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 | 10. For the sequence $\left\{a_{n}\right\}$, if there exists a sequence $\left\{b_{n}\right\}$, such that for any $n \in \mathbf{Z}_{+}$, we have $a_{n} \geqslant b_{n}$, then $\left\{b_{n}\right\}$ is called a "weak sequence" of $\left\{a_{n}\right\}$. Given
$$
\begin{array}{l}
a_{n}=n^{3}-n^{2}-2 t n+t^{2}\left(n \in... | \left(-\infty, \frac{1}{2}\right] \cup\left[\frac{3}{2},+\infty\right) | 233 | 33 |
math | The straight line $l_{1}$ with equation $x-2y+10 = 0$ meets the circle with equation $x^2 + y^2 = 100$ at B in the fi rst quadrant. A line through B, perpendicular to $l_{1}$ cuts the y-axis at P (0, t). Determine the value of $t$. | t = 20 | 81 | 5 |
math | Thirteen, for $\{1,2,, 3 \cdots, n\}$ and each of its non-empty subsets, we define the alternating sum as follows: arrange the numbers in the subset in descending order, then alternately add and subtract the numbers starting from the largest (for example, the alternating sum of $\{1,2,4,6,9\}$ is $9-6+4-2+1=6$, and the... | 448 | 121 | 3 |
math | 3. Solve the system $\left\{\begin{array}{l}3 x \geq 2 y+16, \\ x^{4}+2 x^{2} y^{2}+y^{4}+25-26 x^{2}-26 y^{2}=72 x y .\end{array}\right.$ | (6;1) | 75 | 5 |
math | Two clocks started and finished ringing simultaneously. The first ones ring every 2 seconds, the second ones - every 3 seconds. In total, 13 strikes were made (coinciding strikes were counted as one). How much time passed between the first and last strikes? | 18\mathrm{} | 56 | 5 |
math | 3. Determine the remainder when the natural number $n$ is divided by 45, given that the number $(n+2$ 020) gives a remainder of 43 when divided by 45. | 3 | 47 | 1 |
math | ## 261. Math Puzzle $2 / 87$
The pioneers of class 6a are going on a group trip over the weekend. If each participant pays 12 marks, there will be a surplus of 33 marks compared to the required total amount. If each pays 10 marks, however, 11 marks will be missing. How many people are participating in the trip? How mu... | 22 | 94 | 2 |
math | II. Let the three sides of a triangle be integers $l$, $m$, and $n$, and $l > m > n$. It is known that $\left\{\frac{3^{l}}{10^{4}}\right\}=\left\{\frac{3^{m}}{10^{4}}\right\}=\left\{\frac{3^{n}}{10^{4}}\right\}$, where $\{x\}=x-[x]$, and $[x]$ represents the greatest integer not exceeding $x$. Find the minimum perimet... | 3003 | 128 | 4 |
math | Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says $0$. In each turn except the first the possible moves are determined from the previous number $n$ in the following way: write
\[n =\sum_{m\in O_n}2^m;\]
the valid numbers are the elements $m$ of $O_n$. That ... | 0 | 351 | 3 |
math | B2. Find all natural numbers $n$ and prime numbers $p$ for which $\sqrt[3]{n}+\frac{p}{\sqrt[3]{n}}$ is a square of a natural number. | n=1,p=3n=27,p=3 | 46 | 13 |
math | 1. There are 200 people at the beach, and $65 \%$ of these people are children. If $40 \%$ of the children are swimming, how many children are swimming? | 52 | 43 | 2 |
math | 195. For what values of $x$ will the numbers $\frac{x-3}{7}, \frac{x-2}{5}$, and $\frac{x-4}{3}$ be integers simultaneously? | 105t_{1}+52 | 44 | 10 |
math | Which term in the expansion of $(1+\sqrt{3})^{100}$ by Newton's binomial formula will be the largest? | C_{100}^{64}(\sqrt{3})^{64} | 29 | 19 |
math | 6. In the triangle $A B C, A C=2 B C, \angle C=90^{\circ}$ and $D$ is the foot of the altitude from $C$ onto $A B$. A circle with diameter $A D$ intersects the segment $A C$ at $E$. Find $A E: E C$. | 4 | 73 | 1 |
math | 8.2. Petya came home from school today at $16:45$, looked at the clock and wondered: after what time will the hands of the clock be in the same position for the seventh time since he came home from school? | 435 | 52 | 3 |
math | ## Task 4 - 320614
A cyclist rides from Schnellhausen to Sausedorf, covering 36 kilometers each day. At the same time, another cyclist, who covers 34 kilometers each day, rides from Sausedorf towards him. The distance between Schnellhausen and Sausedorf is 350 km.
