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 | 5. Write the numbers 1 and 2 on the blackboard. New numbers can be added in the following way: if the numbers $a$ and $b$ are on the blackboard, then the number $ab + a + b$ can be added. Can the number 12,131 be obtained using these methods? | 12131 \text{ does not have the form } 2^n \cdot 3^m - 1 | 71 | 26 |
math | 8. In the set of the first ten thousand positive integers $\{1,2$, $\cdots, 10000\}$, the elements that leave a remainder of 2 when divided by 3, a remainder of 3 when divided by 5, and a remainder of 4 when divided by 7 are $\qquad$ in number. | 95 | 76 | 2 |
math | 2. Find all integer solutions $x$ and $y$ that satisfy the equation $x^{2} + xy + y^{2} = 1$. | (1,0),(-1,0),(1,-1),(0,1),(0,-1),(-1,1) | 33 | 27 |
math | 17. If $n$ is a positive integer, find the formula for the sum $\binom{n+1}{1}$
$$
+\binom{n+2}{2} 2^{2}+\binom{n+3}{3} 3^{2}+\cdots+\binom{2 n}{n} n^{2}
$$ | \frac{n(n+1)^{3}\binom{2 n+1}{n+1}}{(n+2)(n+3)} | 75 | 31 |
math | 1. Ana, her mother, and her grandmother are celebrating their birthday on June 23rd. Today, June 23rd, 2021, the sum of their ages is 127. When Ana was 4 years old, her grandmother was twice as old as her mother. Two years ago, her mother was three times as old as Ana. In which year was Ana's mother born? How old will ... | 2036 | 115 | 4 |
math | (a) In how many ways can the number 105 be written as the difference of two perfect squares?
(b) Show that it is not possible to write the number 106 as the difference of two perfect squares.
## | 4 | 48 | 1 |
math | A basket is called "[i]Stuff Basket[/i]" if it includes $10$ kilograms of rice and $30$ number of eggs. A market is to distribute $100$ Stuff Baskets. We know that there is totally $1000$ kilograms of rice and $3000$ number of eggs in the baskets, but some of market's baskets include either more or less amount of rice ... | 99 | 139 | 2 |
math | What percent of the numbers $1, 2, 3, ... 1000$ are divisible by exactly one of the numbers $4$ and $5?$ | 35\% | 36 | 4 |
math | 18. (12 points) To investigate the mathematics exam scores of students citywide, 10 students each from Class A and Class B of a certain middle school were randomly selected, and the following scores (in points) were obtained.
$$
\begin{aligned}
\text { Class A: } & 132,108,112,121,113,121,118, \\
& 128,118,129; \\
\tex... | \frac{33}{20} | 320 | 9 |
math | Problem 2. For what least $n$ do there exist $n$ numbers from the interval $(-1 ; 1)$ such that their sum is 0 and the sum of their squares is 36?
# | 38 | 46 | 2 |
math | 9. Let $x_{1}, x_{2}, \cdots, x_{n}, y_{1}, y_{2}, \cdots, y_{n}, z_{1}, z_{2}, \cdots, z_{n}$ all be 1 or -1, and $x_{1} y_{1}+x_{2} y_{2}+\cdots+$ $x_{n} y_{n}=0, x_{1} z_{1}+x_{2} z_{2}+\cdots+x_{n} z_{n}=0, y_{1} z_{1}+y_{2} z_{2}+\cdots+y_{n} z_{n}=0$, find the value of $n$. | 4k(k\in{N}_{+}) | 161 | 10 |
math | 5. Given the arithmetic sequence $\left\{a_{n}\right\}$, the first term and common difference are both positive numbers, and $a_{2}, a_{5}, a_{9}$ form a geometric sequence in order, then the smallest positive integer $k$ such that $a_{1}+a_{2}+\cdots+a_{k}>100 a_{1}$ is $\qquad$. | 34 | 89 | 2 |
math | 6. For the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{16}=1$, the left and right foci are $A, B$, and $P$ is a point on the hyperbola. If the incenter of $\triangle P A B$ is $(3,1)$, then the radius of the circumcircle of $\triangle P A B$ is $\qquad$ . | \frac{65}{12} | 94 | 9 |
math | 7. (10 points) $\left[x-\frac{1}{2}\right]=3 x-5$, here $[x]$ represents the greatest integer not exceeding $x$, then $x=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 2 | 68 | 1 |
math | 9. (16 points) Let the constant $a \in \mathbf{R}$, and the function
$$
f(x)=(a-x)|x|
$$
has an inverse function $f^{-1}(x)$. If the inequality
$$
f^{-1}\left(x^{2}+m\right)<f(x)
$$
holds for all $x \in[-2,2]$, find the range of the real number $m$. | \in(12,+\infty) | 97 | 10 |
math | 3. Given a function $f(n)$ defined on the set of positive integers satisfies the conditions:
(1) $f(m+n)=f(m)+f(n)+m n\left(m, n \in \mathbf{N}_{+}\right)$;
(2) $f(3)=6$.
