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 | Given that $(1 + \tan 1^{\circ})(1 + \tan 2^{\circ}) \ldots (1 + \tan 45^{\circ}) = 2^n$, find $n$. | 23 | 47 | 2 |
math | 16. Given $\sin \left(\frac{\pi}{4}-x\right)=\frac{5}{13}$, and $x \in\left(0, \frac{\pi}{4}\right)$, find the value of $\frac{\cos 2 x}{\cos \left(\frac{\pi}{4}+x\right)}$. | \frac{24}{13} | 77 | 9 |
math | 10.331. In a rectangle with sides $a$ and $b$, the angle bisectors of all angles are drawn until they intersect each other. Find the area of the quadrilateral formed by the angle bisectors. | \frac{(b-)^{2}}{2} | 48 | 12 |
math | 10. The sequence $a_{0}, a_{1}, a_{2}, \cdots, a_{n}$ satisfies $a_{0}=\sqrt{3}, a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}}\left(\left[a_{n}\right],\left\{a_{n}\right\}\right.$ represent the integer part and fractional part of $a_{n}$, respectively), then $a_{2016}=$ $\qquad$ . | 3024+\sqrt{3} | 120 | 9 |
math | Out of seven Easter eggs, three are red. We placed ten eggs into a larger box and six into a smaller box randomly. What is the probability that there is a red egg in both boxes? | \frac{3}{4} | 40 | 7 |
math | $11 \cdot 31$ Suppose 1987 can be written as a three-digit number $\overline{x y z}$ in base $b$, and $x+y+z=$ $1+9+8+7$, try to determine all possible $x, y, z$ and $b$.
(19th Canadian Mathematics Competition, 1987) | 5,9,11,b=19 | 81 | 10 |
math | 【Example 5】 6 boys and 4 girls are to serve as attendants on 5 buses, with two people per bus. Assuming boys and girls are separated, and the buses are distinguishable, how many ways are there to assign them? | 5400 | 52 | 4 |
math | 2. Let $P$ be a point on the plane of $\triangle A B C$, satisfying $\overrightarrow{P A}+\overrightarrow{P B}+\overrightarrow{P C}=2 \overrightarrow{A B}$.
If $S_{\triangle A B C}=1$, then $S_{\triangle P A B}=$ $\qquad$ | \frac{1}{3} | 77 | 7 |
math | ## Task B-1.3.
Solve the equation in the set of prime numbers
$$
2 p^{3}-q^{2}=2(p+q)^{2}
$$ | (3,2) | 39 | 5 |
math | 351*. Solve the system of equations:
$$
\left\{\begin{array}{l}
x^{2}+y^{2}=1 \\
4 x y\left(2 y^{2}-1\right)=1
\end{array}\right.
$$ | (\cos\pi/8;\sin\pi/8),(\cos5\pi/8;\sin5\pi/8),(\cos9\pi/8;\sin9\pi/8),(\cos13\pi/8;\sin13\pi/8) | 58 | 60 |
math | ## Task 1 - 150621
A Soviet helicopter of the Mi-10 type can transport a payload of 15000 kp.
In a transport of bulky goods with three helicopters of this type, the first helicopter was loaded to $\frac{1}{3}$, the second to $\frac{7}{8}$, and the third to $\frac{3}{5}$ of its capacity.
Determine the total weight of... | 27125 | 107 | 5 |
math | $9.17 C_{x}^{x-1}+C_{x}^{x-2}+C_{x}^{x-3}+\ldots+C_{x}^{x-9}+C_{x}^{x-10}=1023$. | 10 | 62 | 2 |
math | 10th Putnam 1950 Problem A1 a and b are positive reals and a > b. Let C be the plane curve r = a - b cos θ. For what values of b/a is C convex? | 0<k\leq\frac{1}{2} | 50 | 12 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt{\left(n^{4}+1\right)\left(n^{2}-1\right)}-\sqrt{n^{6}-1}}{n}$ | -\frac{1}{2} | 58 | 7 |
math | The coefficients of the polynomial $P(x)$ are nonnegative integers, each less than 100. Given that $P(10) = 331633$ and $P(-10) = 273373$, compute $P(1)$.
