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 | 15. For any positive integer $m$, the set
$$
\{m, m+1, m+2, \cdots, m+99\}
$$
has the property that in any $n(n \geqslant 3)$-element subset, there are always three elements that are pairwise coprime. Find the minimum value of $n$. | 68 | 80 | 2 |
math | We define $(a, b) \diamond(c, d)=(a c-b d, a d+b c)$.
Determine all pairs $(x, y)$ for which $(x, 3) \diamond(x, y)=(6,0)$. | (0,-2) | 52 | 5 |
math | 3. A store received candies at 13 lei/Kg in 50 boxes of $10 \mathrm{Kg}$ and $13 \mathrm{Kg}$. How many boxes are of $10 \mathrm{Kg}$, if after selling the candies, 7007 lei were collected? | 37 | 69 | 2 |
math | 10. Even number of successes. Find the probability that heads will appear an even number of times, in an experiment where:
a) a fair coin is tossed $n$ times;
b) a coin, for which the probability of heads on a single toss is $p(0<p<1)$, is tossed $n$ times. | \frac{1+(1-2p)^{n}}{2} | 70 | 16 |
math | Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$ | f(0) = 1 | 80 | 8 |
math | 1. Without using calculators or other computational tools, determine which of the two numbers is greater:
$$
\frac{1+\sqrt{5}}{2} \text { or } \sqrt[6]{18} ?
$$ | \sqrt[6]{18} | 49 | 8 |
math | Find all pairs of integers, $n$ and $k$, $2 < k < n$, such that the binomial coefficients
\[\binom{n}{k-1}, \binom{n}{k}, \binom{n}{k+1}\]
form an increasing arithmetic series. | (n, k) = (4p^2 - 2, 2p^2 - 1 \pm p) | 60 | 28 |
math | Putnam 1998 Problem B2 Let P be the point (a, b) with 0 < b < a. Find Q on the x-axis and R on y = x, so that PQ + QR + RP is as small as possible. | \sqrt{2a^2+2b^2} | 54 | 13 |
math | \section*{Exercise 1 - 051011}
Find a two-digit number that is equal to the sum of the number in the tens place and the square of the number in the ones place!
Prove that there is only one such number! | 89 | 55 | 2 |
math | Ivan the Tsarevich fought with the three-headed and three-tailed Zmey Gorynych. With one blow, he could cut off either one head, one tail, two heads, or two tails. But if he cut off one tail, two would grow back; if he cut off two tails, a head would grow back; if he cut off a head, a new head would grow back, and if h... | 9 | 126 | 1 |
math | Task 1 - 200521 Two siblings received a total of 6 marks for recycling materials in September. In October, they received 13 marks together. In November, they got 2 marks less than the total of the two previous months.
A third of their total earnings over the three months was donated to solidarity, another third of the... | 6 | 110 | 1 |
math | 3. If the quadratic equation with real coefficients $a x^{2}+b x+c=0$ has two imaginary roots $x_{1} 、 x_{2}$, and $x_{1}^{3} \in \mathbf{R}$, then $\frac{a c}{b^{2}}=$ $\qquad$ . | 1 | 73 | 1 |
math | Question 1 Let $a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}, c_{1}, c_{2},$ $c_{3}, c_{4}$ all be permutations of $1,2,3,4$. Find the minimum value of $\sum_{i=1}^{4} a_{i} b_{i} c_{i}$. | 44 | 96 | 2 |
math | Example 9. Let $x>0$. Try to find the maximum value of $y=\sqrt{x}+\frac{1}{\sqrt{x}}-$ $\sqrt{x+\frac{1}{x}+1}$. | 2-\sqrt{3} | 47 | 6 |
math | ## Task 2 - 250832
Brigade Schulz plays "Tele-Lotto (5 out of 35)" according to a so-called "fully mathematical system with $n$ numbers". This means that if $n$ is a natural number with $5 < n \leq 35$, the system is as follows:
Exactly $n$ numbers from 1, 2, ..., 35 are selected, and then all different subsets of fi... | 880M | 348 | 4 |
math | Example 2. Find the real solutions of the equation
$$
(4+2 i) x+(5-3 i) y=13+i
$$ | 2,\quad1 | 34 | 4 |
math | Éveriste listed all of the positive integers from 1 to 90 . He then crossed out all of the multiples of 3 from the list. Of the remaining numbers, he then crossed out all of the multiples of 5 . How many numbers were not crossed out? | 48 | 57 | 2 |
math | 1. (8 points) The calculation result of the expression $(11 \times 24-23 \times 9) \div 3+3$ is | 22 | 36 | 2 |
math | 9. (16 points) Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
\begin{array}{l}
a_{1}=1, a_{2}=2, a_{3}=4, \\
a_{n}=a_{n-1}+a_{n-2}-a_{n-3}+1(n \geqslant 4) .
