 2. find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $x, y \in \mathbb{R}$ holds for all $x, y \in \mathbb{R}$:
$$
f(x+y f(x))=f(x f(y))-x+f(y+f(x))
$$ | f(y)=1-y | 73 | 5 |
math | 14. Given real numbers $a, \mathbf{b}, \mathbf{c}$ satisfy $a^{2}+b^{2}+c^{2}=1$, find the maximum and minimum values of $M=a^{2} b c+a b^{2} c+a b c^{2}$. | \frac{1}{3} | 67 | 7 |
math | Compute the largest integer that can be expressed in the form $3^{x(3-x)}$ for some real number $x$.
[i]Proposed by James Lin | 11 | 35 | 2 |
math | Bogganov I.I.
Given an infinite supply of white, blue, and red cubes. Any \$N\$ of them are arranged in a circle. A robot, starting at any point on the circle, moves clockwise and, until only one cube remains, repeatedly performs the following operation: it destroys the two nearest cubes in front of it and places a ne... | 2^k | 166 | 3 |
math | In a certain chess tournament, after the 7th round (i.e., 7 games), a player has 5 points. In how many ways could this result have been achieved? (A win is 1 point, a draw is $1 / 2$ point, a loss is 0 points.) | 161 | 64 | 3 |
math | 7. Given $I$ is the incenter of $\triangle A B C$, and
$$
9 \overrightarrow{C I}=4 \overrightarrow{C A}+3 \overrightarrow{C B} \text {. }
$$
Let $R$ and $r$ be the circumradius and inradius of $\triangle A B C$, respectively. Then $\frac{r}{R}=$ $\qquad$ . | \frac{5}{16} | 90 | 8 |
math | From 8 English letters $A, B, C, D, E, X, Y, Z$, any 5 letters (letters can be repeated) form a “word”. All possible “words” are arranged in “dictionary order” (i.e., the order in which English words are arranged in an English-Chinese dictionary), resulting in a “word list”:
Try to find the number of “words” located b... | 9590 | 129 | 4 |
math | Example 4 (to $4^{\circ}$ ). Find $\int \sin ^{2} x \cos ^{4} x d x$. | \frac{1}{16}(x-\frac{1}{12}\sin6x+\frac{1}{4}\sin2x-\frac{1}{4}\sin4x)+C | 32 | 42 |
math | 5. Given that the length of the major axis of an ellipse is 4, the left vertex is on the parabola $y^{2}=x-1$, and the left directrix is the $y$-axis. Then the maximum value of the eccentricity of such an ellipse is $\qquad$ . | \frac{2}{3} | 66 | 7 |
math | In white dwarf stars, no nuclear reactions occur, and their temperature decreases due to heat radiation. Assuming that their surface is a perfect black body and that the temperature distribution inside the star is always homogeneous, determine the time dependence of the temperature! (Data for the van Maanen star: mass ... | \frac{16700\, | 222 | 10 |
math | 2. Given $5 \sin 2 \alpha=\sin 2^{\circ}$, then the value of $\frac{\tan \left(\alpha+1^{\circ}\right)}{\tan \left(\alpha-1^{\circ}\right)}$ is | -\frac{3}{2} | 56 | 7 |
math | ## Task B-4.6.
Determine all natural numbers $x$ for which the equality
$$
3 \cdot\binom{2 x^{2}-10 x+16}{x^{2}-5 x+9}=2 \cdot\binom{2 x^{2}-10 x+17}{x^{2}-5 x+7}
$$
holds. | 1or4 | 83 | 3 |
math | 4・102 Solve the system of equations
$$\left\{\begin{array}{l}
x(x+1)(3 x+5 y)=144, \\
x^{2}+4 x+5 y=24 .
