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 | $5 \cdot 4$ natural numbers $a_{1}, a_{2}, \cdots, a_{49}$ have a sum of 999, let $d$ be the greatest common divisor of $a_{1}, a_{2}, \cdots, a_{49}$, what is the maximum value of $d$?
(Kiev Mathematical Olympiad, 1979) | 9 | 87 | 1 |
math | 11.5. Compare the numbers $X=2019^{\log _{2018} 2017}$ and $Y=2017^{\log _{2019} 2020}$. | X>Y | 56 | 3 |
math | 9.4. It is known that the values of the quadratic trinomial $a x^{2}+b x+c$ on the interval $[-1,1]$ do not exceed 1 in absolute value. Find the maximum possible value of the sum $|a|+|b|+|c|$. Answer. 3. | 3 | 71 | 1 |
math | We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity.
Find all positive integers $n$ such that $n$ has a multiple which is alternating. | 20 \nmid n | 46 | 6 |
math | A2 A point with coordinates $(a, 2 a)$ lies in the 3rd quadrant and on the curve given by the equation $3 x^{2}+y^{2}=28$. Find $a$. | -2 | 46 | 2 |
math | 10.295. The legs of a right triangle are 6 and 8 cm. A circle is drawn through the midpoint of the smaller leg and the midpoint of the hypotenuse, touching the hypotenuse. Find the area of the circle bounded by this circle. | \frac{100\pi}{9} | 58 | 11 |
math | Let $n>1$ be an integer. An $n \times n \times n$ cube is composed of $n^{3}$ unit cubes. Each unit cube is painted with one color. For each $n \times n \times 1$ box consisting of $n^{2}$ unit cubes (of any of the three possible orientations), we consider the set of the colors present in that box (each color is listed... | \frac{n(n+1)(2 n+1)}{6} | 177 | 15 |
math | Zuzka wrote a five-digit number. When she appended a one to the end of this number, she got a number that is three times larger than the number she would get if she wrote a one before the original number.
Which five-digit number did Zuzka write? | 42857 | 56 | 5 |
math | Sergeev I.n.
The sum of the absolute values of the terms of a finite arithmetic progression is 250. If all its terms are increased by 1 or all its terms are increased by 2, then in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to 250. What values can the quantit... | \1000 | 112 | 5 |
math | 3. Suppose that each of $n$ people knows exactly one piece of information, and all $n$ pieces are different. Every time person $A$ phones person $B, A$ tells $B$ everything he knows, while $B$ tells $A$ nothing. What is the minimum of phone calls between pairs of people needed for everyone to know everything? | 2n-2 | 75 | 4 |
math | 20. Let $a$ and $b$ be real numbers such that $17\left(a^{2}+b^{2}\right)-30 a b-16=0$. Find the maximum value of $\sqrt{16 a^{2}+4 b^{2}-16 a b-12 a+6 b+9}$.
(2 marks)
設 $a$ 、 $b$ 為實數 , 使得 $17\left(a^{2}+b^{2}\right)-30 a b-16=0$ 。 求 $\sqrt{16 a^{2}+4 b^{2}-16 a b-12 a+6 b+9}$ 的最大值。 | 7 | 161 | 1 |
math | What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors?
