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 | Example 2 Draw two tangents from point $P(-2,-2)$ to circle $\odot M$:
$$
(x-1)^{2}+(y-2)^{2}=9
$$
Let $A$ and $B$ be the points of tangency. Find the equation of the line containing the chord $AB$. | 3 x+4 y-2=0 | 72 | 9 |
math | For every $n$ in the set $\mathrm{N} = \{1,2,\dots \}$ of positive integers, let $r_n$ be the minimum value of $|c-d\sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for all $n \in \mathbb{N}$. | \frac{1 + \sqrt{3}}{2} | 100 | 13 |
math | In the summer, six grandchildren gathered at their grandmother's and we know about them that
- Martinka sometimes has to take care of her brother Tomášek, who is 8 years younger,
- Věrka, who is 7 years older than Ida, likes to tell ghost stories,
- Martinka often argues with Jaromír, who is a year younger,
- Tomášek ... | Martinka:11,Tomášek:3,Věrka:13,Ida:6,Jaromír:10,Kačka:14 | 138 | 39 |
math | 2. Petya runs down from the fourth floor to the first floor 2 seconds faster than his mother rides the elevator. Mother rides the elevator from the fourth floor to the first floor 2 seconds faster than Petya runs down from the fifth floor to the first floor. How many seconds does it take for Petya to run down from the ... | 12 | 93 | 2 |
math | 5. Find all real numbers $P$ such that the cubic equation $5 x^{3}-5(P+1) x^{2}+(71 P-1) x+1=66 P$ has three roots that are all natural numbers. | 76 | 53 | 2 |
math | If $(x, y)$ is a solution to the system
$$
\left\{\begin{array}{l}
x y=6 \\
x^{2} y+x y^{2}+x+y=63
\end{array}\right.
$$
determine the value of $x^{2}+y^{2}$. | 69 | 72 | 2 |
math | Example 10 Let $p(x)=x^{4}+a x^{3}+b x^{2}+c x+d$, where $a, b, c, d$ are constants, and $p(1)=1993$, $p(2)=3986$, $p(3)=5979$. Try to calculate $\frac{1}{4}[p(11)+p(-7)]$. \quad (1993 Macau Olympiad Question) | 5233 | 107 | 4 |
math | Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$
| 627 | 60 | 3 |
math | 3. Given $a, b, c \in \mathbf{R}$, and
$$
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c} \text {, }
$$
then there exists an integer $k$, such that the following equations hold for
$\qquad$ number of them.
(1) $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^{2 k+1}=\frac{1}{a^{2 k+1}}+\frac{1}{b^{2 k+1}}+\frac{1... | 2 | 344 | 1 |
math | 37. Compute
$$
\sum_{n=1}^{\infty} \frac{2 n+5}{2^{n} \cdot\left(n^{3}+7 n^{2}+14 n+8\right)}
$$ | \frac{137}{24}-8\ln2 | 55 | 14 |
math | Problem 2.3. A square $C$ is completely covered with a whole number of unit squares, without overlaps. If one places as many squares as possible of area 2 inside $C$, with sides parallel to the sides of $C$, without overlaps, it is possible to cover eight ninths of the area of the square. Determine all possible dimensi... | \begin{pmatrix}n^{\}=^{\}=1&n=2&=3\\n^{\}=^{\}=2&n=4&=6\\n^{\}=^{\}=3&n=6&=9\\n^{\}=^{\}=4&n=8&=12\\n^{\}=^{\} | 77 | 78 |
math | ## Task 6B - 241246B
Determine all positive real numbers $k$ for which the sequence $\left(x_{n}\right)$ defined by
$$
x_{1}=1, \quad x_{n+1}=\sqrt{k\left(x_{n}^{2}+x_{n}\right)} \quad(n=1,2,3, \ldots)
$$
is convergent.