\end{array}
$$
(1) Find the general term formula for the sequence $\left\{a_{n}\right\}$;
(2) Prove: $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{... | 3 | 158 | 1 |
math | Proizvolov V.V.
There are 19 weights of $1, 2, 3, \ldots, 19$ grams: nine iron, nine bronze, and one gold. It is known that the total weight of all iron weights is 90 grams more than the total weight of the bronze weights. Find the weight of the gold weight. | 10 | 77 | 2 |
math | I5.1 If the sum of roots of $5 x^{2}+a x-2=0$ is twice the product of roots, find the value of $a$.
15.2 Given that $y=a x^{2}-b x-13$ passes through $(3,8)$, find the value of $b$.
I5.3 If there are $c$ ways of arranging $b$ girls in a circle, find the value of $c$.
I5.4 If $\frac{c}{4}$ straight lines and 3 circles ... | 57 | 147 | 2 |
math | M1. The sum of the squares of two real numbers is equal to fifteen times their sum. The difference of the squares of the same two numbers is equal to three times their difference.
Find all possible pairs of numbers that satisfy the above criteria. | (6,-3),(-3,6),(0,0),(15,15) | 50 | 20 |
math | 1. Given that the real part of $(1+\mathrm{i} x)^{4 n+2}(x \in \mathbf{R})$ is a polynomial in $x$, then the sum of the coefficients of this polynomial is $\qquad$ . | 0 | 54 | 1 |
math | ## Problem Statement
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$$
\left\{\begin{array}{l}
x=\operatorname{ch} t \\
y=\sqrt[3]{\operatorname{sh}^{2} t}
\end{array}\right.
$$ | -\frac{2(3+\operatorname{ch}^{2})}{9\operatorname{sh}^{4}} | 75 | 26 |
math | Carmina has a total of $\$ 3.60$ in nickels and dimes. If her dimes were nickels and her nickels were dimes, then she would have $\$ 5.40$. How many nickels and dimes does Carmina have?
(A) 56
(B) 57
(C) 58
(D) 60
(E) 61 | 60 | 89 | 2 |
math | Fomin D:
The numbers $1, 1/2, 1/3, \ldots, 1/100$ are written on the board. We choose two arbitrary numbers $a$ and $b$ from those written on the board, erase them, and write the number
$a + b + ab$. We perform this operation 99 times until only one number remains. What is this number? Find it and prove that it does ... | 100 | 104 | 3 |
math | 16. Let $n$ be a given positive integer, $S_{n} \subseteq\left\{\alpha \mid \alpha=\left(p_{1}, p_{2}, \cdots, p_{n}\right), p_{k} \in\{0,1\}, k=1,2, \cdots, n\right\}$. For any elements $\beta=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ and $\gamma=\left(y_{1}, y_{2}, \cdots, y_{n}\right)$ in the set $S_{n}$, the follow... | 5 | 259 | 1 |
math | Example 12 (from the 1st American Invitational Mathematics Examination) Find the product of the real roots of the equation $x^{2}+18 x+30=2 \sqrt{x^{2}+18 x+45}$. | 20 | 55 | 2 |
math | $$
\begin{array}{l}
U=\{1,2, \cdots, 8\}, \\
A=\{1,2, \cdots, 5\}, \\
B=\{4,5, \cdots, 8\} .