[i]Ray Li[/i] | 36 | 28 | 2 |
math | Determine all triples of real numbers $(a, b, c)$, that satisfy the following system of equations: $\left\{\begin{array}{l}a^{5}=5 b^{3}-4 c \\ b^{5}=5 c^{3}-4 a . \\ c^{5}=5 a^{3}-4 b\end{array}\right.$
## SOLUTION: | (0,0,0),(1,1,1),(-1,-1,-1),(2,2,2),(-2,-2,-2) | 80 | 33 |
math | 18. Let $A, B, C$ be the three angles of a triangle. Let $L$ be the maximum value of
$$
\sin 3 A+\sin 3 B+\sin 3 C \text {. }
$$
Determine $\lfloor 10 L\rfloor$. | 25 | 65 | 2 |
math | Example 2. Find the total differential of the function $z=z(x, y)$, given by the equation $e^{x y z}-\operatorname{arctg} \frac{x y}{z}=0$. | \frac{1-(x^{2}y^{2}+z^{2})e^{xyz}}{1+(x^{2}y^{2}+z^{2})e^{xyz}}\cdot\frac{z}{xy}(y+x) | 47 | 54 |
math | 354. Give an example of a function that, for $x$ equal to any of the numbers $a_{1}, a_{2}, a_{3} \ldots a_{n}$, would be equal to: 1) zero, 2) a given number $k$. | (x-\alpha_{1})(x-\alpha_{2})^{2}(x-\alpha_{3})^{3}\ldots(x-\alpha_{n} | 62 | 32 |
math | 243. The probability of event $A$ occurring in each trial is 1/2. Using Chebyshev's inequality,
estimate the probability that the number $X$ of occurrences of event $A$ is within the range from 40 to 60, if 100 independent trials are conducted. | 0.75 | 68 | 4 |
math | 28. Write down a four-digit number where each subsequent digit is 1 greater than the previous one, then write the number with the same digits but in reverse order and subtract the smaller number from the larger one. Repeat this several times with different numbers and compare the results. Solve the problem in general t... | 3087 | 68 | 4 |
math | 6. Let $a=\lg z+\lg \left[x(y z)^{-1}+1\right], b=\lg x^{-1}+\lg (x y z+1), c=\lg y+\lg \left[(x y z)^{-1}+1\right]$, and let the maximum of $a, b, c$ be $M$, then the minimum value of $M$ is $\qquad$ . | \lg2 | 92 | 3 |
math | Example 24. Take 3 numbers from $1,3,5,7,9$, and 2 numbers from $2,4,6,8$, to form a five-digit even number without repeated digits. How many such numbers can be formed? | 2880 | 54 | 4 |
math | (3) Let there be a non-empty set $A \subseteq\{1,2,3,4,5,6,7\}$, and when $a \in A$, it must also be that $8-a \in A$. The number of such sets $A$ is $\qquad$. | 15 | 65 | 2 |
math | 3. In the Cartesian coordinate system, $\vec{e}$ is a unit vector, and vector $\vec{a}$ satisfies $\vec{a} \cdot \vec{e}=2$, and $|\vec{a}|^{2} \leqslant 5|\vec{a}+t \vec{e}|$ for any real number $t$, then the range of $|\vec{a}|$ is $\qquad$ . | [\sqrt{5},2\sqrt{5}] | 94 | 11 |
math | How many sides does a regular polygon have if the measure of the exterior angle is 9 degrees less than the number of sides? | 24 | 26 | 2 |
math | Three, the base of the triangular pyramid $S-ABC$ is a regular $\triangle ABC$, with the side length of this triangle being 4. It is also known that $AS=BS=\sqrt{19}$, and $CS=3$. Find the surface area of the circumscribed sphere of this triangular pyramid. | \frac{268}{11} \pi | 68 | 12 |
math | Let $f(x)$ is such function, that $f(x)=1$ for integer $x$ and $f(x)=0$ for non integer $x$.
Build such function using only variable $x$, integer numbers, and operations $+,-,*,/,[.]$(plus, minus, multiply,divide and integer part) | f(x) = \lfloor x \rfloor + \lfloor 1 - x \rfloor | 70 | 23 |
math | 8.5. In an isosceles triangle \(ABC\), the angle \(A\) at the base is \(75^\circ\). The bisector of angle \(A\) intersects side \(BC\) at point \(K\). Find the distance from point \(K\) to the base \(AC\), if \(BK=10\). | 5 | 73 | 1 |
math | 6. (20 points) Let a sequence of non-negative integers be given
$$
k, k+1, k+2, \ldots, k+n
$$
Find the smallest $k$, for which the sum of all numbers in the sequence is equal to 100.
