Fathom-R1 Datasets
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20 | The steamboat "Rarity" travels for three hours at a constant speed after leaving the city, then drifts with the current for an hour, then travels for three hours at the same speed, and so on. If the steamboat starts its journey in city A and goes to city B, it takes it 10 hours. If it starts in city B and goes to city A, it takes 15 hours. How long would it take to travel from city A to city B on a raft? | 60 |
46 | Show that the modified Euclidean algorithm computes the greatest common divisor of the integers \(a\) and \(b\) in time \(\mathcal{O}(\log(a)^2 + \log(b)^2)\). | \mathcal{O}(\log(a)^2 + \log(b)^2) |
58 | Let \( P \) be a point inside the isosceles trapezoid \( ABCD \) where \( AD \) is one of the bases, and let \( PA, PB, PC, \) and \( PD \) bisect angles \( A, B, C, \) and \( D \) respectively. If \( PA = 3 \) and \( \angle APD = 120^\circ \), find the area of trapezoid \( ABCD \). | 6\sqrt{3} |
64 | 4. The sequence $\left(a_{n}\right)_{n=1}^{\infty}$ is defined as
$$
a_{n}=\sin ^{2} \pi \sqrt{n^{2}+n}
$$
Determine
$$
\lim _{n \rightarrow \infty} a_{n}
$$ | 1 |
68 | 16. On Sunday, after Little Fat got up, he found that his alarm clock was broken. He estimated the time and set the alarm clock to 7:00, then he walked to the science museum and saw that the electronic clock on the roof pointed to 8:50. He visited the museum for one and a half hours and returned home at the same speed. When he got home, Little Fat saw that the alarm clock pointed to 11:50. Try to answer: At this time, to what time should the alarm clock be set to be accurate? | 12:00 |
72 | A $1 \times 2014$ grid with 2014 cells is labeled from left to right with $1, 2, \cdots, 2014$. We use three colors $g, r, y$ to color each cell such that even-numbered cells can be colored with any of $g, r, y$ and odd-numbered cells can only be colored with either $g$ or $y$. Additionally, any two adjacent cells must be of different colors. How many possible ways are there to color this grid? | 4 \times 3^{1006} |
100 | 1. 50 students from fifth to ninth grade published a total of 60 photos on Instagram, each not less than one. All students of the same grade (same parallel) published an equal number of photos, while students of different grades (different parallels) published a different number. How many students published only one photo? | 46 |
132 | Senya has three straight sticks, each 24 centimeters long. He broke one of them into two parts such that from these two pieces and the two whole sticks, he was able to form the outline of a right triangle. How many square centimeters is the area of this triangle? | 216 |
140 | 3. In the tournament, each participant was supposed to play exactly one game with each of the remaining participants, but two participants dropped out during the tournament, having played only 4 games each. In the end, the total number of games played turned out to be 62. How many participants were there in total? | 13 |
142 | As shown in Figure 1.8.13, in any quadrilateral $ABCD$, the midpoints of diagonals $AC$ and $BD$ are $M$ and $N$, respectively. The extensions of $BA$ and $CD$ intersect at $O$. Prove that $4 S_{\triangle OMN} = S_{ABCD}$. | 4 S_{\triangle OMN} = S_{ABCD} |
249 | # Task 8.2 (7 points)
The shares of the company "Nu-i-Nu" increase in price by 10 percent every day. Businessman Borya bought shares of the company for 1000 rubles every day for three days in a row, and on the fourth day, he sold them all. How much money did he make from this operation? | 641 |
273 | Given that point \( F \) is the right focus of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a > b > 0\)), and the eccentricity of the ellipse is \(\frac{\sqrt{3}}{2}\), a line \( l \) passing through point \( F \) intersects the ellipse at points \( A \) and \( B \) (point \( A \) is above the \( x \)-axis), and \(\overrightarrow{A F} = 3 \overrightarrow{F B}\). Find the slope of the line \( l \). | -\sqrt{2} |
286 | The sequence \( (a_i) \) is defined as follows: \( a_1 = 0 \), \( a_2 = 2 \), \( a_3 = 3 \), and for \( n = 4, 5, 6, \ldots \), \( a_n = \max_{1 < d < n} \{ a_d \cdot a_{n-d} \} \). Determine the value of \( a_{1998} \). | 3^{666} |
294 |
Let $x, y, z$ be positive numbers satisfying the following system of equations:
$$
\left\{\begin{array}{l}
x^{2} + xy + y^{2} = 12 \\
y^{2} + yz + z^{2} = 9 \\
z^{2} + xz + x^{2} = 21
\end{array}\right.
