task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | ## Task A-3.6.
Let $\overline{BD}$ and $\overline{CE}$ be the altitudes of an acute-angled triangle $ABC$. Determine the smallest measure of angle $\varangle BAC$ for which it is possible that $|AE| \cdot |AD| = |BE| \cdot |CD|$. | 60 | 72 | 2 |
math | 5. Let positive integers $a, b, c, d$ satisfy $a>b>c>d$, and $a+b+c+d=2004, a^{2}-b^{2}+c^{2}-d^{2}=2004$. Then the minimum value of $a$ is . $\qquad$ | 503 | 69 | 3 |
math | What is the largest possible number of subsets of the set $\{1,2, \ldots, 2 n+1\}$ such that the intersection of any two subsets consists of one or several consecutive integers? | (n+1)^2 | 44 | 5 |
math | 19 (12 points) Whether a company invests in a project is decided by three decision-makers, A, B, and C. Each of them has one "agree," one "neutral," and one "disagree" vote. When voting, each person must and can only cast one vote, and the probability of each person casting any one of the three types of votes is $\frac... | \frac{13}{27} | 173 | 9 |
math | Solve the following system of equations:
$$
\begin{aligned}
& x^{2}+x y+y=1 \\
& y^{2}+x y+x=5
\end{aligned}
$$ | -1,3or-2 | 45 | 7 |
math | 139. $16^{x}=1 / 4$.
139. $16^{x}=1 / 4$.
(Note: The translation is the same as the original text because it is a mathematical equation, which is universal and does not change in translation.) | -\frac{1}{2} | 60 | 7 |
math | In the city where the Absent-Minded Scholar lives, telephone numbers consist of 7 digits. The Scholar easily remembers a telephone number if it is a palindrome, that is, it reads the same from left to right as from right to left. For example, the number 4435344 is easily remembered by the Scholar because it is a palind... | 0.001 | 120 | 5 |
math | G4.2 If $x=\frac{\sqrt{5}+1}{2}$ and $y=\frac{\sqrt{5}-1}{2}$, determine the value of $x^{3} y+2 x^{2} y^{2}+x y^{3}$. | 5 | 61 | 1 |
math | Let $a$ and $b$ be two real numbers and let$M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$. Find the values of $a$ and $b$ for which $M(a,b)$ is minimal. | a = -\frac{1}{3}, b = -\frac{1}{3} | 70 | 20 |
math | 8. Given $a, b \in [1,3], a+b=4$. Then
$$
\left|\sqrt{a+\frac{1}{a}}-\sqrt{b+\frac{1}{b}}\right|
$$
the maximum value is $\qquad$. | \sqrt{\frac{10}{3}}-\sqrt{2} | 60 | 15 |
math | 6. Choose two different non-empty subsets $A$ and $B$ of the set $S=\{1,2, \ldots, 10\}$. The probability that the smallest number in $B$ is greater than the largest number in $A$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation res... | \frac{4097}{1045506} | 86 | 16 |
math | 9.054. $5^{2 x+1}>5^{x}+4$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
9.054. $5^{2 x+1}>5^{x}+4$. | x\in(0;\infty) | 68 | 9 |
math | 1.027. $\left(\frac{3.75+2 \frac{1}{2}}{2 \frac{1}{2}-1.875}-\frac{2 \frac{3}{4}+1.5}{2.75-1 \frac{1}{2}}\right) \cdot \frac{10}{11}$. | 6 | 83 | 1 |
math | 15.27. What is the maximum number of parts that $n$ circles can divide a sphere into? | n^2-n+2 | 24 | 6 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{e^{2 x}-e^{-5 x}}{2 \sin x-\tan x}$ | 7 | 43 | 1 |
math | Example 3 Find the number of different integer values taken by the function $f(x)=[x]+[2 x]+[3 x]+\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]$ on $0 \leqslant x \leqslant 100$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directl... | 401 | 92 | 3 |
math | 20. Find all nonnegative integer solutions of the system
$$
\begin{array}{l}
5 x+7 y+5 z=37 \\
6 x-y-10 z=3 .
\end{array}
$$ | (x,y,z)=(4,1,2) | 50 | 10 |
math | Condition of the problem
To find the equations of the tangent and normal to the curve at the point corresponding to the parameter value $t=t_{0}$.