\end{array}
$$
Then the number of subsets of $U$ that are neither subsets of $A$ nor subsets of $B$ is $\qquad$ | 196 | 92 | 3 |
math | 13. Let the intersection locus of two perpendicular tangents to the ellipse $\frac{x^{2}}{5}+\frac{y^{2}}{4}=1$ be $C$. Suppose the tangents $P A, P B$ to curve $C$ intersect at point $P$, and are tangent to $C$ at points $A, B$ respectively. Find the minimum value of $\overrightarrow{P A} \cdot \overrightarrow{P B}$. | 18\sqrt{2}-27 | 100 | 9 |
math | 12. Given four points $O, A, B, C$ on a plane, satisfying $O A=4, O B=3, O C=2, \overrightarrow{O B} \cdot \overrightarrow{O C}=3$, then the maximum value of the area of $\triangle A B C$ is $\qquad$. | 2\sqrt{7}+\frac{3\sqrt{3}}{2} | 73 | 18 |
math | \section*{Problem 5 - 161245}
Determine the number of all pairs \((p, q)\) of natural numbers with \(1 \leq p \leq 100\) and \(1 \leq q \leq 100\) and the property that the equation \(x^{5} + p x + q = 0\) has at least one rational solution. | 133 | 90 | 3 |
math | \section*{Problem 1 - 071241}
Determine all ordered quadruples of real numbers \(\left(x_{1}, x_{2}, x_{3}, x_{4}\right)\) that satisfy the following system of equations:
\[
\begin{aligned}
& x_{1}+a x_{2}+x_{3}=b \\
& x_{2}+a x_{3}+x_{4}=b \\
& x_{3}+a x_{4}+x_{1}=b \\
& x_{4}+a x_{1}+x_{2}=b
\end{aligned}
\]
Here,... | \begin{aligned}&Case1:(,,b-,b-)\\&Case2.1.1:(\frac{b}{2+},\frac{b}{2+},\frac{b}{2+},\frac{b}{2+})\\&Case2.1.2:(,,, | 158 | 64 |
math | 27. Let $\xi$ and $\eta$ be independent random variables having exponential distributions with parameters $\lambda$ and $\mu$ respectively. Find the distribution functions of the variables $\frac{\xi}{\xi+\eta}$ and $\frac{\xi+\eta}{\xi}$. | \begin{aligned}\mathrm{P}(\frac{\xi+\eta}{\xi}\leqslantx)&=1-(\frac{\mu}{\lambda}(x-1)+1)^{-1},\\\mathrm{P}(\frac{\xi}{\xi+\eta}\leqslantx)&=(\frac{\mu(1-x)}{\lambdax}+1 | 58 | 83 |
math | 3. It is known that the ages of A, B, and C
are all positive integers. A's age is twice B's age, B is 7 years younger than C. If the sum of the ages of these three people is a prime number less than 70, and the sum of the digits of this prime number is 13, then the ages of A, B, and C are | x=30, y=15, z=22 | 85 | 14 |
math | 7.226. $2.5^{\log _{3} x}+0.4^{\log _{3} x}=2.9$. | \frac{1}{3};3 | 36 | 8 |
math | Let $ P(z) \equal{} z^3 \plus{} az^2 \plus{} bz \plus{} c$, where $ a$, $ b$, and $ c$ are real. There exists a complex number $ w$ such that the three roots of $ P(z)$ are $ w \plus{} 3i$, $ w \plus{} 9i$, and $ 2w \minus{} 4$, where $ i^2 \equal{} \minus{} 1$. Find $ |a \plus{} b \plus{} c|$. | 136 | 116 | 3 |
math | 2. If $\sqrt{7 x^{2}+9 x+13}+\sqrt{7 x^{2}-5 x+13}=$ $7 x$, then $x=$ $\qquad$ | \frac{12}{7} | 45 | 8 |
math | Problem 2. Determine the pairs of integers $(a, b)$ that have the property that $a^{2}+2 b^{2}+2 a+1$ is a divisor of $2 a b$. | {(-3,-2);(-3,-1);(-1,-1);(-3,2);(-3,1);(-1,1)}\cup{(,0)\mid\in\mathbb{Z}}\cup{(0,b)\midb\in\mathbb{Z}} | 45 | 62 |
math | II. Fill in the Blanks (0 points per question, total 30 points)
7. (10 points) If the units and tens digits of Grandpa's age are swapped, it becomes Dad's age. The age difference between Grandpa and Dad is 5 times Xiaolin's age. Therefore, Xiaolin's age is $\qquad$ years. | 9 | 76 | 1 |
math | Three. (14 points). Let $S=\sqrt{1+\frac{1}{1^{2}}+\frac{1}{2^{2}}}+$ $\sqrt{1+\frac{1}{2^{2}}+\frac{1}{3^{2}}}+\cdots+\sqrt{1+\frac{1}{1999^{2}}+\frac{1}{2000^{2}}}$. Find the greatest integer $[S]$ not exceeding $S$. | 1999-\frac{1}{2000} | 100 | 14 |
math | 5. Let $a, b$ be two positive numbers, and $a>b$. Points $P, Q$ are on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. If the line connecting point $A(-a, 0)$ and $Q$ is parallel to the line $O P$, and intersects the $y$-axis at point $R$, where $O$ is the origin, then $\frac{|A Q| \cdot|A R|}{|O P|^{2}}=$ $\q... | 2 | 128 | 1 |
math | 4. Mary found that when a certain three-digit number is squared, the last three digits of the product are the same as the original three-digit number. What is the sum of all different three-digit numbers that satisfy this property? | 1001 | 46 | 4 |
math | Person A and Person B start from locations $A$ and $B$ respectively at the same time, moving towards each other at uniform speeds. They first meet at a point 8 kilometers away from $A$. After meeting, they continue to move forward, reach the other's starting point, and immediately return. They meet again at a point 6 k... | 15 | 94 | 2 |
math | 14. If the system of equations concerning $x$ and $y$
$$
\left\{\begin{array}{l}
\sin x=m \sin ^{3} y, \\
\cos x=m \cos ^{3} y
\end{array}\right.