\end{array}\right.$$ | \left(-4, \frac{24}{5}\right),\left(3, \frac{3}{5}\right) | 59 | 29 |
math | 3. Solve the equation
$$
2 x^{3}=\left(2 x^{2}+x-1\right) \sqrt{x^{2}-x+1} .
$$ | 1;\frac{-1-\sqrt{13}}{6} | 40 | 14 |
math | 459. Find all values of the greatest common divisor of the numbers $8 a+3$ and $5 a+2$, where $a$ is a natural number. | 1 | 37 | 1 |
math | 16. Evaluate
$$
\frac{1}{\log _{2} 12 \sqrt{5}}+\frac{1}{\log _{3} 12 \sqrt{5}}+\frac{1}{\log _{4} 12 \sqrt{5}}+\frac{1}{\log _{5} 12 \sqrt{5}}+\frac{1}{\log _{6} 12 \sqrt{5}} .
$$ | 2 | 104 | 1 |
math | If $f(1)=1$ and $f(1)+f(2)+\cdots+f(n)=n^{2} f(n)$ for every integer $n \geq 2$, evaluate $f(2008)$. | \frac{2}{2009\cdot2008} | 52 | 16 |
math | Suppose there are $2017$ spies, each with $\frac{1}{2017}$th of a secret code. They communicate by telephone; when two of them talk, they share all information they know with each other. What is the minimum number of telephone calls that are needed for all 2017 people to know all parts of the code? | 4030 | 78 | 4 |
math | Question 1 Can every integer be represented as the sum of the cubes of a finite number of different integers? | (n+7)^{3}-(n+6)^{3}-(n+5)^{3}+(n+4)^{3}-(n+3)^{3}+(n+2)^{3}+(n+1)^{3}-n^{3}=48 | 22 | 61 |
math | 6. Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from eight consecutive natural numbers to some number $a$ is 612, and the sum of the distances from these same eight numbers to some number $b$ is 240. Find all possible values of $a$, given that... | =27,=-3 | 85 | 6 |
math | 62. In the city, there are 10,000 bicycles with various numbers from 1 to 10,000. What is the probability that the number of the first bicycle encountered will not contain the digit 8? | 0.6561 | 52 | 6 |
math | 12. The sum of the longest side and the second longest side of a triangle is 12, the area is $17 \frac{1}{2}$ times the sine value of the angle between these two sides, and one of the interior angles is $120^{\circ}$. Find the lengths of the three sides of this triangle. | 7,5,3 | 74 | 5 |
math | Determine explicit formulas for the following recursively defined sequences:
(1) $u_{0}=2$ and $u_{n+1}=3 u_{n}^{2}-4 u_{n}+2$;
(2) $u_{0}=\frac{5}{2}$ and $u_{n+1}=u_{n}^{2}-2$. | u_{n}=\frac{4^{2^{n}}+2}{3} | 78 | 18 |
math | 11. (40 points) Find all pairs of rational numbers $(a, b)$, for which $\sqrt{a}+\sqrt{b}=$ $=\sqrt{2+\sqrt{3}}$. | (0.5;1.5)(1.5;0.5) | 44 | 17 |
math | 1. Given the sequence $\left\{x_{n}\right\}$ satisfies $x_{n}=\frac{n}{n+2016}$. If $x_{2016}=x_{m} x_{n}$, then a solution for the positive integers $m 、 n$ is $\{m, n\}$ $=$ . $\qquad$ | {4032,6048} | 80 | 11 |
math | 2. $V, W, X, Y, Z$ are 5 digits in base 5. The three three-digit numbers $(V Y Z)_{5},(V Y X)_{5},(V V W)_{5}$ in base 5 increase sequentially with a common difference of 1. What is the three-digit number $(X Y Z)_{5}$ in base 10? | 108 | 85 | 3 |
math | Let $1<k\leq n$ be positive integers and $x_1 , x_2 , \ldots , x_k$ be positive real numbers such that $x_1 \cdot x_2 \cdot \ldots \cdot x_k = x_1 + x_2 + \ldots +x_k.$
a) Show that $x_{1}^{n-1} +x_{2}^{n-1} + \ldots +x_{k}^{n-1} \geq kn.$
b) Find all numbers $k,n$ and $x_1, x_2 ,\ldots , x_k$ for which equality hol... | x_1^{n-1} + x_2^{n-1} + \ldots + x_k^{n-1} \geq kn | 146 | 34 |
math | For a positive integer $n$, let $t_{n}=\frac{n(n+1)}{2}$. Writing down the last digits of $t_{1}=1, t_{2}=3, t_{3}=6, t_{4}=10, t_{5}=15 \cdots \cdots$ can form an infinite repeating decimal: $0.13605 \cdots$. Find the length of the repeating cycle of this decimal. | 20 | 99 | 2 |
math | A sequence of real numbers $a_{0}, a_{1}, \ldots$ is said to be good if the following three conditions hold.
