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 | 1. Given $n>2$ natural numbers, among which there are no three equal, and the sum of any two of them is a prime number. What is the largest possible value of $n$? | 3 | 43 | 1 |
math | 2. In tetrahedron $ABCD$, $\triangle ABC$ is an equilateral triangle, $AD=BD=2, AD \perp BD, AD \perp CD$, then the distance from $D$ to plane $ABC$ is $\qquad$ . | \frac{2\sqrt{3}}{3} | 58 | 12 |
math | (IMO 1990/3) Find all integers $n \geq 1$ such that $n^{2}$ divides $2^{n}+1$.
| n=1n=3 | 38 | 6 |
math | 4. Try to determine all positive integer pairs $(a, b)$ such that $a b^{2}+b+7$ divides $a^{2} b+a+b$. | (,b)=(11,1),(49,1),(7k^{2},7k)(k\in{N}^{*}) | 38 | 31 |
math | Task 3. (12 points)
The arithmetic mean of ten different natural numbers is 20, and the arithmetic mean of any nine of these numbers is not less than 17. Find the maximum possible value of the largest of these numbers.
# | 47 | 53 | 2 |
math | 1. Let $i_{1}, i_{2}, \cdots, i_{10}$ be a permutation of $1,2, \cdots, 10$. Define $S=\left|i_{1}-i_{2}\right|+\left|i_{3}-i_{4}\right|+\cdots+\left|i_{9}-i_{10}\right|$. Find all possible values of $S$.
[2] | 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25 | 94 | 39 |
math | ## Problem Statement
Are the vectors $c_{1 \text { and }} c_{2}$, constructed from vectors $a \text{ and } b$, collinear?
$a=\{3 ;-1 ; 6\}$
$b=\{5 ; 7 ; 10\}$
$c_{1}=4 a-2 b$
$c_{2}=b-2 a$ | c_{1}=-2c_{2} | 81 | 10 |
math | 111 Determine all positive integers $x, y$ that satisfy $x+y^{2}+z^{3}=x y z$, where $z$ is the greatest common divisor of $x$ and $y$. | (5, 2), (5, 3), (4, 2), (4, 6) | 46 | 24 |
math | 290. Excluding the unknown $y$ from the system of equations
$$
\begin{aligned}
& a_{1} x+b_{1} y=c_{1} \\
& a_{2} x+b_{2} y=c_{2}
\end{aligned}
$$
we obtain
$$
x=\frac{c_{1} b_{2}-c_{2} b_{1}}{a_{1} b_{2}-a_{2} b_{1}}
$$
Write, without new calculations, the formula for $y$. | \frac{c_{1}a_{2}-c_{2}a_{1}}{b_{1}a_{2}-b_{2}a_{1}} | 118 | 36 |
math | Let $A$ be the set of positive integers that are the product of two consecutive integers. Let $B$ the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of $A\cap B$. | 216 | 54 | 3 |
math | Evaluate $\textstyle\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$. | \frac{\pi}{2} | 79 | 7 |
math | Let $ f(x)$ be a function such that $ 1\minus{}f(x)\equal{}f(1\minus{}x)$ for $ 0\leq x\leq 1$.