$$
has real solutions, then the range of positive real number $m$ is
$\qquad$ | [1,2] | 82 | 5 |
math | 7.3. It is known that all krakozyabrs have horns or wings (possibly both). According to the results of the world census of krakozyabrs, it turned out that $20 \%$ of the krakozyabrs with horns also have wings, and $25 \%$ of the krakozyabrs with wings also have horns. How many krakozyabrs are left in the world, if it i... | 32 | 112 | 2 |
math | 6. For the numbers $1000^{2}, 1001^{2}, 1002^{2}, \ldots$, the last three digits are discarded. How many of the first terms of the resulting sequence form an arithmetic progression? | 32 | 55 | 2 |
math | ## Task 2 - 080622
During the summer holidays, Monika visited the capital of the USSR. For her mathematics study group, she brought the following problem:
In the "Gorky" ring of the Moscow underground, there are four escalators of different lengths.
The total length of the two middle-length escalators is $136 \mathr... | 84\mathrm{~},72\mathrm{~},64\mathrm{~},60\mathrm{~} | 142 | 28 |
math | 8.4. How many natural numbers less than 1000 are divisible by 4 and do not contain the digits $1,3,4,5,7,9$ in their notation? | 31 | 43 | 2 |
math | If $x, y, z \in \mathbb{R}$ are solutions to the system of equations
$$\begin{cases}
x - y + z - 1 = 0\\
xy + 2z^2 - 6z + 1 = 0\\
\end{cases}$$
what is the greatest value of $(x - 1)^2 + (y + 1)^2$? | 11 | 91 | 2 |
math | B1. We call a sequence of consecutive positive integers balanced if the number of multiples of three in that sequence is equal to the number of multiples of five. For example, the sequence 30, 31, 32, 33, 34, 35, 36 is not balanced, because out of these 7 numbers, 3 are multiples of three (namely 30, 33, and 36) and on... | 11 | 132 | 2 |
math | II. (40 points) Find all positive integers $m, n$ such that $\frac{m^{3}+n^{3}-m^{2} n^{2}}{(m+n)^{2}}$ is a non-negative integer. | (2,2) | 52 | 5 |
math | 3. Let in triangle $A B C$
$$
\cos (\angle A-\angle B)+\sin (\angle A+\angle B)=2
$$
Find the side $B C$, if $A B=4$. | 2\sqrt{2} | 48 | 6 |
math | 9. Let $F(x)=|f(x) \cdot g(x)|$, where $f(x)=a x^{2}+b x+c, g(x)=c x^{2}+b x+a, x \in[-1,1]$. For any $a, b, c$ satisfying $|f(x)| \leqslant 1$. When $a, b, c$ vary, find the maximum value of $F(x)$. | 2 | 97 | 1 |
math | ## Problem Statement
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{n+2}-\sqrt{n^{2}+2}}{\sqrt[4]{4 n^{4}+1}-\sqrt[3]{n^{4}-1}}
$$ | 0 | 59 | 1 |
math | A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}$. Find the difference between ... | 48 | 78 | 2 |
math | 1. Given the set $S=\{1,2,3, \cdots, 2000,2001\}$, where a subset $T$ has the property that for any three elements $x, y, z$ in $T$, $x+y \neq z$, what is the maximum number of elements in $T$? Maximum value | 1001 | 79 | 4 |
math | Pertin A.