| 100 | 62 | 3 |
math | At an international combinatorial conference, a hundred mathematicians are accommodated in a hotel where the rooms are numbered from one to a hundred. The receptionist plans to assign the mathematicians to the rooms corresponding to their arrival order. However, the first arriving guest is forgotten to be given the cor... | 2^{99} | 124 | 5 |
math | Three, as shown, in the triangular prism $A B C A_{1} B_{1} C_{1}$, all nine edges are equal to 1, and $\angle A_{1} A B$ $=\angle A_{1} A C$ $=\angle B A C$. Point $P$ is on the diagonal $A_{1} B$ of the side face $A_{1} A B B_{1}$, with $A_{1} P=\frac{\sqrt{3}}{3}$. Connect $P C_{1}$. Find the degree measure of the a... | 30^{\circ} | 142 | 6 |
math | 2. If $x \in(-\infty,-1]$, the inequality
$$
\left(m-m^{2}\right) 4^{x}+2^{x}+1>0
$$
always holds, then the range of real number $m$ is $\qquad$ | -2 < m < 3 | 63 | 7 |
math | 5. On the edge $C C_{1}$ of a regular triangular prism $A B C A_{1} B_{1} C_{1}$, a point $T$ is taken such that $C T: C_{1} T=1: 3$. The point $T$ is the vertex of a right circular cone such that three vertices of the prism lie on the circumference of its base.
a) Find the ratio of the height of the prism to the edge... | \frac{576\pi\sqrt{3}}{11\sqrt{11}} | 126 | 22 |
math | 2. Let $a<-1$, and the variable $x$ satisfies $x^{2}+a x \leqslant-x$, and the minimum value of $x^{2}+a x$ is $-\frac{1}{2}$. Then $a=$ $\qquad$ . | -\frac{3}{2} | 63 | 7 |
math | 7. (6 points) In the following (1), (2), (3) equations, each has a $\square$. Please indicate in the $\qquad$th equation, $\square=3 \frac{1}{20}$.
(1) $\left.4 \frac{1}{2} \div\left[0.4+\left(6 \frac{1}{3}-\square\right) \div 12 \frac{1}{2} \times 0.75\right]=10 \div 12 \frac{1}{2} \times 0.75\right]=10$
(2) $\frac{\l... | 3 | 257 | 1 |
math | 31. How many ordered pairs of positive integers $(x, y)$ satisfy the equation
$$
x \sqrt{y}+y \sqrt{x}+\sqrt{2006 x y}-\sqrt{2006 x}-\sqrt{2006 y}-2006=0 ?
$$ | 8 | 69 | 1 |
math | 3. Solve the system in integers:
$$
\left\{\begin{array}{c}
a b c d=c+199 \\
51 a^{6}+5 c^{2}+20|b+d|=2016
\end{array}\right.
$$ | (-1,2,1,-100);(-1,-2,1,100);(-1,100,1,-2);(-1,-100,1,2) | 62 | 44 |
math | 116 The four vertices of a regular tetrahedron are $A, B, C, D$, with edge length $1, P \in AB, Q \in CD$, then the range of the distance between points $P, Q$ is $\qquad$ . | [\frac{\sqrt{2}}{2},1] | 58 | 12 |
math | Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have? | 6 | 47 | 1 |
math | \section*{Problem 1 - 271011}
How many ordered pairs of integers \((x, y)\) are there in total for which \(x \cdot y=1987\) holds? | 4 | 48 | 1 |
math | Let's find all right-angled triangles whose sides are integers, and when 6 is added to the hypotenuse, we get the sum of the legs. | (7,24,25),(8,15,17),(9,12,15) | 33 | 25 |
math | 7. The sum of all positive integers $n$ that satisfy $\frac{1}{4}<\sin \frac{\pi}{n}<\frac{1}{3}$ is | 33 | 37 | 2 |
math | The median of a right triangle, drawn to the hypotenuse, divides it into two triangles with perimeters of 8 and 9. Find the sides of the triangle.
# | 3,4,5 | 37 | 5 |
math | 5. Let the complex number $z=(\omega+2)^{3}(\omega-3)^{2}$. Find the maximum value of the modulus of $z$ when $\omega$ takes all complex numbers of modulus 1. | 7^{\frac{5}{2}} | 50 | 9 |
math | 2. A box contains 3 red balls and 3 white balls, all of the same size and shape. Now, a fair die is rolled, and the number of balls taken from the box is equal to the number rolled. What is the probability that the number of red balls taken is greater than the number of white balls taken? $\qquad$ . | \frac{19}{60} | 73 | 9 |
math | 11. The smallest prime \( p \) (where \( p > 3 \)) for which there do not exist non-negative integers \( a, b \) satisfying \( \left|3^{a}-2^{b}\right|=p \) is | 41 | 53 | 2 |
math | 10.29 Try to find the smallest natural number that satisfies the following property: its first digit is 4, but when the first digit is moved to the end, its value becomes $\frac{1}{4}$ of the original.