Evaluate $ \int_0^1 f(x)\ dx$. | \frac{1}{2} | 57 | 7 |
math | 13. (2004 National College Entrance Examination - Jiangsu Paper) Given: $0<\alpha<\frac{\pi}{2}, \tan \frac{\alpha}{2}+\cot \frac{\alpha}{2}=\frac{5}{2}$. Find the value of $\sin \left(\alpha-\frac{\pi}{3}\right)$. | \frac{1}{10}(4-3\sqrt{3}) | 78 | 16 |
math | 10.186. The lengths of the bases of an isosceles trapezoid are in the ratio 5:12, and the length of its height is 17 cm. Calculate the radius of the circle circumscribed around the trapezoid, given that its midline is equal to the height. | 13 | 71 | 2 |
math | 14. Let the larger root of the equation $2002^{2} x^{2}-2003 \times 2001 x-1=0$ be $r$, and the smaller root of the equation $2001 x^{2}-2002 x+1=0$ be $s$. Find the value of $r-s$. | \frac{2000}{2001} | 81 | 13 |
math | 1. (5 points) Wang Yun calculated $325-\square \times 5$ by performing the subtraction first, resulting in 1500. What should the correct result be? $\qquad$ . | 200 | 46 | 3 |
math | 4. (15 points) Two heaters are connected sequentially to the same DC power source. The water in the pot boiled after $t_{1}=120$ s from the first heater. The same water, taken at the same initial temperature, boiled after $t_{2}=180 \mathrm{s}$ from the second heater. How long would it take for the water to boil if the... | 72\mathrm{} | 99 | 5 |
math | Example 6 Given $a+b+c=5$,
$$
\begin{array}{l}
a^{2}+b^{2}+c^{2}=15, a^{3}+b^{3}+c^{3}=47 . \\
\text { Find }\left(a^{2}+a b+b^{2}\right)\left(b^{2}+b c+c^{2}\right)\left(c^{2}+c a+a^{2}\right)
\end{array}
$$
the value. | 625 | 113 | 3 |
math | $8 \cdot 4$ Let the sequence of positive integers $\left\{a_{n}\right\}$ satisfy
$$
a_{n+3}=a_{n+2}\left(a_{n+1}+2 a_{n}\right), n=1,2, \cdots
$$
and $a_{6}=2288$. Find $a_{1}, a_{2}, a_{3}$. | a_{1}=5,a_{2}=1,a_{3}=2 | 93 | 15 |
math | Call a three-digit number $\overline{ABC}$ $\textit{spicy}$ if it satisfies $\overline{ABC}=A^3+B^3+C^3$. Compute the unique $n$ for which both $n$ and $n+1$ are $\textit{spicy}$. | 370 | 64 | 3 |
math | Find all positive integers $x, y$ satisfying the equation
$$
9\left(x^{2}+y^{2}+1\right)+2(3 x y+2)=2005
$$ | (x, y)=(11,7) \text{ or } (x, y)=(7,11) | 46 | 24 |
math | Find the greatest integer $n < 1000$ for which $4n^3 - 3n$ is the product of two consecutive odd integers. | 899 | 34 | 3 |
math | $9.220 \sqrt{x^{2}+3 x+2}-\sqrt{x^{2}-x+1}<1$. | (-\infty;-2]\cup[-1;\frac{\sqrt{13}-1}{6}) | 30 | 22 |
math | 11.130. Find the ratio of the volumes of a regular tetrahedron and an octahedron, given that their total surface areas are equal. | \frac{1}{\sqrt{2}} | 36 | 10 |
math | 182. Grouped Numbers. Consecutive odd numbers are grouped as follows: $1 ;(3,5) ;(7,9,11) ;(13$, $15,17,19) ; \ldots$. Find the sum of the numbers in the $n$-th group.[^10] | n^3 | 73 | 3 |
math | Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$. If $\angle DBC = 15^{\circ}$, then find $AD^2$.
[i]Proposed by Anand Iyer[/i] | 60000 | 64 | 5 |
math | 6. Given the sequence: $\frac{2}{3}, \frac{2}{9}, \frac{4}{9}, \frac{6}{9}, \frac{8}{9}, \frac{2}{27}, \frac{4}{27}, \cdots$, $\frac{26}{27}, \cdots, \frac{2}{3^{n}}, \frac{4}{3^{n}}, \cdots, \frac{3^{n}-1}{3^{n}}, \cdots$. Then $\frac{2018}{2187}$ is the $\qquad$th term of the sequence. | 1552 | 139 | 4 |
math | The eight corners of a cube are cut off, yielding a polyhedron with $6$ octagonal faces and $8$ triangular faces. Given that all polyhedron's edges have length $2$, compute the volume of the polyhedron. | 56 + \frac{112\sqrt{2}}{3} | 51 | 19 |
math | Let's determine the continuous functions $f:(0, \infty) \rightarrow(0, \infty)$ such that for any positive numbers $x, y$,
$$
f\left(\frac{1}{f(x y)}\right)=f(x) f(y)
$$ | f(x)=1orf(x)=\frac{}{x}where>0 | 60 | 16 |
math | 4. From $1,2, \cdots, 10$ choose 3 different numbers $a, b, c$ as the coefficients of the quadratic equation $a x^{2}+b x=c$. Then the number of equations with different solutions is $\qquad$ | 654 | 59 | 3 |
math | Find all four-digit numbers $\overline{abcd}$ such that they are multiples of $3$ and that $\overline{ab}-\overline{cd}=11$.