# | 9 | 63 | 1 |
math | 14. Let $z=\cos \theta+i\sin \theta(0<\theta<\pi)$, given that $\omega=\frac{1-(\bar{z})^{4}}{1+z^{4}},|\omega|=\frac{\sqrt{3}}{3}, \arg \omega<\frac{\pi}{2}$, find $\theta$. | \theta=\frac{\pi}{12} | 78 | 10 |
math | 1. (5 points) Calculate: $\frac{1}{1+2}+\frac{1}{1+2+3}+\cdots+\frac{1}{1+2+\cdots+10}$, get
| \frac{9}{11} | 49 | 8 |
math | 1. In a full container, there are 150 watermelons and melons for a total of 24 thousand rubles, with all the watermelons together costing as much as all the melons. How much does one watermelon cost, given that the container can hold 120 melons (without watermelons) and 160 watermelons (without melons)? | 100 | 85 | 3 |
math | ## Problem Statement
Calculate the definite integral:
$$
\int_{-3}^{0}\left(x^{2}+6 x+9\right) \sin 2 x \, d x
$$ | -\frac{17+\cos6}{4} | 44 | 11 |
math | ## 236. Math Puzzle $1 / 85$
A well has four inflows. The first fills the well in 6 hours, i.e., 1/6 fill per hour, the second in 48 hours, the third in 72 hours, and the fourth in 96 hours. In how many hours will the well be full if all inflows are operating simultaneously? | 4.72 | 85 | 4 |
math | 2. For any natural number $n$, we denote by $p(n)$ the product of the digits of this natural number, written in decimal. Calculate the sum
$$
p(1)+p(2)+\cdots+p(2001)
$$ | 184320 | 55 | 6 |
math | Find all integers $n\ge 3$ for which the following statement is true:
Any arithmetic progression $a_1,\ldots ,a_n$ with $n$ terms for which $a_1+2a_2+\ldots+na_n$ is rational contains at least one rational term. | n \equiv 1 \pmod{3} | 64 | 12 |
math | 5. Arrange the numbers in the set $\left\{2^{x}+2^{y}+2^{z} \mid x 、 y 、 z \in \mathbf{N}, x<y<z\right\}$ in ascending order. The 100th number is $\qquad$ (answer with a number).
| 577 | 73 | 3 |
math | Let $ABC$ be an arbitrary triangle. A regular $n$-gon is constructed outward on the three sides of $\triangle ABC$. Find all $n$ such that the triangle formed by the three centres of the $n$-gons is equilateral. | n = 3 | 53 | 5 |
math | Three. (25 points) From the natural numbers $1, 2, \cdots, 2010$, take $n$ numbers such that the sum of any three of the taken numbers is divisible by 21. Find the maximum value of $n$.
| 96 | 59 | 2 |
math | 8.4. Given a triangle $\mathrm{ABC}$ with angle $\mathrm{BAC}$ equal to $30^{\circ}$. In this triangle, the median $\mathrm{BD}$ was drawn, and it turned out that angle $\mathrm{BDC}$ is $45^{\circ}$. Find angle $\mathrm{ABC}$. | 45 | 73 | 2 |
math | 11.088. A sphere, a cylinder with a square axial section, and a cone are given. The cylinder and the cone have the same bases, and their heights are equal to the diameter of the sphere. How do the volumes of the cylinder, the sphere, and the cone relate? | 3:2:1 | 62 | 5 |
math | 203. Find the derivatives of the following functions:
1) $y=x^{x}$;
2) $r=(\cos \alpha)^{\sin 2 \alpha}$;
3) $s=\frac{2 t}{\sqrt{1-t^{2}}}$;
4) $R=(x-1) \sqrt[3]{(x+1)^{2}(x-2)}$. | \begin{aligned}1)&\quady'=x^x(1+\lnx)\\2)&\quadr'=2(\cos2\alpha\ln\cos\alpha-\sin^2\alpha)(\cos\alpha)^{\sin2\alpha}\\3)&\quad'=\frac{2}{\sqrt{(1-^2)^3}}\\4)&\quad | 86 | 82 |
math | In which acute-angled triangle is the value of $\operatorname{tg} \alpha \cdot \operatorname{tg} \beta \cdot \operatorname{tg} \gamma$ the smallest? | \sqrt{27} | 42 | 6 |
math | 13. Let the sequence $\left\{a_{n}\right\}$ have the first term $a_{1}=1$, and the sum of the first $n$ terms $S_{n}$ and the general term $a_{n}$ satisfy $a_{n}=\frac{2 S_{n}^{2}}{2 S_{n}-1}$ $(n \geqslant 2)$, find: (1) the general formula for the sequence $\left\{S_{n}\right\}$; (2) the general formula for the seque... | S_{n}=\frac{1}{2n-1},\quada_{n}=\begin{cases}1,&n=1\\\frac{-2}{(2n-1)(2n-3)},&n\geqslant2\end{cases} | 132 | 59 |
