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 | Task B-3.5. For which positive real numbers $a$ does the equation
$$
\frac{1}{\sin ^{2} x}+\frac{1}{\cos ^{2} x}+\frac{1}{\operatorname{tg}^{2} x}+\frac{1}{\operatorname{ctg}^{2} x}=a, \quad x \neq \frac{k \pi}{2}, \quad k \in \mathbb{Z}
$$
have real solutions? | \geq6 | 113 | 4 |
math | 10. Evaluate
$$
\int_{-\infty}^{\infty} \frac{1-x^{2}}{1+x^{4}} d x
$$ | 0 | 37 | 1 |
math | Solve the equation $\cos 2 x+3 \cos x=1$. | \\frac{\pi}{3}+2k\pi | 17 | 12 |
math | The polynomial $1976(x+x^2+ \cdots +x^n)$ is decomposed into a sum of polynomials of the form $a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_1, a_2, \ldots , a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible. | 7, 103, 1975 | 95 | 12 |
math | ## Task 4 - 120614
A distance of $168 \mathrm{~m}$ in length is to be divided into three segments, whose lengths are denoted in sequence by $a, b, c$. The second segment is to be three times as long as the first, and the third segment is to be four times as long as the first.
Determine all possibilities for specifyin... | =21\mathrm{~},b=63\mathrm{~},=84\mathrm{~} | 108 | 25 |
math | ## 128. Math Puzzle $1 / 76$
A circus gave 200 performances in the last season, all of which were sold out. The number of seats in the circus tent is three times the fourth part of the number of performances given.
a) How many program leaflets were printed if one fourth of the visitors bought a leaflet?
b) How many ... | 450000 | 113 | 6 |
math | F5 (20-3, UK) Let $f, g: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*}$ be strictly increasing functions, and
$$
f\left(\mathbf{N}^{*}\right) \cup g\left(\mathbf{N}^{*}\right)=\mathbf{N}^{*}, f\left(\mathbf{N}^{*}\right) \cap g\left(\mathbf{N}^{*}\right)=\varnothing, g(n)=f(f(n))+1 .
$$
Find $f(240)$. | 388 | 138 | 3 |
math | Example 2 If $\cos ^{5} \theta-\sin ^{5} \theta<7\left(\sin ^{3} \theta-\cos ^{3} \theta\right)$, where $\theta \in[0,2 \pi)$, then the range of values for $\theta$ is $\qquad$ [2]
(2011, National High School Mathematics Joint Competition) | \left(\frac{\pi}{4}, \frac{5 \pi}{4}\right) | 87 | 20 |
math | Gardener Mr. Malina was selling strawberries. In the last nine crates, he had 28, 51, 135, 67, 123, 29, 56, 38, and 79 strawberry plants, respectively. He sold the crates whole, never removing any plants from the crates. The gardener wanted to sell the crates to three customers so that nothing was left and each of thes... | 202 | 121 | 3 |
math | Example 2 Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with its right focus at $F(c, 0)$, there exists a line $l$ passing through point $F$ that intersects the ellipse at points $A$ and $B$, such that $O A \perp O B$. Find the range of the eccentricity of the ellipse.