$\left\{\begin{array}{l}x=3 \cos t \\ y=4 \sin t\end{array}\right.$
$t_{0}=\frac{\pi}{4}$ | -\frac{4}{3}\cdotx+4\sqrt{2} | 74 | 16 |
math | 2 In a non-decreasing sequence of positive integers $a_{1}, a_{2}, \cdots, a_{m}, \cdots$, for any positive integer $m$, define $b_{m}=$ $\min \left\{n \mid a_{n} \geqslant m\right\}$. It is known that $a_{19}=85$. Find the maximum value of $S=a_{1}+a_{2}+\cdots+a_{19}+b_{1}+b_{2}+\cdots+$ $b_{85}$. (1985 USA Mathemati... | 1700 | 138 | 4 |
math | * Find all positive integers $n$ such that the numbers $n+1, n+3, n+7, n+9, n+13, n+15$ are all prime. | 4 | 43 | 1 |
math |
Problem 4. Let $M$ be a subset of the set of 2021 integers $\{1,2,3, \ldots, 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$ we have $|a+b-c|>10$. Determine the largest possible number of elements of $M$.
| 1006 | 87 | 4 |
math | Task B-4.1. Determine the natural number $N$ for which
$$
\frac{1}{2!11!}+\frac{1}{3!10!}+\frac{1}{4!9!}+\frac{1}{5!8!}+\frac{1}{6!7!}=\frac{N}{1!12!}
$$ | 314 | 83 | 3 |
math | 1. It is known that $m, n, k$ are distinct natural numbers greater than 1, the number $\log _{m} n$ is rational, and, moreover,
$$
k^{\sqrt{\log _{m} n}}=m^{\sqrt{\log _{n} k}}
$$
Find the minimum of the possible values of the sum $k+5 m+n$. | 278 | 85 | 3 |
math | Show that for any positive real numbers $a,b,c$ the following inequality is true:
$$4(a^3+b^3+c^3+3)\geq 3(a+1)(b+1)(c+1)$$
When does equality hold? | 4(a^3 + b^3 + c^3 + 3) \geq 3(a+1)(b+1)(c+1) | 55 | 35 |
math | 7.266. $\left\{\begin{array}{l}\left(x^{2}+y\right) 2^{y-x^{2}}=1 \\ 9\left(x^{2}+y\right)=6^{x^{2}-y}\end{array}\right.$
The system of equations is:
\[
\left\{\begin{array}{l}
\left(x^{2}+y\right) 2^{y-x^{2}}=1 \\
9\left(x^{2}+y\right)=6^{x^{2}-y}
\end{array}\right.
\] | (\sqrt{3};1),(-\sqrt{3};1) | 135 | 15 |
math | ## Task 3 - 040513
The school garden of a city school has an area of 0.15 ha. The garden is divided into 9 plots, each with an area of $150 \mathrm{~m}^{2}$ or $200 \mathrm{~m}^{2}$.
How many plots of each size are there in the garden? | 6\cdot150+3\cdot200=1500 | 85 | 18 |
math | 1. (16 points) Given $a, b \neq 0$, and $\frac{\sin ^{4} x}{a^{2}}+\frac{\cos ^{4} x}{b^{2}}=\frac{1}{a^{2}+b^{2}}$. Find the value of $\frac{\sin ^{100} x}{a^{100}}+\frac{\cos ^{100} x}{b^{100}}$. | \frac{2}{\left(a^{2}+b^{2}\right)^{50}} | 103 | 22 |
math | 2. Express the fraction $\frac{93}{91}$ as the sum of two positive fractions whose denominators are 7 and 13. | \frac{93}{91}=\frac{5}{7}+\frac{4}{13} | 32 | 24 |
math | 6. (3 points) Find the total length of intervals on the number line where the inequalities $|x|<1$ and $\operatorname{tg} \log _{5}|x|<0$ are satisfied. | \frac{2\cdot5^{\frac{\pi}{2}}}{1+5^{\frac{\pi}{2}}} | 47 | 27 |
math | 1A. Solve the equation in $\mathbb{R}$
$$
\left(x^{2}-3\right)^{3}-(4 x+6)^{3}+6^{3}=18(4 x+6)\left(3-x^{2}\right)
$$ | x_{1}=2+\sqrt{7},x_{2}=2-\sqrt{7},x_{3}=-3 | 61 | 26 |
math | 16. Arrange the numbers that can be expressed as the sum of $m$ consecutive positive integers ($m>1$) in ascending order to form a sequence, denoted as $\{a(m, n)\}$, where $a(m, n)$ represents the $n$-th number (in ascending order) that can be expressed as the sum of $m$ consecutive positive integers.