math | 25. If $\cos (2 A)=-\frac{\sqrt{5}}{3}$, find the value of $6 \sin ^{6} A+6 \cos ^{6} A$. | 4 | 45 | 1 |
math | Example 1.6.2. Let \(a, b, c\) be non-negative real numbers such that \(a+b+c=3\). Find the minimum of the expression
\[3^{-a^{2}}+3^{-b^{2}}+3^{-c^{2}}\] | 1 | 61 | 1 |
math | Example 1. Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>$
0 ), find the locus of the midpoints of parallel chords with slope $k$. | b^{2} x + a^{2} k y = 0 | 53 | 15 |
math | # Problem 4. (2 points)
How many negative numbers are there among the numbers of the form $\operatorname{tg}\left(\left(15^{n}\right)^{\circ}\right)$, where $\mathrm{n}$ is a natural number from 1 to 2019? | 1009 | 63 | 4 |
math | 7. Determine the largest even positive integer which cannot be expressed as the sum of two composite odd positive integers. | 38 | 22 | 2 |
math | 4. (13 points) In a dance ensemble, there are 8 boys and 16 girls. Some of them form mixed (boy and girl) dance pairs. It is known that in each pair, at least one of the partners does not belong to any other pair. What is the maximum number of dance pairs that can be formed in this ensemble? | 22 | 74 | 2 |
math | Example 1.32. Form the equation of the plane passing through the line
\[
\begin{aligned}
3 x+2 y+5 z+6 & =0 \\
x+4 y+3 z+4 & =0
\end{aligned}
\]
parallel to the line
\[
\frac{x-1}{3}=\frac{y-5}{2}=\frac{z+1}{-3}
\] | 2x+3y+4z+5=0 | 95 | 12 |
math | Task 11. (16 points)
The Dorokhov family plans to purchase a vacation package to Crimea. The family plans to travel with the mother, father, and their eldest daughter Polina, who is 5 years old. They carefully studied all the offers and chose the "Bristol" hotel. The head of the family approached two travel agencies, ... | 58984 | 320 | 5 |
math | 15. Hydrogen was passed over a heated powder (X1). The resulting red substance (X2) was dissolved in concentrated sulfuric acid. The resulting solution of the substance blue (X3) was neutralized with potassium hydroxide - a blue precipitate (X4) formed, which upon heating turned into a black powder (X1). What substance... | 80 | 98 | 2 |
math | 1. Variant 1. Petya thought of two numbers and wrote down their product. After that, he decreased the first of the thought numbers by 3, and increased the other by 3. It turned out that the product increased by 900. By how much would the product have decreased if Petya had done the opposite: increased the first number ... | 918 | 88 | 3 |
math | A number is guessed from 1 to 144. You are allowed to select one subset of the set of numbers from 1 to 144 and ask whether the guessed number belongs to it. For an answer of "yes," you have to pay 2 rubles, and for an answer of "no" - 1 ruble. What is the smallest amount of money needed to surely guess the number?
# | 11 | 88 | 2 |
math | 8. (10 points) Let for positive numbers $x, y, z$ the following system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=12 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=28
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 16 | 102 | 2 |
math | How many ways are there to insert $+$'s between the digits of $111111111111111$ (fifteen $1$'s) so that the result will be a multiple of $30$? | 2002 | 56 | 4 |
math | 4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\operatorname{GCD}(a ; b ; c)=6 \\
\operatorname{LCM}(a ; b ; c)=2^{15} \cdot 3^{16}
\end{array}\right.