(46th Moscow Mathematical Olympiad, 1983) | 410256 | 65 | 6 |
math | 7. Vanya decided to give Masha a bouquet of an odd number of flowers for her birthday, consisting of yellow and red tulips, so that the number of flowers of one color differs from the number of flowers of the other by exactly one. Yellow tulips cost 50 rubles each, and red ones cost 31 rubles. What is the largest numbe... | 15 | 100 | 2 |
math | Problem 17. The side of a regular triangle $ABC$ is 4. Point $D$ is the midpoint of side $BC$. A line passing through $B$ intersects side $AC$ at point $M$. Perpendiculars from points $D$ and $A$ to line $BM$ are $DH$ and $AK$. Calculate the length of segment $AM$, if
$$
AK^4 - DH^4 = 15
$$ | 2 | 97 | 1 |
math | 4. The sequence $\left\{x_{n}\right\}$ is defined as follows:
$$
x_{1}=\frac{2}{3}, x_{n+1}=\frac{x_{n}}{2(2 n+1) x_{n}+1}\left(n \in \mathbf{Z}_{+}\right) \text {. }
$$
Then $x_{1}+x_{2}+\cdots+x_{2014}=$ | \frac{4028}{4029} | 102 | 13 |
math | 3. A three-stage launch vehicle consists of stages in the form of cylinders. All these cylinders are similar to each other. The length of the middle stage is two times less than the sum of the lengths of the first and third stages. In the fueled state, the mass of the middle stage is $13 / 6$ times less than the total ... | \frac{7}{5} | 111 | 7 |
math | 10,11
Consider a rectangle $A B C D$ and a point $E$ not lying in its plane. Let the planes $A B E$ and $C D E$ intersect along the line $l$, and the planes $B C E$ and $A D E$ - along the line $p$. Find the angle between the lines $l$ and $p$. | 90 | 82 | 2 |
math | 3.2.1 * Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n+1}=\frac{1}{3} a_{n}+1, n=1,2, \cdots$. Find the general term of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{3}{2}-\frac{1}{2}(\frac{1}{3})^{n-1} | 77 | 30 |
math | In trapezoid $A B C D$, the bases $A D=12$ and $B C=8$ are given. On the extension of side $B C$, a point $M$ is chosen such that $C M=2.4$.
In what ratio does the line $A M$ divide the area of trapezoid $A B C D$? | 1:1 | 82 | 3 |
math | $$
\begin{array}{l}
\text { 1. Let } f(x)=x^{2}+a x+b \cos x \text {, and } \\
\{x \mid f(x)=0, x \in \mathbf{R}\} \\
=\{x \mid f(f(x))=0, x \in \mathbf{R}\} \neq \varnothing \text {. }
\end{array}
$$
Then the range of values for $a+b$ is | [0,4) | 109 | 5 |
math | Determine continuous functions $f:\mathbb{R}\to \mathbb{R}$ such that $\left( {{a}^{2}}+ab+{{b}^{2}} \right)\int\limits_{a}^{b}{f\left( x \right)dx=3\int\limits_{a}^{b}{{{x}^{2}}f\left( x \right)dx,}}$ for every $a,b\in \mathbb{R}$ . | f(x) = C | 104 | 6 |
math | 6. Given the complex number $z$ satisfies
$$
(a-2) z^{2018}+a z^{2017} \mathrm{i}+a z \mathrm{i}+2-a=0 \text {, }
$$
where, $a<1, \mathrm{i}=\sqrt{-1}$. Then $|z|=$ $\qquad$ . | 1 | 84 | 1 |
math | Problem 6.4. In a coastal village, 7 people go fishing every day, 8 people go fishing every other day, 3 people go fishing every three days, and the rest do not fish at all. Yesterday, 12 people went fishing, and today, 10 people are fishing. How many people will go fishing tomorrow? | 15 | 73 | 2 |
math | Question 47, Find the maximum value of the function $f(x)=\frac{\sqrt{2} \sin x+\cos x}{\sin x+\sqrt{1-\sin x}}(0 \leq x \leq \pi)$. | \sqrt{2} | 54 | 5 |
math | Task 1. (5 points) Solve the equation $x^{9}-2021 x^{3}+\sqrt{2022}=0$. | {-\sqrt[6]{2022};\sqrt[3]{\frac{\sqrt{2022}\\sqrt{2018}}{2}}} | 34 | 36 |
math | Example 13 (2004-2005 Hungarian Mathematical Olympiad) Find the largest integer $k$ such that $k$ satisfies the following condition: for all integers $x, y$, if $x y+1$ is divisible by $k$ then $x+y$ is also divisible by $k$. | 24 | 69 | 2 |