（1）Write down $a... | 2000=47+48+\cdots+78=68+69+\cdots+92=398+399+400+401+402 | 162 | 48 |
math | Find every real values that $a$ can assume such that
$$\begin{cases}
x^3 + y^2 + z^2 = a\\
x^2 + y^3 + z^2 = a\\
x^2 + y^2 + z^3 = a
\end{cases}$$
has a solution with $x, y, z$ distinct real numbers. | \left(\frac{23}{27}, 1\right) | 85 | 16 |
math | $7 \cdot 73$ Let $n \in N$, and make $37.5^{n}+26.5^{n}$ a positive integer, find the value of $n$.
(Shanghai Mathematical Competition, 1998) | 1,3,5,7 | 57 | 7 |
math | Example 7. From the first machine, 200 parts were sent to assembly, of which 190 are standard; from the second - 300, of which 280 are standard. Find the probability of event $A$, which consists in a randomly taken part being standard, and the conditional probabilities of it relative to events $B$ and $\bar{B}$, if eve... | 0.94,0.95,\frac{14}{15}\approx0.93 | 99 | 23 |
math | [ Properties and characteristics of an isosceles triangle. ] [ Area of a triangle (using height and base). ]
A circle with its center on side $A C$ of the isosceles triangle $A B C (A B = B C)$ touches sides $A B$ and $B C$. Find the radius of the circle if the area of triangle $A B C$ is 25, and the ratio of the heig... | 2\sqrt{3} | 109 | 6 |
math | Example 9. Find all integer values of $x$ that make $y=\frac{x^{2}-2 x+4}{x^{2}-3 x+3}$ an integer. | x = -1, 1, 2 | 39 | 10 |
math | 9. Let $x_{1}, x_{2}, \cdots, x_{n}$ take values 7 or -7, and satisfy
(1) $x_{1}+x_{2}+\cdots+x_{n}=0$;
(2) $x_{1}+2 x_{2}+\cdots+n x_{n}=2009$.
Determine the minimum value of $n$. | 34 | 91 | 2 |
math | 2. We will call a set of distinct natural numbers from 1 to 9 good if the sum of all numbers in it is even. How many good sets are there? | 2^8 | 36 | 3 |
math | Subject (2). Consider the following natural numbers:
$$
a=1 \cdot 3 \cdot 5 \cdot 7 \cdots 27 \cdot 29 \cdot 31 \text { and } b=1 \cdot 3 \cdot 5 \cdot 7 \cdots 27 \cdot 29
$$
a) Prove that the number $a$ is divisible by 2015.
b) Find the largest natural number $n$ such that the number $a+b$ is divisible by $10^n$.
... | 4 | 145 | 1 |
math | \section*{Problem 5 - V1104}
In Moscow, the tallest television tower in the world is being built, which will have a height of approximately \(500 \mathrm{~m}\) upon completion.