(2015, National High School... | e\in[\frac{\sqrt{5}-1}{2},1) | 112 | 16 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 1}\left(\frac{2 x-1}{x}\right)^{1 /(\sqrt[5]{x}-1)}
$$ | e^5 | 49 | 3 |
math | 1. Given the sets $U=\{1,2,3,4,5,6,7,8\}, A=\{1,2,3,4,5\}, B=\{4,5,6,7,8\}$, the number of subsets of set $U$ that are not subsets of set $A$ and are not subsets of set $B$ is | 196 | 83 | 3 |
math | 1. Let $f(x)$ be a function defined on $R$, for any real number $x$ we have $f(x+3) \cdot f(x-4)=-1$. Also, when $0 \leq x<7$, $f(x)=\log _{2}(9-x)$, then the value of $f(-100)$ is . $\qquad$ | -\frac{1}{2} | 83 | 7 |
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 |
math | 375. Find the distance:
1) from the point $(2 ; 3 ;-4)$ to the plane
$$
2 x+6 y-3 z+16=0
$$
2) from the point $(2 ;-4 ; 1)$ to the plane
$$
x-8 y+4 z=0
$$ | 4\frac{2}{9} | 73 | 8 |
math | Problem 19. In triangle $ABC$, the difference between the internal angles $A-B=60^{\circ}$. It is known that the height $CH$ from $C$ to side $AB$ is equal to $CH=BC-AC$. Find the angles of the triangle. | A=90,B=30,C=60 | 62 | 12 |
math | 2. Nikola thought of three different digits, different from zero. Martin wrote down all the two-digit numbers using these digits. The sum of all the numbers that Martin wrote down is equal to 231. Which digits did Nikola think of? | 1,2,4 | 50 | 5 |
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 | [ Pascal's Triangle and Newton's Binomial ]
How many rational terms are contained in the expansion of
a) $(\sqrt{2}+\sqrt[4]{3})^{100}$
b) $(\sqrt{2}+\sqrt[8]{3})^{300}$? | 26 | 63 | 2 |
math | 4. Bivariate function
$$
\begin{array}{l}
f(x, y) \\
=\sqrt{\cos 4 x+7}+\sqrt{\cos 4 y+7}+ \\
\quad \sqrt{\cos 4 x+\cos 4 y-8 \sin ^{2} x \cdot \sin ^{2} y+6}
\end{array}
$$
The maximum value is . $\qquad$ | 6\sqrt{2} | 94 | 6 |
math | Example 1. Find the curve passing through the point $M_{0}(1,4)$ and having the property that the segment of any of its tangents, enclosed between the coordinate axes, is bisected at the point of tangency. | xy=4 | 50 | 3 |
math | Exercise 1. Determine all triplets $(x, y, z)$ of integers such that
$$
x^{2}+y^{2}+z^{2}=16(x+y+z)
$$ | (x,y,z)\in{0,16}^3 | 43 | 13 |
math | 48. The number of trucks passing by a gas station on a highway is to the number of passenger cars passing by the same highway as $3: 2$. It is known that on average, 1 out of 30 trucks and 2 out of 45 passenger cars pull up to the gas station for refueling. What is the probability that a vehicle arriving at the gas sta... | 0.0378 | 87 | 6 |
math | ## 26. Gold Coins
The king is waiting for each of his 30 vassals, as in previous years, to present him with 30 gold coins. But the king knows that one of them has the unfortunate habit of presenting coins not weighing 10 g as required, but 9 g. How can the king, with a single weighing, identify the culprit to have his... | 4650-n=4650- | 100 | 11 |
math | Example 6 Given that the roots of the equation $x^{2}-(k+3) x+k^{2}=0$ are integers. Find the integer value of $k$ and the roots of the equation. | k=-1,0,2,3, x=1,0,3,4 | 45 | 19 |
math | Problem 2.5 Determine the natural numbers $a, b, c$ with the property that $a+b+c=a b c$.
untranslated text remains unchanged. | (1,2,3) | 34 | 7 |
math | 5. (10 points) A natural number that can be expressed as the sum of two consecutive non-zero natural numbers, and also as the sum of three consecutive non-zero natural numbers, is called a "good number". The largest "good number" not exceeding 2011 is . $\qquad$ | 2007 | 64 | 4 |
math | 13. Let $\{a_{n}\}$ be a geometric sequence with the sum of the first $n$ terms denoted as $S_{n}, S_{n}=2^{n}+r$ (where $r$ is a constant), and let $b_{n}=2\left(1+\log _{2} a_{n}\right)\left(n \in \mathbf{N}^{*}\right)$.
(1) Find the sum of the first $n$ terms of the sequence $\{a_{n} b_{n}\}$, denoted as $T_{n}$;
(2... | \frac{3\sqrt{2}}{4} | 210 | 12 |
math | 5. Find all integers n for which the number
$$
2^{n}+n^{2}
$$
is a perfect square of some integer.