($\overline{abcd}$ is a four-digit number; $\overline{ab}$ is a two digit-number as $\overline{cd}$ is). | 1302, 1605, 1908, 2211, 2514, 2817, 3120, 3423, 3726, 4029, 4332, 4635, 4938, 5241, 5544, 5847, 6150, 6453, 6756, 7059, 7362, 7665, 7968, 8271, 8574, 8877, 9180, 9483, 9786, 10089, 10392, 10695, 10998 | 71 | 200 |
math | ## Task 3 - 170723
In an isosceles triangle, it is only known that the sum of the sizes of two interior angles and one exterior angle is exactly $300^{\circ}$. However, it is not specified which of the mentioned interior angles are base angles and whether the mentioned exterior angle belongs to one of these interior a... | 75;75;30or30;30;120 | 97 | 18 |
math | 12. (10 points) Suppose in a certain year, a month has three Sundays that fall on odd dates. What day of the week could the 21st of this month be? | 4or6 | 41 | 3 |
math | 6. Given that the roots of the equation $x^{2}-6 x-4 n^{2}-32 n=0$ are integers. Find the integer value of $n$.
(2004, National Junior High School Mathematics League) | n=-18,-8,0,10 | 53 | 11 |
math | Senderov V.A.
Find all such natural $n$ that for some non-zero real numbers $a, b, c, d$ the polynomial $(a x+b)^{1000}-(c x+d)^{1000}$, after expanding the brackets and combining all like terms, has exactly $n$ non-zero coefficients. | 500,1000,1001 | 72 | 13 |
math | Problem 11.3. The teacher wrote the number 1818 on the board. Vasya noticed that if a multiplication sign is written between the hundreds and tens digits, the value of the resulting expression is a perfect square $\left(18 \times 18=324=18^{2}\right)$. What is the next four-digit number after 1818 that has the same pro... | 1832 | 91 | 4 |
math | 57. If $x, y, z$ are all positive real numbers, and $x^{2}+y^{2}+z^{2}=1$, find the minimum value of $S=\frac{(z+1)^{2}}{2 x y z}$. | 3+2\sqrt{2} | 59 | 8 |
math | 5. (5 points) A rectangular prism with a length of 6 cm, a width of 5 cm, and a height of 4 cm is painted red on all its surfaces, and then the prism is cut into small cubes with an edge length of 1 cm. The number of small cubes that have no faces painted is $\qquad$.
| 24 | 73 | 2 |
math | Solve in the set $R$ the equation
$$2 \cdot [x] \cdot \{x\} = x^2 - \frac32 \cdot x - \frac{11}{16}$$
where $[x]$ and $\{x\}$ represent the integer part and the fractional part of the real number $x$, respectively. | \frac{9}{4} | 78 | 7 |
math | Given $x \geqslant 0, y \geqslant 0$, and $x^{2}+y^{2}=1$, then the maximum value of $x(x+y)$ is
保留源文本的换行和格式，直接输出翻译结果。
Given $x \geqslant 0, y \geqslant 0$, and $x^{2}+y^{2}=1$, then the maximum value of $x(x+y)$ is | \frac{\sqrt{2}+1}{2} | 103 | 12 |
math | Find the height of a triangular pyramid, the lateral edges of which are pairwise perpendicular and equal to 2, 3, and 4.
# | \frac{12}{\sqrt{61}} | 30 | 12 |
math | 23. Find a three-digit number, the square of which is a six-digit number such that each subsequent digit, counting from left to right, is greater than the previous one. | 367 | 37 | 3 |
math | 1. Given that $a, b, x, y$ are positive real numbers, satisfying:
$$
a+b+\frac{1}{a}+\frac{9}{b}=8, a x^{2}+b y^{2}=18 \text {. }
$$
Then the range of values for $a x+b y$ is . $\qquad$ | (3 \sqrt{2}, 6 \sqrt{2}] | 77 | 14 |
math | 3. (3 points) Definition: $a \oplus b=a+b+a b$, then $(2 \oplus 3) \oplus 4$ is
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 59 | 61 | 2 |
math | 5. The sum of all real roots of the equation $x+\frac{x}{\sqrt{x^{2}-1}}=\frac{35}{12}$ is . $\qquad$ | \frac{35}{12} | 40 | 9 |
math | 2.150. $\frac{2-\sqrt{2}-\sqrt{3}}{2+\sqrt{2}-\sqrt{3}}$. | \frac{(2\sqrt{6}+1)(3-4\sqrt{2})}{23} | 33 | 24 |
math | Let's determine the smallest positive integer \( a \) such that \( 47^n + a \cdot 15^n \) is divisible by 1984 for all odd \( n \). | 1055 | 43 | 4 |
math | 5. Randomly draw three different numbers from $1,2, \cdots, 10$, the probability of the sample variance $s^{2} \leqslant 1$ is $\qquad$ | \frac{1}{15} | 46 | 8 |
math | 4. Find the smallest integer $n \geqslant 1$ such that the equation
$$
a^{2}+b^{2}+c^{2}-n d^{2}=0
$$
admits the only integer solution $a=b=c=d=0$.