$$ | 7560 | 89 | 4 |
math | 1. Given the sets $M=\{x, x y, \lg (x y)\}$ and $N=\{0,|x|, y\}$, and $M=N$. Then, $\left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\right.$ $\left.\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right)$ is equal to | -2 | 126 | 2 |
math | Example 4. Calculate the integral
$$
\int_{1-i}^{2+i}\left(3 z^{2}+2 z\right) d z
$$ | 7+19i | 37 | 5 |
math | Let's determine the greatest common divisor of the numbers $\binom{n}{1},\binom{n}{2},\binom{n}{3}, \ldots,\binom{n}{n-1}$. | =1ifnisnotpower,=pifnispowerofthep | 44 | 16 |
math | 4. (15 points) Identical gases are in two thermally insulated vessels with volumes $V_{1}=1$ l and $V_{2}=2$ l. The pressures of the gases are $p_{1}=2$ atm and $p_{2}=3$ atm, and their temperatures are $T_{1}=300$ K and $T_{2}=400$ K, respectively. The gases are mixed. Determine the temperature that will be establishe... | 369\mathrm{~K} | 105 | 9 |
math | # Problem 5. (3 points)
In trapezoid $ABCD$ with base $AD$, the diagonals are the angle bisectors of $\angle B$ and $\angle C=110^{\circ}$. Find the degree measure of $\angle BAC$.
# | 15 | 61 | 2 |
math | 1. A father and son measured the length of the courtyard with their steps in winter, starting from the same place and walking in the same direction. In some places, the father's and son's footprints coincided exactly. In total, there were 61 footprints along the measurement line on the snow. What is the length of the c... | 21.6() | 102 | 5 |
math | 8. The maximum volume of a cone inscribed in a sphere compared to the volume of the sphere is $\qquad$
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | \frac{8}{27} | 50 | 8 |
math | 13.381 Coal mined at point A is sold at q rubles per ton, while coal mined at point B is sold at $p\%$ more expensive. Points A and B are connected by a road $s$ km long. In which zone of this road AB are the consumers of coal located, for whom the purchase and delivery of coal from B is cheaper than from A, if the tra... | \frac{}{2}-\frac{}{200r} | 141 | 15 |
math | Solve the following equation:
$$
x-1=\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]+\left[\frac{x}{6}\right]
$$ | 6k+1,6k+2,6k+3,6k+4 | 40 | 19 |
math | 1. Over the summer, the price of a one-bedroom apartment increased by $21 \%$, a two-bedroom apartment by $11 \%$, and the total cost of the apartments by $15 \%$. How many times cheaper is the one-bedroom apartment compared to the two-bedroom apartment? | 1.5 | 59 | 3 |
math | 2. (5 points) There are three natural numbers, their sum is 2015, the sums of each pair are $m+1, m+2011$ and $m+2012$, then $m=$ $\qquad$ | 2 | 56 | 1 |
math | 5. For a non-empty subset $X$ of the set $\{1,2,3, \ldots, 42\}$, denote by $v(X)$ the sum of the elements of the set $X$. (For example, $v(\{1,3,8\})=12$.)
Calculate the sum of all numbers $v(X)$, as $X$ ranges over all non-empty subsets of the set $\{1,2,3, \ldots, 42\}$.
Solve the problem independently. You have 1... | 21\cdot43\cdot2^{42} | 202 | 13 |
math | Find all triplets $(x, y, z)$ of real numbers for which
$$\begin{cases}x^2- yz = |y-z| +1 \\ y^2 - zx = |z-x| +1 \\ z^2 -xy = |x-y| + 1 \end{cases}$$ | \left(\frac{5}{3}, -\frac{4}{3}, -\frac{4}{3}\right), \left(\frac{4}{3}, \frac{4}{3}, -\frac{5}{3}\right) | 69 | 54 |
math | Find all values of the parameter $a$ such that the equation:
$$
a x^{2}-(a+3) x+2=0
$$
admits two real roots of opposite signs.
Proposition 1.29 (Viète's Relations in the General Case). Let $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+$ $\cdots+a_{1} x+a_{0} \in \mathbb{K}[X]$ with $a_{n} \neq 0$. If $\alpha_{1}, \ldots, \alph... | <0 | 752 | 2 |
math | An ant starts at vertex $A$ in equilateral triangle $\triangle ABC$ and walks around the perimeter of the triangle from $A$ to $B$ to $C$ and back to $A$. When the ant is $42$ percent of its way around the triangle, it stops for a rest. Find the percent of the way from $B$ to $C$ the ant is at that point | 26\% | 84 | 4 |
math | The ages of Mr. and Mrs. Fibonacci are both two-digit numbers. If Mr. Fibonacci’s age can be formed
by reversing the digits of Mrs. Fibonacci’s age, find the smallest possible positive difference between
their ages.