For each such number $k$, determine the limit of the sequence $\left(x_{n}\right)$. | \frac{k}{1-k}for0<k<1 | 116 | 12 |
math | Three, (12 points) The equation $x^{2}+k x+4-k$ $=0$ has two integer roots. Find the value of $k$.
untranslated part:
关于 $x$ 的方程 $x^{2}+k x+4-k$ $=0$ 有两个整数根. 试求 $k$ 的值.
translated part:
The equation $x^{2}+k x+4-k$ $=0$ has two integer roots. Find the value of $k$. | k_{1}=-8 \text{ or } k_{2}=4 | 112 | 16 |
math | Problem 5.3. Irina did poorly in math at the beginning of the school year, so she had 3 threes and 2 twos in her journal. But in mid-October, she pulled herself together and started getting only fives. What is the minimum number of fives Irina needs to get so that her average grade is exactly 4? | 7 | 76 | 1 |
math | 13.049. The first of the unknown numbers is $140\%$ of the second, and the ratio of the first to the third is 14/11. Find these numbers if the difference between the third and the second is 40 units less than the number that is $12.5\%$ of the sum of the first and second numbers. | 280,200,220 | 83 | 11 |
math | Suppose $x,y$ and $z$ are integers that satisfy the system of equations \[x^2y+y^2z+z^2x=2186\] \[xy^2+yz^2+zx^2=2188.\] Evaluate $x^2+y^2+z^2.$ | 245 | 71 | 3 |
math | Example 7 (2003 National High School Competition Question) Given that $a, b, c, d$ are all positive integers, and $\log _{a} b=\frac{3}{2}, \log _{c} d=\frac{5}{4}$, if $a-c=$ 9, then $b-d=$ $\qquad$ . | 93 | 78 | 2 |
math | Three. (20 points) In the pyramid $S-ABC$, $SA=4, SB \geqslant 7, SC \geqslant 9, AB=5, BC \leqslant 6, AC \leqslant 8$. Try to find the maximum volume of the pyramid $S-ABC$. | 8 \sqrt{6} | 74 | 6 |
math | 15th Irish 2002 Problem B2 n = p · q · r · s, where p, q, r, s are distinct primes such that s = p + r, p(p + q + r + s) = r(s - q) and qs = 1 + qr + s. Find n. | 2002 | 69 | 4 |
math | 8,9
Find the height and the radius of the base of the cone of the largest volume inscribed in a sphere of radius $R$.
# | \frac{4}{3}R;\frac{2}{3}R\sqrt{2} | 32 | 21 |
math | Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$
For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$ | 1 | 105 | 1 |
math | 9. For a two-digit number $x$, 6 statements are made:
a) $x$ is divisible by 3; b) $x$ is divisible by 5;
c) $x$ is divisible by 9; d) $x$ is divisible by 15;
e) $x$ is divisible by 25; f) $x$ is divisible by 45.
Find all such $x$ for which exactly three of these statements are true. | 15,30,60 | 101 | 8 |
math | 3. If $[x]$ represents the greatest integer not greater than $x$, $\{x\}=x-[x]$, then the solution to the equation $x+2\{x\}=3[x]$ is $\qquad$ . | \frac{5}{3} | 51 | 7 |
math | 15 . In a $3 \times 3$ grid, fill in the numbers $1,2,3,4,5,6,7,8,9$ so that the sum of the numbers in any three squares in a row, column, or diagonal is equal. | 15 | 59 | 2 |
math | 1st CIS 1992 Problem 4 Given an infinite sheet of square ruled paper. Some of the squares contain a piece. A move consists of a piece jumping over a piece on a neighbouring square (which shares a side) onto an empty square and removing the piece jumped over. Initially, there are no pieces except in an m x n rectangle (... | 2 | 105 | 1 |
math | Problem 11.5. Determine the number of possible values of the product $a \cdot b$, where $a, b-$ are integers satisfying the inequalities
$$
2019^{2} \leqslant a \leqslant b \leqslant 2020^{2}
$$
Answer: $\mathrm{C}_{2 \cdot 2019+2}^{2}+2 \cdot 2019+1=2 \cdot 2019^{2}+5 \cdot 2019+2=8162819$. | 8162819 | 134 | 7 |
math | The residents of a boarding house have to pay their rent. If each of them pays $10 \mathrm{Ft}$, there is a shortage of $88 \mathrm{Ft}$ in the rent, but if each of them pays $10.80 \mathrm{Ft}$, then $2.5 \%$ more money is collected than the rent of the boarding house. How much should each resident pay so that exactly... | 10.54\mathrm{Ft} | 98 | 11 |
math | 8. A person tosses a coin, with the probability of landing heads up and tails up being $\frac{1}{2}$ each. Construct the sequence $\left\{a_{n}\right\}$, such that $a_{n}=\left\{\begin{array}{ll}1, & \text { if the } n \text {th toss is heads; } \\ -1, & \text { if the } n \text {th toss is tails. }\end{array}\right.$ ... | \frac{13}{128} | 165 | 10 |
math | Question 200: Suppose a bag contains one red, one yellow, and one blue ball. Each time a ball is drawn from the bag (the probability of drawing a ball of any color is the same), its color is determined and then it is put back. The process continues until a red ball is drawn twice in a row. Let the number of draws at th... | 12 | 102 | 2 |
math | Three, (25 points) Divide $1,2, \cdots, 9$ into three groups, each containing three numbers, such that the sum of the numbers in each group is a prime number.
