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 | ## Problem Statement
Calculate the lengths of the arcs of the curves given by equations in a rectangular coordinate system.
$$
y=\sqrt{1-x^{2}}+\arcsin x, 0 \leq x \leq \frac{7}{9}
$$ | \frac{2\sqrt{2}}{3} | 56 | 12 |
math | B4. The candy store sells chocolates in the flavors white, milk, and dark. You can buy them in three different colored boxes. The three colored boxes have the following contents:
- Gold: 2 white, 3 milk, 1 dark,
- Silver: 1 white, 2 milk, 4 dark,
- Bronze: 5 white, 1 milk, 2 dark.
Lavinia buys a number of chocolate b... | 20 | 130 | 2 |
math | 4-168 Find the integer solutions of the equation
$$
\left(1+\frac{1}{m}\right)^{m+1}=\left(1+\frac{1}{1988}\right)^{1988} \text {. }
$$ | -1989 | 60 | 5 |
math | Task 1 - 130911 Two points $A$ and $B$ at the same height above the ground are at the same distance and on the same side of a straight, high wall. The segment $A B$ is $51 \mathrm{~m}$ long. A sound generated at $A$ reaches $B$ directly exactly $\frac{1}{10}$ s earlier than via the reflection off the wall.
Determine t... | 34 | 133 | 2 |
math | 9. (10 points) Uncle Zhang and Uncle Li's combined age is 56 years old. When Uncle Zhang was half of Uncle Li's current age, Uncle Li's age at that time was Uncle Zhang's age. How old is Uncle Zhang now? $\qquad$ years old? | 24 | 61 | 2 |
math | For all rational $x$ satisfying $0 \leq x < 1$, the functions $f$ is defined by
\[f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases}\]
Given that $x = 0.b_1b_2b_3 \cdots $ is the binary representation of $x$, find, with proof... | f(x) = 0.b_1b_1b_2b_2b_3b_3 \cdots | 148 | 28 |
math | 12. (5 points) Before New Year's Day, Xiaofang made greeting cards for her five classmates. When putting the cards into envelopes, she made a mistake, and none of the five classmates received the card Xiaofang made for them; instead, they received cards Xiaofang made for others. In total, there are $\qquad$ possible sc... | 44 | 76 | 2 |
math | 26. How many five-digit numbers are there that contain at least one digit 3 and are multiples of 3?
Please retain the original text's line breaks and format, and output the translation result directly. | 12504 | 43 | 5 |
math | [ Varignon Parallelogram ] [ Area of a Quadrilateral ]
The segments connecting the midpoints of opposite sides of a convex quadrilateral are equal to each other. Find the area of the quadrilateral if its diagonals are 8 and 12. | 48 | 54 | 2 |
math | 1. (7 points) Calculate: $123 \times 32 \times 125=$ $\qquad$ | 492000 | 28 | 6 |
math | On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$, it is possible to state that there is at least one rook in each $k\times k$ square ? | 2n+1 | 72 | 4 |
math | 11. Satisfy the equation
$$
\begin{array}{l}
\sqrt{x-2009-2 \sqrt{x-2010}}+ \\
\sqrt{x-2009+2 \sqrt{x-2010}}=2 .
\end{array}
$$
All real solutions are | 2010 \leqslant x \leqslant 2011 | 72 | 20 |
math | Three, (25 points) Given that $m, n, p, q$ satisfy $m n p q = 6(m-1)(n-1)(p-1)(q-1)$.