Then $f(2011)=$ . $\qquad$ | 2023066 | 80 | 7 |
math | 2. The general solution of the equation $\cos \frac{x}{4}=\cos x$ is ( ), and in the interval $(0,24 \pi)$, there are ( ) distinct solutions. | 20 | 43 | 2 |
math | 13. Let the three roots of the cubic equation $x^{3}+p x+1=0$ correspond to points in the complex plane that form an equilateral triangle. Find the area of this triangle. | \frac{3\sqrt{3}}{4} | 45 | 12 |
math | 18. A wire 144 cm long is to be formed into a rectangle with both length and width being integers in centimeters. There are $\qquad$ different ways to do this. | 36 | 41 | 2 |
math | Solve the following system of equations:
$$
\begin{aligned}
& x+y^{2}=a \ldots \\
& y+x^{2}=a \ldots
\end{aligned}
$$
For which values of $a$ do we get real solutions? | \geq-\frac{1}{4} | 57 | 10 |
math | 6. (3 points) Given the cryptarithm: ЖАЛО + ЛОЖА = ОСЕНЬ. Identical letters represent identical digits, different letters represent different digits. Find the value of the letter А. | 8 | 47 | 1 |
math | 3-ча 1. Given 6 digits: $0,1,2,3,4,5$. Find the sum of all four-digit even numbers that can be written using these digits (the same digit can be repeated in a number). | 1769580 | 51 | 7 |
math | Let $P$ be the set of all $2012$ tuples $(x_1, x_2, \dots, x_{2012})$, where $x_i \in \{1,2,\dots 20\}$ for each $1\leq i \leq 2012$. The set $A \subset P$ is said to be decreasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in A$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \leq x_i (1\leq i \leq 2012)$ also belongs t... | \frac{1}{20^{2012}} | 307 | 13 |
math | Suppose $x$ is in the interval $[0, \pi/2]$ and $\log_{24\sin x} (24\cos x)=\frac{3}{2}$. Find $24\cot^2 x$. | 192 | 53 | 3 |
math | Let $f(x)=x^3+ax^2+bx+c$ and $g(x)=x^3+bx^2+cx+a$, where $a,b,c$ are integers with $c\not=0$. Suppose that the following conditions hold:
[list=a][*]$f(1)=0$,
[*]the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$.[/list]
Find the value of $a^{2013}+b^{2013}+c^{2013}$. | -1 | 124 | 2 |
math | 234. $x^{2}-x y+y^{2}-x+3 y-7=0$, if it is known that the solution is: $x=3 ; y=1$. | (3,1),(-1,1),(3,-1),(-3,-1),(-1,-5) | 42 | 24 |
math | 11.002. Calculate the volume of a regular tetrahedron if the radius of the circle circumscribed around its face is $R$. | \frac{R^{3}\sqrt{6}}{4} | 33 | 14 |
math | 50. Given a triangle $ABC$. The tangent to the circumcircle of this triangle at point $B$ intersects the line $AC$ at point $M$. Find the ratio $|AM|:|MC|$, if $|AB|:|BC|=k$. | k^2 | 57 | 3 |
math | 18. (LUX 1) ${ }^{\mathrm{IMO1}}$ Consider two concentric circles of radii $R$ and $r(R>r)$ with center $O$. Fix $P$ on the small circle and consider the variable chord $P A$ of the small circle. Points $B$ and $C$ lie on the large circle; $B, P, C$ are collinear and $B C$ is perpendicular to $A P$.