(1) Prove that there must be two groups with equal sums;
(2) Find the number of all different ways to divide them.
| 12 | 73 | 2 |
math | 9. Given $z \in \mathbf{C}$. If the equation in $x$
$$
4 x^{2}-8 z x+4 i+3=0
$$
has real roots. Then the minimum value of $|z|$ is $\qquad$ | 1 | 59 | 1 |
math | 24. A bag contains 999 balls, marked with the 999 numbers $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{1000}$ respectively. Each ball is either red or blue, and the number of red balls is a positive even number. Let $S$ denote the product of the numbers on all red balls. Find the sum of all possible values ... | \frac{498501}{2000} | 204 | 15 |
math | 1. Let $A B C D E F$ be a regular hexagon, and let $P$ be a point inside quadrilateral $A B C D$. If the area of triangle $P B C$ is 20 , and the area of triangle $P A D$ is 23 , compute the area of hexagon $A B C D E F$. | 189 | 78 | 3 |
math | 3. In Rt $\triangle A B C$, it is known that $\angle C=90^{\circ}, B C$ $=6, C A=3, C D$ is the angle bisector of $\angle C$. Then $C D=$ $\qquad$ . | 2 \sqrt{2} | 59 | 6 |
math | 12. The sum of a set of numbers is the sum of all its elements. Let $S$ be a set composed of positive integers not exceeding 15, such that the sums of any two disjoint subsets of $S$ are not equal, and among all sets with the above property, the sum of $S$ is the largest. Find the set $S$ and its sum. | 61 | 81 | 2 |
math | ## 16. Navigation
A ship is sailing along a river. After 6 hours, it returns to the starting point, having traveled a distance of 36 km on the map (naturally, the ship had to move in different directions at different times).
What is the speed of the ship, assuming it did not spend any time turning around, and the spe... | 7.24 | 96 | 4 |
math | Example 11 (1996 National High School League Question) Find the range of real numbers $a$ such that for any real number $x$ and any $\theta \in\left[0, \frac{\pi}{2}\right]$, we have $(x+3+2 \sin \theta \cos \theta)^{2}+(x+a \sin \theta+a \cos \theta)^{2} \geqslant \frac{1}{8}$.
| \geqslant\frac{7}{2} | 103 | 12 |
math | Example. Find the solution to the Cauchy problem
$$
x y^{\prime}+y=x y^{2}
$$
with the initial condition
$$
y(1)=1
$$ | \frac{1}{x(1-\lnx)} | 43 | 12 |
math | B2. Petra found an attractive offer for her vacation in a tourist catalog. The promotional offer read: "When you buy a seven or more day package, we will give you two days of vacation for free." Petra chose accommodation from the catalog for €50 per day.
(a) How much would she pay for a five-day package?
(b) How many... | 250€,10,400€,8,5,28,32 | 277 | 22 |
math | Find, with proof, all irrational numbers $x$ such that both $x^3-6x$ and $x^4-8x^2$ are rational. | \pm \sqrt{6}, \pm (1 + \sqrt{3}), \pm (1 - \sqrt{3}) | 36 | 27 |
math | Given positive integers $N$ and $k$, we counted how many ways the number $N$ can be written in the form $a+b+c$, where $1 \leq a, b, c \leq k$, and the order of the addends matters. Could we have gotten 2007 as a result? | 2007 | 69 | 4 |
math | 4. Given the sequence $a_{0}, a_{1}, \cdots, a_{n}, \cdots$ satisfies the relation $\left(3-a_{n+1}\right)\left(6+a_{n}\right)=18$, and $a_{0}=3$. Then
$$
\sum_{i=0}^{n} \frac{1}{a_{i}}=
$$
$\qquad$ | \frac{1}{3}(2^{n+2}-n-3) | 91 | 17 |
math | ## Task B-4.3.