(1) If $m, n, p, q$ are all positive integers, find the values of $m, n, p, q$;
(2) If $m, n, p, q$ are all greater than 1, find the minimum value of $m+n+p+q$. | (9,4,2,2),(6,5,2,2),(4,3,3,2) | 108 | 25 |
math | 2. Let $n$ be a natural number and $C$ be a non-negative real number. Determine the number of sequences of real numbers $1, x_{2}, \ldots, x_{n}, 1$ such that the absolute value of the difference between any two consecutive terms of the sequence is equal to $C$. | r=\begin{cases}1,&C=0\\0,&\text{nisodd}C>0\\2\binom{n-1}{n/2},&\text{niseven}C>00\end{cases} | 69 | 52 |
math | 5】 Given $x_{i}$ are non-negative real numbers, $i=1,2,3,4 . x_{1}+x_{2}+x_{3}+x_{4}=1$. Let $S=1-$ $\sum_{i=1}^{4} x_{i}^{3}-6 \sum_{1 \leqslant i<j<k \leqslant 4} x_{i} x_{j} x_{k}$, find the range of $S$. | S \in \left[0, \frac{3}{4}\right] | 110 | 17 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(2 ; 1 ; 4)$
$M_{2}(3 ; 5 ;-2)$
$M_{3}(-7 ;-3 ; 2)$
$M_{0}(-3 ; 1 ; 8)$ | 4 | 90 | 1 |
math | 117 Let the function $f(x)=x^{2}+x+\frac{1}{2}$ have the domain $[n, n+1]$ (where $n$ is a natural number), then the range of $f(x)$ contains $\qquad$ integers. | 2(n+1) | 59 | 5 |
math | Find all integers $x$, $y$ and $z$ that satisfy $x+y+z+xy+yz+zx+xyz=2017$. | (0,1,1008) | 33 | 10 |
math | 12. Given that $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ are all positive numbers, determine the largest real number $C$ such that the inequality $C\left(x_{1}^{2005}+x_{2}^{2005}+x_{3}^{2005}+x_{4}^{2005}+x_{5}^{2005}\right) \geqslant x_{1} x_{2} x_{3} x_{4} x_{5}\left(x_{1}^{125}+x_{2}^{125}+x_{3}^{125}+x_{4}^{125}+x_{5}^{... | 5^{15} | 199 | 5 |
math | Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$. | 8 | 45 | 1 |
math | 217. Solve the equation: a) $x^{2}-6 x+13=0$; b) $9 x^{2}+12 x+$ $+29=0$ | x_1=3-2i,x_2=3+2i | 44 | 16 |
math | 3. Given that the length of the major axis of an ellipse is 4, the focal distance is 2, and the sum of the lengths of two perpendicular chords passing through the left focus is $\frac{48}{7}$. Try to find the product of the lengths of these two chords.
untranslated text remains unchanged. | \frac{576}{49} | 67 | 10 |
math | I1.1 The obtuse angle formed by the hands of a clock at $10: 30$ is $(100+a)^{\circ}$. Find $a$.
11.2 The lines $a x+b y=0$ and $x-5 y+1=0$ are perpendicular to each other. Find $b$.
I1.3 If $(b+1)^{4}=2^{c+2}$, find $c$.
I1.4 If $c-9=\log _{c}(6 d-2)$, find $d$. | 35,7,10,2 | 126 | 9 |
math | Let $t(A)$ denote the sum of elements of a nonempty set $A$ of integers, and define $t(\emptyset)=0$. Find a set $X$ of positive integers such that for every integers $k$ there is a unique ordered pair of disjoint subsets $(A_{k},B_{k})$ of $X$ such that $t(A_{k})-t(B_{k}) = k$. | X = \{3^0, 3^1, 3^2, \ldots\} | 88 | 24 |
math | Example 9 Let $S=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ be a permutation of the first $n$ natural numbers $1,2, \cdots, n$. $f(S)$ is defined as the minimum value of the absolute differences between every two adjacent elements in $S$. Find the maximum value of $f(S)$. | [\frac{n}{2}] | 81 | 6 |
math | 13.412 Three state farms are not located on the same straight line. The distance from the first to the third via the second is four times longer than the direct route between them; the distance from the first to the second via the third is $a$ km longer than the direct route; the distance from the second to the third v... | \frac{425-}{7},\frac{255+5}{7},\frac{170+}{7}for0<<68 | 115 | 36 |