## Solutions | 7 | 63 | 1 |
math | 8. (1981 Kyiv Mathematical Olympiad) If the numbers $p, p+10, p+14$ are all prime, find $p$.
保留了源文本的换行和格式。 | 3 | 48 | 1 |
math | 14. The Markov chain is named after the Russian mathematician Andrey Markov, and its process has the property of being "memoryless," meaning the probability distribution of the state at time $n+1$ depends only on the state at time $n$, and is "unrelated" to the states at times $n-1, n-2, n-3, \cdots$. There are two box... | 1 | 257 | 1 |
math | 3. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=1, a_{2}=2, a_{2 k+1}=\frac{a_{2 k}^{2}}{a_{2 k-1}}$, and $a_{2 k+2}=2 a_{2 k+1}-a_{2 k}\left(k \in \mathbf{N}^{*}\right)$, then the last two digits of $a_{2022}$ are | 32 | 111 | 2 |
math | (BXMO 2014) Let $a, b, c$ and $d$ be strictly positive integers. Determine the smallest value that the expression:
$$
S=\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{a+b+d}{c}\right\rfloor+\left\lfloor\frac{a+c+d}{b}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor
$$
can take. | 9 | 115 | 1 |
math | (7) Let the function $f(x)=\left\{\begin{array}{l}\frac{1}{p}\left(x=\frac{q}{p}\right), \\ 0\left(x \neq \frac{q}{p}\right),\end{array}\right.$ where $p, q$ are coprime (prime), and $p \geqslant 2$. Then the number of $x$ values that satisfy $x \in[0,1]$, and $f(x)>\frac{1}{5}$ is $\qquad$ | 5 | 123 | 1 |
math | PROBLEM 4. a) Consider the collinear points $A_{0}, A_{1}, A_{2}, \ldots, A_{2015}$ (in this order), such that $A_{0} A_{1}=1 \text{ cm}, A_{1} A_{2}=2 \text{ cm}, \ldots, A_{2014} A_{2015}=2015 \text{ cm}$. Calculate the distance between the points $A_{1000}$ and $A_{2015}$.
b) Let $A, B, C, D, E$ and $F$ be six poin... | 1530620 | 239 | 7 |
math | In a certain sequence of 80 numbers, any term, except for the two endpoints, is equal to the product of its neighboring terms. The product of the first 40 terms of the sequence is 8 and the product of all the terms is also 8. Determine the terms of the sequence. | 2,4,2,\frac{1}{2},\frac{1}{4},\frac{1}{2},2,4,2,\frac{1}{2},\frac{1}{4},\frac{1}{2},\ldots | 63 | 55 |
math | If $a=2^{3}$ and $b=3^{2}$ evaluate $\frac{(a-b)^{2015}+1^{2015}}{(a-b)^{2015}-1^{2015}}$. | 0 | 55 | 1 |
math | 1. Given 6000 cards, on which natural numbers from 1 to 6000 are written (each card has exactly one number, and the numbers do not repeat). It is required to choose two cards such that the sum of the numbers written on them is divisible by 100. In how many ways can this be done? | 179940 | 74 | 6 |
math | Let $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by $1000$. | 100 | 86 | 3 |
math | Compute the smallest positive integer that is $3$ more than a multiple of $5$, and twice a multiple of $6$. | 48 | 26 | 2 |
math | 3. Solve the inequality $2^{\frac{1-x}{3+x}} \cdot 7^{3+x} \geq 56$. | x\in(-3;-2]\cup[4\log_{7}2-3;+\infty) | 32 | 24 |
math | Problem 3. One notebook, 3 notepads, and 2 pens cost 98 rubles, while 3 notebooks and a notepad are 36 rubles cheaper than 5 pens. How much does each item cost, if the notebook costs an even number of rubles? (Each of these items costs a whole number of rubles.) | 4,22,14 | 75 | 7 |
math | 10. $\tan 20^{\circ} \tan 30^{\circ}+\tan 30^{\circ} \tan 40^{\circ}+\tan 40^{\circ} \tan 20^{\circ}=$ | 1 | 58 | 1 |
math | Sally rolls an $8$-sided die with faces numbered $1$ through $8$. Compute the probability that she gets a power of $2$. | \frac{1}{2} | 33 | 7 |
math | Fix positive integers $k$ and $n$.Derive a simple expression involving Fibonacci numbers for the number of sequences $(T_1,T_2,\ldots,T_k)$ of subsets $T_i$ of $[n]$ such that $T_1\subseteq T_2\supseteq T_3\subseteq T_4\supseteq\ldots$.