Determine all three-digit numbers that in base 9 have the representation $\overline{x y z}_{9}$, and in base 11 have the representation $\overline{z y x}_{11}$. | 302_{9}=203_{11},604_{9}=406_{11} | 53 | 26 |
math | 2. Non-negative real numbers $a_{1}, a_{2}, \cdots, a_{n}$ satisfy $a_{1}+a_{2}+\cdots+a_{n}=1$.
Find the minimum value of $\frac{a_{1}}{1+a_{2}+\cdots+a_{n}}+\frac{a_{2}}{1+a_{1}+a_{3}+\cdots+a_{n}}+\cdots+$ $\frac{a_{n}}{1+a_{1}+\cdots+a_{n-1}}$. (1982, China Mathematical Competition) | \frac{n}{2 n-1} | 131 | 9 |
math | 9. (2004 Shanghai High School Competition Problem) The sequence $\left\{a_{n}\right\}$ satisfies $(n-1) a_{n+1}=(n+1) a_{n}-2(n-1), n=$ $1,2, \cdots$ and $a_{100}=10098$. Find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=(n-1)(n+2) | 97 | 13 |
math | Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$. Find $\angle BCA$. | 90^\circ | 82 | 4 |
math | 7. If three points are randomly chosen on a circle, the probability that the triangle formed by these three points is an acute triangle is $\qquad$ .
| \frac{1}{4} | 32 | 7 |
math | 14. Let $n(\geqslant 3)$ be a positive integer, and $M$ be an $n$-element set. Find the maximum positive integer $k$ such that there exists a family of $k$ distinct 3-element subsets of $M$, where the intersection of any two 3-element subsets is non-empty. | C_{n-1}^{2} | 73 | 9 |
math | 9.222 For what values of $p$ is the inequality $-9<\frac{3 x^{2}+p x-6}{x^{2}-x+1}<6$ satisfied for all values of $x$? | p\in(-3;6) | 52 | 8 |
math | $\left[\begin{array}{l}\text { Equations in integers } \\ {[\text { Case enumeration }}\end{array}\right]$
Let $S(x)$ be the sum of the digits of a natural number $x$. Solve the equation $x+S(x)=2001$. | 1977 | 63 | 4 |
math | 4. Let $x, y, z$ be non-negative real numbers, and satisfy the equation
$$
4^{\sqrt{5 x+9 y+4 z}}-68 \times 2^{\sqrt{5 x+9 y+4 x}}+256=0 \text {, }
$$
Then the product of the maximum and minimum values of $x+y+z$ is $\qquad$ . | 4 | 91 | 1 |
math | 5.70 It is known that 12 theater troupes are participating in a 7-day drama festival, and it is required that each troupe can see the performances of all other troupes (actors who do not have a performance on that day watch the performances from the audience). How many performances are needed at a minimum?
将上面的文本翻译成英文... | 22 | 92 | 2 |
math | 3. (16 points) In a garden plot, it was decided to create a rectangular flower bed. Due to a lack of space, the length of the flower bed was reduced by $10 \%$, and the width was reduced by $20 \%$. As a result, the perimeter of the flower bed decreased by $12 \%$. However, this was not enough, so it was decided to red... | 18 | 118 | 2 |
math | 9. (16 points) Let the function $f(x)$ satisfy $f\left(2^{x}\right)=x^{2}-2 a x+a^{2}-1$, and $f(x)$ has a range of $[-1,0]$ on $\left[2^{a-1}, 2^{a^{2}-2 a+2}\right]$, find the range of values for $a$.