In how many days will the two cyclists meet? Also, pe... | 5 | 91 | 1 |
math | Problem 4. Determine the four-digit number $\overline{x y z t}$ for which
$$
\overline{x y z t}+4 \cdot \overline{y z t}+2 \cdot \overline{z t}=2018
$$ | 1174 | 59 | 4 |
math | 42 Let $k$ be a natural number. Try to determine the smallest natural number $n$ such that: in any $n$ integers, there must be two numbers whose sum or difference is divisible by $2 k+1$.
| k+2 | 50 | 3 |
math | The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 125 | 68 | 3 |
math | 98 For $i=1,2, \cdots, n$, we have $\left|x_{i}\right|<1$, and assume $\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|=19+$ $\left|x_{1}+x_{2}+\cdots+x_{n}\right|$, then the minimum value of the integer $n$ is $\qquad$ | 20 | 97 | 2 |
math | 3. Positive real numbers $u, v, w$ are not equal to 1. If $\log _{u} v w+\log _{v} w=5, \log _{v} u+\log _{w} v=3$, then the value of $\log _{w} u$ is $\qquad$ . | \frac{4}{5} | 73 | 7 |
math | ## Zadatak B-4.2.
Realne funkcije $f \circ g$ i $f$ zadane su pravilima pridruživanja
$$
(f \circ g)(x)=2^{4^{4^{\sin x}}} \quad \text { i } \quad f(x)=4^{8^{-2^{x}}}
$$
Odredite pravilo pridruživanja kojim je zadana funkcija $g$ i njezino područje definicije.
| (x)=\log_{2}\frac{1-2^{1+2\sinx}}{3},\quadx\in\bigcup_{k\in\mathbb{Z}}\langle\frac{7\pi}{6}+2k\pi,\frac{11\pi}{6}+2k\pi\rangle | 116 | 74 |
math | 3. It is known that the equations $x^{2}+a x+b=0$ and $x^{3}+b x+a=0$ have a common root and $a>b>0$. Find it. | -1 | 47 | 2 |
math | ## 35. Being Late for Work
Maurice goes to work either by his own car (and then due to traffic jams, he is late half of the time), or by subway (and then he is late only one time out of four). If on any given day Maurice arrives at work on time, he always uses the same mode of transport the next day as he did the prev... | \frac{1}{3} | 130 | 7 |
math | A [i]T-tetromino[/i] is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square.
Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes. | 4 | 84 | 1 |
math | 1. What is the smallest number that is greater than 2015 and divisible by both 6 and 35 ? | 2100 | 27 | 4 |
math | ## Task 6 - 150736
If $z$ is a natural number, let $a$ be the cross sum of $z$, $b$ be the cross sum of $a$, and $c$ be the cross sum of $b$.
Determine $c$ for every 1000000000-digit number $z$ that is divisible by 9! | 9 | 88 | 1 |
math | In trapezoid $ABCD$ with $AD\parallel BC$, $AB=6$, $AD=9$, and $BD=12$. If $\angle ABD=\angle DCB$, find the perimeter of the trapezoid. | 39 | 53 | 2 |
math | 5. Six musicians gathered at a chamber music festival. . At each scheduled concert some of the musicians played while the others listened as members of the audience. What is the least number of such concerts which would need to be scheduled so that for every two musicians each must play for the other in some concert? | 4 | 61 | 1 |
math | 3. Given the set $M=\{(a, b) \mid a \leqslant-1, b \leqslant m\}$. If for any $(a, b) \in M$, it always holds that $a \cdot 2^{b}-b-3 a \geqslant 0$, then the maximum value of the real number $m$ is $\qquad$. | 1 | 87 | 1 |
math | Example 7 When is $x$ a rational number such that the algebraic expression $9 x^{2}$ $+23 x-2$ is exactly the product of two consecutive positive even numbers?