math | Find the positive integer $n$ such that a convex polygon with $3n + 2$ sides has $61.5$ percent fewer diagonals than a convex polygon with $5n - 2$ sides. | 26 | 46 | 2 |
math | 12. Given that $a, b, c$ are positive real numbers, and $a+b+c=1$, find the minimum value of $u=\frac{3 a^{2}-a}{1+a^{2}}+\frac{3 b^{2}-b}{1+b^{2}}+\frac{3 c^{2}-c}{1+c^{2}}$. | 0 | 78 | 1 |
math | Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}=2 a_{n}-1(n=1,2, \cdots)$, and the sequence $\left\{b_{n}\right\}$ satisfies $b_{1}=3, b_{k+1}=a_{k}+b_{k}(k=1,2, \cdots)$. Find the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$. | 2^{n}+2n-1 | 119 | 9 |
math | 2. If $x-y=1, x^{3}-y^{3}=2$, then $x^{4}+y^{4}=$
$\qquad$ $x^{5}-y^{5}=$ $\qquad$ . | \frac{23}{9}, \frac{29}{9} | 51 | 16 |
math | 105. In a football championship, 16 teams participated. A team receives 2 points for a win; in the case of a draw in regular time, both teams take a series of penalty kicks, and the team scoring more goals gets one point. After 16 rounds, all teams accumulated 222 points. How many matches ended in a draw in regular tim... | 34 | 80 | 2 |
math | Let the sequence $(a_{n})$ be defined by $a_{1} = t$ and $a_{n+1} = 4a_{n}(1 - a_{n})$ for $n \geq 1$. How many possible values of t are there, if $a_{1998} = 0$? | 2^{1996} + 1 | 76 | 10 |
math | [Example 4.1.1] There is a square piece of paper, and inside the paper, there are $n$ different points. Now, let $M$ represent the set of points consisting of the four vertices of the square and these $n$ points, totaling $n+4$ points. The task is to cut this square piece of paper into some triangles according to the f... | 2n+2 | 158 | 4 |
math | B4 Let $a$ be the largest real value of $x$ for which $x^{3}-8 x^{2}-2 x+3=0$. Determine the integer closest to $a^{2}$. | 67 | 45 | 2 |
math | Let $ABCD$ be a rectangle. We consider the points $E\in CA,F\in AB,G\in BC$ such that $DC\perp CA,EF\perp AB$ and $EG\perp BC$. Solve in the set of rational numbers the equation $AC^x=EF^x+EG^x$. | \frac{2}{3} | 72 | 7 |
math | 33. In a constructed school boarding house, several rooms were double-occupancy, and the rest were triple-occupancy. In total, the boarding house is designed for more than 30 but fewer than 70 places. When $\frac{1}{5}$ of the available places were occupied, the number of unoccupied places turned out to be the same as ... | 50 | 140 | 2 |
math | 5. Let the sequence $\left\{a_{n}\right\}$ satisfy
$$
\begin{array}{l}
\left(2-a_{n+1}\right)\left(4+a_{n}\right)=8(n \geqslant 1), \text { and } a_{1}=2 . \\
\text { Then } \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=
\end{array}
$$ | 2^{n}-\frac{n}{2}-1 | 112 | 11 |
math | Compute the $\textit{number}$ of ordered quadruples $(w,x,y,z)$ of complex numbers (not necessarily nonreal) such that the following system is satisfied:
\begin{align*}
wxyz &= 1\\
wxy^2 + wx^2z + w^2yz + xyz^2 &=2\\
wx^2y + w^2y^2 + w^2xz + xy^2z + x^2z^2 + ywz^2 &= -3 \\
w^2xy + x^2yz + wy^2z + wxz^2 &= -1\end{align*... | 24 | 136 | 2 |
math | Example 5 Find all integers $n$, such that
$$
n^{4}+6 n^{3}+11 n^{2}+3 n+31
$$
is a perfect square.
