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 | Subject (3). Determine the natural numbers $m, n$ such that $85^{m}-n^{4}=4$. | =1,n=3 | 27 | 5 |
math | [ Coordinate method in space ] [ Parametric equations of a line]
Given points $A(1 ; 0 ; 1), B(-2 ; 2 ; 1), C(2 ; 0 ; 3)$ and $D(0 ; 4 ;-2)$. Formulate the parametric equations of the line passing through the origin and intersecting the lines $A B$ and $C D$. | 8,2,11 | 86 | 6 |
math | Example 3 If $p$ and $p+2$ are both prime numbers, then these two prime numbers are called "twin primes". Consider the following two sequences.
Fibonacci sequence: $1,1,2,3,5,8, \cdots$ (the sequence satisfying $F_{1}=1, F_{2}=1$, $F_{n+2}=F_{n+1}+F_{n}, n=1,2, \cdots$).
Twin prime sequence: $3,5,7,11,13,17,19, \cdot... | 3,5,13 | 159 | 6 |
math | Example 9 Find the least common multiple of 24871 and 3468.
The text above is translated into English, preserving the original text's line breaks and format. | 5073684 | 40 | 7 |
math | A clockmaker wants to design a clock such that the area swept by each hand (second, minute, and hour) in one minute is the same (all hands move continuously). What is the length of the hour hand divided by the length of the second hand? | 12\sqrt{5} | 53 | 7 |
math | Problem 8.3. Four children were walking along an alley and decided to count the number of firs planted along it.
- Anya said: "There are a total of 15 firs along the alley."
- Borya said: "The number of firs is divisible by 11."
- Vера said: "There are definitely fewer than 25 firs."
- Gena said: "I am sure that their... | 11 | 125 | 2 |
math | 14. Let $x$ be the smaller of the two solutions of the equation $x^{2}-4 x+2=0$. What are the first three digits after the decimal point in the base 10 representation of the number
$$
x+x^{2}+x^{3}+\cdots+x^{2009} ?
$$ | 414 | 74 | 3 |
math | 17 Find all positive integer triples $(x, y, z)$ that satisfy: $\left\{\begin{array}{l}x+y=z, \\ x^{2} y=z^{2}+1 .\end{array}\right.$ | (5,2,7)(5,13,18) | 51 | 15 |
math | The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\tfrac{m+\sqrt{n}}{k}$ where $m$, $n$, and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$. | 6 | 75 | 1 |
math | $86 \times 87 \times 88 \times 89 \times 90 \times 91 \times 92 \div 7$ The remainder is $\qquad$ | 0 | 45 | 1 |
math | Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When ... | 114 | 111 | 3 |
math | 5.086 In how many ways can the players of two football teams be arranged in a single row so that no two players from the same team stand next to each other? | 2\cdot(11!)^2 | 37 | 9 |
math | II. (25 points) Several boxes are unloaded from a cargo ship, with a total weight of 10 tons, and the weight of each box does not exceed 1 ton. To ensure that these boxes can be transported away in one go, the question is: what is the minimum number of trucks with a carrying capacity of 3 tons needed? | 5 | 73 | 1 |
math | The sequence ${a_{n}}$ $n\in \mathbb{N}$ is given in a recursive way with $a_{1}=1$,
$a_{n}=\prod_{i=1}^{n-1} a_{i}+1$, for all $n\geq 2$.