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | x=2, -\frac{41}{9}, -17, \frac{130}{9} | 67 | 26 |
math | 2+ $[$ Prime numbers and their properties $]$
Four kids were discussing the answer to a problem.
Kolya said: "The number is 9."
Roman: "It's a prime number."
Katya: "It's an even number."
And Natasha said that the number is divisible by 15.
One boy and one girl answered correctly, while the other two were wrong. ... | 2 | 90 | 1 |
math | 5. Let $\sin x+\cos x=\frac{1}{2}$, then $\sin ^{3} x+\cos ^{3} x=$ | \frac{11}{16} | 33 | 9 |
math | 9.1. Find the smallest six-digit number that is a multiple of 11, where the sum of the first and fourth digits is equal to the sum of the second and fifth digits and is equal to the sum of the third and sixth digits. | 100122 | 52 | 6 |
math | 6. Inside a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, there is a small ball that is tangent to the diagonal segment $A C_{1}$. Then the maximum radius of the ball is $\qquad$ . | \frac{4-\sqrt{6}}{5} | 64 | 12 |
math | 4. Find all positive integers $n$, such that there exists a convex $n$-gon, whose interior angles are consecutive positive integers. | 3,5,9,15,16 | 29 | 11 |
math | (solved by Sébastien Miquel). The natural number $A$ has the following property: the number $1+2+\cdots+A$ is written (in base 10) as the number $A$ followed by three other digits. Find $A$. | 1999 | 57 | 4 |
math | 3 Given a positive integer $n$, find the smallest positive number $\lambda$, such that for any $\theta_{i} \in \left(0, \frac{\pi}{2}\right)(i=1,2, \cdots, n)$, if $\tan \theta_{1} \cdot \tan \theta_{2} \cdots \cdots \cdot \tan \theta_{n} = 2^{\frac{n}{2}}$, then $\cos \theta_{1}+\cos \theta_{2}+\cdots+\cos \theta_{n}$... | n-1 | 137 | 3 |
math | 2. Houses on the left side of the street have odd, and houses on the right side of the street have even house numbers. The sum of all house numbers on one side of the street is 1369, and on the other side 2162. How many houses are there on that street? | 83 | 66 | 2 |
math | Let $P(X)$ be a monic polynomial of degree 2017 such that $P(1)=1, P(2)=2, \ldots, P(2017)=$ 2017. What is the value of $P(2018)$? | 2017!+2018 | 64 | 10 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{4^{x}(\ln 4 \cdot \sin 4 x-4 \cos 4 x)}{16+\ln ^{2} 4}
$$ | 4^{x}\cdot\sin4x | 50 | 9 |
math | 4. In rectangle $A B C D$, $A B=2, A D=1$, point $P$ on side $D C$ (including points $D, C$) and point $Q$ on the extension of $C B$ (including point $B$) satisfy $|\overrightarrow{D P}|=|\overrightarrow{B Q}|$. Then the dot product $\overrightarrow{P A} \cdot \overrightarrow{P Q}$ of vectors $\overrightarrow{P A}$ and... | \frac{3}{4} | 124 | 7 |
math | A rectangle with area $n$ with $n$ positive integer, can be divided in $n$ squares(this squares are equal) and the rectangle also can be divided in $n + 98$ squares (the squares are equal). Find the sides of this rectangle | 3 \times 42 | 55 | 8 |
math | The sequence $a_{n}$ is defined as follows: $a_{1}=1, a_{2 n}=a_{n}, a_{2 n+1}+a_{n}=1$. What is $a_{2006}$? | 0 | 53 | 1 |
math | At the top of a piece of paper is written a list of distinctive natural numbers. To continue the list you must choose 2 numbers from the existent ones and write in the list the least common multiple of them, on the condition that it isn’t written yet. We can say that the list is closed if there are no other solutions l... | 2^{10} - 1 = 1023 | 124 | 16 |
math | 6. 28 Given the value of $\sin \alpha$. Try to find: (a) $\sin \frac{\alpha}{2}$, (b) $\sin \frac{\alpha}{3}$, respectively, how many different values can they have at most? | 4, 3 | 56 | 4 |
math | 13. Two parabolas param1 and param2 touch at a point lying on the Ox axis. A vertical line through point $D$ - the second intersection point of the first parabola with the Ox axis - intersects the second parabola at point $A$, and the common tangent to the parabolas at point $B$. Find the ratio $B D: A B$.