$(2004$, Western Mathematical Olympiad) | 10 | 56 | 2 |
math | Three lines are drawn parallel to each of the three sides of $\triangle ABC$ so that the three lines intersect in the interior of $ABC$. The resulting three smaller triangles have areas $1$, $4$, and $9$. Find the area of $\triangle ABC$.
[asy]
defaultpen(linewidth(0.7)); size(120);
pair relpt(pair P, pair Q, real a, ... | 36 | 279 | 2 |
math | Example 5 Calculate the coefficient of $x^{100}$ in the expansion of $\left(1+x+x^{2}+\cdots+x^{100}\right)^{3}$ after combining like terms.
(7th All-Russian High School Mathematics Olympiad, Third Round Competition) | 5151 | 61 | 4 |
math | 14. Find the domain of the function $y=x^{2}$. | x\in\mathbb{R} | 16 | 9 |
math | Let $k$ be a positive integer. $12k$ persons have participated in a party and everyone shake hands with $3k+6$ other persons. We know that the number of persons who shake hands with every two persons is a fixed number. Find $k.$ | k = 3 | 57 | 5 |
math | Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions :
i) $a_0=1$, $a_1=3$
ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$
to be true that
$$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$. | M = 2 | 125 | 5 |
math | 1. Find all injective functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for every real number $x$ and natural number $n$,
$$
\left|\sum_{i=1}^{n} i(f(x+i+1)-f(f(x+i)))\right|<2016
$$
(Macedonia) | f(x)=x+1 | 81 | 6 |
math | Example 6. Solve the inequality
$$
3|x-1|+x^{2}>7
$$ | (-\infty;-1)\cup(2;+\infty) | 23 | 15 |
math | 3. (10 points) $\overline{\mathrm{abc}}$ is a three-digit number. If $a$ is an odd number, and $\overline{\mathrm{abc}}$ is a multiple of 3, then the smallest $\overline{\mathrm{abc}}$ is | 102 | 61 | 3 |
math | # 3.2. Solve the system of equations:
$$
\left\{\begin{array}{l}
2^{x+y}-2^{x-y}=4 \\
2^{x+y}-8 \cdot 2^{y-x}=6
\end{array}\right.
$$ | \frac{5}{2},\frac{1}{2} | 60 | 14 |
math | $1 \cdot 109$ Choose 3 numbers from $0,1,2, \cdots \cdots, 9$ such that their sum is an even number not less than 10. How many different ways are there to do this? | 51 | 56 | 2 |
math | Example 2. Form the equation of the plane that intercepts segment $O A=3$ on the $O x$ axis and is perpendicular to the vector $\vec{N}=\{2,-3,1\}$. | 2x-3y+z-6=0 | 48 | 10 |
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 | Find all ordered triples $(x, y, z)$ of integers satisfying the following system of equations:
$$
\begin{aligned}
x^{2}-y^{2} & =z \\
3 x y+(x-y) z & =z^{2}
\end{aligned}
$$ | (0,0,0),(1,0,1),(0,1,-1),(1,2,-3),(2,1,3) | 59 | 31 |
math | A cricket can jump two distances: 9 and 8 meters. It competes in a 100-meter race to the edge of a cliff. How many jumps must the cricket make to reach the end of the race without going past the finish line and falling off the cliff?