$$
Find the value of the expression $xy + yz + xz$. | 12 |
297 | In \(\triangle PMO\), \(PM = 6\sqrt{3}\), \(PO = 12\sqrt{3}\), and \(S\) is a point on \(MO\) such that \(PS\) is the angle bisector of \(\angle MPO\). Let \(T\) be the reflection of \(S\) across \(PM\). If \(PO\) is parallel to \(MT\), find the length of \(OT\). | 2\sqrt{183} |
299 | A Christmas garland hanging along a school corridor consists of red and blue bulbs. Next to each red bulb, there must be a blue one. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs? | 33 |
301 | Example 2 Choose five different numbers from $1,2, \cdots, 20$, the probability that at least two of them are consecutive numbers is $\qquad$ . [1] | \dfrac{232}{323} |
320 | 5. The six edges of tetrahedron $ABCD$ have lengths $7, 13, 18, 27, 36, 41$, and it is known that $AB=41$, then $CD=$ | 13 |
331 | Find all four-digit numbers that have exactly five four-digit and exactly nine one-digit divisors.
( $S$. Bednářová) | 5040 |
338 | Given that \( E, F, D \) are the midpoints of the sides \([AB],[BC],[CA]\) of a triangle \(ABC\) respectively. A perpendicular height \([BG]\) is drawn. Show that \(\widehat{EGF} = \widehat{EDF}\). | \widehat{EGF} = \widehat{EDF} |
359 | ## 6. Pretty Numbers
A natural number is called pretty if its unit digit is equal to the product of all the remaining digits. How many four-digit pretty numbers are there?
## Result: $\quad 215$ | 215 |
368 | N1) Determine all integer values that the expression
$$
\frac{p q+p^{p}+q^{q}}{p+q}
$$
can take, where $p$ and $q$ are both prime numbers.
Answer: The only possible integer value is 3 . | 3 |
371 | Problem 2. Rectangle $ABCD$ with side lengths $8 \mathrm{~cm}$ and $5 \mathrm{~cm}$ is divided by segment $MN$ into two quadrilaterals with equal perimeters of $20 \mathrm{~cm}$ each. Determine the length of segment $MN$. | 7 |
403 | Show that the characteristic function $\varphi=\varphi(t)$ of any absolutely continuous distribution with density $f=f(x)$ can be represented in the form
$$
\varphi(t)=\int_{\mathbb{R}} \phi(t+s) \overline{\phi(s)} \, d s, \quad t \in \mathbb{R}
$$
for some complex-valued function $\phi$ that satisfies the condition $\int_{\mathbb{R}}|\phi(s)|^{2} \, d s=1$. Assume for simplicity that
$$
\int_{\mathbb{R}} \sqrt{f(x)} \, d x<\infty, \quad \int_{\mathbb{R}} d t \left|\int_{\mathbb{R}} e^{i t x} \sqrt{f(x)} \, d x\right|<\infty.
$$ | \varphi(t)=\int_{\mathbb{R}} \phi(t+s) \overline{\phi(s)} \, d s |
422 |
Svetlana takes a triplet of numbers and transforms it by the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and smallest numbers in the triplet on the 1580th step of applying this rule, if the initial triplet of numbers was $\{80, 71, 20\}$? If the problem allows for multiple answers, list them all as a set. | 60 |
424 | [ [Decimal numeral system ]
The numbers $2^{2000}$ and $5^{2000}$ are written in sequence. How many digits are written in total?