| param1 | ... | 1.5 | 227 | 3 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(-1 ; 2 ; 4)$
$M_{2}(-1 ;-2 ;-4)$
$M_{3}(3 ; 0 ;-1)$
$M_{0}(-2 ; 3 ; 5)$ | \frac{5}{9} | 90 | 7 |
math | 2. Given the parabola $y=-x^{2}+m x-1$, points $A(3,0), B(0,3)$, find the range of $m$ when the parabola intersects the line segment $A B$ at two distinct points. | [3,\frac{10}{3}] | 60 | 10 |
math | 3. Given that $\triangle A B C$ is an equilateral triangle, the ellipse $\Gamma$ has one focus at $A$, and the other focus $F$ lies on the line segment $B C$. If the ellipse $\Gamma$ passes exactly through points $B$ and $C$, then its eccentricity is | \frac{\sqrt{3}}{3} | 66 | 10 |
math | In $\triangle ABC$ let $I$ be the center of the inscribed circle, and let the bisector of $\angle ACB$ intersect $AB$ at $L$. The line through $C$ and $L$ intersects the circumscribed circle of $\triangle ABC$ at the two points $C$ and $D$. If $LI=2$ and $LD=3$, then $IC=\tfrac{m}{n}$, where $m$ and $n$ are relatively ... | 13 | 112 | 2 |
math | 5. In the cash register of the commercial director Vasily, money appeared for the first time in a long time - 2019 coins. Unfortunately, it is known that one of them is counterfeit, differing in weight from the genuine ones. Enraged workers employed by Vasily are demanding immediate payment of their salaries, and only ... | no | 138 | 1 |
math | 2.050. $\frac{2\left(x^{4}+4 x^{2}-12\right)+x^{4}+11 x^{2}+30}{x^{2}+6}$. | 1+3x^{2} | 50 | 7 |
math | Example 1. Calculate approximately the value of the function $f(x)=$ $=\sqrt{x^{2}+x+3}$ at $x=1.97$ | 2.975 | 37 | 5 |
math | 29. In the parking lot, there were passenger cars and motorcycles. The number of motorcycles with sidecars was half the number of those without sidecars. What could be the maximum number of cars if the total number of wheels on these cars and motorcycles was $115?$ | 27 | 57 | 2 |
math | 4. In the expansion of $(\sqrt[5]{3}+\sqrt[3]{5})^{100}$, there are $\qquad$ terms that are rational numbers. | 7 | 39 | 1 |
math | 4. Find the limit of the variable quantity $x=\frac{a z+1}{z}$ as $z \rightarrow \infty$. | a | 30 | 1 |
math | 2. Simplify the fraction: $\frac{x^{14}+x^{13}+\ldots+x+1}{x^{5}+x^{4}+x^{3}+x^{2}+x}$. | \frac{x^{10}+x^{5}+1}{x} | 50 | 17 |
math | 14. A regular 201-sided polygon is inscribed inside a circle of center $C$. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point $C$ lying inside the triangle?