# | 12 | 59 | 2 |
math | 1. [3 points] Find the number of eight-digit numbers, the product of the digits of each of which is equal to 16875. The answer should be presented as an integer. | 1120 | 42 | 4 |
math | 5. (20 points) Alexei came up with the following game. First, he chooses a number $x$ such that $2017 \leqslant x \leqslant 2117$. Then he checks if $x$ is divisible by 3, 5, 7, 9, and 11 without a remainder. If $x$ is divisible by 3, Alexei awards the number 3 points, if by 5 - then 5 points, ..., if by 11 - then 11 p... | 2079 | 146 | 4 |
math | Given the set $A=\{1,2,3,4,5,6\}$, find:
(1) The number of all subsets of $A$;
(2) The number of all non-empty proper subsets of $A$;
(3) The number of all non-empty subsets of $A$;
(4) The sum of all elements in all non-empty subsets of $A$. | 672 | 84 | 3 |
math | 8. For $i=0,1, \ldots, 5$ let $l_{i}$ be the ray on the Cartesian plane starting at the origin, an angle $\theta=i \frac{\pi}{3}$ counterclockwise from the positive $x$-axis. For each $i$, point $P_{i}$ is chosen uniformly at random from the intersection of $l_{i}$ with the unit disk. Consider the convex hull of the po... | 2+4\ln(2) | 135 | 8 |
math | 5. (6 points) Two identical air capacitors are charged to a voltage $U$ each. One of them is submerged in a dielectric liquid with permittivity $\varepsilon$ while charged, after which the capacitors are connected in parallel. The amount of heat released upon connecting the capacitors is $Q$. Determine the capacitance ... | \frac{2\varepsilon(\varepsilon+1)Q}{U^{2}(\varepsilon-1)^{2}} | 434 | 30 |
math | A school admits students with artistic talents, and based on the entrance examination scores, it determines the admission score line and admits $\frac{2}{5}$ of the candidates. The average score of all admitted students is 15 points higher than the admission score line, and the average score of the candidates who were ... | 96 | 101 | 2 |
math | Let $n \geq 2$ be a given integer
$a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$
$b)$ Determine all possible values of... | 0 | 226 | 1 |
math | 12-42 Find all positive integers $w, x, y, z$ that satisfy the equation $w!=x!+y!+z!$.
(15th Canadian Mathematics Competition, 1983) | 2,2,2,w=3 | 49 | 8 |
math | It is known that the point symmetric to the center of the inscribed circle of triangle $ABC$ with respect to side $BC$ lies on the circumcircle of this triangle. Find angle $A$.
# | 60 | 43 | 2 |
math | 27. At a fun fair, coupons can be used to purchsse food. Each coupon is worth $\$ 5, \$ 8$ or $\$ 12$. For example, for a $\$ 15$ purchase you can use three coupons of $\$ 5$, or use one coupon of $\$ 5$ and one coupon of $\$ 8$ and pay $\$ 2$ by cash. Suppose the prices in the fun fair are all whole dollars. What is t... | 19 | 114 | 2 |
math | 1. Determine the smallest natural number whose half is a perfect square, a third is a perfect cube, and a fifth is a perfect fifth power. | 30233088000000 | 30 | 14 |
math | 7. (10 points) Xiao Ming is preparing to make dumplings. He added 5 kilograms of water to 1.5 kilograms of flour and found the dough too runny. His grandmother told him that the dough for making dumplings should be mixed with 3 parts flour and 2 parts water. So, Xiao Ming added the same amount of flour three times, and... | 2 | 106 | 1 |
math | Example 9 Given a positive integer $k$, when $x^{k}+y^{k}+z^{k}=1$, find the minimum value of $x^{k+1}+y^{k+1}+z^{k+1}$. | 3^{-\frac{1}{k}} | 55 | 9 |
math | Task 4. (20 points) For the numerical sequence $\left\{x_{n}\right\}$, all terms of which, starting from $n \geq 2$, are distinct, the relation $x_{n}=\frac{x_{n-1}+98 x_{n}+x_{n+1}}{100}$ holds. Find $\sqrt{\frac{x_{2023}-x_{1}}{2022} \cdot \frac{2021}{x_{2023}-x_{2}}}+2021$. | 2022 | 128 | 4 |
math | 1. Given that $x$ and $y$ are real numbers, and satisfy
$$
\left(x+\sqrt{x^{2}+2008}\right)\left(y+\sqrt{y^{2}+2008}\right)=2008 \text {. }
$$
Then the value of $x^{2}-3 x y-4 y^{2}-6 x-6 y+2008$
is $\qquad$ | 2008 | 98 | 4 |
math | Sprilkov N.P.