How far can a point on the Earth's surface be from the top of the tower at most, so that it is still visible from there (without considering ... | 79.8\mathrm{~} | 104 | 9 |
math | Example 9. Find the range of $y=\frac{4 \sin x+5 \cos x}{\sin x+\cos x+3}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \left[\frac{-9-4 \sqrt{23}}{7}, \frac{-9+4 \sqrt{23}}{7}\right] | 58 | 34 |
math | Four, (Full marks 14 points) Given that the roots of the equation $x^{2}+p x+q=0$ are 1997 and 1998, when $x$ takes the integer values $0,1,2, \cdots, 1999$, the corresponding values of the quadratic trinomial $y=x^{2}+p x+q$ are $y_{0}$. Find the number of these values that are divisible by 6. | 1333 | 109 | 4 |
math | 3. [4] Three real numbers $x, y$, and $z$ are such that $(x+4) / 2=(y+9) /(z-3)=(x+5) /(z-5)$. Determine the value of $x / y$. | \frac{1}{2} | 57 | 7 |
math | Find all functions $f: \mathbb N \to \mathbb N$ Such that:
1.for all $x,y\in N$:$x+y|f(x)+f(y)$
2.for all $x\geq 1395$:$x^3\geq 2f(x)$ | f(n) = kn | 67 | 6 |
math | 3. The numbers $1,2,3, \ldots, 29,30$ were written in a row in a random order, and partial sums were calculated: the first sum $S_{1}$ equals the first number, the second sum $S_{2}$ equals the sum of the first and second numbers, $S_{3}$ equals the sum of the first, second, and third numbers, and so on. The last sum $... | 23 | 139 | 2 |
math | Given that
$$
\begin{aligned}
a+b & =23 \\
b+c & =25 \\
c+a & =30
\end{aligned}
$$
determine (with proof) the value of $a b c$. | 2016 | 52 | 4 |
math | # 5. Option 1.
An apple, three pears, and two bananas together weigh 920 g; two apples, four bananas, and five pears together weigh 1 kg 710 g. How many grams does a pear weigh? | 130 | 55 | 3 |
math | 9. (3 points) At Dongfang Primary School, Grade 6 selected $\frac{1}{11}$ of the girls and 22 boys to participate in the "Spring Welcome Cup" Math Competition. The remaining number of girls is twice the remaining number of boys. It is also known that the number of girls is 2 more than the number of boys. How many stude... | 86 | 90 | 2 |
math | ## PROBLEM 30. SYSTEM OF DIOPHANTINE EQUATIONS
Solve the following system of equations in natural numbers:
$$
\begin{aligned}
a^{3}-b^{3}-c^{3} & =3 a b c \\
a^{2} & =2(b+c)
\end{aligned}
$$ | 2,1,1 | 71 | 5 |
math | 7. In a company, several employees have a total monthly salary of 10000 dollars. A kind manager proposes to triple the salary for those earning up to 500 dollars, and increase the salary by 1000 dollars for the rest, so the total salary will become 24000 dollars. A mean manager proposes to reduce the salary to 500 doll... | 7000 | 112 | 4 |
math | 4. There are three alloys of nickel, copper, and manganese. In the first, $30\%$ is nickel and $70\%$ is copper, in the second - $10\%$ is copper and $90\%$ is manganese, and in the third - $15\%$ is nickel, $25\%$ is copper, and $60\%$ is manganese. A new alloy of these three metals is needed with $40\%$ manganese. Wh... | 40to43\frac{1}{3} | 123 | 12 |
math | 20. Find the value of $\frac{1}{\sin 10^{\circ}}-4 \sin 70^{\circ}$. | 2 | 33 | 1 |
math | 2. The cold water tap fills the bathtub in 17 minutes, and the hot water tap in 23 minutes. The hot water tap was opened. After how many minutes should the cold water tap be opened so that by the time the bathtub is completely filled, there is an equal amount of cold and hot water in it? | 3 | 68 | 1 |
math | Let $T$ be a triangle with side lengths $26$, $51$, and $73$. Let $S$ be the set of points inside $T$ which do not lie within a distance of $5$ of any side of $T$. Find the area of $S$. | \frac{135}{28} | 61 | 10 |
math | 4. If $x, y$ are two different real numbers, and
$$
x^{2}=2 x+1, y^{2}=2 y+1 \text {, }
$$
then $x^{6}+y^{6}=$ $\qquad$ . | 198 | 59 | 3 |
math | In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ De... | \beta = \frac{1}{2(1 - \alpha)} | 166 | 16 |
math | 4. 3. 11 $\star \star$ Find the maximum and minimum values of the function $y=\sqrt{3 x+4}+\sqrt{3-4 x}$. | \frac{5}{2} | 41 | 7 |