# | 2001 |
433 | If \( x, y \in \mathbf{R} \) and \(3x^{2} + 2y^{2} = 2x\), then the maximum value of \(x^{2} + y^{2}\) is? | \dfrac{4}{9} |
437 | Grandpa prepared a pile of hazelnuts for his six grandchildren, asking them to divide them somehow. First, Adam came, counted half for himself, took one more nut, and left. The second, Bob, the third, Cyril, the fourth, Dan, and the fifth, Eda, all did the same. Only Franta sadly looked at the empty table; no nuts were left for him.
How many nuts were there in the pile originally?
(M. Volfová)
Hint. How many nuts did Eda take? | 62 |
438 | 10. The notation $[a]$ represents the greatest integer not exceeding $a$, for example $\left[\frac{17}{5}\right]=3,[0.888 \cdots]=0$, then $\left[\frac{1}{100}\right]+\left[\frac{2}{100}\right]+\left[\frac{3}{100}\right]+\cdots+\left[\frac{865}{100}\right]=$. | 3328 |
467 | Find all the roots of the equation
\[ 1 - \frac{x}{1} + \frac{x(x-1)}{2!} - \frac{x(x-1)(x-2)}{3!} + \frac{x(x-1)(x-2)(x-3)}{4!} - \frac{x(x-1)(x-2)(x-3)(x-4)}{5!} + \frac{x(x-1)(x-2)(x-3)(x-4)(x-5)}{6!} = 0 \]
(Where \( n! = 1 \cdot 2 \cdot 3 \cdots n \))
In the answer, specify the sum of the found roots. | 21 |
470 | 2. How many integers $x, -1000 < x < 1000$ are divisible by 3, and how many integers $y, -444 < y < 444$ are not divisible by 4? Which is greater? | 667 |
485 | Given the sequence $\{a_n\}$ satisfying
\[
a_{n+1} + (-1)^n a_n = 2n - 1,
\]
and the sum of the first 2019 terms of the sequence $\{a_n - n\}$ is 2019, find the value of $a_{2020}$. | 1 |
500 | Given a moving point \( P \) on the \( x \)-axis, \( M \) and \( N \) lie on the circles \((x-1)^{2}+(y-2)^{2}=1\) and \((x-3)^{2}+(y-4)^{2}=3\) respectively. Find the minimum value of \( |PM| + |PN| \). | 2\sqrt{10} - 1 - \sqrt{3} |
502 | Let the following system of equations hold for positive numbers \(x, y, z\):
\[ \left\{\begin{array}{l}
x^{2}+x y+y^{2}=48 \\
y^{2}+y z+z^{2}=25 \\
z^{2}+x z+x^{2}=73
\end{array}\right. \]
Find the value of the expression \(x y + y z + x z\). | 40 |
505 | In a $4 \times 4$ grid, numbers are written in such a way that the sum of the numbers in the neighboring cells (cells that share a side) of each number is equal to 1.
Find the sum of all the numbers in the grid. | 6 |
549 | Show that a $\sigma$-algebra $\mathscr{G}$ is countably generated if and only if $\mathscr{G} = \sigma(X)$ for some random variable $X$. | \mathscr{G} \text{ is countably generated if and only if } \mathscr{G} = \sigma(X) \text{ for some random variable } X |
565 | ## Task 6
Calculate the numbers.