(i) The value of $a_{0}$ is a positive integer.
(ii) For each non-negative integer $i$ we have $a_{i+1}=2 a_{i}+1$ or $a_{i+1}=\frac{a_{i}}{a_{i}+2}$.
(iii) There exists a positive integer $k$ such that $a_{k}=2... | 60 | 172 | 2 |
math | Let $x,y$ and $z$ be positive real numbers such that $xy+z^2=8$. Determine the smallest possible value of the expression $$\frac{x+y}{z}+\frac{y+z}{x^2}+\frac{z+x}{y^2}.$$ | 4 | 60 | 1 |
math | 1. Let $x, y, z$ satisfy:
$$
\left\{\begin{array}{l}
\log _{2}\left(x y z-3+\log _{5} x\right)=5, \\
\log _{3}\left(x y z-3+\log _{5} y\right)=4, \\
\log _{4}\left(x y z-3+\log _{5} z\right)=4 .
\end{array}\right.
$$
Then $\log _{5} x y z=$ $\qquad$ | 3 | 122 | 1 |
math | 138. Calculate the sum for any $\alpha$
$$
\sin ^{2} \alpha+\sin ^{2}\left(\alpha+1^{\circ}\right)+\sin ^{2}\left(\alpha+2^{\circ}\right)+\ldots+\sin ^{2}\left(\alpha+179^{\circ}\right)
$$ | 90 | 79 | 2 |
math | 12.54 Find the function $S(x)$, if its derivative $S^{\prime}(x)=\frac{2}{\sqrt{5-x}}$ and $S(1)=-1$.
Calculate the integrals (12.55-12.58): | S(x)=7-4\sqrt{5-x} | 63 | 12 |
math | 3. Solve the system of equations
$$
\begin{cases}x^{y} & =y^{x} \\ a^{x} & =b^{y}\end{cases}
$$
$a \neq 1, b \neq 1$. Perform a discussion. | \begin{cases}(\frac{\logb}{\log})^{\frac{\logb}{\logb-\log}}\\(\frac{\logb}{\log})^{\frac{\log}{\logb-\log}}\end{cases}if\logb\neq\log,yif\logb | 60 | 68 |
math | 6. In a right triangular prism $A B C-A_{1} B_{1} C_{1}$, $A B=1, B C=C C_{1}=\sqrt{3}, \angle A B C=90^{\circ}$, point $P$ is a moving point on the plane $A B C$, then the minimum value of $A_{1} P+\frac{1}{2} P C$ is $\qquad$ . | \frac{5}{2} | 98 | 7 |
math | Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S$, the probability that it is divisible by $9$ is $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 913 | 78 | 3 |
math | Suppose $x$ and $y$ are nonzero real numbers simultaneously satisfying the equations
$x + \frac{2018}{y}= 1000$ and $ \frac{9}{x}+ y = 1$.