(a) For what value(... | 6R^{2}+2r^{2} | 175 | 11 |
math | [ Special cases of parallelepipeds (other). ] [ Skew lines, angle between them ]
On the diagonals $A B 1$ and $B C 1$ of the faces of the parallelepiped $A B C D A 1 B 1 C 1 D 1$, points $M$ and $N$ are taken, such that the segments $M N$ and $A 1 C$ are parallel. Find the ratio of these segments. | 1:3 | 100 | 3 |
math | 22(1150). How many hours can each of the three workers complete the work if the productivity of the third worker is equal to half the sum of the productivities of the first and second workers? It is known that if the third worker worked for 48 hours, the first would need 10 hours to finish the work, and the second woul... | 50 | 82 | 2 |
math | 10. For any real numbers $x, y$, define the operation $x * y$ as $x * y=a x+b y+c x y$, where $a, b, c$ are constants, and the operations on the right side of the equation are the usual real number addition and multiplication. It is known that $1 * 2=3, 2 * 3=4$, and there is a non-zero real number $d$, such that for a... | 4 | 119 | 1 |
math | Let $n \geqslant 2$ be a positive integer, and let $a_{1}, a_{2}, \cdots, a_{n}$ be positive real numbers and $b_{1}, b_{2}, \cdots, b_{n}$ be non-negative real numbers satisfying
(a) $a_{1}+a_{2}+\cdots+a_{n}+b_{1}+b_{2}+\cdots+b_{n}=n$;
(b) $a_{1} a_{2} \cdots a_{n}+b_{1} b_{2} \cdots b_{n}=\frac{1}{2}$.
Find the ma... | \frac{1}{2} | 225 | 7 |
math | Let $f(x) = x^2 + 6x + c$ for all real number s$x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots? | c = \frac{11 - \sqrt{13}}{2} | 56 | 17 |
math | $1.54 \sqrt[3]{38+\sqrt{1445}}+\sqrt[3]{38-\sqrt{1445}}=4$. | 4 | 38 | 1 |
math | IX OM - II - Task 2
Six equal disks are placed on a plane in such a way that their centers lie at the vertices of a regular hexagon with a side equal to the diameter of the disks. How many rotations will a seventh disk of the same size make while rolling externally on the same plane along the disks until it returns to... | 4 | 74 | 1 |
math | 20. In $\triangle A B C$, $\sin \frac{\angle A B C}{2}=\frac{\sqrt{3}}{3}, A B=2, D$ is a point on line segment $A C$, and $A D=2 D C, B D=\frac{4 \sqrt{3}}{3}$.
(1) Find the length of $B C$;
(2) Find the area of $\triangle D B C$. | BC=3,S_{\triangleDBC}=\frac{2\sqrt{2}}{3} | 98 | 21 |
math | 1. Solve the equation
$$
\sqrt{x^{2}+x}+\sqrt{1+\frac{1}{x^{2}}}=\sqrt{x+3}
$$ | -1 | 37 | 2 |
math | 2. [5 points] Solve the equation $\frac{\cos 4 x}{\cos 3 x - \sin 3 x} + \frac{\sin 4 x}{\cos 3 x + \sin 3 x} = \sqrt{2}$. | \frac{\pi}{52}+\frac{2\pik}{13},k\neq13p-5,k\in\mathbb{Z},p\in\mathbb{Z} | 57 | 46 |
math | 14, 43 students, each carrying a different amount of money ranging from 8 cents to 5 yuan. Each student spent all their money on picture cards. There are only two types of picture cards, 3 cents each and 5 cents each, and each student tried to buy as many 5-cent cards as possible. How many 3-cent cards did they buy in ... | 84 | 81 | 2 |
math | 46. Calculate in the most rational way:
$$
333\left(\frac{71}{111111}+\frac{573}{222222}-\frac{2}{7 \cdot 37 \cdot 3}\right)
$$ | \frac{3}{14} | 63 | 8 |
math | Example 3. Find the derivative of the function
$$
y=\operatorname{arctg} \frac{x}{a}+\frac{1}{2} \ln \left(x^{2}+a^{2}\right)
$$ | \frac{x+}{x^{2}+^{2}} | 51 | 13 |
math | 10.118. Determine the side of the rhombus, knowing that its area is $S$, and the lengths of the diagonals are in the ratio $m: n$. | \sqrt{\frac{S(^{2}+n^{2})}{2n}} | 40 | 19 |
math | Let $X$ be a set containing $n$ elements. Find the number of ordered triples $(A,B, C)$ of subsets of $X$ such that $A$ is a subset of $B$ and $B$ is a proper subset of $C$. | 4^n - 3^n | 55 | 6 |
math | Find all positive integers $n$ such that $-5^4 + 5^5 + 5^n$ is a perfect square. Do the same for $2^4 + 2^7 + 2^n.$ | n = 5 | 47 | 5 |
math |
Problem 8.3. Given $n$ points, $n \geq 5$, in the plane such that no three lie on a line. John and Peter play the following game: On his turn each of them draws a segment between any two points which are not connected. The winner is the one after whose move every point is an end of at least one segment. If John is to ... | n=4k+1orn=4k+2 | 105 | 12 |
math | Find the largest real number $a$ such that \[\left\{ \begin{array}{l}
x - 4y = 1 \\
ax + 3y = 1\\
\end{array} \right.