Determine the least number $M$, such that $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$ for all $m\in \mathbb{N}$
| 2 | 112 | 1 |
math | 120. Три водителя. Три водителя грузовиков зашли в придорожное кафе. Один водитель купил четыре сандвича, чашку кофе и десять пончиков на общую сумму 1 доллар 69 центов. Второй водитель купил три сандвича, чашку кофе и семь пончиков за 1 доллар 26 центов. Сколько заплатил третий водитель за сандвич, чашку кофе и пончик... | 40 | 127 | 2 |
math | 170. In a chess tournament, students from the 9th and 10th grades participated. Each participant played against every other participant once. There were 10 times more 10th graders than 9th graders, and they scored 4.5 times more points in total than all the 9th graders. How many 9th grade students participated in the t... | n=1,=10 | 92 | 7 |
math | 9. (40 points) The numbers $s_{1}, s_{2}, \ldots, s_{1008}$ are such that their sum is $2016^{2}$. It is known that
$$
\frac{s_{1}}{s_{1}+1}=\frac{s_{2}}{s_{2}+3}=\frac{s_{3}}{s_{3}+5}=\ldots=\frac{s_{1008}}{s_{1008}+2015}
$$
Find $s_{17}$. | 132 | 129 | 3 |
math | Example 2 In $\triangle A B C$, $\angle A B C=40^{\circ}, \angle A C B=20^{\circ}, N$ is a point inside $\triangle A B C$, $\angle N B C=30^{\circ}$, $\angle N A B=20^{\circ}$. Find the degree measure of $\angle N C B$. | 10 | 83 | 2 |
math | Example 5. Solve the inequality $2 \sqrt{x} \geq 3-\frac{1}{x}$. | x>0 | 26 | 3 |
math | Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$. | (30, 21) \text{ and } (-21, -30) | 37 | 21 |
math | 1. Calculate $\frac{(1+i)^{1989}}{(1-i)^{1989}-(1+i)^{1989}} \quad\left(i^{2}=-1\right)$. | \frac{i-1}{2} | 49 | 8 |
math | Ada and Luisa train every day, each always at the same speed, for the big race that will take place at the end of the year at school. The training starts at point $A$ and ends at point $B$, which are $3000 \mathrm{~m}$ apart. They start at the same time, but when Luisa finishes the race, Ada still has $120 \mathrm{~m}$... | 125 | 148 | 3 |
math | 11. In an isosceles right $\triangle ABC$, it is known that $\angle ABC=90^{\circ}$, and the coordinates of points $A, B$ are $A(1,0), B(3,1)$, then the coordinates of vertex $C$ are $\qquad$ | (2,3)or(4,-1) | 67 | 11 |
math | For a given positive integer $m$, the series
$$\sum_{k=1,k\neq m}^{\infty}\frac{1}{(k+m)(k-m)}$$
evaluates to $\frac{a}{bm^2}$, where $a$ and $b$ are positive integers. Compute $a+b$. | 7 | 72 | 1 |
math | Let $a<b<c$ be the solutions of the equation $2016 x^{3}-4 x+\frac{3}{\sqrt{2016}}=0$. Determine the value of $-1 /\left(a b^{2} c\right)$. | 1354752 | 58 | 7 |
math | 4. In Rt $\triangle A B C$, $C D$ is the altitude on the hypotenuse $A B$, the three sides $a, b, c$ of $\triangle A B C$ are all positive integers, $B D$ $=27$. Then $\cos B=$ $\qquad$ | \frac{3}{5} | 66 | 7 |
math | Let's determine those prime numbers $p$ for which $p^{2}+8$ is also prime. | 3 | 23 | 1 |
math | 50th Putnam 1989 Problem B5 A quadrilateral is inscribed in a circle radius 1. Two opposite sides are parallel. The difference between their lengths is d > 0. The distance from the intersection of the diagonals to the center of the circle is h. Find sup d/h and describe the cases in which it is attained. | 2 | 75 | 1 |
math | I1.2 If $\left\{\begin{array}{c}x+y=2 \\ x y-z^{2}=a \\ b=x+y+z\end{array}\right.$, find the value of $b$. | 2 | 47 | 1 |
math | For numerous technical applications (e.g., electronic computing), numbers need to be expressed in the binary system - that is, as the sum of different powers of 2. Express the number 413 in binary. | 413=(110011101)_{2} | 44 | 17 |
math | 7. In the polar coordinate system, the distance from the point $\left(2,-\frac{\pi}{6}\right)$ to the line $\rho \sin \left(\theta-30^{\circ}\right)=1$ is | \sqrt{3}+1 | 50 | 7 |
math | Find the sum of all integers $x$ satisfying $1 + 8x \le 358 - 2x \le 6x + 94$. | 102 | 36 | 3 |
math | 4. 122 Solve the system of equations $\left\{\begin{array}{l}\sqrt{x-1}+\sqrt{y-3}=\sqrt{x+y}, \\ \lg (x-10)+\lg (y-6)=1 .\end{array}\right.$ | no solution | 63 | 2 |
math | Example 6 Solve the equation
\[
\begin{array}{l}
a^{4} \cdot \frac{(x-b)(x-c)}{(a-b)(a-c)}+b^{4} \cdot \frac{(x-c)(x-a)}{(b-c)} \\
+\epsilon^{4} \cdot \frac{(x-a)(x-b)}{(c-a)(c-b)}=x^{4} .