| [\frac{3-\sqrt{5}}{2},1]\cup[2,\frac{3+\sqrt{5}}{2}] | 89 | 29 |
math | Let $ (a_n)^{\infty}_{n\equal{}1}$ be a sequence of integers with $ a_{n} < a_{n\plus{}1}, \quad \forall n \geq 1.$ For all quadruple $ (i,j,k,l)$ of indices such that $ 1 \leq i < j \leq k < l$ and $ i \plus{} l \equal{} j \plus{} k$ we have the inequality $ a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}.$ Determine the le... | 2015029 | 138 | 7 |
math | 5. In the number $2016 * * * * 02 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,2,4,5,7,9$ (digits can be repeated) so that the resulting 11-digit number is divisible by 15. In how many ways can this be done? | 864 | 79 | 3 |
math | Find the value of $\operatorname{ctg} x$ if $\operatorname{ctg} x = \sin x?$ | \operatorname{ctg}\sqrt{\frac{\sqrt{5}-1}{2}}\text{or}\operatorname{ctg}-\sqrt{\frac{\sqrt{5}-1}{2}} | 27 | 43 |
math | Find all triplets of three strictly positive integers such that:
$$
x+\frac{1}{y+\frac{1}{z}}=\frac{10}{7}
$$ | (1,2,3) | 37 | 7 |
math | Let $0 \leq k < n$ be integers and $A=\{a \: : \: a \equiv k \pmod n \}.$ Find the smallest value of $n$ for which the expression
\[ \frac{a^m+3^m}{a^2-3a+1} \]
does not take any integer values for $(a,m) \in A \times \mathbb{Z^+}.$ | n = 11 | 96 | 6 |
math | Seven, given a sequence whose terms are 1 or 2, the first term is 1, and there are $2^{k-1}$ 2s between the $k$-th 1 and the $(k+1)$-th 1, i.e., $1,2,1,2,2,1,2,2,2,2,1,2,2,2,2,2,2,2,2,1, \cdots$
(1) Find the sum of the first 1998 terms of the sequence $S_{1998}$;
(2) Does there exist a positive integer $n$ such that th... | 3985 | 193 | 4 |
math | Let $f$ be a function defined for the non-negative integers, such that:
a) $f(n)=0$ if $n=2^{j}-1$ for some $j \geq 0$.
b) $f(n+1)=f(n)-1$ otherwise.
i) Show that for every $n \geq 0$ there exists $k \geq 0$ such that $f(n)+n=2^{k}-1$.
ii) Find $f(2^{1990})$. | f(2^{1990}) = -1 | 114 | 13 |
math | 4・194 A book has page numbers from 1 to $n$. When the page numbers of this book were added up, one page number was mistakenly added one extra time, resulting in an incorrect sum of 1986. What is the page number that was added one extra time? | 33 | 62 | 2 |
math | 455*. From point A to point B, which is 40 km away from A, two tourists set off simultaneously: the first on foot at a speed of 6 km/h, and the second on a bicycle. When the second tourist overtook the first by 5 km, the first tourist got into a passing car traveling at a speed of 24 km/h. Two hours after leaving A, th... | 9 | 112 | 1 |
math | 3. (4 points) Solve the equation of the form $f(f(x))=x$, given that $f(x)=x^{2}+5 x+1$.
# | -2\\sqrt{3},-3\\sqrt{2} | 38 | 14 |
math | 3. Given a sphere that is tangent to all six edges of a regular tetrahedron with edge length $a$. Then the volume of this sphere is $\qquad$ . | \frac{\sqrt{2}}{24} \pi a^{3} | 37 | 17 |
math | Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$ | 0, 1, 2 | 42 | 7 |
math | 4. In the acute triangle $\triangle A B C$, the lengths of the three sides $a, b, c$ are all integers, and $a<b<c, a+b+c=20$. Then $\angle A$ $+\angle C=$ $\qquad$ . | 120^{\circ} | 57 | 7 |
math | Problem 10.8. Real numbers $x$ and $y$ are such that $x^{3}+21 x y+y^{3}=343$. What can $x+y$ be? List all possible options. | 7,-14 | 50 | 4 |
math | The lines with the two equations below intersect at the point $(2,-3)$.
$$
\begin{array}{r}
\left(a^{2}+1\right) x-2 b y=4 \\
(1-a) x+b y=9
\end{array}
$$
What are the possible ordered pairs $(a, b)$ ? | (4,-5)\text{}(-2,-1) | 74 | 12 |
math | 8. Given real numbers $x, y, z$ satisfy $x^{2}+y^{2}+z^{2}=1$. Then the maximum value of $x y+y z$ is $\qquad$ . | \frac{\sqrt{2}}{2} | 47 | 10 |
math | Let $f(n)$ and $g(n)$ be functions satisfying
\[f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}\]and
\[g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}\]for positive integers $n$.... | 258 | 153 | 3 |
math | 4. It is known that the number $\cos 6^{0}$ is a root of the equation $32 t^{5}-40 t^{3}+10 t-\sqrt{3}=0$. Find the other four roots of this equation. (Answers in the problem should be compact expressions, not containing summation signs, ellipses, and radicals.)