Find the maximum possible value of $x + 1000y$. | 1991 | 68 | 4 |
math | 7. Given a triangle with sides as three consecutive natural numbers, the largest angle is twice the smallest angle. Then the perimeter of the triangle is $\qquad$ . | 15 | 34 | 2 |
math | Suppose that $\alpha$ and $\beta$ are the two positive roots of the equation
$$
x^{2}-\sqrt{13} x^{\log _{13} x}=0
$$
Determine the value of $\alpha \beta$. | 169 | 56 | 3 |
math | 4.5.3 $\star \star$ Find all real numbers $k$ such that the inequality
$$
a^{3}+b^{3}+c^{3}+d^{3}+1 \geqslant k(a+b+c+d)
$$
holds for all $a, b, c, d \in[-1,+\infty$ ). | \frac{3}{4} | 79 | 7 |
math | 7. Four boys and three girls want to sit on the same bench. Is it more likely that the people at both ends of the bench will be of the same or opposite gender? Determine both probabilities.
The use of a pocket calculator or any manuals is not allowed. | \frac{4}{7} | 54 | 7 |
math | 4.47 Let $x$ be a natural number. If a sequence of natural numbers
$$
x_{0}=1, \quad x_{1}, \quad x_{2}, \cdots, \quad x_{l}=x,
$$
satisfies
$$
x_{i-1}<x_{i}, x_{i-1} \mid x_{i}, \quad i=1,2, \cdots, l .
$$
then $\left\{x_{0}, x_{1}, x_{2}, \cdots, x_{l}\right\}$ is called a divisor chain of $x$, and $l$ is the lengt... | L(x)=3n+k+,\quadR(x)=\frac{(3n+k+)!}{(n!)^2!(k} | 244 | 29 |
math | Question 190: Person A and Person B each roll a six-sided die, stopping when they roll a "6". Let the probability that the number of times they roll the dice differs by at most one be $\mathrm{P}$, then $\mathrm{P}=$ $\qquad$ - | \frac{8}{33} | 62 | 8 |
math | 5. Find all cubic polynomials $x^{3}+a x^{2}+b x+c$ with rational roots $a, b, c$.
(1985 Turkish Competition Problem) | x^3+x^2-2x=0x^3+x^2-x-1=0 | 43 | 22 |
math | 9.5 $n$ is the smallest integer with the following property: it is a multiple of 15, and each of its digits is 0 or 8. Find $\frac{n}{15}$.
(2nd American Invitational Mathematics Examination, 1984) | 592 | 60 | 3 |
math | 16. Two cars are driving on a highway, 100 meters apart, both traveling at 60 kilometers per hour. The highway has different speed points (the speed points are far apart). After each car passes the first speed point, their speed immediately increases to 80 kilometers per hour; after passing the second speed point, thei... | 200 | 123 | 3 |
math | How many sequences $ a_1,a_2,...,a{}_2{}_0{}_0{}_8$ are there such that each of the numbers $ 1,2,...,2008$ occurs once in the sequence, and $ i \in (a_1,a_2,...,a_i)$ for each $ i$ such that $ 2\le i \le2008$? | 2^{2007} | 86 | 7 |
math | 2. The barrel is filled with 100% alcohol. From the barrel, two liters of alcohol are removed and the same amount of distilled water is added. The procedure is repeated once more, i.e., two liters of the solution are removed and two liters of distilled water are added. In this way, the barrel contains 36% alcohol. How ... | 5 | 83 | 1 |
math | 5. Find the first decimal digit (immediately to the right of the decimal point) and the last digit before the decimal point (the units digit) of the number $(\sqrt{2}+\sqrt{3})^{1980}$, and prove your conclusion. | 7,9 | 57 | 3 |
math | 8. (10 points) On the right is an equation, where 9 Chinese characters represent the numbers 1 to 9, and different characters represent different numbers. The maximum possible value of the equation is $\qquad$.