(Tomáš Jurík) | n=0n=6 | 40 | 6 |
math | 9. The common ratio of the geometric sequence $a+\log _{2} 3, a+\log _{4} 3, a+\log _{8} 3$ is | \frac{1}{3} | 41 | 7 |
math | 1. First solution: Since the Rabbit ran at a speed twice that of Alice, by the time Alice arrived at the Duchess's, the Rabbit was again halfway. Since he was 10 minutes late, Alice spent 20 minutes on half the journey, and 40 minutes on the entire journey. | 40 | 63 | 2 |
math | 4. Let $n$ be a positive integer, $\left(1+x+x^{2}\right)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2 n} x^{2 n}$, find the value of $a_{0}+a_{3}+a_{6}+a_{9}+\cdots$.
untranslated text remains the same as the source text. | 3^{n-1} | 96 | 6 |
math | There are integers $m$ and $n$ so that $9 +\sqrt{11}$ is a root of the polynomial $x^2 + mx + n.$ Find $m + n.$ | 52 | 43 | 2 |
math | If we increase the sides of an isosceles right triangle by $4 \mathrm{~cm}$, then the area of the triangle increases by $112 \mathrm{~cm}^{2}$. What are the sides of the original triangle? | 26 | 54 | 2 |
math | Example 4 Let positive numbers $a, b, c, x, y, z$ satisfy
$$
c y+b z=a, a z+c x=b, b x+a y=c .
$$
Find the minimum value of the function $f(x, y, z)=\frac{x^{2}}{1+x}+\frac{y^{2}}{1+y}+\frac{z^{2}}{1+z}$.
(2005, National High School Mathematics Competition) | \frac{1}{2} | 103 | 7 |
math | Given the parabola $C: y^{2}-2 p x(p>0)$, and a fixed point $A\left(\frac{p}{2}, p\right)$, does there exist a point $T$ in the coordinate plane such that any line $l$ passing through $T$ intersects the parabola $C$ at points $B$ and $C$, and as long as $A$, $B$, and $C$ are not collinear, $\triangle A B C$ can always ... | (\frac{5}{2}p,-p) | 135 | 11 |
math | 7th Putnam 1947 Problem B3 Let O be the origin (0, 0) and C the line segment { (x, y) : x ∈ [1, 3], y = 1 }. Let K be the curve { P : for some Q ∈ C, P lies on OQ and PQ = 0.01 }. Let k be the length of the curve K. Is k greater or less than 2? Solution | k<2 | 96 | 3 |
math | 7. Let the side length of the equilateral triangle $\triangle A B C$ be $1, t$ be any real number. Then the minimum value of $|\overrightarrow{A B}+t \overrightarrow{A C}|$ is $\qquad$ | \frac{\sqrt{3}}{2} | 56 | 10 |
math | \section*{Problem 1 - 161221}
Let \(R\) be a rectangle with area \(A\), side lengths \(a, b\), and diagonal length \(d\). Furthermore, let \(a\) be the arithmetic mean of \(b\) and \(d\).
Determine \(a, b\), and \(d\) for this rectangle in terms of \(A\). | =\frac{2\sqrt{3}}{3}\sqrt{A},\,b=\frac{\sqrt{3}}{2}\sqrt{A},\,=\frac{5\sqrt{3}}{6}\sqrt{A} | 85 | 51 |
math | 5. Represent the number $\frac{3}{7}$ as the sum of several different common fractions, the numerators of which are equal to one.
# | \frac{1}{7}+\frac{1}{8}+\frac{1}{56}+\frac{1}{9}+\frac{1}{72}+\frac{1}{57}+\frac{1}{56\cdot57} | 32 | 57 |
math | $4 \cdot 50$ Solve the equation
$$\frac{1}{x^{2}-10 x-29}+\frac{1}{x^{2}-10 x-45}-\frac{2}{x^{2}-10 x-69}=0$$ | x^2 - 10x - 39 = 0 | 63 | 15 |
math | ## 293. Math Puzzle 10/89
A wooden cube has a mass of $5000 \mathrm{~g}$. What is the mass of a cube made of the same material, but with all edges being only $\frac{1}{5}$ of the edge lengths of the first block? | 40\mathrm{~} | 69 | 7 |
math | 2. Points $X, Y$, and $Z$ lie on a circle with center $O$ such that $X Y=12$. Points $A$ and $B$ lie on segment $X Y$ such that $O A=A Z=Z B=B O=5$. Compute $A B$. | 2\sqrt{13} | 65 | 7 |
math | 9.109. Find the domain of the function $f$, if
$f(x)=\sqrt[6]{4^{\frac{x+1}{x}}-17 \cdot 2^{\frac{1}{x}}+4}$ | x\in[-\frac{1}{2};0)\cup(0;\frac{1}{2}] | 53 | 23 |
math | Let $k$ be a positive integer and $S$ be a set of sets which have $k$ elements. For every $A,B \in S$ and $A\neq B$ we have $A \Delta B \in S$. Find all values of $k$ when $|S|=1023$ and $|S|=2023$.