Nезнayka does not know about the existence of multiplication and exponentiation operations. However, he has mastered addition, subtraction, division, and square root extraction, and he also knows how to use parentheses.
Practicing, Nезнayka chose three numbers 20, 2, and 2 and formed the expression $\sq... | 20+10\sqrt{2} | 118 | 10 |
math | 11.182. The radius of the base of the cone is $R$, and the lateral surface area is equal to the sum of the areas of the base and the axial section. Determine the volume of the cone. | \frac{2\pi^{2}R^{3}}{3(\pi^{2}-1)} | 47 | 22 |
math | Problem 3. A four-digit number $X$ is not divisible by 10. The sum of the number $X$ and the number written with the same digits in reverse order is equal to $N$. It turns out that the number $N$ is divisible by 100. Find $N$. | 11000 | 65 | 5 |
math | Find all functions $ f: \mathbb{N} \to \mathbb{Z} $ satisfying $$ n \mid f\left(m\right) \Longleftrightarrow m \mid \sum\limits_{d \mid n}{f\left(d\right)} $$ holds for all positive integers $ m,n $ | f(n) = 0 | 66 | 7 |
math | 30. In $\triangle A B C \angle B A C=45^{\circ} . D$ is a point on $B C$ such that $A D$ is perpendicular to $B C$. If $B D=3$ $\mathrm{cm}$ and $D C=2 \mathrm{~cm}$, and the area of the $\triangle A B C$ is $x \mathrm{~cm}^{2}$. Find the value of $x$. | 15 | 101 | 2 |
math | 2. Find all pairs of positive integers $a, b$, for which the number $b$ is divisible by the number $a$ and at the same time the number $3a+4$ is divisible by the number $b+1$.
| (1,6),(2,4),(3,12) | 52 | 14 |
math | A triangle has vertex $A=(3,0), B=(0,3)$ and $C$, where $C$ is on the line $x+y=7$. What is the area of the triangle? | 6 | 43 | 1 |
math | 4th Centromerican 2002 Problem B1 ABC is a triangle. D is the midpoint of BC. E is a point on the side AC such that BE = 2AD. BE and AD meet at F and ∠FAE = 60 o . Find ∠FEA. | 60 | 65 | 2 |
math | 5. How many diagonals in a regular 32-sided polygon are not parallel to any of its sides | 240 | 22 | 3 |
math | On an island live 34 chameleons, which can take on 3 colors: yellow, red, and green. At the beginning, 7 are yellow, 10 are red, and 17 are green. When two chameleons of different colors meet, they simultaneously take on the third color. It turns out that after a certain amount of time, all the chameleons on the island... | green | 114 | 1 |
math | 2. A natural number minus 69 is a perfect square, and this natural number plus 20 is still a perfect square. Then this natural number is $\qquad$ . | 2005 | 38 | 4 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{6^{2 x}-7^{-2 x}}{\sin 3 x-2 x}$ | \ln(6^{2}\cdot7^{2}) | 43 | 12 |
math | Find all subsets $A$ of $\left\{ 1, 2, 3, 4, \ldots \right\}$, with $|A| \geq 2$, such that for all $x,y \in A, \, x \neq y$, we have that $\frac{x+y}{\gcd (x,y)}\in A$.