- A is four times the smallest three-digit number,
- B is half of the smallest four-digit number,
- C is the sum of D and E,
- D is $\mathrm{B}$ minus $\mathrm{A}$,
- E is the sum of B and D. | 700 |
584 | Problem 7. How many solutions in integers does the equation
$6 y^{2}+3 x y+x+2 y-72=0$ have? | 4 |
606 | Find the minimum of the function
$$
f(x)=\frac{\cos (x)^{3}}{\sin (x)}+\frac{\sin (x)^{3}}{\cos (x)}
$$
on $] 0, \pi / 2[$.
Theorem 1.70 (Tchebycheff's Inequality). With the same notations as before, if the $a_{i}$ and the $b_{i}$ are arranged in the same order, then
$$
\sum a_{i} b_{i} \geq \frac{1}{n}\left(\sum a_{i}\right)\left(\sum b_{i}\right)
$$
Proof. We use the rearrangement inequality to find the following $n$ relations
$$
\begin{aligned}
a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n} & \geq a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n} \\
a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n} & \geq a_{1} b_{2}+a_{2} b_{3}+\cdots+a_{n} b_{1} \\
a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n} & \geq a_{1} b_{3}+a_{2} b_{4}+\cdots+a_{n} b_{2} \\
\cdots & \cdots \\
a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n} & \geq a_{1} b_{n}+a_{2} b_{1}+\cdots+a_{n} b_{n-1}
\end{aligned}
$$
And by adding these relations we obtain the desired inequality. | 1 |
633 | Variant 8.1.4. Dima and Seryozha were picking berries from a raspberry bush, on which 450 berries grew. Seryozha alternated actions while picking berries: he put one berry in the basket and ate the next one. Dima also alternated: he put two berries in the basket and ate the next one. It is known that Seryozha picks berries twice as fast as Dima. At some point, the boys had picked all the raspberries from the bush.
Who of them ended up putting more berries in the basket? What will the difference be? | 50 |
646 | Let \( n > 6 \) and \( a_1 < a_2 < \cdots < a_k \) be all natural numbers that are less than \( n \) and relatively prime to \( n \). Show that if \( \alpha_1, \alpha_2, \ldots, \alpha_k \) is an arithmetic progression, then \( n \) is a prime number or a natural power of two. | n \text{ is a prime number or a natural power of two} |
655 | Let \( p \) and \( q \) be complex numbers (\( q \neq 0 \)). If the magnitudes of the roots of the equation \( x^{2} + p x + q^{2} = 0 \) are equal, prove that \( \frac{p}{q} \) is a real number. | \frac{p}{q} \text{ is a real number} |
665 | Let $A$ be a ten-billion digit, positive number divisible by nine. The sum of the digits of $A$ is $B$, the sum of the digits of $B$ is $C$. What is the sum of the digits of $C$? | 9 |
666 | 4. Construct on the plane the set of points whose coordinates satisfy the inequality $|3 x+4|+|4 y-3| \leq 12$. In your answer, specify the area of the resulting figure.
ANSWER: 24. | 24 |
672 | 6. (10 points) A beam of light with a diameter of $d_{1}=10 \, \text{cm}$ falls on a thin diverging lens with an optical power of $D_{p}=-6$ Diopters. On a screen positioned parallel to the lens, a bright spot with a diameter of $d_{2}=20 \, \text{cm}$ is observed. After replacing the thin diverging lens with a thin converging lens, the size of the spot on the screen remains unchanged. Determine the optical power $D_{c}$ of the converging lens.