\] has an integer solution. | 1 | 56 | 1 |
math | Example 1 (to $1^{\circ}$). Investigate the function $z=$ $=x^{3}+y^{3}-3 x y$ for extremum. | z_{\}=z(1,1)=-1 | 38 | 12 |
math | [ Motion problems ]
A cyclist rode from point A to point B, where he stayed for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and 2 hours later met him on the return trip. The pedestrian arrived in B at the same time the cyclist returned to A. How much time did it take the pedestrian t... | 10 | 96 | 2 |
math | 4. In a box, there are a thousand balls numbered $1,2,3, \ldots, 999,1000$. What is the probability that a number drawn in one draw is not divisible by either 4 or 6? Write the solution as a percentage. | 66.7 | 62 | 4 |
math | 3. Do there exist integers $x$ and $y$ such that $x^{2}+2012=y^{2}$? Justify your claim. If such numbers exist, determine all of them. | (502,504),(-502,504),(502,-504),(-502,-504) | 45 | 35 |
math | Example 6 (2006 National Training Team Test) Find all positive integer pairs $(a, n)$ such that $\frac{(a+1)^{n}-a^{n}}{n}$ is an integer.
Find all positive integer pairs $(a, n)$ such that $\frac{(a+1)^{n}-a^{n}}{n}$ is an integer. | (a, 1) | 79 | 5 |
math | 1. Three athletes start from the same point on a closed running track that is 400 meters long and run in the same direction. The first runs at a speed of 155 m/min, the second at 200 m/min, and the third at 275 m/min. After what least amount of time will they all be at the same point again? How many overtakes will occu... | \frac{80}{3} | 99 | 8 |
math | 2. Find all integer solutions $(x, y)$ of the following equation:
$$
7 x^{2}-40 x y+7 y^{2}=(|x-y|+2)^{3} \text {. }
$$ | (2,-2),(-2,2) | 49 | 10 |
math | 17. (3 points) Given that $a$, $b$, and $c$ are three distinct prime numbers, if $a + b \times c = 37$, then the maximum value of $a + b - c$ is | 32 | 51 | 2 |
math | 3.18. Two touching circles with centers $O_{1}$ and $O_{2}$ touch internally a circle of radius $R$ with center $O$. Find the perimeter of triangle $O O_{1} O_{2}$. | 2R | 51 | 2 |
math | 5. Now arrange for seven students to participate in five sports events, requiring that students A and B cannot participate in the same event, each event must have participants, and each person only participates in one event. Then the number of different schemes that meet the above requirements is $\qquad$ | 15000 | 58 | 5 |
math | 3. On a line, consider the distinct points $A_{0}, A_{1}, A_{2}, A_{3}, \ldots, A_{2014}$, in this order, such that $M$ is the midpoint of segment $A_{0} A_{10}$ and $A_{0} A_{1}=3 \text{~cm}, A_{1} A_{2}=7 \text{~cm}, A_{2} A_{3}=11 \text{~cm}$, $A_{3} A_{4}=15 \text{~cm}$, and so on.