# | \cos66,\cos78,\cos138,\cos150 | 78 | 18 |
math | ## Task Condition
Find the derivative.
$y=\arccos \frac{x^{2}-4}{\sqrt{x^{4}+16}}$ | -\frac{2\sqrt{2}(4+x^{2})}{x^{4}+16} | 33 | 23 |
math | 12.55 Find the integer solutions of the equation $\frac{x y}{z}+\frac{x z}{y}+\frac{y z}{x}=3$.
(26th Moscow Mathematical Olympiad, 1963) | (1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1) | 52 | 27 |
math | Example 6 Let $f(x)$ be a function defined on $\mathbf{R}$. If $f(0)=2008$, and for any $x \in \mathbf{R}$, it satisfies
$f(x+2)-f(x) \leqslant 3 \times 2^{x}$,
$f(x+6)-f(x) \geqslant 63 \times 2^{x}$,
then $f(2008)=$ $\qquad$
(2008, National High School Mathematics Joint Competition) | 2^{2008}+2007 | 122 | 12 |
math | 5. Given that the 2017 roots of the equation $x^{2017}=1$ are 1, $x_{1}, x_{2}, \cdots, x_{2016}$. Then $\sum_{k=1}^{2016} \frac{1}{1+x_{k}}=$ $\qquad$ . | 1008 | 79 | 4 |
math | G3.3If $c$ is the largest slope of the tangents from the point $A\left(\frac{\sqrt{10}}{2}, \frac{\sqrt{10}}{2}\right)$ to the circle $C: x^{2}+y^{2}=1$, find the value of $c$. | 3 | 71 | 1 |
math | 1. If the 5th term of the expansion of $\left(x \sqrt{x}-\frac{1}{x}\right)^{6}$ is $\frac{15}{2}$, then $\lim _{n \rightarrow \infty}\left(x^{-1}+x^{-2}+\cdots+x^{-a}\right)=$ | 1 | 73 | 1 |
math | 10.4.1. (12 points) Find all pairs of integers $(x, y)$ that are solutions to the equation
$$
7 x y-13 x+15 y-37=0 \text {. }
$$
In your answer, indicate the sum of the found values of $x$. | 4 | 68 | 1 |
math | # Problem 1.
Three electric generators have powers $x_{1}, x_{2}, x_{3}$, the total power of all three does not exceed 2 MW. In the power system with such generators, a certain process is described by the function
$$
f\left(x_{1}, x_{2}, x_{3}\right)=\sqrt{x_{1}^{2}+x_{2} x_{3}}+\sqrt{x_{2}^{2}+x_{1} x_{3}}+\sqrt{x_{... | 3 | 140 | 1 |
math | 7.071. $3 \cdot 5^{2 x-1}-2 \cdot 5^{x-1}=0.2$. | 0 | 32 | 1 |
math | 2.8. Let $n>1$ be a natural number. Find all positive solutions of the equation $x^{n}-n x+n-1=0$.
## 2.3. Equations with Radicals
In problems $2.9-2.15$, it is assumed that the values of square roots are non-negative. We are interested only in the real roots of the equations. | 1 | 85 | 1 |
math | Task 8. For what values of the parameter $a$ does the equation
$$
5^{x^{2}-6 a x+9 a^{2}}=a x^{2}-6 a^{2} x+9 a^{3}+a^{2}-6 a+6
$$
have exactly one solution? | 1 | 69 | 1 |
math | ## Task 3 - V00703
The tenth part of a number is increased by 3. The same value results when one hundredth of this number is decreased by 6!