# | 18 |
683 | Let \( n \geq 5 \) be a natural number and let \( n \) distinct natural numbers \( a_{1}, a_{2}, \ldots, a_{n} \) have the following property: for any two different non-empty subsets \( A \) and \( B \) of the set
\[ S = \{a_{1}, a_{2}, \ldots, a_{n} \} \]
the sums of all the numbers in \( A \) and \( B \) are not equal. Under these conditions, find the maximum value of
\[ \frac{1}{a_{1}} + \frac{1}{a_{2}} + \cdots + \frac{1}{a_{n}}. \] | 2 - \frac{1}{2^{n-1}} |
690 | Let \( x, y, z \in [0, 1] \). The maximum value of \( M = \sqrt{|x-y|} + \sqrt{|y-z|} + \sqrt{|z-x|} \) is ______ | 1 + \sqrt{2} |
705 | Calculate the surface area of the part of the paraboloid of revolution \( 3y = x^2 + z^2 \) that is located in the first octant and bounded by the plane \( y = 6 \). | \dfrac{39}{4}\pi |
720 | Problem 4.3. Zhenya drew a square with a side of 3 cm, and then erased one of these sides. A figure in the shape of the letter "P" was obtained. The teacher asked Zhenya to place dots along this letter "P", starting from the edge, so that the next dot was 1 cm away from the previous one, as shown in the picture, and then count how many dots he got. He got 10 dots.
Then the teacher decided to complicate the task and asked to count the number of dots, but for the letter "P" obtained in the same way from a square with a side of 10 cm. How many dots will Zhenya have this time? | 31 |
743 | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(\frac{x^{2}+4}{x+2}\right)^{x^{2}+3}$ | 8 |
759 | Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i > 1$ for $1 \leq i \leq n$, let $R = \frac{1}{n} \sum_{i=1}^{n} r_i$, $S = \frac{1}{n} \sum_{i=1}^{n} s_i$, $T = \frac{1}{n} \sum_{i=1}^{n} t_i$, $U = \frac{1}{n} \sum_{i=1}^{n} u_i$, and $V = \frac{1}{n} \sum_{i=1}^{n} v_i$. Prove that:
\[
\prod_{i=1}^{n} \left(\frac{r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \right) \geq \left( \frac{R S T U V + 1}{R S T U V - 1} \right)^{n}.
\] | \prod_{i=1}^{n} \left(\frac{r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \right) \geq \left( \frac{R S T U V + 1}{R S T U V - 1} \right)^{n} |
766 | Let the set
\[ S=\{1, 2, \cdots, 12\}, \quad A=\{a_{1}, a_{2}, a_{3}\} \]
where \( a_{1} < a_{2} < a_{3}, \quad a_{3} - a_{2} \leq 5, \quad A \subseteq S \). Find the number of sets \( A \) that satisfy these conditions. | 185 |
771 | A rectangular cuboid \(A B C D-A_{1} B_{1} C_{1} D_{1}\) has \(A A_{1} = 2\), \(A D = 3\), and \(A B = 251\). The plane \(A_{1} B D\) intersects the lines \(C C_{1}\), \(C_{1} B_{1}\), and \(C_{1} D_{1}\) at points \(L\), \(M\), and \(N\) respectively. What is the volume of tetrahedron \(C_{1} L M N\)? | 2008 |
776 | Two spheres touch the plane of triangle \(ABC\) at points \(A\) and \(B\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 9, and the distance between their centers is \(\sqrt{305}\). The center of a third sphere with a radius of 7 is at point \(C\), and it externally touches each of the first two spheres. Find the radius of the circumcircle of triangle \(ABC\). | 2\sqrt{14} |
796 | Let G, O, D, I, and T be digits that satisfy the following equation:
\begin{tabular}{ccccc}
&G&O&G&O\\
+&D&I&D&I\\
\hline
G&O&D&O&T
\end{tabular}
(Note that G and D cannot be $0$, and that the five variables are not necessarily different.)
Compute the value of GODOT. | 10908 |
810 | 19.6. In the institute, there are truth-lovers, who always tell the truth, and liars, who always lie. One day, each employee made two statements.
1) There are not even ten people in the institute who work more than I do.
2) At least a hundred people in the institute earn more than I do.
