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 | Let ${(a_n)_{n\ge1}} $ be a sequence with ${a_1 = 1} $ and ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$ for all ${n \ge 1}$, where ${\lfloor x \rfloor}$ denotes the greatest integer less than or equal to ${x}$. Find all ${n \le 2013}$ such that ${a_n}$ is a perfect square | n = 1 | 115 | 5 |
math | In the $xyz$ space, prove that the sphere $S$ centered the origin $O$ with radius $\sqrt{6}$ has common points with the plane $\alpha$ passing through 3 points $(4,\ 0,\ 0),\ (0,\ 4,\ 0),\ (0,\ 0,\ 4)$. Then when the point $(x,\ y,\ z)$ moves on the whole set of the common points, find the range of $xyz$.
[i]2011 Kyot... | \frac{50}{27} \leq xyz \leq 2 | 120 | 18 |
math | Valakinek (expressed in whole years) age at death was the 31st part of their birth year. How old was this person in 1930? | 39 | 38 | 2 |
math | Example 4 (53rd Romanian Mathematical Olympiad (Final)) Find all real numbers $a, b, c, d, e \in[-2,2]$, such that $a+b+c+d+e=0, a^{3}+b^{3}+c^{3}+d^{3}+e^{3}=0, a^{5}+b^{5}+c^{5}+d^{5}+e^{5}=10$.
| ,b,,,e\in{2,\frac{\sqrt{5}-1}{2},-\frac{\sqrt{5}+1}{2}} | 102 | 30 |
math | 2. Find the smallest positive integer $a$, such that there exists a positive odd integer $n$, satisfying
$$2001 \mid\left(55^{n}+a \cdot 32^{n}\right)$$ | 436 | 51 | 3 |
math | 11.4. (7 points)
Solve the equation $(\sqrt[5]{7+4 \sqrt{3}})^{x}+(\sqrt[5]{7-4 \sqrt{3}})^{x}=194$. | -10;10 | 52 | 6 |
math | The base and height of a right circular cylinder and a right circular cone are equal. The lateral surface areas of these solids are in the ratio $8: 5$. The area of the axial section of the cone is $588 \mathrm{~cm}^{2}$. What are the heights and the radii of the bases of these solids? | =28\mathrm{~}\text{}r=21\mathrm{~} | 73 | 19 |
math | 1. In the set of natural numbers, solve the equation
$$
y^{3}=x^{3}+8 x^{2}-6 x+8
$$ | 9,11 | 35 | 4 |
math | a) Find the value of the sum
$$
\frac{1}{1+1 / x}+\frac{1}{1+x}
$$
b) Find the value of the sum
$$
\frac{1}{2019^{-2019}+1}+\ldots+\frac{1}{2019^{-1}+1}+\frac{1}{2019^{0}+1}+\frac{1}{2019^{1}+1}+\ldots+\frac{1}{2019^{2019}+1}
$$ | \frac{4039}{2} | 131 | 10 |
math | # Task 7. (14 points)
In a company, there are 168 employees. Among any four people, at least one can be chosen who is acquainted with the other three. What is the minimum possible number of people who are acquainted with everyone? # | 165 | 56 | 3 |
math | 5.3. In a seven-story building, domovoi (Russian house spirits) live. The elevator travels between the first and the last floors, stopping at every floor. On each floor, starting from the first, one domovoi entered the elevator, but no one exited. When the thousandth domovoi entered the elevator, it stopped. On which f... | 4 | 83 | 1 |
math | 12. $A B C D$ and $E F G H$ are squares of side length 1, and $A B / / E F$. The overlapped region of the two squares has area $\frac{1}{16}$. Find the minimum distance between the centres of the two squares.