14. 一個正 201 邊形內接於一個以 $C$ 為圓心的圓形內。把多邊形 201 個頂點中的任意三個頂點連起, 可以組成不同的三角形。有多少個這樣的三角形, 將 $C$ 點包含在... | 338350 | 140 | 6 |
math | A sequence of positive integers is defined by $a_0=1$ and $a_{n+1}=a_n^2+1$ for each $n\ge0$. Find $\text{gcd}(a_{999},a_{2004})$. | 677 | 58 | 3 |
math | 4. When a die is rolled three times, the probability that the three numbers facing up can form the side lengths of a triangle whose perimeter is divisible by 3 is $\qquad$ .
| \frac{11}{72} | 39 | 9 |
math | 11. Let $f(x)$ and $g(x)$ be odd and even functions defined on $R$, respectively, and $f(x) + g(x) = 2^x$. If for $x \in [1,2]$, the inequality $a f(x) + g(2x) \geq 0$ always holds, then the range of the real number $a$ is $\qquad$ | -\frac{17}{6} | 89 | 8 |
math | Example 2 (to item $6^{\circ}$). Solve the system
$$
\left\{\begin{aligned}
3 x_{1}-x_{2}+3 x_{3} & =5 \\
2 x_{1}-x_{2}+4 x_{3} & =5 \\
x_{1}+2 x_{2}-3 x_{3} & =0
\end{aligned}\right.
$$ | {\begin{pmatrix}x_{1}=1\\x_{2}=1\\x_{3}=1\end{pmatrix}.} | 92 | 30 |
math | $\underline{\text { Khamtsov }}$ D:
A cube with side $n$ ( $n \geq 3$ ) is divided by partitions into unit cubes. What is the minimum number of partitions between unit cubes that need to be removed so that from each cube it is possible to reach the boundary of the cube? | (n-2)^{3} | 69 | 7 |
math | 3. (20 points) At the Journalism Faculty of the University of Enchanted Commonwealth, 4 chickens are applying. The faculty has 2 places in the daytime program and 3 places in the evening program. Assuming all 4 chickens will be admitted to the faculty, determine the number of outcomes in which exactly two chickens will... | 6 | 75 | 1 |
math | Do there exist two irrational numbers $a$ and $b$ ($a > b$) such that their sum and product are equal to the same integer, and both $a$ and $b$ are greater than $\frac{1}{2}$ and less than 4? If they exist, find these two numbers; if not, explain the reason. | a=\frac{5+\sqrt{5}}{2}, b=\frac{5-\sqrt{5}}{2} | 73 | 26 |
math | 7. Let $f(x)=x-[x]-\tan x$, where $[x]$ is the greatest integer not exceeding $x$, then $\{T \mid f(x+T)=f(x), T \neq 0\}=$ | \varnothing | 52 | 3 |
math | Example 1. The random variable $\xi$ is distributed according to the normal law with parameters $a$ and $\sigma^{2}$. From the sample $x_{1}, \ldots, x_{n}$ of values of $\xi$, the empirical moments $M_{1}^{*}=\bar{x}=2.3$ and $M_{2}^{*}=\overline{x^{2}}=8.7$ have been determined. Using these moments, find the paramete... | =2.3,\sigma^{2}=3.41 | 112 | 13 |
math | # Task 2.
What is the last digit of the value of the sum $5^{2020}+6^{2019} ?$ | 1 | 34 | 1 |
math | Let $a_{1}, a_{2}, \cdots, a_{n}$ represent any permutation of the integers 1, 2, $\cdots, n$. Let $f(n)$ be the number of such permutations that satisfy: (1) $a_{1}=1$; (2) $\left|a_{i}-a_{i+1}\right| \leqslant 2, i=1$, $2, \cdots, n-1$. Determine whether $f(1996)$ is divisible by 3. | 1 | 118 | 1 |
math | Let $N = 2^{\left(2^2\right)}$ and $x$ be a real number such that $N^{\left(N^N\right)} = 2^{(2^x)}$. Find $x$. | 66 | 52 | 2 |
math | Can a circle be circumscribed around quadrilateral $A B C D$, if $\angle A D C=30^{\circ}, A B=3, B C=4, A C=6$? | No | 45 | 1 |
math | Initial 281 Given that there exist $k(k \in \mathbf{N}, k \geqslant 2)$ consecutive positive integers, the mean of their squares is a perfect square. Try to find the minimum value of $k$.
| 31 | 54 | 2 |
math | 6・17 Function $f$ is defined on the set of integers, and satisfies
$$
f(n)=\left\{\begin{array}{l}
n-3, \quad \text { when } n \geqslant 1000 ; \\
f(f(n+5)), \quad \text { when } n<1000 .