(3p) a) Calculate the lengths of... | 8114406 | 204 | 7 |
math | 12*. In how many ways can milk be transferred from a 12-liter barrel, filled with milk, to another empty barrel of the same volume using two empty cans of 1 liter and 2 liters? Transferring milk from one can to another is not allowed.
Note that the question in this problem is different from the previous problems. | 233 | 71 | 3 |
math | ## Task Condition
Find the derivative.
$$
y=3 x-\ln \left(1+\sqrt{1-e^{6 x}}\right)-e^{-3 x} \cdot \arcsin \left(e^{3 x}\right)
$$ | 3e^{-3x}\cdot\arcsin(e^{3x}) | 53 | 16 |
math | Example 8 Determine the largest real number $z$, such that $x+y+z=5, xy+yz+zx=3$, and $x, y$ are also real numbers.
(7th Canadian Competition Question) | \frac{13}{3} | 46 | 8 |
math | A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\frac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive ... | 113 | 122 | 3 |
math | ## Task 4.
Determine all pairs of natural numbers $(a, b)$ for which there exist natural numbers $x, y$ and $z$ such that
$$
a^{x}=(a+b)^{y}=(5 a+11 b)^{z}
$$ | (,b)=(2^{n},2^{n}) | 60 | 12 |
math | 8. (10 points) In the expression $(x+y+z)^{2020}+(x-y-z)^{2020}$, the brackets were expanded and like terms were combined. How many monomials $x^{a} y^{b} z^{c}$ with a non-zero coefficient were obtained? | 1022121 | 69 | 7 |
math | 72. Find the product:
$$
\left(1-\frac{1}{4}\right) \cdot\left(1-\frac{1}{9}\right) \cdot\left(1-\frac{1}{16}\right) \ldots\left(1-\frac{1}{225}\right)
$$ | \frac{8}{15} | 72 | 8 |
math | 3.1. Find all real values of $\alpha$ for which the following holds: if $a, b, c$ are the sides of a triangle, then
$$
a^{2}+b^{2}+c^{2} \leq \alpha(a b+a c+b c)
$$ | \alpha\geq2 | 64 | 6 |
math | 15. A piece of lead wire of length $2 n$ (where $n$ is a natural number and $n \geqslant 4$) is folded into a triangle with integer side lengths. Let $(a, b, c)$ represent a triangle with side lengths $a, b, c$ such that $a \leqslant b \leqslant c$.
(1) For the cases $n=4, 5, 6$, write down all the $(a, b, c)$ that sat... | 12 | 257 | 2 |
math | 11. There are 11 students who have signed up for the volunteer tour guide activity at the museum. The activity runs from 9 AM to 5 PM, with a public mini-lecture every hour. Each session requires 1 student to provide tour guide services to visitors. To avoid overworking the students, the museum will not schedule the sa... | 100000010 | 102 | 9 |
math | 2. To make the function $f(x)=\frac{1}{(x-2)^{2}}-2 x+\cos 2 \theta-3 \sin \theta+2$ always positive for $x \in(-\infty, 2)$, the range of the parameter $\theta$ in the interval $(0, \pi)$ is $\qquad$ . | 0<\theta<\frac{\pi}{6}or\frac{5\pi}{6}<\theta<\pi | 80 | 27 |
math | 27. If $S=\sum_{k=1}^{99} \frac{(-1)^{k+1}}{\sqrt{k(k+1)}(\sqrt{k+1}-\sqrt{k})}$, find the value of $1000 S$. | 1100 | 58 | 4 |
math | For which integers $n$ is the expression $n^{2}-6 n-2$ divisible by a) 8 ; b) 9; c) 11 ; d) 121 ?
# | 3+11k(k\in{Z}) | 45 | 11 |
math | ## Task Condition
Approximately calculate using the differential.