[color=#008000]Moderator says: and the original source for this one is Richard Sta... | F_{k+2}^n | 134 | 8 |
math | ## Task Condition
Write the canonical equations of the line.
$6 x-7 y-4 z-2=0$
$x+7 y-z-5=0$ | \frac{x-1}{35}=\frac{y-\frac{4}{7}}{2}=\frac{z}{49} | 36 | 31 |
math | Let $ S \equal{} \{1,2,\ldots,2008\}$. For any nonempty subset $ A\in S$, define $ m(A)$ to be the median of $ A$ (when $ A$ has an even number of elements, $ m(A)$ is the average of the middle two elements). Determine the average of $ m(A)$, when $ A$ is taken over all nonempty subsets of $ S$. | \frac{2009}{2} | 95 | 10 |
math |
6. Let $S=\{1,2, \ldots, n\}$ and let $T$ be the set of all ordered triples of subsets of $S$, say $\left(A_{1}, A_{2}, A_{3}\right)$, such that $A_{1} \cup A_{2} \cup A_{3}=S$. Determine, in terms of $n$,
$$
\sum_{\left(A_{1}, A_{2}, A_{3}\right) \in T}\left|A_{1} \cap A_{2} \cap A_{3}\right|
$$
where $|X|$ denotes... | n\cdot7^{n-1} | 229 | 9 |
math | 1. The maximum value of $y=2 x+\sqrt{6-x}$ is | 12 \frac{1}{8} | 18 | 9 |
math | 1. Each of the six houses on one side of the street is connected by cable lines to each of the eight houses on the opposite side. How many pairwise intersections do the shadows of these cables form on the surface of the street, if no three of them intersect at the same point? Assume that the light causing these shadows... | 420 | 69 | 3 |
math | 4. In the stands of the hockey arena, there are several rows with 168 seats in each row. For the final match, 2016 students from several sports schools were invited as spectators, with no more than 45 from each school. Students from any school need to be seated in one row. What is the minimum number of rows that must b... | 16 | 88 | 2 |
math | 13. Given a right triangle, one of whose acute angles is equal to $\alpha$. Find the ratio of the radii of the circumscribed and inscribed circles and determine for which $\alpha$ this ratio will be the smallest? | \sqrt{2}+1 | 49 | 7 |
math | Find all functions $f : \mathbb N \mapsto \mathbb N$ such that the following identity
$$f^{x+1}(y)+f^{y+1}(x)=2f(x+y)$$
holds for all $x,y \in \mathbb N$ | f(f(n)) = f(n+1) | 61 | 11 |
math | 19. (5 points) There are 4 pieces of part $A$ each weighing 5 tons, 6 pieces of part $B$ each weighing 4 tons, 11 pieces of part $C$ each weighing 3 tons, and 7 pieces of part $D$ each weighing 1 ton. If all the parts need to be transported in one go, at least how many trucks with a capacity of 6 tons are required? $\q... | 16 | 98 | 2 |
math | 1. Let positive real numbers $a, b, c$ satisfy $a b c=10^{11}$, and $\lg a \cdot \lg (b c)+\lg b \cdot \lg (c a)+\lg c \cdot \lg (a b)=40$, then $\sqrt{\lg ^{2} a+\lg ^{2} b+\lg ^{2} c}=$ $\qquad$ . | 9 | 94 | 1 |
math | $7 \cdot 58$ For the set $S=\left\{\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right) \mid a_{i}=0\right.$ or $1, i=1,2,3$, $4,5\}$, for any two elements $\left(\bar{a}_{1}, \bar{a}_{2}, \bar{a}_{3}, \bar{a}_{4}, \bar{a}_{5}\right)$ and $\left(\bar{b}_{1}, \bar{b}_{2}, \bar{b}_{3}, \bar{b}_{4}, \bar{b}_{5}\right)$, define ... | 4 | 292 | 1 |
math | Let $a,b, c$ denote the real numbers such that $1 \le a, b, c\le 2$.