(2 marks)
$A B C D$ 和 $E F G H$ 皆是邊長為 1 的正方形, 且 $A B / / E F$ 。兩個正方形的重疊部分的面積為 $\frac{1}{16}$ 。求兩個正方形的中心的距離的最小可能... | \frac{\sqrt{14}}{4} | 142 | 11 |
math | Given $1962$ -digit number. It is divisible by $9$. Let $x$ be the sum of its digits. Let the sum of the digits of $x$ be $y$. Let the sum of the digits of $y$ be $z$. Find $z$. | 9 | 61 | 1 |
math | $ABCD$ is a cyclic quadrilateral such that $AB = BC = CA$. Diagonals $AC$ and $BD$ intersect at $E$. Given that $BE = 19$ and $ED = 6$, find the possible values of $AD$. | 10 \text{ or } 15 | 57 | 10 |
math | 17 Let $n \geqslant 2, x_{1}, x_{2}, \cdots, x_{n}$ be real numbers, and $\sum_{i=1}^{n} x_{i}^{2}+\sum_{i=1}^{n-1} x_{i} x_{i+1}=1$, for each given positive integer $k, 1 \leqslant k \leqslant n$, find the maximum value of $\left|x_{k}\right|$. | \sqrt{\frac{2 k (n+1-k)}{n+1}} | 112 | 18 |
math | 13.051. Two cylinders roll down an inclined board 6 m long, one of which has a circumference of 3 dm, and the other 2 dm. Can the circumferences of both cylinders be increased by the same amount so that on the same path one of them makes 3 more revolutions than the other? | 2 | 68 | 1 |
math | 12. (15th American Mathematical Invitation Contest) Given that $a, b, c, d$ are non-zero real numbers, $f(x)=\frac{a x+b}{c x+d} (x \in \mathbf{R})$, and $f(19)=19, f(97)=97$. If for any real number $x$, when $x \neq-\frac{d}{c}$, $f[f(x)]=x$, find the only number that is not in the range of $f(x)$. | 58 | 119 | 2 |
math | 6. (5 points) 2015 minus its $\frac{1}{2}$, then minus the remaining $\frac{1}{3}$, then minus the remaining $\frac{1}{4}, \cdots$, and finally minus the remaining $\frac{1}{2015}$, the final number obtained is $\qquad$ . | 1 | 74 | 1 |
math | ## Task $2 / 81$
Let $n=\sum_{i=0}^{k} 10^{i} a_{i}>0$ with $0 \leq a_{i} \leq 9 ; a_{i} \in N$ be a $(k+1)$-digit natural number and $Q(n)=\prod_{i=0}^{k} a_{i}$ its "cross product". How large is the number $r$ of at most ( $\mathrm{k}+1$ )-digit natural numbers $n$, for which $Q(n)$ is a prime number? | 2(k+1)(k+2) | 131 | 9 |
math | If $\mathrm{P}$ and $\mathrm{Q}$ are two points in the plane, let $\mathrm{m}(\mathrm{PQ})$ be the perpendicular bisector of $\mathrm{PQ} . \mathrm{S}$ is a finite set of $n>1$ points such that: (1) if $P$ and $Q$ belong to $S$, then some point of $m(P Q)$ belongs to S, (2) if PQ, $\mathrm{P}^{\prime} \mathrm{Q}^{\prim... | n=3,5 | 269 | 5 |
math | ## Task 13/88
Determine all pairs $(p ; q)$ of prime numbers $p$ and $q$ for which the following holds:
$$
3 p^{2}+6 p=2 q^{2}+7 q
$$ | (11,13) | 55 | 7 |
math | Given the equation $x^2+ax+1=0$, determine:
a) The interval of possible values for $a$ where the solutions to the previous equation are not real.