\end{array}
\] | x_{1}=a, x_{2}=b, x_{3}=c, x_{4}=-(a+b+c) | 95 | 27 |
math | How many positive integers $n$ exist such that $\frac{2 n^{2}+4 n+18}{3 n+3}$ is an integer? | 4 | 34 | 1 |
math | Task B-4.2. If $z+z^{-1}=2 \cos \frac{\alpha}{2012}$, determine $\alpha$ for which $z^{2012}+z^{-2012}=1$ | \alpha=\\frac{\pi}{3}+2k\pi,\quadk\in\mathbb{Z} | 52 | 26 |
math | Let $a$ and $b$ be two distinct roots of the polynomial $X^{3}+3 X^{2}+X+1$. Calculate $a^{2} b+a b^{2}+3 a b$. | 1 | 48 | 1 |
math | 16. Let positive real numbers $x, y$ satisfy $x y=1$, find the range of the function
$$
f(x, y)=\frac{x+y}{[x][y]+[x]+[y]+1}
$$
(where $[x]$ denotes the greatest integer less than or equal to $x$). | {\frac{1}{2}}\cup[\frac{5}{6},\frac{5}{4}) | 71 | 23 |
math | 8. Let $A=\left\{a_{1}, a_{2}, \cdots, a_{7}\right\}$. Here $a_{i} \in \mathbf{Z}^{\prime}$, and let $n_{A}$ denote the number of triples $(x, y, z)$ such that: $x<y$, $x+y=z, x, y, z \in A$. Then the maximum possible value of $n_{A}$ is $\qquad$. | 9 | 104 | 1 |
math | 3. (CUB 3) ${ }^{\mathrm{IMO} 1}$ Let $n>m \geq 1$ be natural numbers such that the groups of the last three digits in the decimal representation of $1978^{m}, 1978^{n}$ coincide. Find the ordered pair $(m, n)$ of such $m, n$ for which $m+n$ is minimal. | (3, 103) | 89 | 8 |
math | 6. Calculate
$$
\frac{2 a b\left(a^{3}-b^{3}\right)}{a^{2}+a b+b^{2}}-\frac{(a-b)\left(a^{4}-b^{4}\right)}{a^{2}-b^{2}} \quad \text { for } \quad a=-1, \underbrace{5 \ldots 5}_{2010} 6, \quad b=5, \underbrace{4 \ldots 44}_{2011}
$$
Answer: 343. | 343 | 126 | 3 |
math | Several irrational numbers are written on a blackboard. It is known that for every two numbers $ a$ and $ b$ on the blackboard, at least one of the numbers $ a\over b\plus{}1$ and $ b\over a\plus{}1$ is rational. What maximum number of irrational numbers can be on the blackboard?
[i]Author: Alexander Golovanov[/i] | 3 | 85 | 1 |
math | Problem 5. (Option 2). Find all values of the parameters $a, b$, and $c$, for which the set of real roots of the equation $x^{5}+4 x^{4}+a x=b x^{2}+4 c$ consists exactly of the two numbers 2 and -2. | =-16,b=48,=-32 | 69 | 11 |
math | 4. Arrange $\frac{131}{250}, \frac{21}{40}, 0.5 \dot{2} \dot{3}, 0.52 \dot{3}, 0.5 \dot{2}$ in ascending order, the third number is $\qquad$ | 0.52\dot{3} | 68 | 9 |
math | 7. Given that the equations of the asymptotes of a hyperbola are $y= \pm \frac{2}{3} x$, and it passes through the point $(3,4)$, the equation of this hyperbola is . $\qquad$ | \frac{y^{2}}{12}-\frac{x^{2}}{27}=1 | 55 | 22 |
math | Example 11. When $x \geqslant 1, x \neq 2$, simplify
$$
y=(x+2 \sqrt{x-1})^{-\frac{1}{2}}+(x-2 \sqrt{x-1})^{-\frac{1}{2}}
$$ | \frac{2}{2-x} \text{ for } 1 \leqslant x < 2 \text{ and } \frac{2 \sqrt{x-1}}{x-2} \text{ for } x > 2 | 65 | 53 |
math | 4. Sharik and Matroskin are skiing on a circular track, half of which is an uphill climb and the other half is a downhill descent. Their speeds on the climb are the same and four times slower than their speeds on the descent. The minimum distance by which Sharik lags behind Matroskin is 4 km, and the maximum distance i... | 24 | 85 | 2 |
math | Marek is playing with a calculator. He wrote down one number on a piece of paper. He entered it into the calculator and then pressed the buttons in sequence: plus, four, divide, four, minus, four, times, four. He wrote down the result on the paper. Then he repeated the same process with this number, again: plus, four, ... | 38;26;14;2 | 159 | 10 |