math | Problem 1. The sum of the first four terms of an arithmetic progression, as well as the sum of the first nine terms, are natural numbers. In addition, its first term $b_{1}$ satisfies the inequality $b_{1} \leqslant \frac{3}{4}$. What is the greatest value that $b_{1}$ can take? | \frac{11}{15} | 77 | 9 |
math | 7. Given $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ are arithmetic sequences (i.e. $a_{2}-a_{1}=a_{3}-a_{2}=a_{4}-a_{3}=\cdots$ and $\left.b_{2}-b_{1}=b_{3}-b_{2}=b_{4}-b_{3}=\cdots\right)$ such that $(3 n+1) a_{n}=(2 n-1) b_{n}$ for all positive integers $n$. If $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$ and $T_{n}=b_{1}... | \frac{27}{23} | 386 | 9 |
math | 330. Find $y^{\prime}$, if $y=\sqrt{x}+\cos ^{2} 3 x$. | \frac{1}{2\sqrt{x}}-3\sin6x | 29 | 16 |
math | The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.... | 6 | 365 | 1 |
math | 18. (6 points) There are 300 balls in total, including white balls and red balls, and 100 boxes. Each box contains 3 balls, with 27 boxes containing 1 white ball, 42 boxes containing 2 or 3 red balls, and the number of boxes containing 3 white balls is the same as the number of boxes containing 3 red balls. Therefore, ... | 158 | 98 | 3 |
math | 8. The system of equations $\left\{\begin{array}{l}x+x y+y=1, \\ x^{2}+x^{2} y^{2}+y^{2}=17\end{array}\right.$ has the real solution $(x, y)=$ | \left(\frac{3+\sqrt{17}}{2}, \frac{3-\sqrt{17}}{2}\right) \text{ or } \left(\frac{3-\sqrt{17}}{2}, \frac{3+\sqrt{17}}{2}\right) | 61 | 65 |
math | Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$.
I.Voronovich
| a \in \mathbb{R} \setminus \{1\} | 55 | 18 |
math | 1-17 Let $A$ be the sum of the digits of the decimal number $4444^{4444}$, and let $B$ be the sum of the digits of $A$. Find the sum of the digits of $B$ (all numbers here are in decimal). | 7 | 63 | 1 |
math | Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$, $OG = 1$ and $OG \parallel BC$. (As usual $O$ is the circumcenter and $G$ is the centroid.) | 12 | 59 | 2 |
math | ## Task 6 - 210736
A liquid is sold in small, medium, and large bottles. Each small bottle holds exactly $200 \mathrm{~g}$, each medium bottle exactly $500 \mathrm{~g}$, and each large bottle exactly $1000 \mathrm{~g}$ of the liquid. Each filled $200 \mathrm{~g}$ bottle costs $1.20 \mathrm{M}$, and each filled $500 \m... | 0.55 | 248 | 4 |
math | Problem 3. Determine how many roots of the equation
$$
4 \sin 2 x + 3 \cos 2 x - 2 \sin x - 4 \cos x + 1 = 0
$$
are located on the interval $\left[10^{2014!} \pi ; 10^{2014!+2015} \pi\right]$. In your answer, write the sum of all digits of the found number. | 18135 | 105 | 5 |
math | Example 12 In space, there are 1989 points, where no three points are collinear. Divide them into 30 groups with different numbers of points. For any three different groups, take one point from each as the vertex to form a triangle. Question: To maximize the total number of such triangles, how many points should be in ... | 51,52,\cdots,56,57,\cdots81 | 76 | 19 |
math | Example 9. Simplify $1-\frac{1}{4} \sin ^{2} 2 \alpha-\sin ^{2} \beta-\cos ^{4} \alpha$ into a product form of trigonometric functions. (84 College Entrance Examination for Liberal Arts)
(84年高考文科试题)
Note: The last line is kept in Chinese as it is a reference to the source of the problem and does not need to be trans... | \sin (\alpha+\beta) \sin (\alpha-\beta) | 120 | 14 |
math | 9.4. In triangle $ABC$ with sides $AB=c, BC=a, AC=b$, the median $BM$ is drawn. Incircles are inscribed in triangles $ABM$ and $BCM$. Find the distance between the points of tangency of these incircles with the median $BM$. | \frac{|-|}{2} | 63 | 8 |
math | 18. (6 points) In a certain exam, the average score of 11 students, rounded to the first decimal place, is 85.3. It is known that each student's score is an integer. Therefore, the total score of these 11 students is $\qquad$ points. | 938 | 65 | 3 |
math | ## Task Condition
Find the derivative of the specified order.