b) The loci of the roots of the polynomial, when $a$ is in the previous interval. | a \in (-2, 2) | 60 | 10 |
math | 9.1. Solve the equation:
$$
\left(x^{2}-20\right)^{2}+\left(x^{2}-19\right)^{2}=2019
$$ | \\sqrt{\frac{39+\sqrt{4037}}{2}} | 44 | 18 |
math | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{2 ; 7 ; 5\}$
$p=\{1 ; 0 ; 1\}$
$q=\{1 ;-2 ; 0\}$
$r=\{0 ; 3 ; 1\}$ | 4p-2q+r | 74 | 6 |
math | 2.16. Find the volume of an oblique triangular prism, the base of which is an equilateral triangle with side $a$, if the lateral edge of the prism is equal to the side of the base and is inclined to the base plane at an angle of $60^{\circ}$. | \frac{3^{3}}{8} | 63 | 10 |
math | 3.449 $A=\operatorname{tg}\left(\arccos \frac{1}{\sqrt{1+a^{2}}}+\arccos \frac{a}{\sqrt{1+a^{2}}}\right) a<0$.
3.449 $A=\tan\left(\arccos \frac{1}{\sqrt{1+a^{2}}}+\arccos \frac{a}{\sqrt{1+a^{2}}}\right) a<0$. | \frac{1-^{2}}{2} | 108 | 11 |
math | Example 1: A and B participate in an election, A gets $m$ votes, B gets $n$ votes, $m>n$. Question: In the process of counting the $m+n$ votes one by one, how many possible vote counting records are there where A's vote count is always leading? | \frac{-n}{+n}C_{+n}^{} | 64 | 15 |
math | 1. Viktor was driving to the airport of a neighboring city. After half an hour of driving at a speed of 60 km/h, he realized that if he did not change his speed, he would be 15 minutes late. Then he increased his speed, as a result of which he covered the remaining part of the journey at an average speed of 80 km/h and... | 150 | 104 | 3 |
math | The hyperbola $3 x^{2}-y^{2}+24 x+36=0$ has a focus $F$ and a right directrix $l$. The ellipse $C$ has $F$ and $l$ as its corresponding focus and directrix. A line parallel to $y=x$ is drawn through $F$, intersecting the ellipse $C$ at points $A$ and $B$. It is known that the center of $C$ is inside the circle with $A ... | 0<\mathrm{e}<\sqrt{2-\sqrt{2}} | 125 | 16 |
math | 3. Once, a detective had to interrogate three witnesses of a robbery: John White, Sam Gray, and Bob Black. John insisted that all of Sam's statements were lies, while Sam kept repeating that Bob was lying. Bob, for his part, tried to convince the detective not to believe either White or, especially, Gray. The detective... | SamGray | 110 | 2 |
math | 2. In a right-angled tetrahedron $ABCD$, if the sum of the six edge lengths is 6, then the maximum value of its volume is $\qquad$ . | \frac{4}{3(1+\sqrt{2})^{3}} | 40 | 16 |
math | Let's determine the base of the number system in which 12551 can be written as 30407. | 8 | 28 | 1 |
math |
7.008. $\frac{81^{\frac{1}{\log _{5} 9}}+3^{\frac{3}{\log _{\sqrt{6}} 3}}}{409} \cdot\left((\sqrt{7})^{\frac{2}{\log _{25} 7}}-125^{\log _{25} 6}\right)$
| 1 | 95 | 1 |
math | Problem 3. The set of non-zero natural numbers is divided into subsets as follows:
$$
\{1,2\},\{3,4,5\},\{6,7,8,9\}, \ldots
$$
a) Find the smallest element in the 100th subset.
b) Is 2015 the largest element of one of these subsets?