Find all numbers of the form $2^{n}$ ($n$ is natural) such that when the first digit of their decimal representation is erased, the result is again a power of two. | 32,64 | 44 | 5 |
math | 5. Find all pairs of natural numbers $(m, n)$ for which the equality
$$
n \cdot(n-1) \cdot(n-2) \cdot(n-3)=m \cdot(m-1)
$$
holds. | (1,1),(2,1);(3,1) | 50 | 14 |
math | In square $ABCD$ with side length $2$, let $M$ be the midpoint of $AB$. Let $N$ be a point on $AD$ such that $AN = 2ND$. Let point $P$ be the intersection of segment $MN$ and diagonal $AC$. Find the area of triangle $BPM$.
[i]Proposed by Jacob Xu[/i] | \frac{2}{7} | 81 | 7 |
math | 22 Find all positive integer tuples $(x, y, z, w)$, such that $x!+y!+z!=w!$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | (2,2,2,3) | 57 | 9 |
math | ## Task Condition
Find the derivative.
$y=\frac{x^{4}-8 x^{2}}{2\left(x^{2}-4\right)}$ | x+\frac{16x}{(x^{2}-4)^{2}} | 34 | 18 |
math | 3. Given a constant $a>0$ and $a \neq 1$, if the function $f(x)=\log _{a}\left(x^{3}-a x\right)(a>0, a \neq 1)$ is monotonically increasing in the interval $\left(-\frac{1}{2}, 0\right)$, then the range of values for $a$ is $\qquad$ . | \frac{3}{4}\leqslant1 | 92 | 12 |
math | Can we put intengers $1,2,\cdots,12$ on a circle, number them $a_1,a_2,\cdots,a_{12}$ in order. For any $1\leq i<j\leq12$, $|a_i-a_j|\neq|i-j|$? | \text{yes} | 68 | 5 |
math | \section*{Task 1 - 011131}
A motor vehicle traveling at a speed of \(90 \frac{\mathrm{km}}{\mathrm{h}}\) is braked and comes to a stop after 70 \(\mathrm{m}\).
Has the prescribed deceleration of at least \(4.0 \frac{\mathrm{m}}{\mathrm{s}^{2}}\) in the Road Traffic Regulations been observed or not? Justify your concl... | 4.464\frac{\mathrm{}}{\mathrm{}^{2}} | 102 | 17 |
math | 10.4. Let's call a snake in a convex $n$-gon an open, non-self-intersecting broken line with $n-1$ segments, the set of vertices of which coincides with the set of all vertices of the $n$-gon. Find the number of different snakes in an $n$-gon. (Snakes are equal if they coincide as geometric loci of points. For example,... | n\cdot2^{n-3} | 101 | 9 |
math | Problem 2. (Option 1). Given an acute triangle $\mathrm{ABC}(\mathrm{AB}=\mathrm{BC})$ and $\mathrm{BC}=12$. $A N \perp B C$. On the side $\mathrm{BC}$, a point $M$ (M lies between B and $\mathrm{N}$) is marked such that $\mathrm{AN}=\mathrm{MN}$ and $\angle \mathrm{BAM}=\angle \mathrm{NAC}$. Find $\mathrm{BN}$. | 6\sqrt{3} | 112 | 6 |
math | Calculate the value of
$$
A=\frac{1001 \cdot 1002 \cdot 1003 \cdot \ldots \cdot 2000}{1 \cdot 3 \cdot 5 \cdot \ldots \cdot 1999}
$$
# | 2^{1000} | 67 | 7 |
math | a) Prove that, whatever the real number x would be, the following inequality takes place
${{x}^{4}}-{{x}^{3}}-x+1\ge 0.$
b) Solve the following system in the set of real numbers:
${{x}_{1}}+{{x}_{2}}+{{x}_{3}}=3,x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}$.
The Mathematical Gazette | (1, 1, 1) | 127 | 9 |
math | 799. Solve the equation in integers
$$
x^{2}-y^{2}=1997
$$ | (999;998),(999;-998),(-999;-998),(-999;998) | 26 | 35 |
math | 1. Let the polynomial $f(x)$ satisfy for any $x \in \mathbf{R}$, $f(x+1)+f(x-1)=2 x^{2}-4 x$, then the minimum value of $f(x)$ is $\qquad$ . | -2 | 57 | 2 |
math | Question 132, Let the sequence $\left\{a_{n}\right\}$ satisfy: $a_{1}=1, a_{n+1}=\frac{a_{n}}{(n+1)\left(a_{n}+1\right)}\left(n \in Z^{+}\right)$, try to find the value of $\lim _{n \rightarrow+\infty} n!\cdot a_{n}$. | \frac{1}{e} | 94 | 7 |
math | The distance from point $P(x, y)$ to the origin $O(0,0)$ is 17 .