math | 17. From a square with a side length of 20, remove a rectangle with an area of 36, where the two side lengths of the rectangle are integers, and one side of the rectangle is part of one side of the square. The maximum perimeter of the remaining shape is $\qquad$. | 116 | 64 | 3 |
math | 【Question 30】
There are two piles of stones, one with 11 stones and the other with 13 stones, denoted as $(11,13)$. Two players, A and B, take turns to remove stones, following these rules: one can take any number of stones from one pile or the same number of stones from both piles, but must take at least one stone. Th... | (3,5)or(8,13) | 117 | 12 |
math | Problem 8.5. A field was partially planted with corn, oats, and millet. If the remaining part is completely planted with millet, then millet will occupy half of the entire field. If the remaining part is equally divided between oats and corn, then oats will occupy half of the entire field. By what factor will the amoun... | 3 | 83 | 1 |
math | In how many different ways can three knights be placed on a chessboard so that the number of squares attacked would be maximal? | 64 | 25 | 2 |
math | 9. The sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$ satisfy $a_{1}=2, b_{1}=1,\left\{\begin{array}{l}a_{n+1}=5 a_{n}+3 b_{n}+7 \\ b_{n+1}=3 a_{n}+5 b_{n}\end{array} \quad\left(n \in \mathbf{N}^{*}\right)\right.$, then their general terms are $a_{n}=$ $\qquad$ $b_{n}=$ $\qquad$ . | a_{n}=2^{3n-2}+2^{n+1}-4,b_{n}=2^{3n-2}-2^{n+1}+3 | 137 | 38 |
math | 5. If the equation with respect to $x$
$$
4^{x}+(a+3) 2^{x}+5=0
$$
has at least one real root in the interval $[1,2]$, then the range of the real number $a$ is $\qquad$ . | -\frac{33}{4} \leqslant a \leqslant-3-2 \sqrt{5} | 66 | 28 |
math | 6. The maximum value of the function $y=\sin x+\sqrt{3} \cos x-2 \sin 3 x$ is $\qquad$ | \frac{16\sqrt{3}}{9} | 34 | 13 |
math | Find all real values of $ x>1$ which satisfy:
$ \frac{x^2}{x\minus{}1} \plus{} \sqrt{x\minus{}1} \plus{}\frac{\sqrt{x\minus{}1}}{x^2} \equal{} \frac{x\minus{}1}{x^2} \plus{} \frac{1}{\sqrt{x\minus{}1}} \plus{} \frac{x^2}{\sqrt{x\minus{}1}}$ | x = 2 | 104 | 5 |
math | Sam dumps tea for $6$ hours at a constant rate of $60$ tea crates per hour. Eddie takes $4$ hours to dump the same
amount of tea at a different constant rate. How many tea crates does Eddie dump per hour?
[i]Proposed by Samuel Tsui[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{90}$
Sam dumps a total of $6 \cdot 60 = 3... | 90 | 145 | 2 |
math | 7. If the union of two sets $A$ and $B$ has two elements, and $f(A)$ denotes the number of elements in $A$, $f(B)$ denotes the number of elements in $B$, then the number of such pairs $(f(A), f(B))$ is $\qquad$ pairs. | 6 | 67 | 1 |
math | Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided up into exactly $2007$ smaller cubes, each of whose sides is of integer length. | n = 13 | 43 | 6 |
math | 3. Given $x, y \in \mathbf{R}$, and $x^{2}+y^{2} \leqslant 1$. Then the maximum value of $x+y-x y$ is $\qquad$ . | 1 | 52 | 1 |
math | 27. Person A and Person B start from points 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, reaching each other's starting points and immediately turning back. They meet for the second ... | 15 | 87 | 2 |
math | Exercise 1. Solve in real numbers the system of equations
$$
\begin{gathered}
x_{1}\left(x_{1}-1\right)=x_{2}-1 \\
x_{2}\left(x_{2}-1\right)=x_{3}-1 \\
\cdots \\
x_{2016}\left(x_{2016}-1\right)=x_{2017}-1 \\
x_{2017}\left(x_{2017}-1\right)=x_{1}-1
\end{gathered}
$$ | x_{i}=1 | 123 | 5 |
math | \section*{Problem 4 - 111244}
a) Determine all ordered triples \((x, y, z)\) of real numbers that satisfy the equation \(x^{3} z + x^{2} y + x z + y = x^{5} + x^{3}\).