Mathematical Gazette | 5050 | 89 | 4 |
math | $9.251 \quad \frac{x^{2}-|x|-12}{x-3} \geq 2 x$. | x\in(-\infty;3) | 31 | 10 |
math | 14. (15 points) In the four idioms “虚有其表”, “表里如一”, “一见如故”, “故弄玄虚”, each Chinese character represents one of 11 consecutive non-zero natural numbers. The same character represents the same number, and different characters represent different numbers. Additionally, “表” > “一” > “故” > “如” > “虚”, and the sum of the numbers ... | 9 | 120 | 1 |
math | Example 8 Solve the equation
$$
\frac{13 x-x^{2}}{x+1}\left(x+\frac{13-x}{x+1}\right)=42 \text {. }
$$
(9th Zu Chongzhi Cup Mathematics Invitational Competition) | x_{1}=1, x_{2}=6, x_{3}=3+\sqrt{2}, x_{4}=3-\sqrt{2} | 61 | 32 |
math | Example 8 (2005 Romanian Mathematical Olympiad) Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, satisfying the following two conditions:
(1) $x[f(x+1)-f(x)]=f(x)$, for all $x, y \in \mathbf{R}$;
(2) $|f(x)-f(y)| \leqslant|x-y|$, for all $x, y \in \mathbf{R}$. | f(x)=kx,wherek\in{R}|k|\leqslant1 | 108 | 20 |
math | 3.079. $\frac{1+\operatorname{cot} 2 \alpha \operatorname{cot} \alpha}{\operatorname{tan} \alpha+\operatorname{cot} \alpha}$. | \frac{\operatorname{ctg}\alpha}{2} | 47 | 13 |
math | Senderov V.A.
1999 numbers stand in a row. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of its two neighbors.
Find the last number.
# | 1 | 52 | 1 |
math | 1. Riješite sustav jednadžbi
$$
\begin{array}{r}
2\left(x^{2}+y^{2}\right)-3 x y+2(x+y)-39=0 \\
3\left(x^{2}+y^{2}\right)-4 x y+(x+y)-50=0
\end{array}
$$ | \begin{pmatrix}x_{1,2}=\frac{5\\sqrt{13}}{2},\quady_{1,2}=\frac{5\\sqrt{13}}{2}\\x_{3}=3,\quady_{3}=5\\x_{4}=5,\quady_{4}=3\end{pmatrix} | 81 | 77 |
math | 5. In a row, the squares of the first 2022 natural numbers are written: $1, 4, 9, \ldots, 4088484$. For each written number, except the first and the last, the arithmetic mean of its left and right neighbors was calculated and written below it (for example, under the number 4, $\left.\frac{1+9}{2}=5\right)$ was written... | 10231311025154 | 134 | 14 |
math | In an international meeting of $n \geq 3$ participants, 14 languages are spoken. We know that:
- Any 3 participants speak a common language.
- No language is spoken more that by the half of the participants.
What is the least value of $n$? | 8 | 60 | 1 |
math | Let $S$ be a subset of $\{0,1,2,\ldots,98 \}$ with exactly $m\geq 3$ (distinct) elements, such that for any $x,y\in S$ there exists $z\in S$ satisfying $x+y \equiv 2z \pmod{99}$. Determine all possible values of $m$. | m = 3, 9, 11, 33, 99 | 86 | 20 |
math | 3. Find all pairs of numbers (a, b) for which the equality $(\mathrm{a}+\mathrm{b}-1)^{2}=\mathrm{a}^{2}+\mathrm{b}^{2}-1$ holds. | (1,),(,1) | 52 | 7 |
math | For an upcoming international mathematics contest, the participating countries were asked to choose from nine combinatorics problems. Given how hard it usually is to agree, nobody was surprised that the following happened:
- Every country voted for exactly three problems.
- Any two countries voted for different sets o... | 56 | 89 | 2 |
math | 5. Aneta and Viktor are playing a game. The game starts with Aneta saying a number from 1 to 7. Then Viktor adds that number to another number from 1 to 7 and says the sum. Aneta, in turn, adds the number Viktor said to another number from 1 to 7 and says the sum, and this process alternates until one of them says the ... | 4 | 123 | 1 |
math | 10. Let the monotonic increasing sequence $\left\{a_{n}\right\}$ consist of positive integers, and $a_{7}=120, a_{n+2}=a_{n}+a_{n+1}\left(n \in \mathbf{Z}_{+}\right)$. Then $a_{8}=$ . $\qquad$ | 194 | 79 | 3 |
math | 10,11 | |
Point $M$ lies on the edge $CD$ of the parallelepiped $ABCD A1 B1 C1 D1$, with $CM: MD=1: 2$. Construct the section of the parallelepiped by a plane passing through point $M$ and parallel to the lines $DB$ and $AC1$. In what ratio does this plane divide the diagonal $A1C$ of the parallelepiped? | 1:11 | 97 | 4 |
math | How should a rook move across the chessboard to visit each square exactly once and make the fewest number of turns?