The distance from point $P(x, y)$ to $A(16,0)$ is also 17 .
What are the two possible pairs of coordinates $(x, y)$ for $P$ ? | (8,15)(8,-15) | 68 | 11 |
math | 13. There are two identical cups, A and B. Cup A contains half a cup of clear water, and Cup B is full of a $50 \%$ alcohol solution. First, half of the alcohol solution from Cup B is poured into Cup A, then the mixture in Cup A is stirred evenly, and half of the alcohol solution from Cup A is poured back into Cup B. W... | \frac{3}{8} | 94 | 7 |
math | (10) (15 points) Given the sequence of positive numbers $\left\{a_{n}\right\}(n \geqslant 0)$ satisfies $a_{n}=\frac{a_{n-1}}{m a_{n-2}}, n=2$,
$3, \cdots$, where $m$ is a real parameter. If $a_{2009}=\frac{a_{0}}{a_{1}}$, find the value of $m$. | 1 | 108 | 1 |
math | A positive integer $n > 1$ is juicy if its divisors $d_1 < d_2 < \dots < d_k$ satisfy $d_i - d_{i-1} \mid n$ for all $2 \leq i \leq k$. Find all squarefree juicy integers. | 2, 6, 42, 1806 | 65 | 14 |
math | Example 1.17 Find the coefficient of $x^{5}$ in the expansion of $\left(1+2 x+3 x^{2}+4 x^{3}\right)^{5}$. | 1772 | 44 | 4 |
math | Let $\mathbb{N}$ denote the set of positive integers. Let $\varphi: \mathbb{N} \rightarrow \mathbb{N}$ be a bijective function and assume that there exists a finite limit
$$
\lim _{n \rightarrow \infty} \frac{\varphi(n)}{n}=L
$$
What are the possible values of $L$ ? | 1 | 84 | 1 |
math | 4.1. (13 points) The numerator of an irreducible fraction was cubed, and the denominator was increased by 3. As a result, the fraction doubled. Find the value of the original fraction, if its numerator and denominator are natural numbers. | \frac{2}{3} | 54 | 7 |
math | 1. Determine all real solutions of the following equations:
a) $\sqrt[3]{x+1}+\sqrt{x+1}=2$;
b) $\frac{x^{2}+1}{x}=2^{x(2-x)}$. | )0,\,b)1 | 52 | 7 |
math | 1. The range of the function $f(x)=\sin x+\cos x+\tan x+$ $\arcsin x+\arccos x+\arctan x$ is $\qquad$ . | [-\sin 1+\cos 1-\tan 1+\frac{\pi}{4}, \sin 1+\cos 1+\tan 1+\frac{3 \pi}{4}] | 43 | 41 |
math | $$
x^{2}+p x+q=0 \quad \text { and } \quad x^{2}+p^{\prime} x+q^{\prime}=0
$$
are equations with roots $\alpha, \beta, \alpha^{\prime}$ and $\beta^{\prime}$. Express
$$
\left(\alpha-\alpha^{\prime}\right)\left(\alpha-\beta^{\prime}\right)\left(\beta-\alpha^{\prime}\right)\left(\beta-\beta^{\prime}\right)
$$
as a fun... | [q-q^{\}]^{2}+[p-p^{\}][q^{\}p-p^{\}q] | 140 | 26 |
math | 15. (6 points) The number of simplest proper fractions where the sum of the numerator and denominator is 2013 is. $\qquad$ | 600 | 33 | 3 |
math | 6. Given $f(x)=\frac{1+x}{2-x}$, for $n \in \mathbf{N}$, define: $f_{1}(x)=f(x), f_{n+1}(x)=f\left[f_{n}(x)\right]$. If $f_{13}(x)=f_{31}(x)$, then the analytical expression of $f_{16}(x)$ is . $\qquad$ | \frac{x-1}{x} | 97 | 8 |
math | 7.096. $\lg \left(3 x^{2}+12 x+19\right)-\lg (3 x+4)=1$. | -1;7 | 36 | 4 |
math | Three, (20 points) Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that for any $x, y \in \mathbf{R}$, we have
$$
\begin{array}{l}
f(4 x-y-1)+4 f(x+y)-16 f(x) \\
=(6 x-2)^{2}+45 y^{2}
\end{array}
$$ | f(x)=(3 x-1)^{2} | 97 | 11 |
math | 2. (1992 Ukrainian Mathematical Olympiad) Find all prime numbers $p$ not exceeding 1000 such that $2p+1$ is a power of a natural number (i.e., there exist natural numbers $m$ and $n, n \geqslant 2$, such that $2p+1=m^n$). | 13 | 76 | 2 |
math | 3. Divide the numbers $1,2, \cdots, 30$ into $k$ groups (each number can only appear in one group) such that the sum of any two different numbers in each group is not a perfect square. Find the minimum value of $k$.