What is it? | -100 | 45 | 4 |
math | 4. Given that the three interior angles $A, B, C$ of $\triangle A B C$ form an arithmetic sequence, and the corresponding sides are $a, b, c$, and $a, c, \frac{4}{\sqrt{3}} b$ form a geometric sequence, then $S_{\triangle A B C}: a^{2}=$ $\qquad$ . | \frac{\sqrt{3}}{2} | 82 | 10 |
math | 29th Putnam 1968 Problem B1 The random variables X, Y can each take a finite number of integer values. They are not necessarily independent. Express prob( min(X, Y) = k) in terms of p 1 = prob( X = k), p 2 = prob(Y = k) and p 3 = prob( max(X, Y) = k). | p_1+p_2-p_3 | 83 | 9 |
math | 143. Find $\lim _{x \rightarrow 0} \frac{\sin k x}{x}(k-$ a constant). | k | 29 | 1 |
math | 1. Find all values of $m$ in the interval ($-1,1$) for which the equation
$$
4^{\sin x}+m 2^{\sin x}+m^{2}-1=0
$$
has solutions. | (-1,\frac{-1+\sqrt{13}}{4}] | 55 | 15 |
math | Four unit circles are placed on a square of side length $2$, with each circle centered on one of the four corners of the square. Compute the area of the square which is not contained within any of the four circles. | 4 - \pi | 45 | 4 |
math | In $\triangle A B C$,
$$
\frac{\overrightarrow{A B} \cdot \overrightarrow{B C}}{3}=\frac{\overrightarrow{B C} \cdot \overrightarrow{C A}}{2}=\frac{\overrightarrow{C A} \cdot \overrightarrow{A B}}{1} \text {. }
$$
Then $\tan A=$ $\qquad$ . | \sqrt{11} | 89 | 6 |
math | Exercise 4. Let $n \geqslant 3$ be an integer. For each pair of prime numbers $p$ and $q$ such that $p<q \leqslant n$, Morgane has written the sum $p+q$ on the board. She then notes $\mathcal{P}(n)$ as the product of all these sums. For example, $\mathcal{P}(5)=(2+3) \times(2+5) \times(3+5)=280$.
Find all values of $n... | 7 | 197 | 1 |
math | 2.1. A metal weight has a mass of 20 kg and is an alloy of four metals. The first metal in this alloy is one and a half times more than the second, the mass of the second metal is to the mass of the third as $3: 4$, and the mass of the third to the mass of the fourth - as $5: 6$. Determine the mass of the fourth metal.... | 5.89 | 109 | 4 |
math | [ Motion task ]
Two hunters set out at the same time towards each other from two villages, the distance between which is 18 km. The first walked at a speed of 5 km/h, and the second at 4 km/h. The first hunter took a dog with him, which ran at a speed of 8 km/h. The dog immediately ran towards the second hunter, met h... | 16 | 116 | 2 |
math | 2. Let the sets be
$$
\begin{array}{l}
A=\left\{x \left\lvert\, x+\frac{5-x}{x-2}=2 \sqrt{x+1}\right., x \in \mathbf{R}\right\}, \\
B=\{x \mid x>2, x \in \mathbf{R}\} .
\end{array}
$$
Then the elements of the set $A \cap B$ are | \frac{5+\sqrt{13}}{2} | 103 | 13 |
math | ## Task 4 - 040624
Fritz gives Heinz the following riddle:
"In our class, 26 students can ride a bike and 12 students can swim. Every student can do at least one of these. If you multiply the number of students by 5, the cross sum of this product is twice as large as the cross sum of the number of students. Additiona... | 30 | 102 | 2 |
math | Let $S$ be the locus of all points $(x,y)$ in the first quadrant such that $\dfrac{x}{t}+\dfrac{y}{1-t}=1$ for some $t$ with $0<t<1$. Find the area of $S$. | \frac{1}{6} | 58 | 7 |
math | 34. A bag of fruit contains 10 fruits, with an even number of apples, at most two oranges, a multiple of 3 bananas, and at most one pear. How many types of these bagged fruits are there? | 11 | 49 | 2 |
math | A function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ has the property that $ \lim_{x\to\infty } \frac{1}{x^2}\int_0^x f(t)dt=1. $
[b]a)[/b] Give an example of what $ f $ could be if it's continuous and $ f/\text{id.} $ doesn't have a limit at $ \infty . $
[b]b)[/b] Prove that if $ f $ is nondecreasing then $ f... | 2 | 143 | 3 |
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 | Let $m$ be an integer greater than 3. In a party with more than $m$ participants, every group of $m$ people has exactly one common friend. How many friends does the person with the most friends have? | m | 48 | 1 |
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