# | 14 | 26 | 2 |
math | Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of $10$ mph. If he completes the first three laps at a constant speed of only $9$ mph, what speed does he need to maintain in miles per hour on the fourth lap to achieve his goal? | 15 \text{ mph} | 65 | 7 |
math | 11.1. Angles $\alpha$ and $\beta$ are such that $\operatorname{tg} \alpha+\operatorname{tg} \beta=2$, and $\operatorname{ctg} \alpha+\operatorname{ctg} \beta=5$. Find the value of $\operatorname{tg}(\alpha+\beta)$. | \frac{10}{3} | 74 | 8 |
math | Let $a, b, c, d, e, f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of
$$
a b c+b c d+c d e+d e f+e f a+f a b
$$
and determine all 6-tuples $(a, b, c, d, e, f)$ for which this maximal value is achieved.
Answer: 8 . | 8 | 95 | 1 |
math | 1.48 How many real numbers $a$ are there such that $x^{2}+a x+6 a=0$ has only integer solutions.
(9th American Invitational Mathematics Examination, 1991) | 10 | 49 | 2 |
math | Calculate the following indefinite integrals.
[1] $\int (2x+1)\sqrt{x+2}\ dx$
[2] $\int \frac{1+\cos x}{x+\sin x}\ dx$
[3] $\int \sin ^ 5 x \cos ^ 3 x \ dx$
[4] $\int \frac{(x-3)^2}{x^4}\ dx$
[5] $\int \frac{dx}{\tan x}\ dx$ | \frac{2}{5} (x+2) (2x-1) \sqrt{x+2} + C | 105 | 26 |
math | 121. Find $\lim _{x \rightarrow 2} \frac{x^{2}+x+2}{x^{2}+2 x+8}$. | \frac{1}{2} | 37 | 7 |
math | Example 7 A positive integer $n$ cannot be divisible by 2 or 3, and there do not exist non-negative integers $a, b$ such that $\left|2^{a}-3^{b}\right|=n$. Find the minimum value of $n$.
(2003 China Training Team Test) | 35 | 68 | 2 |
math | On the plane, there is an angle of $60^{\circ}$. A circle touches one side of this angle, intersects the other side at points $A$ and $B$, and intersects the angle bisector at points $C$ and $D$. $A B = C D = \sqrt{6}$. Find the area of the circle bounded by this circle. | \pi\sqrt{3} | 78 | 7 |
math | 3. How many solutions are there to the equation
$$
m^{4}+8 n^{2}+425=n^{4}+42 m^{2},
$$
where $m$ and $n$ are integers? | 16 | 51 | 2 |
math | 10. Let the function $f(x)=4 x^{3}+b x+1(b \in \mathbf{R})$, for any $x \in[-1,1]$, it holds that $f(x) \geqslant 0$. Find the range of the real number $b$. | -3 | 67 | 2 |
math | Determine all pairs $(m, n)$ of positive integers such that $2^{m}-1=3^{n}$ | (2,1) | 25 | 5 |
math | 22. (12 points) Given the function
$$
f(x)=\ln (a x+1)+\frac{1-x}{1+x}(x \geqslant 0, a>0) \text {. }
$$
(1) If $f(x)$ has an extremum at $x=1$, find the value of $a$;
(2) If $f(x) \geqslant \ln 2$ always holds, find the range of values for $a$. | \geqslant1 | 109 | 6 |
math | Example 11. In $\triangle \mathrm{A} B \mathrm{C}$, find the extremum of $\sin \mathrm{A}+\sin \mathrm{B}+\sin \mathrm{C}$.
untranslated text remains the same as it is a mathematical expression. | \frac{3 \sqrt{3}}{2} | 59 | 12 |
math | Example 1. (47th Putnam Competition) Find and prove the maximum value of $f(x)=x^{3}-3 x$, where $x$ is a real number satisfying $x^{4}+36 \leqslant 13 x^{2}$.