Consider $T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}$.
Determine the largest possible value of $T$. | 2 | 80 | 1 |
math | 2. Solve the inequality $x^{\log _{13} x}+7(\sqrt[3]{x})^{\log _{13} x} \leqslant 7+(\sqrt[3]{13})^{\log _{\sqrt{13}}^{2} x}$ | x\in(0;13^{-\sqrt{\log_{13}7}}]\cup{1}\cup[13\sqrt{\log_{13}7};+\infty) | 66 | 42 |
math | 3. (48th Slovenian Mathematical Olympiad) We write a positive integer on each face of a cube, and then, at each vertex, we write a number that is the product of the integers on the three faces meeting at that vertex. It is known that the sum of the numbers at the vertices of the cube is 70. Find the sum of the numbers ... | 14 | 84 | 2 |
math | 4. The board has the number 5555 written in an even base $r$ ($r \geqslant 18$). Petya found out that the $r$-ary representation of $x^{2}$ is an eight-digit palindrome, where the difference between the fourth and third digits is 2. (A palindrome is a number that reads the same from left to right and from right to left... | 24 | 99 | 2 |
math | 2. In an airplane with 62 rows, each having 6 seats, passengers have seated themselves in such a way that in no 2 rows are seats occupied in the same positions. What is the maximum number of passengers that can be in the airplane? | 191 | 53 | 3 |
math | 7. At the ends of a vertically positioned homogeneous spring, two small loads are fixed. Above is a load with mass $m_{1}$, and below is $-m_{2}$. A person grabbed the middle of the spring and held it vertically in the air. In this case, the upper half of the spring was deformed by $x_{1}=8 \mathrm{~cm}$, and the lower... | 30\, | 161 | 4 |
math | 8. In the sequence $\left\{a_{n}\right\}$, $a_{1}=1, a_{n+1}>a_{n}$, and $a_{n+1}^{2}+a_{n}^{2}+1=2\left(a_{n+1} a_{n}+a_{n+1}+\right.$ $\left.a_{n}\right), \lim _{n \rightarrow+\infty} \frac{S_{n}}{n a_{n}}=$ $\qquad$ . | \frac{1}{3} | 118 | 7 |
math | 3. The quadratic function $f(x)$ reaches its minimum value of -2 at $x=1$. Determine $f(x)$ such that $f(-3)+4 f(0)=0$. Calculate $f(1+\sqrt{5})$. | \frac{1}{2} | 52 | 7 |
math | Source: 2018 Canadian Open Math Challenge Part A Problem 2
-----
Let $v$, $w$, $x$, $y$, and $z$ be five distinct integers such that $45 = v\times w\times x\times y\times z.$ What is the sum of the integers? | 5 | 67 | 1 |
math | Netrusova N.M.
In the equation TIKH + TIGR = SPIT, replace identical letters with identical digits, and different letters with different digits so that TIGR is as small as possible (there are no zeros among the digits).
# | 1386+1345=2731 | 53 | 14 |
math | 31.11. a) Find the remainder of the division of $171^{2147}$ by 52.
b) Find the remainder of the division of $126^{1020}$ by 138. | 54 | 55 | 2 |
math | 9. (16 points) Given $x, y \in \mathbf{R}$, and satisfying $(x+1)(y-1)=1$.