$y=\sqrt[3]{x}, x=8,24$ | 2.02 | 27 | 4 |
math | Kornél found two rings of the same size, and the side of each ring is divided into 36 equal parts. On both rings, 18 parts are painted yellow, and 18 are painted green. Can the rings be placed on top of each other so that the dividing lines on the side of the rings align, and the overlapping side parts are of the same ... | 18 | 87 | 2 |
math | Problem 8.8. Masha wrote on the board in ascending order all natural divisors of some number $N$ (the very first divisor written is 1, the largest divisor written is the number $N$ itself). It turned out that the third from the end divisor is 21 times greater than the second from the beginning. What is the largest valu... | 441 | 82 | 3 |
math | All the points on the line with equation $y=3 x+6$ are translated up 3 units, then translated left 4 units, and then reflected in the line with equation $y=x$. Determine the $y$-intercept of the resulting line. | -7 | 55 | 2 |
math | 10. (5 points) It is known that the denominators of three simplest proper fractions are 6, 15, and 20, respectively, and their product is $\frac{1}{30}$. Among these three simplest proper fractions, the largest number is $\qquad$ | \frac{5}{6} | 62 | 7 |
math | 3. Given a periodic sequence $\left\{x_{n}\right\}$ satisfying $x_{n}=\mid x_{n-1}-$ $x_{n-2} \mid(n \geqslant 3)$, if $x_{1}=1, x_{2}=a \geqslant 0$, then when the period of the sequence is the smallest, find the sum of the first 2008 terms of the sequence. | 1339 | 99 | 4 |
math | Problem 1. In two tubs, there are $4 \frac{1}{6}$ and $3 \frac{1}{2}$ liters of water respectively. How much water needs to be transferred from the first tub to the second tub so that after the transfer, both tubs have the same amount of water? Explain your answer! | \frac{1}{3} | 70 | 7 |
math | Example 2.60. Calculate the surface area formed by the rotation of the arc of the circle $x^{2}+(y-a)^{2}=R^{2}$ around the $O Y$ axis over the segment $0<y_{1} \leqslant y \leqslant y_{2}<R$. | 2\piR(y_{2}-y_{1}) | 70 | 12 |
math | Betty Lou and Peggy Sue take turns flipping switches on a $100 \times 100$ grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and col... | 9802 | 95 | 4 |
math | 3. [6 points] On the plane $O x y$, there is a point $A$, the coordinates $(x ; y)$ of which satisfy the equation $5 a^{2}-4 a x+6 a y+x^{2}-2 x y+2 y^{2}=0$, and a circle with center at point $B$, given by the equation $a^{2} x^{2}+$ $a^{2} y^{2}-4 a^{3} x-2 a x+2 a^{2} y+4 a^{4}+1=0$. Find all values of the parameter... | (0;\frac{1}{2})\cup(1;3) | 169 | 16 |
math | 3. Let $[x]$ denote the greatest integer not exceeding the real number $x$ (for example, $[3.1]=3,[-3.1]=-4$). Suppose the real number $x$ is not an integer, and $x+\frac{113}{x}=[x]+\frac{113}{[x]}$. Then the value of $x$ is | -10 \frac{3}{11} | 84 | 11 |
math | Example 7 How many positive integer factors does the number 20! have?
保留源文本的换行和格式，所以翻译结果应该是：
Example 7
How many positive integer factors does the number 20! have? | 41040 | 49 | 5 |
math | Example 4 Draw the following lines on the coordinate plane $y=k, y=\sqrt{3} x+2 k, y=-\sqrt{3} x+2 k$, where $k=$ $0, \pm 1, \pm 2, \cdots, \pm 10$. These 63 lines can divide the plane into several equilateral triangles. Find the number of equilateral triangles with side length $\frac{2}{\sqrt{3}}$.
(1994 12th American... | 660 | 118 | 3 |
math | Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8,4$ and let $n = (10a + b)^{10c+d}$. Find the probability that $1984!$ is divisible by $n.$ | \frac{5}{6} | 63 | 7 |
math | $(a)$ Compute -
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg]
\end{align*}
$(b)$ For $x > 0 $ define $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t . $\\
\\$1.$ Determine the open interval(s) (if any) where $F ( x )$ is decreasing and all the open inter... | e^x x \cos^4(e^x) | 194 | 12 |
math | 1. Isaac writes down a three-digit number. None of the digits is a zero. Isaac gives his sheet with the number to Dilara, and she writes under Isaac's number all the three-digit numbers that can be obtained by rearranging the digits of Isaac's number. Then she adds up all the numbers on the sheet. The result is 1221. W... | 911 | 88 | 3 |
math | Example 15 (2000 National High School Competition Question) There are $n$ people, and it is known that any 2 of them make at most one phone call. The total number of calls made among any $n-2$ of them is equal, and is equal to $3^{k}$ ($k$ is a positive integer). Find all possible values of $n$.