保留源文本的换行和格式，翻译结果如下：
Example 1. (47th Putnam Competition) Find and prove the maximum value of $f(x)=x^{3}-3 x$, where $x$ is a real number satisfying $x^{4}+36... | 18 | 133 | 2 |
math | 12. Chris planned a $210 \mathrm{~km}$ bike ride. However, he rode $5 \mathrm{~km} / \mathrm{h}$ faster than he planned and finished his ride 1 hour earlier than he planned. His average speed for the ride was $x \mathrm{~km} / \mathrm{h}$. What is the value of $x$ ? | 35 | 84 | 2 |
math | 4. Consider an $n \times n$ matrix, where each element is a real number with absolute value not exceeding 1, and the sum of all elements is 0. Let $n$ be a positive even number. Find the minimum value of $c$ such that for every such matrix, there exists a row or a column whose sum of elements has an absolute value not ... | \frac{n}{2} | 83 | 6 |
math | Kovács had a dinner party for four couples. After the introductions, Mr. Kovács noted that apart from himself, each of the other nine people present had shaken hands with a different number of people.
How many people did Mrs. Kovács introduce herself to? | 4 | 58 | 1 |
math | 【Question 29】
How many five-digit numbers are divisible by 3 and have at least one digit as '3'? | 12504 | 27 | 5 |
math | 5. Person A and Person B simultaneously solve the same math competition problem. If they solve it correctly, the teacher will give a reward of 190 yuan, with the rule that the reward goes to whoever solves it correctly. If both solve it correctly, the reward is split equally. It is known that the probability of A solvi... | 90 | 110 | 2 |
math | 2. For any point $A(x, y)$ in the plane region $D$:
$$
\left\{\begin{array}{l}
x+y \leqslant 1, \\
2 x-y \geqslant-1, \\
x-2 y \leqslant 1
\end{array}\right.
$$
and a fixed point $B(a, b)$ satisfying $\overrightarrow{O A} \cdot \overrightarrow{O B} \leqslant 1$. Then the maximum value of $a+b$ is $\qquad$ | 2 | 124 | 1 |
math | 19.2.3 * For a positive integer $k$, there exist positive integers $n, m$, such that $\frac{1}{n^{2}}+\frac{1}{m^{2}}=\frac{k}{n^{2}+m^{2}}$, find all such $k$. | 4 | 63 | 1 |
math |
92.1. Determine all real numbers $x>1, y>1$, and $z>1$, satisfying the equation
$$
\begin{aligned}
x+y+z+\frac{3}{x-1} & +\frac{3}{y-1}+\frac{3}{z-1} \\
& =2(\sqrt{x+2}+\sqrt{y+2}+\sqrt{z+2})
\end{aligned}
$$
| \frac{3+\sqrt{13}}{2} | 99 | 13 |
math | To celebrate her birthday, Ana is going to prepare pear and apple pies. At the market, an apple weighs $300 \mathrm{~g}$ and a pear, $200 \mathrm{~g}$. Ana's bag can hold a maximum weight of $7 \mathrm{~kg}$. What is the maximum number of fruits she can buy to be able to make pies of both fruits? | 34 | 85 | 2 |
math | José tore out some consecutive leaves from a book with pages numbered with consecutive integers and written on both sides of each leaf. The sum of the numbers on the torn pages is 344.
a) Determine the prime factorization of the number 344.
b) Find the sum of the first and the last number among those written on the t... | 16 | 84 | 2 |
math | ## Problem Statement
Calculate the lengths of the arcs of the curves given by equations in a rectangular coordinate system.