$y=\left(x^{2}+3 x+1\right) e^{3 x+2}, y^{V}=?$ | 3^{3}\cdot(9x^{2}+57x+74)e^{3x+2} | 41 | 25 |
math | For any positive integer $n$, define $a_n$ to be the product of the digits of $n$.
(a) Prove that $n \geq a(n)$ for all positive integers $n$.
(b) Find all $n$ for which $n^2-17n+56 = a(n)$. | n = 4 | 70 | 5 |
math | 5. Ante, Bruno, Ciprijan, Davor, Emanuel, and Franko need to line up in a row.
a) In how many different ways can the boys line up if Bruno stands to the left of Emanuel?
b) In how many different ways can the boys line up if there is no one standing between Ciprijan and Davor?
The use of a pocket calculator or any re... | 360 | 135 | 3 |
math | Find all positive integers $n, n>1$ for wich holds :
If $a_1, a_2 ,\dots ,a_k$ are all numbers less than $n$ and relatively prime to $n$ , and holds $a_1<a_2<\dots <a_k $, then none of sums $a_i+a_{i+1}$ for $i=1,2,3,\dots k-1 $ are divisible by $3$. | n = 2, 4, 10 | 99 | 11 |
math | 5. (5 points) There are three simplest true fractions, the ratio of their numerators is $3: 2: 4$, and the ratio of their denominators is $5: 9: 15$. After adding these three fractions and simplifying, the result is $\frac{28}{45}$. What is the sum of the denominators of the three fractions? $\qquad$ . | 203 | 87 | 3 |
math | Question 45: Given $x \in R$, find the maximum value of $\frac{\sin x(2-\cos x)}{5-4 \cos x}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{\sqrt{3}}{4} | 62 | 10 |
math | ## Task 4 - 320734
Determine the number of all six-digit natural numbers that are divisible by 5 and whose cross sum is divisible by 9! | 20000 | 39 | 5 |
math | 24. The lengths of the sides of pentagon $A B C D E$ are as follows: $A B=16 \mathrm{~cm}, B C=14 \mathrm{~cm}$, $C D=17 \mathrm{~cm}, D E=13 \mathrm{~cm}, A E=14 \mathrm{~cm}$. Five circles with centres at the points $A$, $B, C, D, E$ are drawn so that each circle touches both of its immediate neighbours. Which point ... | A | 132 | 1 |
math | 13. (i) (Grade 11) In the arithmetic sequence $\left\{a_{n}\right\}: a_{n}=4 n -1\left(n \in \mathbf{N}_{+}\right)$, after deleting all numbers that can be divided by 3 or 5, the remaining numbers are arranged in ascending order to form a sequence $\left\{b_{n}\right\}$. Find the value of $b_{2006}$.
(ii) (Grade 12) Gi... | 15043 | 217 | 5 |
math | 12. Let $n$ be a natural number, write $n$ as a sum of powers of $p$ (where $p$ is a positive integer greater than 1) and each power of $p$ can appear at most $p^{2}-1$ times, denote the total number of such decompositions as $C(n, p)$. For example: $8=4+4=4+2+2=4+2+1+1=2+2+2+1+1=8$, then $C(8,2)=5$. Note that in $8=4+... | 118 | 184 | 3 |
math | Example 9: Given $\triangle A B C$ intersects with a line $P Q$ parallel to $A C$ and the area of $\triangle A P Q$ equals a constant $k^{2}$, what is the relationship between $b^{2}$ and the area of $\triangle A B C$? Is there a unique solution? | S \geqslant 4 k^{2} | 71 | 12 |
math | Find all natural numbers $n$ for which $2^{8}+2^{11}+2^{n}$ is equal to the square of an integer. | 12 | 34 | 2 |
math | 8. Find all sequences $\left\{a_{1}, a_{2}, \cdots\right\}$ that satisfy the following conditions:
$a_{1}=1$ and $\left|a_{n}-a_{m}\right| \leqslant \frac{2 m n}{m^{2}+n^{2}}$ (for all positive integers $m, n$). | a_{1}=a_{2}=\cdots=1 | 83 | 13 |
math | 1. Let $x$ and $y$ be real numbers, and satisfy
$$
\left\{\begin{array}{l}
(x-1)^{3}+1997(x-1)=-1 \\
(y-1)^{3}+1997(y-1)=1 .