Hope $\times$ Longing + Tree $\times$ Green + Sky $\times$ Blue | 8569 | 66 | 4 |
math | 7. The function
$$
f(x)=\frac{\sin x-1}{\sqrt{3-2 \cos x-2 \sin x}}(0 \leqslant x \leqslant 2 \pi)
$$
has the range . $\qquad$ | [-1,0] | 61 | 5 |
math | From the number 215, we can create a four-digit number by inserting any other digit between its digits. This way, we created two four-digit numbers, the sum of which was 4360. What could these two four-digit numbers be? Determine all possibilities.
(L. Simünek) | 2195+2165,2185+2175,2215+2145 | 64 | 29 |
math | 14. [9] Real numbers $x$ and $y$ satisfy the following equations:
$$
\begin{array}{l}
x=\log _{10}\left(10^{y-1}+1\right)-1 \\
y=\log _{10}\left(10^{x}+1\right)-1
\end{array}
$$
Compute $10^{x-y}$. | \frac{101}{110} | 91 | 11 |
math | 6. 144 Find all infinitely differentiable functions \( f: R \rightarrow R \) that satisfy
$$f(x+y) \equiv f(x)+f(y)+2 x y, x, y \in R$$ | f(x) = x^2 + ax | 48 | 9 |
math | 5. The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=p, a_{n+1}=a_{n}^{2}+2 a_{n}$. Then the general term $a_{n}=$ $\qquad$ . | a_{n}=(p+1)^{2^{n-1}}-1 | 58 | 18 |
math | 7. Given a regular quadrilateral pyramid $P-ABCD$ with base edge length $AB=2$, height $PO=3$. $O'$ is a point on the line segment $PO$, and a plane parallel to the base of the regular quadrilateral pyramid $P-ABCD$ is drawn through $O'$, intersecting the edges $PA, PB, PC, PD$ at points $A', B', C', D'$ respectively. ... | \frac{16}{27} | 116 | 9 |
math | 6. Find the smallest positive integer $k$ such that for any $k$-element subset $A$ of the set $S=\{1,2, \cdots, 2012\}$, there exist three distinct elements $a$, $b$, and $c$ in $S$ such that $a+b$, $b+c$, and $c+a$ are all in the set $A$. | 1008 | 88 | 4 |
math | 26. (2004 Western China Mathematical Olympiad) Find all positive integer triples \((a, b, c)\) satisfying \(a^{2}+b^{2}+c^{2}=2005\), and \(a \leqslant b \leqslant c\). | (23,24,30),(12,30,31),(9,18,40),(9,30,32),(4,15,42),(15,22,36),(4,30,33) | 67 | 60 |
math | 7. Denote by $\langle x\rangle$ the fractional part of the real number $x$ (for instance, $\langle 3.2\rangle=0.2$ ). A positive integer $N$ is selected randomly from the set $\{1,2,3, \ldots, M\}$, with each integer having the same probability of being picked, and $\left\langle\frac{87}{303} N\right\rangle$ is calcula... | \frac{50}{101} | 138 | 10 |
math | $p$ is a prime number such that its remainder divided by 8 is 3. Find all pairs of rational numbers $(x,y)$ that satisfy the following equation.