$$
y=\sqrt{1-x^{2}}+\arccos x, 0 \leq x \leq \frac{8}{9}
$$ | \frac{4\sqrt{2}}{3} | 56 | 12 |
math | 1. Find all values of $x$, for each of which one of the three given numbers $\log _{x^{2}}\left(x^{2}-7 x+10\right)$, $\log _{x^{2}} \frac{x^{2}}{x-2}$, and $\log _{x^{2}} \frac{x^{2}}{x-5}$ is equal to the sum of the other two. | 6 | 92 | 1 |
math | 8. (i) (Grade 11) Given the function
$$
f(x)=2 \cos \left(\frac{k}{4} x+\frac{\pi}{3}\right)
$$
the smallest positive period is no greater than 2. Then the smallest positive integer value of $k$ is $\qquad$ .
(ii) (Grade 12) The line $y=k x-2$ intersects the parabola $y^{2}$ $=8 x$ at points $A$ and $B$. If the x-coor... | 2 \sqrt{15} | 138 | 7 |
math | 1. [2] Let $S=\{1,2,3,4,5,6,7,8,9,10\}$. How many (potentially empty) subsets $T$ of $S$ are there such that, for all $x$, if $x$ is in $T$ and $2 x$ is in $S$ then $2 x$ is also in $T$ ? | 180 | 89 | 3 |
math | 17. It is known that the number $a$ satisfies the equation param1, and the number $b$ satisfies the equation param2. Find the maximum possible value of the sum $a+b$.
| param1 | param2 | |
| :---: | :---: | :---: |
| $x^{3}-3 x^{2}+5 x-17=0$ | $x^{3}-3 x^{2}+5 x+11=0$ | |
| $x^{3}+3 x^{2}+6 x-9=0$ | $x^{3}+3 x^{2}+6... | 4 | 242 | 1 |
math | Sally's salary in 2006 was $\$37,500$. For 2007 she got a salary increase of $x$ percent. For 2008 she got another salary increase of $x$ percent. For 2009 she got a salary decrease of $2x$ percent. Her 2009 salary is $\$34,825$. Suppose instead, Sally had gotten a $2x$ percent salary decrease for 2007, an $x$ percent ... | 34,825 | 151 | 6 |
math | Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. At least how many problems are needed? | n = 13 | 41 | 6 |
math | 2. Simplify $\left(\log _{3} 4+\log _{2} 9\right)^{2}-\left(\log _{3} 4-\log _{2} 9\right)^{2}$ $=$ $\qquad$ | 16 | 58 | 2 |
math | 3. Tourists from the USA, when traveling to Europe, often use an approximate formula to convert temperatures in degrees Celsius $C$ to the familiar degrees Fahrenheit $F$: $\mathrm{F}=2 \mathrm{C}+30$. Indicate the range of temperatures (in degrees Celsius) for which the deviation of the temperature in degrees Fahrenhe... | 1\frac{11}{29}\leqC\leq32\frac{8}{11} | 121 | 26 |
math | The generatrix of a cone forms an angle $\alpha$ with the plane of its base, $\cos \alpha=\frac{1}{4}$. A sphere is inscribed in the cone, and a plane is drawn through the circle of contact between the sphere and the lateral surface of the cone. The volume of the part of the cone enclosed between this plane and the bas... | 27 | 96 | 2 |
math |
Opgave 2. Vind alle functies $f: \mathbb{R} \rightarrow \mathbb{R}$ waarvoor
$$
f\left(x^{2}\right)-f\left(y^{2}\right) \leq(f(x)+y)(x-f(y))
$$
voor alle $x, y \in \mathbb{R}$.
| f(x)=x | 81 | 4 |
math | Find a positive integrer number $n$ such that, if yor put a number $2$ on the left and a number $1$ on the right, the new number is equal to $33n$. | 87 | 45 | 2 |
math | 2. (3 points) There are five gifts priced at 2 yuan, 5 yuan, 8 yuan, 11 yuan, and 14 yuan, and five gift boxes priced at 3 yuan, 6 yuan, 9 yuan, 12 yuan, and 15 yuan. One gift is paired with one gift box, resulting in $\qquad$ different prices. | 9 | 82 | 1 |
math | Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\
ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\] | (5, 5, 5, 5) | 84 | 13 |
math | Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be convex 8-gon (no three diagonals concruent).