Note:$A \Delta B = (A \setminus B) \cup (B \setminus A)$ | k = 2^9 \times m | 103 | 10 |
math | ## Task 4 - 221224
Let $n \neq 0$ be a natural number. On a circle, there are $2 n$ pairwise distinct points $P_{1}, P_{2}, \ldots, P_{2 n}$ given.
We are looking for the number $A_{n}$ of all different ways to draw a set of $n$ chords such that the following conditions are met:
Each chord connects one of the points... | 42 | 295 | 2 |
math | Fifty points are chosen inside a convex polygon having eighty sides such that no three of the fifty points lie on the same straight line. The polygon is cut into triangles such that the vertices of the triangles are just the fifty points and the eighty vertices of the polygon. How many triangles are there? | 178 | 59 | 3 |
math | 19. In $\triangle A B C$, $A B=A C, \angle A=100^{\circ}, I$ is the incenter, $D$ is a point on $A B$ such that $B D=B I$. Find the measure of $\angle B C D$.
(Problem 1073 from Mathematical Bulletin) | 30 | 75 | 2 |
math | Suppose $S_n$ is the set of positive divisors of $n$, and denote $|X|$ as the number of elements in a set $X$. Let $\xi$ be the set of positive integers $n$ where $|S_n| = 2m$ is even, and $S_n$ can be partitioned evenly into pairs $\{a_i, b_i\}$ for integers $1 \le i \le m$ such that the following conditions hold:
$\b... | 64 | 196 | 2 |
math | 1. A1 (GBR 3) ${ }^{\mathrm{IMO}}$ The function $f(n)$ is defined for all positive integers $n$ and takes on nonnegative integer values. Also, for all $m, n$,
$$ \begin{gathered} f(m+n)-f(m)-f(n)=0 \text { or } 1 \\ f(2)=0, \quad f(3)>0, \quad \text { and } \quad f(9999)=3333 \end{gathered} $$
Determine $f(1982)$. | 660 | 131 | 3 |
math | 102. Deck of Cards. The cards in the deck are sequentially numbered from 1 to 7, and then thoroughly shuffled. Five cards are randomly drawn sequentially from the deck. What is the probability that the numbers on these cards will be in increasing order? | \frac{1}{120} | 54 | 9 |
math | \section*{Problem 9 - V01209}
What percentage of
a) all 2-digit numbers
b) all 3-digit numbers
c) all 5-digit numbers
d) all 10-digit numbers
e) all 20-digit numbers
f) all 50-digit numbers
do not contain the digit 0? | )90,b)81,)65.61,)38.7,e)13.5,f)0.57 | 78 | 30 |
math | ## Task B-3.7.
A square with an area of $P=7-\log _{2} x$ and a cube with a volume of $V=\log _{2} x-2$ are given. Calculate the real number $x$ if the side length of the square is 1 unit greater than the edge length of the cube. What are the side length of the square and the edge length of the cube? | 8,b=2,=1 | 91 | 7 |
math | 2. Let the set $M=\{1,99,-1,0,25,-36,-91,19,-2,11\}$, and denote all non-empty subsets of $M$ as $M_{i}, i=1,2, \cdots$, 2013. The product of all elements in each $M_{i}$ is $m_{i}$. Then $\sum_{i=1}^{2013} m_{i}=$ $\qquad$ . | -1 | 112 | 2 |
math | Solve the inequality
$$
\log _{x}\left(2.5-\frac{1}{x}\right)>1
$$ | (\frac{2}{5};\frac{1}{2})\cup(1;2) | 30 | 21 |
math | $9.279 \log _{x} 10-0.5 \log _{a} 10>0, \ 0<a<1$. | x\in(0;^{2})\cup(1;+\infty) | 38 | 18 |
math | 12 Use red, yellow, and blue to color the vertices of a regular pentagon, then the probability that no two adjacent vertices have the same color is $\qquad$ . | \frac{10}{81} | 37 | 9 |
math | How many $(n ; k)$ number pairs are there for which $n>k$, and the difference between the interior angles of the $n$-sided and $k$-sided regular polygons is $1^{\circ}$? | 52 | 49 | 2 |
math | ## Task Condition
Find the derivative.