$$p^2 x^4-6px^2+1=y^2$$ | (0, \pm 1) | 52 | 8 |
math | The manager of a store went to check what had been the selling price in 2006 of a television from the VejoTudo brand. He found a faded invoice, which read: "lot of 72 VejoTudo TVs sold for $R \$ \ldots 679 \ldots$ reais", where the digits in the units and ten-thousands place were illegible. What was the selling price i... | 511 | 103 | 3 |
math | 4. Determine all pairs $(p, q), p, q \in \mathbb{N}$ such that
$$
(p+1)^{p-1}+(p-1)^{p+1}=q^{q}
$$ | (p,q)=(1,1)\text{}(p,q)=(2,2) | 50 | 17 |
math | 10.375 An equilateral triangle $ABC$, inscribed in a circle of radius $R$, is rotated around the center of the circle by $90^{\circ}$ to the position $A_{1} B_{1} C_{1}$. Calculate the area of the hexagon $A A_{1} B B_{1} C C_{1}$. | \frac{9R^{2}}{4} | 80 | 11 |
math | A triangle $A B C$ has an area equal to 944. Let $D$ be the midpoint of $[A B]$, $E$ the midpoint of $[B C]$, and $F$ the midpoint of $[A E]$. What is the area of $D E F$? | 118 | 67 | 3 |
math | 12. For any positive integer $n$, define the function $\mu(n)$:
$$
\mu(1)=1 \text {, }
$$
and when $n=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{t}^{\alpha_{4}} \geqslant 2$,
$$
\mu(n)=\left\{\begin{array}{ll}
(-1)^{t}, & \alpha_{1}=\alpha_{2}=\cdots=\alpha_{t}=1 ; \\
0, & \text { otherwise, }
\end{array}\right.
... | 0 | 241 | 1 |
math | Example 8 Let $n$ be a given integer greater than 5, solve the system of equations
$$
\left\{\begin{array}{l}
x_{1}+x_{2}+\cdots+x_{n}=n+2, \\
x_{1}+2 x_{2}+\cdots+n x_{n}=2 n+2, \\
x_{1}+2^{2} x_{2}+\cdots+n^{2} x_{n}=n^{2}+n+4, \\
x_{1}+2^{3} x_{2}+\cdots+n^{3} x_{n}=n^{3}+n+8 .
\end{array}\right.
$$
where $x_{i} \g... | x_1=n,x_2=1,x_n=1,x_i=0\text | 181 | 19 |
math | Example 12. Map the upper half-plane $\operatorname{Im} z>0$ onto the unit disk $|w|<1$ so that the point $z=i$ (where $\operatorname{Im} z>0$) is mapped to the center $\boldsymbol{w}=0$ of the disk. | e^{i\alpha}\frac{z-z_{0}}{z-\bar{z}_{0}} | 69 | 22 |
math | Determine all positive integers $n{}$ which can be expressed as $d_1+d_2+d_3$ where $d_1,d_2,d_3$ are distinct positive divisors of $n{}$. | n = 6k | 47 | 6 |
math | Example 5 Find all positive integer arrays $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$, such that
$$\left\{\begin{array}{l}
a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}, \\
a_{1}+a_{2}+\cdots+a_{n}=26, \\
a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=62, \\
a_{1}^{3}+a_{2}^{3}+\cdots+a_{n}^{3}=164 .
\end{array}\right.$... | (1,1,1,1,1,2,2,2,3,3,3,3,3) \text{ and } (1,2,2,2,2,2,2,2,2,2,3,4) | 159 | 57 |
math | 12. (6 points) The natural number $a$ is a multiple of 3, $a-1$ is a multiple of 4, $a-2$ is a multiple of 5, then the smallest $a$ is $\qquad$ | 57 | 55 | 2 |
math | 12 different items are distributed among 3 people so that each person gets 4 items. In how many ways is this possible? | 34650 | 27 | 5 |
math | 6. Find at least one solution to the equation
$$
567 x^{3}+171 x^{2}+15 x-777 \ldots 7555 \ldots 5333 \ldots 3=0
$$
where the constant term contains 2023 sevens, 2023 fives, and 2023 threes. | 111\ldots1_{w}2023 | 92 | 14 |
math | 700*. In what number system is the number $11111_{d}$ a perfect square? | 3 | 24 | 1 |
math | 6. 89 Determine the maximum value of $m^{2}+n^{2}$. Where $m, n$ are integers, and $m, n \in\{1$, $2, \cdots, 1981\},\left(n^{2}-m n-m^{2}\right)^{2}=1$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
6. 89 Determine the max... | 3524578 | 174 | 7 |
math | Find those numbers which, when their decimal representation is appropriately split into two parts and these parts are considered as independent numbers, the product of these two parts equals half of the original number (e.g., $1352=2 \cdot 13 \cdot 52$). Which of these are perfect squares? | 6^2,252^2 | 66 | 9 |