| 9 | 47 | 1 |
math | Determine all natural numbers $m$ and $n$ such that
\[
n \cdot (n + 1) = 3^m + s(n) + 1182,
\]
where $s(n)$ represents the sum of the digits of the natural number $n$. | (m, n) = (0, 34) | 67 | 14 |
math | 18th Putnam 1958 Problem B4 Let S be a spherical shell radius 1. Find the average straight line distance between two points of S. [In other words S is the set of points (x, y, z) with x 2 + y 2 + z 2 = 1). Solution | \frac{4}{3} | 69 | 7 |
math | 2. Let $a, b, c \geqslant 1$, and positive real numbers $x, y, z$ satisfy
$$
\left\{\begin{array}{l}
a^{x}+b^{y}+c^{z}=4, \\
x a^{x}+y b^{y}+z x^{z}=6, \\
x^{2} a^{x}+y^{2} b^{y}+z^{2} c^{z}=9 .
\end{array}\right.
$$
Then the maximum possible value of $c$ is $\qquad$ | \sqrt[3]{4} | 130 | 7 |
math | Example 5 Simplify $\frac{\sin x+\sin 3 x+\sin 5 x+\cdots+\sin (2 n-1) x}{\cos x+\cos 3 x+\cos 5 x+\cdots+\cos (2 n-1) x}$. | \tannx | 60 | 3 |
math | 3A. The numbers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ are consecutive terms of an arithmetic progression, and the numbers $\cos \alpha_{1}, \cos \alpha_{2}, \ldots, \cos \alpha_{n}$ are also consecutive terms of an arithmetic progression. Determine $n$, if $\cos \alpha_{1}=\frac{1}{2}$, and $\cos \alpha_{n}=-\fr... | 3 | 106 | 1 |
math | [b]p10.[/b] Square $ABCD$ has side length $n > 1$. Points $E$ and $F$ lie on $\overline{AB}$ and $\overline{BC}$ such that $AE = BF = 1$. Suppose $\overline{DE}$ and $\overline{AF}$ intersect at $X$ and $\frac{AX}{XF} = \frac{11}{111}$ . What is $n$?
[b]p11.[/b] Let $x$ be the positive root of $x^2 - 10x - 10 = 0$. C... | -50 | 230 | 3 |
math | Task A-2.4. (8 points)
Given are complex numbers $z=7-i, w=-3+4 i$. Determine $\left|\frac{z^{20}}{\bar{w}^{10}}\right|$. | 10^{10} | 52 | 6 |
math | Problem 1. Vasya thought of a two-digit number, then added the digit 1 to the left of it, and the digit 8 to the right, which increased the number by 28 times. What number could Vasya have thought of? (Find all options and prove that there are no others.) | 56 | 67 | 2 |
math | How many triangles are there in which the measures of the angles - measured in degrees - are integers? | 2700 | 20 | 4 |
math | 10.244. A circle with a radius of 3 cm is inscribed in a triangle. Calculate the lengths of the sides of the triangle if one of them is divided by the point of tangency into segments of 4 and $3 \mathrm{~cm}$. | 24 | 59 | 2 |
math | Find all functions $f:\mathbb{Q}^{+} \to \mathbb{Q}^{+}$ such that for all $x\in \mathbb{Q}^+$: [list] [*] $f(x+1)=f(x)+1$, [*] $f(x^2)=f(x)^2$. [/list] | f(x) = x | 74 | 6 |
math | Find the triplets of real numbers $x, y, z$ satisfying:
$$
\begin{aligned}
x+y+z & =2 \\
x^{2}+y^{2}+z^{2} & =26 \\
x^{3}+y^{3}+z^{3} & =38
\end{aligned}
$$
## Snacking | 1,-3,4 | 78 | 5 |
math | Find the sum of all positive integers whose largest proper divisor is $55$. (A proper divisor of $n$ is a divisor that is strictly less than $n$.)