[i]Dan Schwarz[/i] | \{d, d(d-1)\} | 87 | 11 |
math | 2. The function
$$
f(x)=\sqrt{3} \sin ^{2} x+\sin x \cdot \cos x-\frac{\sqrt{3}}{2}\left(x \in\left[\frac{\pi}{12}, \frac{\pi}{2}\right]\right)
$$
has the range . $\qquad$ | \left[-\frac{1}{2}, 1\right] | 75 | 15 |
math | 90. Find the mathematical expectation of the random variable $Z=2 X+4 Y+5$, if the mathematical expectations of $X$ and $Y$ are known: $M(X)=3, M(Y)=5$. | 31 | 48 | 2 |
math | 1. On the board, the numbers $1, 2, \ldots, 2009$ are written. Several of them are erased and instead, the remainder of the sum of the erased numbers when divided by 13 is written on the board. After a certain number of repetitions of this procedure, only three numbers remain on the board, two of which are 99 and 999. ... | 9 | 97 | 1 |
math | Find all positive integers $n$ such that $2^{n}+3$ is a perfect square. | 0 | 22 | 1 |
math | 11.1. Find all solutions in natural numbers for the equation: $x!+9=y^{3}$. | 6,9 | 25 | 3 |
math | 4. Zhang Hua wrote a five-digit number, which can be divided by 9 and 11. If the first, third, and fifth digits are removed, the resulting number is 35; if the first three digits are removed, the resulting number can be divided by 9; if the last three digits are removed, the resulting number can also be divided by 9. T... | 63954 | 89 | 5 |
math | Point $A,B$ are marked on the right branch of the hyperbola $y=\frac{1}{x},x>0$. The straight line $l$ passing through the origin $O$ is perpendicular to $AB$ and meets $AB$ and given branch of the hyperbola at points $D$ and $C$ respectively. The circle through $A,B,C$ meets $l$ at $F$.
Find $OD:CF$ | 1:2 | 98 | 5 |
math | Solve the equation $\underbrace{\sqrt{n+\sqrt{n+\ldots \sqrt{n}}}}_{1964 \text { times }}=m$ in integers. | (0,0) | 37 | 5 |
math | 10.2 Find the number conjugate to the number $z=\frac{2 \sqrt{3}-i}{\sqrt{3}+i}+3$. | 4.25+0.75\sqrt{3}i | 36 | 15 |
math | Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$,
$$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$.
(b) Calculate $f(n)$ for any natural number $n$. | f(n) = n | 102 | 6 |
math | 7-0. The number $n$ has exactly six divisors (including 1 and itself). They were arranged in ascending order. It turned out that the third divisor is seven times greater than the second, and the fourth is 10 more than the third. What is $n$? | 2891 | 61 | 4 |
math | Question 60, Given real numbers $a$, $b$, and $c$ satisfy: $f(x)=a \cos x+b \cos 2 x+c \cos 3 x \geq-1$ for any real number $x$, try to find the maximum value of $a+b+c$.
untranslated part:
已知实数 $a 、 b$ 、 c满足: $f(x)=a \cos x+b \cos 2 x+c \cos 3 x \geq-1$ 对任意实数 $\mathrm{x}$ 均成立, 试求 $\mathrm{a}+\mathrm{b}+\mathrm{c}$ ... | 3 | 209 | 1 |
math | Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon. | 1506 | 60 | 4 |
math | 4. How many zeros does the product of numbers from 1 to 100 inclusive end with? | 24 | 22 | 2 |
math | 5.1. At the vertices of a cube, numbers $\pm 1$ are placed, and on its faces - numbers equal to the product of the numbers at the vertices of that face. Find all possible values that the sum of these 14 numbers can take. In your answer, specify their product. | -20160 | 64 | 6 |
math | 5.5. The hedgehogs collected 65 mushrooms and divided them so that each hedgehog got at least one mushroom, but no two hedgehogs had the same number of mushrooms. What is the maximum number of hedgehogs that could be | 10 | 52 | 2 |
math | 23. Represent the number 231 as the sum of several natural numbers so that the product of these addends also equals 231. | 231=3+7+11+\underbrace{1+1+\ldots+1}_{210\text{ones}} | 32 | 31 |
math | If point $M(x,y)$ lies on the line with equation $y=x+2$ and $1<y<3$, calculate the value of
$A=\sqrt{y^2-8x}+\sqrt{y^2+2x+5}$ | 5 | 55 | 1 |
math | 11.3. Find the roots of the equation: $\left(x^{3}-2\right)\left(2^{\sin x}-1\right)+\left(2^{x^{3}}-4\right) \sin x=0$. | \sqrt[3]{2},\pin(n\in{Z}) | 54 | 15 |
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