\end{array}\right.
$$
Find $f(84)$. | 997 | 95 | 3 |
math | 8. Find the linear combination $3 A-2 B$, if
$$
A=\left(\begin{array}{rrr}
2 & -4 & 0 \\
-1 & 5 & 1 \\
0 & 3 & -7
\end{array}\right), \quad B=\left(\begin{array}{rrr}
4 & -1 & -2 \\
0 & -3 & 5 \\
2 & 0 & -4
\end{array}\right)
$$ | (\begin{pmatrix}-2&-10&4\\-3&21&-7\\-4&9&-13\end{pmatrix}) | 105 | 37 |
math | 22nd ASU 1988 Problem 15 What is the minimal value of b/(c + d) + c/(a + b) for positive real numbers b and c and non-negative real numbers a and d such that b + c ≥ a + d? Solution | \sqrt{2}-\frac{1}{2} | 59 | 12 |
math | 1. Given real numbers $x, y, z$ satisfy
$$
x+y=4,|z+1|=x y+2 y-9 \text {. }
$$
then $x+2 y+3 z=$ $\qquad$ | 4 | 53 | 1 |
math | Three, given the system of equations $\left\{\begin{array}{l}x-y=2, \\ m x+y=6 .\end{array}\right.$ If the system has non-negative integer solutions, find the value of the positive integer $m$, and solve the system of equations. | m=1, (x,y)=(4,2); m=3, (x,y)=(2,0) | 62 | 24 |
math | 8. Let $a_{n}$ be the coefficient of the $x$ term in the expansion of $(3-\sqrt{x})^{n}$ $(n=2,3,4, \cdots)$. Then $\lim _{n \rightarrow \infty}\left(\frac{3^{2}}{a_{2}}+\frac{3^{3}}{a_{3}}+\cdots+\frac{3^{n}}{a_{n}}\right)=$ $\qquad$ . | 18 | 105 | 2 |
math | ## Task 1 - 010521
In 1961, 70000 tons of livestock and poultry, 115000 tons of milk, and 300000000 eggs were brought to the market in the GDR more than in 1960. The population of our republic is about 17000000.
How much more livestock and poultry, milk, and eggs could each citizen of our republic consume in 1961? Ro... | 4\mathrm{~} | 123 | 6 |
math | 9.104. For what values of $a$ does the quadratic trinomial $a x^{2}-7 x+4 a$ take negative values for any real values of $x$? | \in(-\infty;-\frac{7}{4}) | 43 | 14 |
math | 14. Given that $A B C-A_{1} B_{1} C_{1}$ is a regular triangular prism, $A B=B C$ $=C A=2, A A_{1}=\sqrt{2}, D$ and $E$ are the midpoints of $A C$ and $B C$ respectively. Then the angle formed by $A_{1} D$ and $C_{1} E$ is $\qquad$ . | 60^{\circ} | 98 | 6 |
math | 6. Given the complex number $z=x+y \mathrm{i}(x, y \in \mathbf{R})$, satisfying that the ratio of the real part to the imaginary part of $\frac{z+1}{z+2}$ is $\sqrt{3}$. Then the maximum value of $\frac{y}{x}$ is | \frac{4 \sqrt{2}-3 \sqrt{3}}{5} | 70 | 18 |
math | ## Task 21/73
Determine a polynomial $P(x)$ of the smallest degree with integer coefficients that is always divisible by 8 for odd $x$. The polynomial $P(x)$ should not be divisible by 2 for every integer $x$. | P(x)=x^2-^2withodd | 55 | 11 |
math | ## Task Condition
Find the angle between the planes:
$2 x-z+5=0$
$2 x+3 y-7=0$ | \arccos\frac{4}{\sqrt{65}}\approx60^{0}15^{\}18^{\\} | 31 | 32 |
math | ## Task 1 - 090831
The age specifications (expressed in full years of life) of a family - father, mother, and their two children - have the following properties:
The product of all four ages is 44950; the father is 2 years older than the mother.