math | Let $ f(t)$ be a continuous function on the interval $ 0 \leq t \leq 1$, and define the two sets of points \[ A_t\equal{}\{(t,0): t\in[0,1]\} , B_t\equal{}\{(f(t),1): t\in [0,1]\}.\] Show that the union of all segments $ \overline{A_tB_t}$ is Lebesgue-measurable, and find the minimum of its measure with respect to all ... | \sqrt{2} - 1 | 122 | 9 |
math | 4th Putnam 1941 Problem B2 Find: (1) lim n→∞ ∑ 1≤i≤n 1/√(n 2 + i 2 ); (2) lim n→∞ ∑ 1≤i≤n 1/√(n 2 + i); (3) lim n→∞ ∑ 1≤i≤n 2 1/√(n 2 + i); | (1)\ln(1+\sqrt{2})\approx0.8814,(2)1,(3)\infty | 96 | 28 |
math | $[\underline{\text { equations in integers }}]$
Solve the equation $\underbrace{\sqrt{n+\sqrt{n+\ldots \sqrt{n}}}}_{1964 \text { times }}=m$ in integers
# | 0 | 50 | 1 |
math | Solve the following equation:
$$
8^{x}+27^{x}+64^{x}+125^{x}=24^{x}+30^{x}+40^{x}+60^{x} .
$$ | 0 | 56 | 1 |
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 | 1. (4p) a) Determine the non-zero digits $a, b, c$ that simultaneously satisfy the conditions:
(i) $\overline{a b}+\overline{b a}=\overline{c c}$;
(ii) $\overline{a b}-\overline{b a}=c$;
(iii) $(\overline{c c}-c-c): c=c$.
(3p) b) Determine the number $\overline{a b}$, which satisfies the equation: $\overline{a b c d}-... | =5,b=4,\overline{}=21 | 150 | 12 |
math | Example 2 Use $1$, $2$, and $3$ to write $n$-digit numbers, with the requirement that no two $1$s are adjacent. How many $n$-digit numbers can be formed? | a_{n}=\frac{\sqrt{3}+2}{2 \sqrt{3}}(1+\sqrt{3})^{n}+\frac{\sqrt{3}-2}{2 \sqrt{3}}(1-\sqrt{3})^{n} | 47 | 55 |
math | 3. Find all functions $f: \mathbb{Q} \rightarrow \mathbb{R}$ that satisfy the following conditions:
(1) $f(x+y) - y f(x) - x f(y) = f(x) f(y) - x - y + x y$ for all $x, y \in \mathbb{Q}$;
(2) $f(x) = 2 f(x+1) + 2 + x$ for all $x \in \mathbb{Q}$;
(3) $f(1) + 1 > 0$.
(Kunap) | f(x)=2^{-x}-x | 130 | 8 |
math | 2. Let $\alpha, \beta$ be acute angles. When
$$
-\frac{1}{\cos ^{2} \alpha}+\frac{1}{\sin ^{2} \alpha \sin ^{2} \beta \cos ^{2} \beta}
$$
takes the minimum value, the value of $\operatorname{tg}^{2} \alpha+\operatorname{tg}^{2} \beta$ is | 3 | 95 | 1 |
math | 13.108. On the plots allocated by the agrolaboratory for experiments, $c$ from two plots, 14.7 tons of grain were collected. The next year, after the application of new agricultural techniques, the yield on the first plot increased by $80 \%$, and on the second - by $24 \%$, as a result of which from these same plots, ... | 10.26 | 114 | 5 |
math | Let $a_n$ be a sequence de fined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot 3^n$ for $n \ge 0$.
Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$. | a_0 = 1 | 96 | 6 |
math | Consider the $10$-digit number $M=9876543210$. We obtain a new $10$-digit number from $M$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $M$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For example, from $M=9\underl... | 88 | 173 | 2 |
math | ## Task Condition
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$$
\left\{\begin{array}{l}
x=\cos ^{2} t \\
y=\operatorname{tg}^{2} t
\end{array}\right.
$$ | \frac{2}{\cos^{6}} | 70 | 10 |
math | ## 16. How many of you were there, children?