\end{array}\right.
$$
Then $x+y=$ | x+2 | 80 | 3 |
math | . Find all positive integers $n$ such that the decimal representation of $n^{2}$ consists of odd digits only. | n=1n=3 | 25 | 6 |
math | ## Task 5 - 341245
Determine all pairs $(x ; y)$ of non-negative integers $x, y$ for which the following holds:
$$
x^{3}+8 x^{2}-6 x+8=y^{3}
$$ | (x,y)\in{(0,2),(9,11)} | 58 | 14 |
math | 1. Find all values of $p$, for each of which the numbers $9 p+10, 3 p$ and $|p-8|$ are respectively the first, second, and third terms of some geometric progression. | p=-1,p=\frac{40}{9} | 48 | 12 |
math | 5. For a set, the difference between the maximum and minimum elements is called the "capacity" of the set. Let $2 \leqslant r \leqslant n$, and let $F(n, r)$ denote the arithmetic mean of the capacities of all $r$-element subsets of the set $M=\{1,2, \cdots, n\}$. Then $F(n, r)=$ | \frac{(r-1)(n+1)}{r+1} | 90 | 16 |
math |
4. Find all pairs of prime numbers $(p, q)$ for which
$$
7 p q^{2}+p=q^{3}+43 p^{3}+1
$$
| (2,7) | 43 | 5 |
math | Consider a positive real number $a$ and a positive integer $m$. The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as:
$x_1=1$, $x_2=a$, $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$.
$a)$ Prove that the sequence is converging.
$b)$ Find $\lim_{n\rightarrow \infty}{x_n}$. | a^{\frac{m+1}{m+2}} | 103 | 13 |
math | Problem 11.1. The sequence $\left\{a_{n}\right\}_{n=1}^{\infty}$ is defined by $a_{1}=0$ and $a_{n+1}=$ $a_{n}+4 n+3, n \geq 1$.
a) Express $a_{n}$ as a function of $n$.
b) Find the limit
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{a_{n}}+\sqrt{a_{4 n}}+\sqrt{a_{4^{2} n}}+\cdots+\sqrt{a_{4}{ }^{10} n}}{\sqrt{a_{n}}+... | 683 | 196 | 3 |
math | 1. The solution set of the inequality $(x+1)^{3}\left(x^{3}+5 x\right)<10(x+1)^{2}$ +8 is | (-2,1) | 39 | 5 |
math | Let $n$ be an integer. Determine the largest constant $C$ possible so that for all $a_{1} \ldots, a_{n} \geqslant 0$ we have $\sum a_{i}^{2} \geqslant C \sum_{i<j} a_{i} a_{j}$. | \frac{2}{n-1} | 73 | 9 |
math | (French-Slovak Competition 1996) Find all strictly positive integers $x, y, p$ such that $p^{x}-y^{p}=1$ with $p$ prime. | 2 | 42 | 1 |
math | 2. How many positive values can the expression
$$
a_{0}+3 a_{1}+3^{2} a_{2}+3^{3} a_{3}+3^{4} a_{4}
$$
take if the numbers $a_{0}, a_{1}, a_{2}, a_{3}$ and $a_{4}$ are from the set $\{-1,0,1\}$? | 121 | 92 | 3 |
math | Define $m(n)$ to be the greatest proper natural divisor of $n\in \mathbb{N}$. Find all $n \in \mathbb{N} $ such that $n+m(n) $ is a power of $10$.
[i]N. Agakhanov[/i] | 75 | 65 | 2 |
math | 6.12 The sum of an infinite geometric progression with a common ratio $|q|<1$ is 16, and the sum of the squares of the terms of this progression is 153.6. Find the fourth term and the common ratio of the progression. | \frac{3}{16} | 59 | 8 |
math | 7. Arrange all proper fractions into such a sequence $\left\{a_{n}\right\}: \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \cdots$, the sorting method is: from left to right, first arrange the denominators in ascending order, for fractions with the same denominator, then arrange them in a... | \frac{1}{65} | 127 | 8 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.