Find the minimum value of $\left(x^{2}+1\right)\left(y^{2}+1\right)$. | \frac{9}{2} | 62 | 7 |
math | 10. (20 points) Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1$, $a_{2}=9$, and for any positive integer $n$,
$$
n a_{n+2}-6(n+1) a_{n+1}+9(n+2) a_{n}=0 \text {. }
$$
Find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=3^{n-2}\left(n^{3}-n+3\right) | 103 | 21 |
math | Siva has the following expression, which is missing operations:
$$\frac12 \,\, \_ \,\,\frac14 \,\, \_ \,\, \frac18 \,\, \_ \,\,\frac{1}{16} \,\, \_ \,\,\frac{1}{32}.$$ For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, ... | 39 | 186 | 2 |
math | 5. The train is $400 \mathrm{~m}$ long, and it takes 10 minutes to pass through the tunnel (from the front of the train entering the tunnel to the rear of the train leaving the tunnel). If the speed increases by 0.1 kilometers per minute, then it will take 9 minutes, and the length of the tunnel is $\qquad$ $ـ$. | 8600 | 84 | 4 |
math | 12. Let $a$ be the largest positive root of the equation $x^{3}-3 x^{2}+1=0$, then among the 2006 numbers $\left[a^{n}\right](1 \leqslant n \leqslant 2006)$, there are $\qquad$ numbers that can be divided by 17. | 251 | 82 | 3 |
math | Example 1 Given an acute triangle $\triangle A B C$ with three interior angles satisfying $A>B>C$, let $\alpha$ represent the minimum of $A-B$, $B-C$, and $90^{\circ}-A$. Then the maximum value of $\alpha$ is $\qquad$ $(2005$, National Junior High School Mathematics League) | 15^{\circ} | 75 | 6 |
math | 2. Let $D$ be a point inside acute $\triangle A B C$, $\angle A D B=\angle A C B+90^{\circ}$, and $A C \cdot B D=A D \cdot B C$. Find the value of $\frac{A B \cdot C D}{A C \cdot B D}$. | \sqrt{2} | 72 | 5 |
math | Find the largest $k$ such that $1991^{k}$ divides $A=1990^{1991^{1992}}+1991^{1990^{1992}}+1992^{1991^{1990}}$. | 1991 | 68 | 4 |
math | [b]Q14.[/b] Let be given a trinagle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$. Suppose that $IH$ is perpendicular to $BC$ ($H$ belongs to $BC$). If $HB=5 \text{cm}, \; HC=8 \text{cm}$, compute the area of $\triangle ABC$. | 40 | 97 | 2 |
math | Question 33: Let the function $f(x)=a x^{2}+b x+c$ ($a$, $b$, $c \in \mathbb{R}$), satisfying $\max _{x \in[0,1]}|f(x)| \leq 1$. Let $g(x)=$ $cx^{2}+bx+a$. Try to find the maximum possible value of $\max _{x \in[0,1]}|g(x)|$.
保留源文本的换行和格式如下：
Question 33: Let the function $f(x)=a x^{2}+b x+c$ ($a$, $b$, $c \in \mathbb{... | 8 | 215 | 1 |
math | 4. Let $ABC$ be a triangle, $M \in (AB)$ such that $\overrightarrow{AM} = \frac{1}{2} \overrightarrow{AB}$ and $N \in (AC)$ such that $\overrightarrow{AN} = \frac{3}{4} \overrightarrow{AC}$. Denote by $P$ the midpoint of $(BC)$ and by $R$ the intersection point of the lines $AP$ and $MN$. If $\overrightarrow{AR} = k \c... | \frac{3}{5} | 129 | 7 |
math | Bob and Bill's history class has $32$ people in it, but only $30$ people are allowed per class. Two people will be randomly selected for transfer to a random one of two history classes. What is the probability that Bob and Bill are both transferred, and that they are placed in the same class? Write your answer as a fra... | \frac{1}{992} | 76 | 9 |
math | Problem 2.3. A square $C$ is completely covered with a whole number of unit squares, without overlaps. If one places as many squares as possible of area 2 inside $C$, with sides parallel to the sides of $C$, without overlaps, it is possible to cover eight ninths of the area of the square. Determine all possible dimensi... | \begin{pmatrix}n^{\}=^{\}=1&n=2&=3\\n^{\}=^{\}=2&n=4&=6\\n^{\}=^{\}=3&n=6&=9\\n^{\}=^{\}=4&n=8&=12\\n^{\}=^{\} | 77 | 78 |
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