b) Among the triples found in a), identify those in which exactly one of the three numbers \(x, y, z\) is positive, exactly one is negative, and exact... | (x,0,x^2)wherex<0 | 106 | 12 |
math | 4. Determine the maximum product of natural numbers whose sum is equal to a given natural number $n$. | P=\begin{cases}3^{k},&n=3k,k\in\mathbb{N}\\2^{2}3^{k-1},&n=3k+1,k\in\mathbb{N}\\2\cdot3^{k},&n=3k+2,k\in\mathbb{N}\cup{0}\end{cases} | 21 | 81 |
math | 3. Suppose that $y=a x+(1-a)$ for some unknown number $a$. If $x=3$, the value of $y$ is 7 . Then, if $x=8$, what is the value of $y$ ? | 22 | 52 | 2 |
math | Problem 6. For $x=\frac{\pi}{2 n}$, find the value of the sum
$$
\cos ^{2}(x)+\cos ^{2}(2 x)+\cos ^{2}(3 x)+\ldots+\cos ^{2}(n x)
$$ | \frac{n-1}{2} | 64 | 8 |
math | [ Formulas for abbreviated multiplication (other).]
Calculate: $\frac{(2001 \cdot 2021+100)(1991 \cdot 2031+400)}{2011^{4}}$ | 1 | 56 | 1 |
math | [ Angles between bisectors ]
Point $O$ is the center of the circle inscribed in triangle $A B C$. It is known that $B C=a, A C=b, \angle A O B=120^{\circ}$. Find the side $A B$. | \sqrt{^{2}+b^{2}-} | 60 | 12 |
math | 379. Form the equation of the tangent to the parabola $y=x^{2}$ $-4 x$ at the point with abscissa $x_{0}=1$. | -2x-1 | 40 | 5 |
math | Given that $a_1, a_2, a_3, . . . , a_{99}$ is a permutation of $1, 2, 3, . . . , 99,$ find the maximum possible value of
$$|a_1 - 1| + |a_2 - 2| + |a_3 - 3| + \dots + |a_{99} - 99|.$$
| 4900 | 96 | 4 |
math | An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.
a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$;
b) Calculate the distance of $p$ from either base;
c) Determine under what conditions such points $P$ actu... | h^2 \leq ac | 99 | 8 |
math | ## Task A-1.5. (4 points)
In a bag, there is a sufficiently large number of red, white, and blue balls. Each student randomly takes three balls from the bag. How many students must there be at a minimum to ensure that at least one pair of them has the same combination of balls, i.e., the same number of balls of each c... | 11 | 78 | 2 |
math | \section*{Task 1 - 211011}
A traveler covered the first part of a business trip by car and the rest by train.
When he had driven the car part of the journey and exactly one fifth of the train part, he realized that at that point he had covered exactly one third of the total distance. Later, when he had covered exactl... | 120\mathrm{~} | 117 | 8 |
math | 3. On the extensions of sides $\boldsymbol{A B}, \boldsymbol{B C}, \boldsymbol{C D}$ and $\boldsymbol{A}$ of the convex quadrilateral $\boldsymbol{A} \boldsymbol{B C D}$, points $\boldsymbol{B}_{1}, \boldsymbol{C}_{1}, \boldsymbol{D}_{1}$ and $\boldsymbol{A}_{1}$ are taken such that $\boldsymbol{B} \boldsymbol{B}_{1}=\... | 5 | 245 | 1 |
math | Determine the number of ten-digit positive integers with the following properties:
$\bullet$ Each of the digits $0, 1, 2, . . . , 8$ and $9$ is contained exactly once.