$$
y=\sin \sqrt{3}+\frac{1}{3} \cdot \frac{\sin ^{2} 3 x}{\cos 6 x}
$$ | \frac{\sin6x}{\cos^{2}6x}=\frac{\tan6x}{\cos6x} | 46 | 27 |
math | 15. Given a quadratic equation in $x$
$$
\dot{x}^{2}-2 x-a^{2}-a=0 \quad (a>0) \text {. }
$$
(1) Prove: one root of this equation is greater than 2, and the other root is less than 2;
(2) If for $a=1,2, \cdots, 2004$, the two roots of the corresponding quadratic equations are $\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{... | -\frac{4008}{2005} | 222 | 13 |
math | Problem 6.3. Three merchants: Foma, Yerema, and Julius met in Novgorod. If Foma gives Yerema 70 gold coins, then Yerema and Julius will have the same amount of money. If Foma gives Yerema 40 gold coins, then Foma and Julius will have the same amount of money. How many gold coins should Foma give Yerema so that they bot... | 55 | 99 | 2 |
math | The height of a regular quadrilateral pyramid $S A B C D$ ($S$ - the vertex) is $\sqrt{3}$ times the length of the base edge. Point $E$ is the midpoint of the apothem lying in the face $A S B$. Find the angle between the line $D E$ and the plane $A S C$.
# | 45 | 77 | 2 |
math | 60. In what ratio does a plane, parallel to two skew edges of a triangular pyramid and dividing one of the other edges in the ratio $2: 1$, divide the volume of the pyramid? | \frac{20}{7} | 42 | 8 |
math | 7. Let $f(x)$ be a polynomial with integer coefficients, $f(0)=11$, and there exist $n$ distinct integers $x_{1}, x_{2}, \cdots, x_{n}$, such that
$$
f\left(x_{1}\right)=f\left(x_{2}\right)=\cdots=f\left(x_{n}\right)=2010 .
$$
Then the maximum value of $n$ is | 3 | 99 | 1 |
math | 1. (2 points) Solve the equation $11 p=q^{3}-r^{3}$, where $p, q, r-$ are prime numbers. | q=13,r=2,p=199 | 34 | 12 |
math | 1. Simplify:
$$
\sum_{k=1}^{2016}(k \sqrt{k+1}+(k+1) \sqrt{k})^{-1}=
$$ | 1-\frac{1}{\sqrt{2017}} | 41 | 14 |
math | 9. Given $F_{1} 、 F_{2}$ are the two foci of the ellipse $\frac{x^{2}}{4}+y^{2}=1$, $A 、 B$ are the left vertex and the upper vertex of the ellipse, respectively, and point $P$ lies on the line segment $A B$. Then the minimum value of $\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}$ is | -\frac{11}{5} | 99 | 8 |
math | ## Task B-3.6.
Mr. Algebrić decided to encourage his two sons to solve a problem, so he said that they would determine their own pocket money for going out in the following way. From the set $\left\{5,5^{2}, 5^{3}, \ldots, 5^{33}\right\}$, they need to choose two different numbers, $a$ and $b$, such that $\log _{a} b$... | 90 | 137 | 2 |
math | 2. (6 points) The remainder when $439 \times 319 \times 2012 + 2013$ is divided by 7 is $\qquad$ . | 1 | 44 | 1 |
math | \section*{Problem 3}
A circle radius 100 is drawn on squared paper with unit squares. It does not touch any of the grid lines or pass through any of the lattice points. What is the maximum number of squares can it pass through?
| 800 | 54 | 3 |
math | 2. (BEL 4) From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process contimues until the bag is empty or there are 4 socks of different colors held outside the ... | \frac{8}{15} | 86 | 8 |
math | 13.351. The road from post office $A$ to village $B$ goes uphill for 2 km, then on flat ground for 4 km, and finally downhill for 3 km. The postman takes 2 hours and 16 minutes to travel from $A$ to $B$, and 2 hours and 24 minutes to return. If the final destination of his route were located on the same road but twice ... | )3\mathrm{}/\mathrm{};b)4\mathrm{}/\mathrm{};)5\mathrm{}/\mathrm{} | 148 | 29 |
math | Example 3 Solve the inequality $(\lg x+3)^{7}+\lg ^{7} x+\lg x^{2}+3 \geqslant 0$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | [10^{-\frac{3}{2}},+\infty) | 65 | 15 |
math | [ Area of a quadrilateral ]
The area of a quadrilateral is 3 cm², and the lengths of its diagonals are 6 cm and 2 cm. Find the angle between the diagonals.