If you had asked me such a question, I would have answered you only that my mother dreamed of having no fewer than 19 children, but she did not manage to fulfill her dream; however, I had three times as many sisters as cousins, and brothers - half as many as sisters. How ma... | 10 | 82 | 2 |
math | Example 4 Let $x>0$, find the range of the function
$$
f(x)=\frac{x+\frac{1}{x}}{[x] \cdot\left[\frac{1}{x}\right]+[x]+\left[\frac{1}{x}\right]+1}
$$
where $[x]$ denotes the greatest integer not exceeding $x$. | {\frac{1}{2}}\cup[\frac{5}{6},\frac{5}{4}) | 79 | 23 |
math | B-4. $p(x)$ is a non-zero polynomial of degree less than 1992, and $p(x)$ has no common factor with $x^{3}-x$. Let
$$
\frac{d^{1992}}{d x^{1992}}\left(\frac{p(x)}{x^{3}-x}\right)=\frac{f(x)}{g(x)},
$$
where $f(x), g(x)$ are polynomials. Find the smallest possible degree of $f(x)$. | 3984 | 115 | 4 |
math | 2. Let $x_{i} \geqslant 0, i=1,2, \cdots, n$, and $\sum_{i=1}^{n} x_{i}=1, n \geqslant 2$. Find the maximum value of $\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}+x_{j}\right)$. (32nd IMO Shortlist Problem) | \frac{1}{4} | 107 | 7 |
math | ## Task Condition
Find the derivative.
$y=\frac{1}{\sin \alpha} \ln (\tan x+\cot \alpha)$ | \frac{1}{\cosx\cdot\cos(\alpha-x)} | 30 | 16 |
math | Example 5. Find the mathematical expectation of the random variable $Y=2X+7$, given that $M(X)=4$. | 15 | 28 | 2 |
math | 8. Given the sequence $\left.\mid a_{n}\right\}$, where $a_{n}$ is an integer, and for $n \geqslant 3, n \in \mathbf{N}$, we have $a_{n}=a_{n-1}-$ $a_{n-2}$, if the sum of the first 1985 terms is 1000, and the sum of the first 1995 terms is 4000, then the sum of the first 2002 terms is $\qquad$ _. | 3000 | 127 | 4 |
math | ## Task B-3.2.
Determine the zeros of the function $f: \mathbf{R} \rightarrow \mathbf{R}, f(x)=\log _{2}\left(18 \cdot 4^{x}-8 \cdot 2^{x}+1\right)-2 x-1$. | -2 | 70 | 2 |
math | 1.25. Find the general solution of the non-homogeneous system
\[
\left\{\begin{array}{l}
x_{1}+7 x_{2}-8 x_{3}+9 x_{4}=1 \\
2 x_{1}-3 x_{2}+3 x_{3}-2 x_{4}=1 \\
4 x_{1}+11 x_{2}-13 x_{3}+16 x_{4}=3 \\
7 x_{1}-2 x_{2}+x_{3}+3 x_{4}=4
\end{array}\right.
\] | (\begin{pmatrix}\alpha\cdot\frac{3}{17}+\beta\cdot(-\frac{13}{17})+\frac{10}{17}\\\alpha\cdot(\frac{19}{17})+\beta\cdot(-\frac{20}{17})+\frac{1}{17}\\\alpha | 134 | 77 |
math | Write down the first $n$ natural numbers in decimal form on a (fairly long) strip of paper, then cut the strip so that each piece contains only one digit. Put these pieces in a box, mix them up, and draw one at random. Let $p_{n}$ denote the probability that the digit 0 is on the piece of paper drawn. Determine the lim... | \frac{1}{10} | 97 | 8 |
math | $\boxed{A1}$ Find all ordered triplets of $(x,y,z)$ real numbers that satisfy the following system of equation $x^3=\frac{z}{y}-\frac {2y}{z}$ $y^3=\frac{x}{z}-\frac{2z}{x}$ $z^3=\frac{y}{x}-\frac{2x}{y}$ | (1, 1, -1), (1, -1, 1), (-1, 1, 1), (-1, -1, -1) | 82 | 37 |
math | 3. Four friends rent a cottage for a total of $£ 300$ for the weekend. The first friend pays half of the sum of the amounts paid by the other three friends. The second friend pays one third of the sum of the amounts paid by the other three friends. The third friend pays one quarter of the sum of the amounts paid by the... | 65 | 87 | 2 |
math |
Problem 8'.3. Find all natural numbers $n$ such that there exists an integer number $x$ for which $499\left(1997^{n}+1\right)=x^{2}+x$.