The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of $A_1A_2A_3A_4A_5A_6A_7A_8$ and such convex quadrilaterals will be called "sub quadrilaterals".Find the smalle... | 14 | 203 | 2 |
math | In the $xy$-coordinate plane, the $x$-axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$-axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$. They are $(126, 0)$, $(105, 0)$,... | 111 | 154 | 3 |
math | Example 40 (1992 Canadian Mathematical Olympiad Training Problem) Find all integer solutions $x$ and $y$ that satisfy the equation $x^{2}=1+4 y^{3}(y+2)$. | (x,y)=(1,0),(1,-2),(-1,0),(-1,-2) | 48 | 21 |
math | The natural number $n$ was multiplied by $3$, resulting in the number $999^{1000}$. Find the unity digit of $n$. | 7 | 35 | 1 |
math | 9. (20 points) Find all values of $x$ and $y$ for which the following equality holds:
$$
(x-8)^{2}+(y-9)^{2}+(x-y)^{2}=\frac{1}{3}
$$ | 8\frac{1}{3},8\frac{2}{3} | 57 | 16 |
math | 7. Given that $x$, $y$, $z$ are positive real numbers, and $x+y+z=1$. If $\frac{a}{x y z}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}-2$, then the range of the real number $a$ is $\qquad$. | (0,\frac{7}{27}] | 74 | 10 |
math | 2-110 Let the sequence of positive real numbers $x_{0}, x_{1}, \cdots, x_{1995}$ satisfy the following two conditions:
(1) $x_{0}=x_{1995}$;
(2) $x_{i-1}+\frac{2}{x_{i-1}}=2 x_{i}+\frac{1}{x_{i}}, i=1,2, \cdots, 1995$.
Find the maximum value of $x_{0}$ for all sequences satisfying the above conditions. | 2^{997} | 125 | 6 |
math | The lateral side of an isosceles trapezoid is equal to $a$, the midline is equal to $b$, and one angle at the larger base is $30^{\circ}$. Find the radius of the circle circumscribed around the trapezoid. | \sqrt{b^2+\frac{^2}{4}} | 60 | 14 |
math | Example 4. The cubic function $\mathrm{f}(\mathrm{x})$ passes through points $\mathrm{A}(1,4), \mathrm{B}(3,8)$, and has an extremum of 0 at $x=2$, find $f(x)$. | f(x)=2 x^{3}-6 x^{2}+8 | 59 | 15 |
math | $\begin{array}{l}\text { 1. In } \triangle A B C, A B=4, B C=7, C A=5, \\ \text { let } \angle B A C=\alpha. \text { Find } \sin ^{6} \frac{\alpha}{2}+\cos ^{6} \frac{\alpha}{2} \text {. }\end{array}$ | \frac{7}{25} | 88 | 8 |
math | 2. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
\frac{a_{n+1}+a_{n}-1}{a_{n+1}-a_{n}+1}=n\left(n \in \mathbf{Z}_{+}\right) \text {, and } a_{2}=6 \text {. }
$$
Then the general term formula of the sequence $\left\{a_{n}\right\}$ is $a_{n}=$ $\qquad$ | a_{n}=n(2n-1) | 112 | 11 |
math | Let $M$ be an interior point of tetrahedron $V ABC$. Denote by $A_1,B_1, C_1$ the points of intersection of lines $MA,MB,MC$ with the planes $VBC,V CA,V AB$, and by $A_2,B_2, C_2$ the points of intersection of lines $V A_1, VB_1, V C_1$ with the sides $BC,CA,AB$.
[b](a)[/b] Prove that the volume of the tetrahedron $V ... | \frac{1}{4} V_{VABC} | 225 | 13 |
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