| 550 | 36 | 3 |
math | A certain right-angled triangle has an area $t=121.5 \mathrm{~cm}^{2}$; one of its angles $\alpha=36^{\circ} 52^{\prime} 10.7^{\prime \prime}$; calculate the surface area and volume of the double cone that is formed by the triangle's rotation around its hypotenuse. $\pi=3.14159$. | 1068.75 | 94 | 7 |
math | Example 2.1.4 Two organizations, A and B, have a total of 11 members, with 7 from organization A and 4 from organization B. A 5-person team is to be formed from these members. (1) If the team must include 2 members from organization B; (2) If the team must include at least 2 members from organization B; (3) If a specif... | 378 | 121 | 3 |
math | Let $\mathcal E$ be an ellipse with foci $F_1$ and $F_2$. Parabola $\mathcal P$, having vertex $F_1$ and focus $F_2$, intersects $\mathcal E$ at two points $X$ and $Y$. Suppose the tangents to $\mathcal E$ at $X$ and $Y$ intersect on the directrix of $\mathcal P$. Compute the eccentricity of $\mathcal E$.
(A [i]par... | \frac{2 + \sqrt{13}}{9} | 270 | 14 |
math | \section*{Problem 4 - 141244}
Determine all pairs \((x, y)\) of real numbers \(x\) and \(y\) for which the equations hold:
\[
\begin{array}{r}
24 x^{2}-25 x y-73 x+25 y-35=0 \\
x^{2}-y^{2}-2 x-2 y-7=0
\end{array}
\] | (5,2)(-3,-4) | 101 | 10 |
math | 16. A store received fewer than 600 but more than 500 plates. When they started arranging them in tens, they were short of three plates to make a complete number of tens, and when they started arranging them in dozens (12 plates), there were 7 plates left. How many plates were there? | 547 | 69 | 3 |
math | 1000. Determine the domain of convergence of the functional series:
$$
\text { 1) } \sum_{n=1}^{+\infty} \frac{1}{n(x+2)^{n}} ; \quad \text { 2) } \sum_{n=1}^{+\infty} n \sqrt[3]{\sin ^{n} x}
$$ | \mathbb{R}\setminus{\frac{\pi}{2}+\pik\midk\in\mathbb{Z}} | 86 | 29 |
math | II If one side of the square $A B C D$ lies on the line $y=2 x-17$, and the other two vertices are on the parabola $y=x^{2}$. Then the minimum value of the area of the square is $\qquad$ . | 80 | 60 | 2 |
math | $4 \cdot 74$ If $\alpha, \beta, \gamma$ are the roots of the equation $x^{3}-x-1=0$, find
$$\frac{1+\alpha}{1-\alpha}+\frac{1+\beta}{1-\beta}+\frac{1+\gamma}{1-\gamma}$$
the value. | -7 | 76 | 2 |
math | 2A. Determine how many ordered pairs of natural numbers $(x, y)$ exist for which $\operatorname{LCM}(x, y)=6$! | 135 | 33 | 3 |
math | 1. Determine the greatest value of the natural number $n$ such that the number $39^{n}$ is a divisor of the number 39! Explain your answer. | 3 | 37 | 1 |
math | ## Problem Statement
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$\left\{\begin{array}{l}x=\cos t+\sin t \\ y=\sin 2 t\end{array}\right.$ | 2 | 60 | 1 |
math | 7.3. When the passengers entered the empty tram, half of them took seats. How many passengers entered at the very beginning, if after the first stop their number increased by exactly $8 \%$ and it is known that the tram can accommodate no more than 70 people? | 50 | 58 | 2 |
math | G3.1 If $a$ is the smallest real root of the equation $\sqrt{x(x+1)(x+2)(x+3)+1}=71$, find the value of $a$. | -10 | 43 | 3 |
math | Task 1 - 150821 The weighing of a container filled with water resulted in a total mass (container and water mass) of 2000 g. If 20% of the water is poured out, the weighed total mass decreases to 88%.
Calculate the mass of the empty container! | 800\mathrm{~} | 68 | 8 |
math | 6.18. $f(x)=\frac{x^{4}(1+x)^{5}}{\sqrt{(x+8)(x-2)}}$. | f'(x)=\frac{x^4(1+x)^5}{\sqrt{(x+8)(x-2)}}\cdot(\frac{4}{x}+\frac{5}{1+x}-\frac{1}{2}\cdot\frac{1}{x+8}-\frac{1}{2}\cdot\frac{1}{x-2}) | 33 | 77 |
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