$\bullet$ Each digit, except $9$, has a neighbouring digit that is larger than it.
(Note. For example, in the number $1230$, the digits $1$ and $3$ are ... | 256 | 141 | 3 |
math | 10. Given the function $f(x)$ satisfies for any real numbers $x, y$,
$$
f(x+y)=f(x)+f(y)+6xy,
$$
and $f(-1) f(1) \geqslant 9$.
Then $f\left(\frac{2}{3}\right)=$ $\qquad$ | \frac{4}{3} | 76 | 7 |
math | 7. If the system of equations concerning $x$ and $y$ $\left\{\begin{array}{l}a x+b y=1, \\ x^{2}+y^{2}=10\end{array}\right.$ has solutions, and all solutions are integers, then the number of ordered pairs $(a, b)$ is $\qquad$ | 32 | 76 | 2 |
math | Gapochkin A.i.
How many integers from 1 to 1997 have a sum of digits that is divisible by 5? | 399 | 30 | 3 |
math | Given a sequence $\left\{a_{n}\right\}$ with all terms no less than 1, it satisfies:
$$
a_{1}=1, \quad a_{2}=1+\frac{\sqrt{2}}{2}, \quad\left(\frac{a_{n}}{a_{n+1}-1}\right)^{2}+\left(\frac{a_{n}-1}{a_{n-1}}\right)^{2}=2 .
$$
Try to find:
(1) The general term formula of the sequence $\left\{a_{n}\right\}$;
(2) The valu... | \frac{2}{3} | 157 | 7 |
math | 2. Let positive real numbers $a, b, c, d, e$ satisfy $a<b<c<d$ $<e$, and the smallest three of the 10 products of any two numbers are $28, 32, 56$, and the largest two are 128, 240. Then $e=$ $\qquad$ | 16 | 78 | 2 |
math | Let $a$, $b$, $c$ be the sides of an acute triangle $\triangle ABC$ , then for any $x, y, z \geq 0$, such that $xy+yz+zx=1$ holds inequality:$$a^2x + b^2y + c^2z \geq 4F$$ where $F$ is the area of the triangle $\triangle ABC$ | a^2x + b^2y + c^2z \geq 4F | 87 | 21 |
math | I. Fill-in-the-blank Questions (8 points each, total 64 points)
1. Let the sequence $\left\{\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}\right\}$ have the sum of its first $n$ terms as $S_{n}$. Then the number of rational numbers in the first 2016 terms of the sequence $\left\{S_{n}\right\}$ is $\qquad$. | 43 | 104 | 2 |
math | 7. The quadratic equation $(1-\mathrm{i}) x^{2}+(\lambda+\mathrm{i}) x+(1+\mathrm{i} \lambda)=0(\mathrm{i}$ is the imaginary unit, $\lambda \in \mathbf{R})$ has two imaginary roots if and only if the range of $\lambda$ is $\qquad$ . | \lambda\neq2 | 72 | 6 |
math | 6. Given the equation in $x$
$$
x^{3}-4 x^{2}+5 x+a=0(a \in \mathbf{R})
$$
has three real roots $x_{1}, x_{2}, x_{3}$. Then the maximum value of $\max \left\{x_{1}, x_{2}, x_{3}\right\}$ is $\qquad$ . | 2 | 87 | 1 |
math | 8. Try to divide 2004 into the sum of several distinct positive integers, and make the product of these positive integers the largest possible. Find this maximum value. | \frac{63!}{11} | 36 | 10 |
math | Tokarev S.i.
In a line, all integers from 1 to 100 are written in an unknown order. With one question about any 50 numbers, you can find out the order of these 50 numbers relative to each other. What is the minimum number of questions needed to definitely determine the order of all 100 numbers? | 5 | 75 | 1 |
math | Example 3. Solve the integral equation:
$$
\int_{0}^{+\infty} \varphi(t) \sin x t \, d t=e^{-x} \quad(x>0)
$$ | \varphi()=\frac{2}{\pi}\frac{}{1+^{2}} | 45 | 20 |
math | Example 4 Find all pairs of positive integers $(m, n)$ such that $m^{6}=n^{n+1}+n-1 .{ }^{[6]}$
(2013, Turkey National Team Selection Exam) | m=n=1 | 51 | 4 |
math | # Problem 5.