| 1 | 53 | 1 |
math | 8.2. Petya and three of his classmates started a 100-meter race at the same time, and Petya came in first. After 12 seconds from the start of the race, no one had finished yet, and the four participants had run a total of 288 meters. When Petya finished the race, the other three participants had 40 meters left to run i... | 80 | 122 | 2 |
math | 4. (10 points) Given that $a$ is a two-digit number with all digits the same, $b$ is a two-digit number with all digits the same, and $c$ is a four-digit number with all digits the same, and $a^{2}+b=c$. Find all $(a, b, c)$ that satisfy the conditions. | (,b,)=(33,22,1111),(66,88,4444),(88,33,7777) | 76 | 38 |
math | ## Task B-3.3.
To access a certain webpage, Matko has to choose a 4-digit PIN. Leading zeros are allowed, but there are some requirements (restrictions) on the PIN. No digit can be repeated three or more times in a row. For example, 0006 or 6666 are not allowed PINs, but 0030 is an allowed PIN. Additionally, no pair o... | 9720 | 140 | 4 |
math | 2. [4 points] Find all pairs of real parameters $a$ and $b$, for each of which the system of equations
$$
\left\{\begin{array}{l}
3(a+b) x+12 y=a \\
4 b x+(a+b) b y=1
\end{array}\right.
$$
has infinitely many solutions. | (1;3),(3;1),(-2-\sqrt{7};\sqrt{7}-2),(\sqrt{7}-2;-2-\sqrt{7}) | 77 | 36 |
math |
3. A finite non-empty set $S$ of integers is called 3 -good if the the sum of the elements of $S$ is divisble by 3 . Find the number of 3 -good non-empty subsets of $\{0,1,2, \ldots, 9\}$.
| 351 | 66 | 3 |
math | 1. Given a triangle $\triangle A B C$ with side lengths $4,5,6$ respectively, the circumcircle of $\triangle A B C$ is a great circle of sphere $O$, and $P$ is a point on the sphere. If the distances from point $P$ to the three vertices of $\triangle A B C$ are all equal, then the volume of the tetrahedron $P-A B C$ is... | 10 | 97 | 2 |
math | 1. In the equation $\overline{x 5} \cdot \overline{3 y} \bar{z}=7850$, restore the digits $x, y, z$ | x=2, y=1, z=4 | 41 | 11 |
math | ## Task 4 - 240614
Rita multiplies a number $z$ by 9 and gets the result 111111111.
(a) Which number $z$ is it?
(b) Determine a number $x$ that has the following property!
If you multiply $x$ by the number $z$ determined in (a), then the product is a number written entirely with the digit 8 (in the usual decimal no... | 72000000072 | 136 | 11 |
math | Consider the polynomial
\[P(x)=x^3+3x^2+6x+10.\]
Let its three roots be $a$, $b$, $c$. Define $Q(x)$ to be the monic cubic polynomial with roots $ab$, $bc$, $ca$. Compute $|Q(1)|$.
[i]Proposed by Nathan Xiong[/i] | 75 | 82 | 2 |
math | 46.Cargo was delivered to three warehouses. 400 tons were delivered to the first and second warehouses, 300 tons were delivered to the second and third warehouses together, and 440 tons were delivered to the first and third warehouses. How many tons of cargo were delivered to each warehouse separately? | 270,130,170 | 66 | 11 |
math | Determine all non-constant polynomials $X^{n}+a_{1} X^{n-1}+\cdots+a_{n-1} X+a_{n}$, with integer coefficients, whose roots are exactly the numbers $a_{1}, \ldots, a_{n-1}, a_{n}$ (with multiplicity). | P(X)=X^{n}, P(X)=X^{n}(X^{2}+X-2), P(X)=X^{n}(X^{3}+X^{2}-X-X) | 73 | 42 |
math | [ Volume of a Tetrahedron and Pyramid
Given a regular quadrilateral pyramid PABCD ( $P$ - vertex) with the side of the base $a$ and the lateral edge $a$. A sphere with center at point $O$ passes through point $A$ and touches the edges $P B$ and $P D$ at their midpoints. Find the volume of the pyramid $O P C D$.