# | 30 | 44 | 2 |
math | 3. In a right rectangular parallelepiped, the sum of all edges is $76 \mathrm{~cm}$, and the length of the diagonal is $13 \mathrm{~cm}$.
Calculate $(a+b-c) \cdot(a-b+c)+(a+b-c) \cdot(-a+b+c)+(a-b+c) \cdot(-a+b+c)$, where $a, b$ and c are the dimensions of the parallelepiped. | 23 | 95 | 2 |
math | 44th Putnam 1983 Problem B1 Let C be a cube side 4, center O. Let S be the sphere center O radius 2. Let A be one of the vertices of the cube. Let R be the set of points in C but not S, which are closer to A than to any other vertex of C. Find the volume of R. Solution | 8-\frac{4\pi}{3} | 80 | 10 |
math | Simplify the following expression as much as possible:
$$
\frac{(x+1)^{\frac{1}{2}}+1}{(x+1)^{\frac{1}{2}}-1} \cdot \frac{\left[(x+1)^{\frac{1}{2}}+1\right](x+1)^{-\frac{1}{2}}-\left[(x+1)^{\frac{1}{2}}-1\right](x+1)^{-\frac{1}{2}}}{2\left[(x+1)^{\frac{1}{2}}+1\right]^{2}}
$$ | \frac{1}{x(x+1)^{\frac{1}{2}}} | 135 | 17 |
math | 4. Several different numbers are written on the blackboard. The sum of any three of them is a rational number, while the sum of any two is an irrational number. The maximum number of numbers that can be written on the blackboard is $\qquad$ . | 3 | 54 | 1 |
math | For which numbers $n$ is it possible to put marks on a stick such that all distances $1$ cm, $2$ cm, . . . , $n$ cm each appear exactly once as the distance between two of the marks, and no other distance appears as such a distance?
| n = 3 | 60 | 5 |
math | 9-46 Let real numbers $a, b, c, d$ satisfy $a^{2}+b^{2}+c^{2}+d^{2} \leqslant 1$, find
$$
S=(a+b)^{4}+(a+c)^{4}+(a+d)^{4}+(b+c)^{4}+(b+d)^{4}+(c+d)^{4}
$$
the maximum value. | 6 | 98 | 1 |
math | Question 102: Let the complex number $z$ satisfy $z+\frac{1}{z} \in[1,2]$, then the minimum value of the real part of $z$ is $\qquad$
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | \frac{1}{2} | 73 | 7 |
math | 5. The force with which the airflow acts on the sail can be calculated using the formula
$F=\frac{A S \rho\left(v_{0}-v\right)^{2}}{2}$, where $A-$ is the aerodynamic force coefficient, $S-$ is the area of the sail $S$ $=4 \mathrm{M}^{2} ; \rho-$ is the density of air, $v_{0}$ - is the wind speed $v_{0}=4.8 \mathrm{~m... | 1.6\mathrm{M}/\mathrm{} | 229 | 11 |
math | ## Cooperative algorithms $\quad]$ Dirichlet's Principle (etc.). $\quad]$ [Pairings and groupings; bijections]
Before the clairvoyant lies a deck of 36 cards face down (4 suits, 9 cards of each suit). He names the suit of the top card, after which the card is revealed to him. Then he names the suit of the next card, a... | 23 | 215 | 2 |
math | Example 8 If $a, b, c$ are the length, width, and height of a rectangular prism, and $a+b-$ $c=1$, it is known that the length of the diagonal of the rectangular prism is 1, and $a>b$, try to find the range of values for the height $c$.
The above text is translated into English, please retain the original text's line ... | 0<c<\frac{1}{3} | 96 | 10 |
math | 1. [2] Jacob flips five coins, exactly three of which land heads. What is the probability that the first two are both heads? | \frac{3}{10} | 29 | 8 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{7^{3 x}-3^{2 x}}{\tan x+x^{3}}$ | \ln\frac{7^{3}}{3^{2}} | 42 | 14 |
math | 2. A group of four friends were collecting plastic bottles. Nikola collected 72 bottles, Vlado collected a third less than Nikola, and Petar collected a third more than Vlado. Marko concluded that he, along with Vlado and Petar, collected twice as many bottles as Nikola. How many bottles did Marko collect? | 32 | 72 | 2 |
math | 2. Find the number of natural numbers $k$, not exceeding 333300, such that $k^{2}-2 k$ is divisible by 303. Answer: 4400. | 4400 | 47 | 4 |
math | Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$.