The segments of two lines, enclosed between two parallel planes, are in the ratio $5: 9$, and the acute angles between these lines and one of the planes are in the ratio $2: 1$. Find the cosine of the smaller angle. | 0.9 | 57 | 3 |
math | 5. Find all values of the parameter $b$, for each of which there exists a number $a$ such that the system
$$
\left\{\begin{array}{l}
x=\frac{7}{b}-|y+b| \\
x^{2}+y^{2}+96=-a(2 y+a)-20 x
\end{array}\right.
$$
has at least one solution $(x ; y)$. | b\in(-\infty;-\frac{7}{12}]\cup(0;+\infty) | 95 | 25 |
math | ## Task B-2.5.
Complex numbers $z_{1}, z_{2}$, $z_{3}$ are associated with points $A, B, C$ in the complex plane that are 2016 units away from the origin. If for the complex numbers $z_{1}, z_{2}, z_{3}$ it holds that $z_{1}+z_{2}+z_{3}=0$, calculate the lengths of the sides of triangle $ABC$. | 2016\sqrt{3} | 102 | 9 |
math | Let $p_{1}, p_{2}, \ldots, p_{2005}$ be different prime numbers. Let $\mathrm{S}$ be a set of natural numbers which elements have the property that their simple divisors are some of the numbers $p_{1}, p_{2}, \ldots, p_{2005}$ and product of any two elements from $\mathrm{S}$ is not a perfect square.
What is the maxim... | 2^{2005} | 105 | 7 |
math | 14. Given the function $f(x)=4 x+a x^{2}-\frac{2}{3} x^{3}$, where $a \in[-1,1]$ is a constant. Let the two non-zero real roots of the equation $f(x)=2 x+\frac{1}{3} x^{3}$ with respect to $x$ be $x_{1}, x_{2}$. Is there a real number $m$ such that the inequality $m^{2}+t m+1 \geqslant \left|x_{1}-x_{2}\right|$ holds f... | \geqslant2or\leqslant-2 | 168 | 14 |
math | 6. In a match without ties, the game ends when one person wins 2 more games than the other, and the one with more wins is the winner. It is known that in the odd-numbered games, the probability of A winning is $\frac{3}{5}$; in the even-numbered games, the probability of B winning is $\frac{3}{5}$. Then the expected nu... | \frac{25}{6} | 95 | 8 |
math | II. (50 points)
Real numbers $a, b, c$ and a positive number $\lambda$ make $f(x)=x^{3}+a x^{2}+b x+c$ have 3 real roots $x_{1}, x_{2}$, $x_{3}$, and satisfy
(1) $x_{2}-x_{1}=\lambda$;
(2) $x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)$.
Find the maximum value of $\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}$. | \frac{3\sqrt{3}}{2} | 139 | 12 |
math | G6.2 If $b=\log _{3}\left[2(3+1)\left(3^{2}+1\right)\left(3^{4}+1\right)\left(3^{8}+1\right)+1\right]$, find $b$. | 16 | 63 | 2 |
math | 7. Let $a, b, c$ be three distinct real numbers such that the equations $x^{2}+a x+1=0$ and $x^{2}+b x+c=0$ have a common real root, and the equations $x^{2}+x+a=0$ and $x^{2}+c x+b=0$ also have a common real root, then the value of $a+b+c$ is $\qquad$. | -3 | 99 | 2 |
math | Solve the following equation:
$$
(x-2)^{4}+(x-1)^{4}=\frac{17}{4}(x-2)^{2}(x-1)^{2}
$$ | x_{1}=0,x_{2}=3,x_{3}=\frac{5}{3},x_{4}=\frac{4}{3} | 46 | 32 |
math | 9.3. Find all prime numbers $p$ for which $p^{2}+200$ is a perfect square of an integer. | p=5orp=23 | 31 | 7 |
math | 11. Given
$$
(1+\sqrt{3})^{n}=a_{n}+b_{n} \sqrt{3} \text {, }
$$
where $a_{n}$ and $b_{n}$ are integers. Then $\lim _{n \rightarrow+\infty} \frac{a_{n}}{b_{n}}=$ $\qquad$ . | \sqrt{3} | 83 | 5 |
math | 13. (15 points) From the sequence of consecutive natural numbers $1,2,3, \cdots, 2014$, select $n$ numbers such that these $n$ numbers satisfy: taking any two of them, one number will not be 7 times the other. Find the maximum value of $n$, and explain your reasoning. | 1763 | 76 | 4 |
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