# | \frac{5a^3\sqrt{2}}{96} | 90 | 16 |
math | 2. In the sequence $\left\{a_{n}\right\}$, it is known that $a_{n+2}=3 a_{n+1}-2 a_{n}, a_{1}=1, a_{2}=3$, then the general term formula of the sequence $\left\{a_{n}\right\}$ is $a_{n}=$ | a_{n}=2^{n}-1 | 78 | 9 |
math | (9) In the Cartesian coordinate system, the "rectangular distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as $d(P, Q)=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|$. If the "rectangular distance" from $C(x, y)$ to points $A(1,3)$ and $B(6,9)$ is equal, where real num... | 5(\sqrt{2}+1) | 184 | 9 |
math | 13.413 An alloy consists of tin, copper, and zinc. If 20 g is separated from this alloy and melted with 2 g of tin, then in the newly obtained alloy, the mass of copper will be equal to the mass of tin. If, however, 30 g is separated from the original alloy and 9 g of zinc is added, then in this new alloy, the mass of ... | 50,40,10 | 107 | 8 |
math | The bases of the trapezoid are 8 and 2. The angles adjacent to the larger base are each $45^{\circ}$. Find the volume of the solid formed by rotating the trapezoid around the larger base. | 36\pi | 51 | 4 |
math |
Problem 3. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{3}+y^{3}+x y\right)=x^{2} f(x)+y^{2} f(y)+f(x y)
$$
for all $x, y \in \mathbb{R}$.
| f(x)=xf(1) | 83 | 7 |
math | A regular $n$-sided polygon has an area of $t$; what is the area of the annulus formed by the circles inscribed in and circumscribed around the $n$-sided polygon? For example, $n=11, t=107.05 \mathrm{dm}^{2}$. | 8.9772\mathrm{}^{2} | 72 | 12 |
math | $6 \cdot 53$ Find all positive integers $M$ less than 10 such that 5 divides $1989^{M}+$ $M^{1989}$.
(China National Training Team Problem, 1990) | 1or4 | 56 | 3 |
math | Calculate $S_{n}=\sum_{k=1}^{n} k^{2}\binom{n}{k}$ | n(n+1)2^{n-2} | 26 | 11 |
math | Example 8. Find the prime number $p$ such that $8 p^{2}+1$ is also a prime number.
The text above is translated into English, preserving the original text's line breaks and format. | p=3 | 46 | 3 |
math | 54. Let $x_{1}, x_{2}, \cdots, x_{5}$ be real numbers, and $\sum_{i=1}^{5} x_{i}=0$, let $x_{6}=x_{1}$, find the smallest real number $k$ such that the inequality $\sum_{i=1}^{5} x_{i} x_{i+1} \leqslant k\left(\sum_{i=1}^{5} a_{i}^{2}\right) .(2001$ Czech and Slovak Olympiad problem) | \frac{\sqrt{5}-1}{4} | 126 | 11 |
math | 7. Calculate:
$$
\begin{array}{l}
\frac{1}{\sin 45^{\circ} \cdot \sin 46^{\circ}}+\frac{1}{\sin 46^{\circ} \cdot \sin 47^{\circ}}+ \\
\cdots+\frac{1}{\sin 89^{\circ} \cdot \sin 90^{\circ}} \\
=
\end{array}
$$ | \frac{1}{\sin 1^{\circ}} | 101 | 13 |
math | 2. Let positive integers $m, n$ satisfy
$$
m(n-m)=-11 n+8 \text {. }
$$
Then the sum of all possible values of $m-n$ is $\qquad$ | 18 | 47 | 2 |
math | 1. A circular coin $A$ is rolled, without sliding, along the circumference of another stationary circular coin $B$ with radius twice the radius of coin $A$. Let $x$ be the number of degrees that the coin $A$ makes around its centre until it first returns to its initial position. Find the value of $x$. | 1080 | 70 | 4 |
math | Example 34 Let $n$ be a positive integer, $a, b$ be positive real numbers, and satisfy $a+b=2$, then the minimum value of $\frac{1}{1+a^{n}}+\frac{1}{1+b^{n}}$ is $\qquad$ . | 1 | 63 | 1 |
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