Titu Andreescu, Mathematics Department, College of Texas, USA | (2, 3, 5) | 54 | 10 |
math | [Example 2.4.1] There is a rectangular iron sheet with dimensions $80 \times 50$. We need to cut out a square of the same size from each of the four corners and then fold it to form an open-top box. What should be the side length of the square cut out from each corner to maximize the volume of the open-top box? | 10 | 78 | 2 |
math | 1. Determine all pairs $(m, n)$ of natural numbers for which
$$
m+s(n)=n+s(m)=70
$$
where $s(a)$ denotes the sum of the digits of the natural number $a$.
(Jaroslav Švrček) | (56,59),(57,58),(58,57),(59,56),(60,64),(61,63),(62,62),(63,61),(64,60) | 60 | 55 |
math | 6. There are two docks, $\mathrm{A}$ and $\mathrm{B}$, on a river, with $\mathrm{A}$ upstream and $\mathrm{B}$ downstream. Two people, Jia and Yi, start from $\mathrm{A}$ and $\mathrm{B}$ respectively at the same time, rowing towards each other, and meet after 4 hours. If Jia and Yi start from $\mathrm{A}$ and $\mathrm... | 10 | 159 | 2 |
math | [ Decimal numeral system ]
Try to find all natural numbers that are 5 times larger than their last digit.
# | 25 | 23 | 2 |
math | 3. Let $a=256$. Find the unique real number $x>a^{2}$ such that
$$
\log _{a} \log _{a} \log _{a} x=\log _{a^{2}} \log _{a^{2}} \log _{a^{2}} x .
$$ | 2^{32} | 72 | 5 |
math | 15 Given that $\alpha, \beta$ are two distinct real roots of the equation $4 x^{2}-4 t x-1=0(t \in \mathbf{R})$, and the function $f(x)=\frac{2 x-t}{x^{2}+1}$ has a domain of $[\alpha, \beta]$.
(1) Find $g(t)=\max f(x)-\min f(x)$;
(2) Prove: For $u_{i} \in\left(0, \frac{\pi}{2}\right)(i=1,2,3)$, if $\sin u_{1}+\sin u_{... | \frac{3}{4}\sqrt{6} | 215 | 11 |
math | 1. Let $p$ be a given odd prime, and the positive integer $k$ makes $\sqrt{k^{2}-p k}$ also a positive integer. Then $k=$ $\qquad$
(2004, National High School Mathematics Competition) | \frac{(p+1)^{2}}{4} | 54 | 13 |
math | ## Zadatak B-4.2.
Riješite sustav jednadžbi:
$$
\begin{aligned}
& \log _{3}|\pi x|+2 \log _{|\pi x|} 3=3 \\
& \sin ^{2}(x+y)+1=2 \sin (x+y)
\end{aligned}
$$
| (x,y)\in{(\\frac{9}{\pi},\frac{\pi}{2}\\frac{9}{\pi}+2k\pi),(\\frac{3}{\pi},\frac{\pi}{2}\\frac{3}{\pi}+2k\pi),k\in\mathbb{Z}} | 81 | 70 |
math | 15. Let $a=\frac{1+\sqrt{2009}}{2}$. Find the value of $\left(a^{3}-503 a-500\right)^{10}$. | 1024 | 48 | 4 |
math | G3.1 In $\triangle A B C, \angle A B C=2 \angle A C B, B C=2 A B$. If $\angle B A C=a^{\circ}$, find the value of $a$.
G3.2 Given that $x+\frac{1}{x}=\sqrt{2}, \frac{x^{2}}{x^{4}+x^{2}+1}=b$, find the value of $b$.
G3.3 If the number of positive integral root(s) of the equation $x+y+2 x y=141$ is $c$, find the value o... | 90 | 200 | 2 |
math | 9.143. $\sqrt{x+3}<\sqrt{x-1}+\sqrt{x-2}$. | x\in(\sqrt{\frac{28}{3}};\infty) | 25 | 17 |
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