math | Example 14. In a batch of 12 parts, 8 are standard. Find the probability that among 5 randomly selected parts, 3 will be standard. | \frac{14}{33} | 36 | 9 |
math | The Pythagorean school believed that numbers are the origin of all things, and they called numbers such as $1, 3, 6, 10, \cdots$ triangular numbers. Therefore, arranging the triangular numbers in ascending order, the sum of the first 100 triangular numbers is $\qquad$. | 171700 | 67 | 6 |
math | \section*{Problem 4 - 321024}
Determine whether it is possible to inscribe more than \(64\) circles, each with a diameter of \(1 \mathrm{~cm}\), in a square with a side length of \(8 \mathrm{~cm}\), such that no two circles overlap and no point of any circle lies outside the square! | 68 | 81 | 2 |
math | 9.203. $5^{\log _{5}^{2} x}+x^{\log _{5} x}<10$. | x\in(\frac{1}{5};5) | 34 | 12 |
math | Let $A \text{ :}= \mathbb{Q}\setminus \{0,1\}$ denote the set of all rationals other than $0$ and $1$. A function $f:A\to \mathbb{R}$ has the property that for all $x\in A$, \[f(x)+f\left(1-\dfrac{1}{x}\right)=\log |x|.\] Compute the value of $f(2007)$. | \log \frac{2007}{2006} | 106 | 15 |
math | 16. (25 points) Let the hyperbola $\Gamma: \frac{x^{2}}{4}-\frac{y^{2}}{5}=1$ have its left and right foci as $F_{1}$ and $F_{2}$, respectively. Let $P$ be a moving point on the hyperbola and in the first quadrant, and let the centroid and incenter of $\triangle P F_{1} F_{2}$ be $G$ and $I$, respectively.
(1) Does the... | -2x+6 | 258 | 5 |
math | Let $S\,$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S\,$ so that the union of the two subsets is $S\,$? The order of selection does not matter; for example, the pair of subsets $\{a, c\},\{b, c, d, e, f\}$ represents the same selection as the pair $\... | 365 | 108 | 3 |
math | 1. (16 points) Suppose the equation $\left|x^{2}+a x\right|=4$ has only 3 distinct real roots. Find the value of $a$ and the corresponding 3 roots. | a=4, \text{ roots: } -2, -2 \pm 2\sqrt{2}; \text{ or } a=-4, \text{ roots: } 2, 2 \pm 2\sqrt{2} | 46 | 53 |
math | Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get. | P(2005) = 0 | 57 | 11 |
math | 169. Let $x, y$ be positive numbers, $s$ be the smallest of the numbers $x, y+\frac{1}{x}, \frac{1}{y}$. Find the greatest possible value of $s$. For which $x$ and $y$ is it achieved? | \sqrt{2} | 64 | 5 |
math | 3. (10 points) $a_{1}, a_{2}, a_{3}, \cdots, a_{n}$ are natural numbers satisfying $0<a_{1}<a_{2}<a_{3} \cdots<a_{n}$, and $\frac{13}{14}=\frac{1}{\mathrm{a}_{1}}, \frac{1}{\mathrm{a}_{2}}, \frac{1}{\mathrm{a}_{3}}+\cdots$ $+\frac{1}{a_{n}}$, then the minimum value of $n$ is . $\qquad$ | 4 | 129 | 1 |
math | Example 12 Let $a, b, c, d, e$ be real numbers, and $a+b+c+d+e=8, a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16$. Then the maximum value of $e$ is . $\qquad$ | \frac{16}{5} | 72 | 8 |
math | 6. In a perfectly competitive market, the demand function for a certain good is $\mathrm{Q}_{\mathrm{d}}(\mathrm{p})=150-\mathrm{p}$, and the supply function for this good is: $\mathrm{Q}_{\mathrm{s}}(\mathrm{p})=3 \mathrm{p}-10$. As a result of a sharp increase in the number of consumers of this good, under all other ... | 1.4 | 141 | 3 |
math | 4.3. Two balls of one radius and two of another are arranged so that each ball touches three others and a given plane. Find the ratio of the radii of the balls. | 2+\sqrt{3} | 38 | 6 |
math | Seven, let the sequence $\left\{a_{n}\right\}$ satisfy
$$
\begin{array}{l}
a_{1}=1, \\
a_{n+1}=\left(1+\frac{k}{n}\right) a_{n}+1(n=1,2, \cdots) .