How old are the four family members? | 31,29,10,5 | 76 | 10 |
math | 865*. Solve the equation in natural numbers $x, y$ and $z$
$$
x y + y z + z x = 2(x + y + z).
$$ | (1;2;4),(1;4;2),(2;1;4),(2;4;1),(2;2;2),(4;1;2),(4;2;1) | 39 | 43 |
math | 11. (3 points) Cutting a cake, cutting 1 time can make at most 2 pieces, cutting 2 times can make at most 4 pieces, cutting 3 times can make at most 7 pieces, continuing this way, cutting 5 times can make at most $\qquad$ pieces. | 16 | 65 | 2 |
math | 9.3. The inscribed circle of triangle $A B C$ with center $O$ touches the sides $A B, B C$ and $A C$ at points $M, N$ and $K$ respectively. It turns out that angle $A O C$ is four times the angle $M K N$. Find angle $B$. | 108 | 73 | 3 |
math | 7. A company has 16 machines in inventory at locations $A$ and $B$. Now, these machines need to be transported to locations 甲 and 乙, with 15 machines required at 甲 and 13 machines required at 乙. It is known that the transportation cost from $A$ to 甲 is 500 yuan per machine, and to 乙 is 400 yuan per machine; the transpo... | 10300 | 136 | 5 |
math | 8. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=3, a_{n+1}=$ $9 \sqrt[3]{a_{n}}(n \geqslant 1)$. Then $\lim _{n \rightarrow \infty} a_{n}=$ $\qquad$ . | 27 | 75 | 2 |
math | Example 9 The permutation of integers $1,2, \cdots, n$ satisfies: each number is either greater than all the numbers before it, or less than all the numbers before it. How many such permutations are there?
(21st Canadian High School Mathematics Competition) | 2^{n-1} | 58 | 6 |
math | We built a cuboid from small cubes with an edge of $1 \mathrm{~cm}$. If we removed one column from the cuboid, the remaining structure would consist of 602 cubes. If we removed one row from the top layer instead, the remaining structure would consist of 605 cubes. What are the dimensions of the cuboid? | 14\, | 75 | 4 |
math | 5. Find all natural numbers $n$ such that $n$ has as many digits as it has distinct prime divisors, and the sum of the distinct prime divisors is equal to the sum of the powers of the same divisors. | 48,72,4 | 49 | 7 |
math | If $(2^x - 4^x) + (2^{-x} - 4^{-x}) = 3$, find the numerical value of the expression $$(8^x + 3\cdot 2^x) + (8^{-x} + 3\cdot 2^{-x}).$$ | -1 | 66 | 2 |
math | A week-old set's three-element subsets need to be colored such that if the intersection of two subsets is empty, then their colors differ. How many colors do we need at least? | 3 | 37 | 1 |
math | [ Covering with percentages and ratios ] [ Inclusion-exclusion principle ]
Three crazy painters started painting the floor each in their own color. One managed to paint $75 \%$ of the floor red, another $70 \%$ green, and the third $65 \%$ blue. What part of the floor is definitely painted with all three colors? | 10 | 73 | 2 |
math | Problem 2. Determine the general term of the sequence $\left(x_{n}\right)_{n \geq 1}$ defined by:
$$
x_{1}=1, x_{2}=2 \text { and } x_{n+2}=\frac{x_{n+1}^{2}}{\left[\sqrt{x_{n}^{2}+x_{n+1}}\right]}-\frac{1}{n}, n \geq 1
$$
Marius Perianu, Slatina | a_{n}=n | 111 | 5 |
math | A train approaching at a speed of $20 \mathrm{~m} / \mathrm{s}$ sounded its horn at the railway crossing. We heard the horn 4 seconds before the train arrived. How far was the train when it started to sound the horn? (The speed of sound is $340 \mathrm{~m} / \mathrm{s}$.) | 85\mathrm{~} | 76 | 7 |
math | 1. Using the vertices of a regular dodecagon as the vertices of triangles, the total number of acute and obtuse triangles that can be formed is $\qquad$.
untranslated part: $\qquad$ (This part is typically left as a blank space for the answer to be filled in.) | 160 | 62 | 3 |
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