---
The above text is... | 5 | 104 | 1 |
math | 4. Given $P(1,4,5)$ is a fixed point in the rectangular coordinate system $O-x y z$, a plane is drawn through $P$ intersecting the positive half-axes of the three coordinate axes at points $A$, $B$, and $C$ respectively. Then the minimum value of the volume $V$ of all such tetrahedrons $O-A B C$ is $\qquad$ . | 90 | 90 | 2 |
math | Problem 8.1. The numbers $x, y, z$ are such that $x \in[-3,7], y \in[-2,5], z \in[-5,3]$.
(a) (1 point) Find the smallest possible value of the quantity $x^{2}+y^{2}$.
(b) (3 points) Find the smallest possible value of the quantity $x y z - z^{2}$. | 0 | 93 | 1 |
math | 15. $\triangle A B C$ is a right-angled triangle with $\angle A B C=90^{\circ}$. A circle $C_{1}$ is drawn with $A B$ as diameter, and another circle $C_{2}$ is drawn with $B C$ as diameter. The circles $C_{1}$. and $C_{2}$ meet at the points $B$ and $P$. If $A B=5 \mathrm{~cm}, B C=12 \mathrm{~cm}$ and $B P=x \mathrm{... | 520 | 139 | 3 |
math | 18.3.47 Find the integer solutions of the equation $x^{2}+x=y^{4}+y^{3}+y^{2}+y$.
untranslated text:
18.3.47 求方程 $x^{2}+x=y^{4}+y^{3}+y^{2}+y$ 的整数解. | (0,-1),(-1,-1),(0,0),(-1,0),(5,2),(-6,2) | 83 | 28 |
math | Example 5 Find all positive integers $k$ such that for any positive numbers $a, b, c$ satisfying the inequality
$$
k(a b+b c+c a)>5\left(a^{2}+b^{2}+c^{2}\right)
$$
there must exist a triangle with side lengths $a, b, c$.
(First China Girls Mathematical Olympiad)
Analysis: To find $k$, we can first determine the upper... | 6 | 118 | 1 |
math | In $\triangle ABC$, the area is $1, DE \parallel AB$, connect $BD$, let the largest area among $\triangle DCE$, $\triangle ABD$, and $\triangle BDE$ be $y$. Find the minimum value of $y$.
The area of $\triangle ABC$ is $1, DE \parallel AB$, connect $BD$, and let the largest area among $\triangle DCE$, $\triangle ABD$,... | \frac{3-\sqrt{5}}{2} | 108 | 12 |
math | 13. How many polynomials $P$ with integer coefficients and degree at most 5 satisfy $0 \leq P(x)<120$ for all $x \in\{0,1,2,3,4,5\} ?$ | 86400000 | 55 | 8 |
math | Example 11 Rationalize the denominator:
$$
\frac{3+2 \sqrt{2}-\sqrt{3}-\sqrt{6}}{1+\sqrt{2}-\sqrt{3}}=
$$
$\qquad$
(Fifth National Partial Provinces and Cities Junior High School Mathematics Competition) | 1+\sqrt{2} | 65 | 6 |
math | 1. (3 points) $20092009 \times 2010-20102010 \times 2009$; | 0 | 40 | 1 |
math | Question 215, Let the sum of all positive divisors of the positive integer $\mathrm{n}$ be $\sigma(\mathrm{n})$, find the value: $\left[\sqrt[3]{\sum_{\mathrm{n}=1}^{2020} \frac{\sigma(\mathrm{n}}{\mathrm{n}}}\right]$, here $[\mathrm{x}]$ represents the greatest integer not exceeding $x$.
保留源文本的换行和格式，翻译结果如下：
Question... | 14 | 190 | 2 |
math | ## Task 1 - 090731
Imagine all natural numbers from 1 to 2555, each written exactly once. Determine the total number of the digit 9 that would need to be written! | 705 | 48 | 3 |
math | 6.189. $\left\{\begin{array}{l}(x-y)\left(x^{2}+y^{2}\right)=5, \\ (x+y)\left(x^{2}-y^{2}\right)=9 .\end{array}\right.$ | (2;1),(-1;-2) | 58 | 10 |
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