\end{array}
$$
Find all positive integers $k$ such that every term in the sequence $\left\{a_{n}\right\}$ is an integer.
(Zhang Lei) | 2 | 109 | 1 |
math | Example 6. In the sequence $\left\{a_{n}\right\}$, for any natural number $n(n \geqslant 2)$, we have $a_{n}=3 a_{n-1}-2 a_{n-2}$, and $a_{0}=2, a_{1}=3$, find the general term formula of this sequence. | a_n = 2^n + 1 | 80 | 9 |
math | . Positive integers $x_{1}, \ldots, x_{m}$ (not necessarily distinct) are written on a blackboard. It is known that each of the numbers $F_{1}, \ldots, F_{2018}$ can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of $m$ ?
(Here $F_{1}, \ldots, F_{2018}$ are... | 1009 | 146 | 4 |
math | (9) If the real numbers $x, \alpha, \beta$ satisfy $x=\log _{3} \tan \alpha=-\log _{3} \tan \beta$, and $\alpha-\beta=$ $\frac{\pi}{6}$, then the value of $x$ is $\qquad$. | \frac{1}{2} | 67 | 7 |
math | Three. (50 points) A subset of the set $S_{n}=\{1,2, \cdots, n\}$ is called a "good subset" if it does not contain two consecutive natural numbers. How many good subsets are there in $S_{n}$? | a_{n}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+2}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+2}\right] | 60 | 60 |
math | [b]Q11.[/b] Let be given a sequense $a_1=5, \; a_2=8$ and $a_{n+1}=a_n+3a_{n-1}, \qquad n=1,2,3,...$ Calculate the greatest common divisor of $a_{2011}$ and $a_{2012}$. | 1 | 84 | 1 |
math | Let $f(x)=x+\frac{1}{2 x+\frac{1}{2 x+\frac{1}{2 x+} \ddots}}$ for $x>0$. Find $f(99) f^{\prime}(99)$. | 99 | 56 | 2 |
math | 253. Find all possible systems of four real numbers such that the sum of each of them with the product of the others is 2. | x=y=z==1orx=y=z=-1,=3 | 30 | 14 |
math | Example 7 Given the set $A=\left\{z \mid z^{2 n-1}=\bar{z}, z \in \mathbf{C}, n \in \mathbf{N}, n \geqslant 2\right\}$, on the complex plane, how many right triangles can be formed with the points corresponding to the complex numbers in $A$ as vertices? | 2n(n-1) | 85 | 6 |
math | ## 166. Math Puzzle $3 / 79$
Frank needs four minutes to cover the $300 \mathrm{~m}$ to the street corner, then runs $50 \mathrm{~m}$ up the stairs in five minutes, and completes the remaining $600 \mathrm{~m}$ of the journey in ten minutes.
What is his average speed?
Can he be considered a "fast walker" if we allow... | 3\mathrm{~}/\mathrm{} | 135 | 9 |
math | ## Task A-2.7.
Determine the positive rational numbers $x$ and $y$ for which $x+\frac{1}{y}$ and $y+\frac{1}{x}$ are natural numbers. | (1,1),(\frac{1}{2},2),(2,\frac{1}{2}) | 46 | 22 |
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