Put the above text into English, please keep the original text's line breaks and format, and output the... | 3 | 85 | 1 |
math | 4. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Then the set
$$
\{[x]+[2 x]+[3 x] \mid x \in \mathbf{R}\} \cap\{1,2, \cdots, 100\}
$$
has $\qquad$ elements. | 67 | 77 | 2 |
math | $\boxed{\text{A5}}$ Determine all positive integers $a,b$ such that $a^{2}b^{2}+208=4([a,b]+(a,b))^2$ where $[a,b]$-lcm of $a,b$ and $(a,b)$-gcd of $a,b$. | (4, 6), (6, 4), (2, 12), (12, 2) | 71 | 26 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 2}\left(2 e^{x-2}-1\right)^{\frac{3 x+2}{x-2}}$ | e^{16} | 47 | 5 |
math | 13. Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}=2 a_{n}-1(n=1,2, \cdots)$, and the sequence $\left\{b_{n}\right\}$ satisfies $b_{1}=3, b_{k+1}=a_{k}+b_{k}$ $(k=1,2,3, \cdots)$, find the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$. | 2^{n}+2n-1 | 125 | 9 |
math | Example 2 Color each vertex of a square pyramid with one color, and make the endpoints of the same edge have different colors. If only 5 colors are available, then the total number of different coloring methods is $\qquad$ | 420 | 47 | 3 |
math | 1A. Solve the irrational equation
$$
\frac{\sqrt{x^{2}+x+6}+\sqrt{x^{2}-x-4}}{\sqrt{x^{2}+x+6}-\sqrt{x^{2}-x-4}}=5
$$ | x_{1}=5,x_{2}=-\frac{12}{5} | 58 | 18 |
math | $\begin{array}{l} \text { Given } a>0, b>0, a+b=1, \\ \sqrt{2}<\sqrt{a+\frac{1}{2}}+\sqrt{b+\frac{1}{2}} \leqslant 2 .\end{array}$ | \sqrt{a+\frac{1}{2}}+\sqrt{b+\frac{1}{2}}>\frac{\sqrt{2}}{2}+\frac{\sqrt{6}}{2} | 66 | 42 |
math | Let $a$ be the sum of the numbers:
$99 \times 0.9$
$999 \times 0.9$
$9999 \times 0.9$
$\vdots$
$999\cdots 9 \times 0.9$
where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$.
Find the sum of the digits in the number $a$. | 891 | 110 | 3 |
math | 13. (6 points) A fraction, if the denominator is reduced by 1, simplifies to $\frac{1}{3}$; if the numerator is increased by 4, it simplifies to $\frac{1}{2}$. This fraction is $\qquad$ | \frac{7}{22} | 58 | 8 |
math | 11. (16 points) Let $A$ and $B$ be two different subsets of the set $\left\{a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right\}$, such that $A$ is not a subset of $B$, and $B$ is not a subset of $A$. Find the number of different ordered pairs $(A, B)$. | 570 | 89 | 3 |
math | ## Task 11/73
A train passes a kilometer post with a two-digit kilometer number. After the time $\Delta t_{1}$, it passes another kilometer post with the same digits but in reversed order.
Finally, after the additional time $\Delta t_{2}=\Delta t_{2}$, it encounters a third kilometer post, whose number is the same as... | 16,61,106 | 149 | 9 |
math | 8. If there exists a sufficiently large positive integer $n$, which, apart from not being divisible by two consecutive positive integers $k$, $k+1$ from 1 to 250, is divisible by all other integers from 1 to 250, then $k=$ $\qquad$ | 127 | 65 | 3 |
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