It is known that the workload of all employees is different, and so are their salaries. How many people work in the institute? | 110 |
821 | 1. Determine all prime numbers of the form $1+2^{p}+3^{p}+\ldots+p^{p}$ where $p$ is a prime number. | 5 |
828 | 32. Find the remainder when the 2017-digit number $\underbrace{7777}_{2017 \text { 7s }}$ is divided by 30. | 7 |
832 | Given the vectors \(\boldsymbol{a} = (x, 1)\), \(\boldsymbol{b} = (2, y)\), and \(\boldsymbol{c} = (1,1)\), and knowing that \(\boldsymbol{a} - \boldsymbol{b}\) is collinear with \(\boldsymbol{c}\). Find the minimum value of \( |\boldsymbol{a}| + 2|\boldsymbol{b}| \). | 3\sqrt{5} |
844 | Margot wants to prepare a jogging timetable. She wants to jog exactly twice a week, and on the same days every week. She does not want to jog on two consecutive days. How many different timetables could Margot prepare? | 14 |
847 | 19. The airplane robot Roly is leading all the airplane robots to prepare for an air show. If each performance team consists of 12 airplane robots, there are still 5 airplane robots short; if each performance team consists of 15 airplane robots, there are still 8 airplane robots short. It is known that the total number of airplane robots is greater than 15 and less than 100. Therefore, there are $\qquad$ airplane robots. | 67 |
850 | Find the envelope of a family of straight lines that form a triangle with a right angle and have an area of $a^{2} / 2$. | xy = \dfrac{a^2}{4} |
862 | 7. What is the minimum number of cells that need to be painted in a square with a side of 35 cells (a total of $35 \times 35$ cells, which is 1225 cells in the square), so that from any unpainted cell it is impossible to move to another unpainted cell with a knight's move in chess? | 612 |
922 | 52. Xiao Fei's class forms a rectangular array for morning exercises. There is 1 classmate to Xiao Fei's left, and 2 classmates to his right. Counting from the front, he is the 6th, and counting from the back, he is the 5th. Therefore, the total number of students in Xiao Fei's class is. $\qquad$ | 40 |
936 | In a chess tournament, 12 participants played. After the tournament, each participant compiled 12 lists. The first list includes only the participant himself, the second list includes himself and those he won against, the third list includes everyone from the second list and those they won against, and so on. The twelfth list includes everyone from the eleventh list and those they won against. It is known that for any participant, there is a person in their twelfth list who was not in their eleventh list. How many drawn games were played in the tournament? | 54 |
948 | 1. Vasya's dad is good at math, but on the way to the garage, he forgot the code for the digital lock on the garage. In his memory, he recalls that all the digits of the code are different and their sum is 28. How many different codes does dad need to try to definitely open the garage, if the opening mechanism of the lock consists of four disks with a complete set of digits on each? | 48 |
967 | Problem 4.8. If in the number 79777 the digit 9 is crossed out, the number 7777 is obtained. How many different five-digit numbers exist from which 7777 can be obtained by crossing out one digit? | 45 |
984 | Example 1 Allocate 24 volunteer slots to 3 schools. Then the number of allocation methods where each school gets at least one slot and the number of slots for each school is different is $\qquad$ kinds. ${ }^{[2]}$ | 222 |
1,008 | In a trapezoid $ABCD$ with $\angle A = \angle B = 90^{\circ}$, $|AB| = 5 \text{cm}$, $|BC| = 1 \text{cm}$, and $|AD| = 4 \text{cm}$, point $M$ is taken on side $AB$ such that $2 \angle BMC = \angle AMD$. Find the ratio $|AM| : |BM|$. | \dfrac{3}{2} |
1,012 | 36. Find the sum of all positive three-digit numbers that are not divisible by 2 or 3. | 164700 |
1,043 | Given that the ellipse \(x^{2} + 4(y - a)^{2} = 4\) and the parabola \(x^{2} = 2y\) have a common point, determine the range of values for \(a\). | [-1, \dfrac{17}{8}] |
1,067 | 4. The diagram below shows $\triangle A B C$, which is isoceles with $A B=A C$ and $\angle A=20^{\circ}$. The point $D$ lies on $A C$ such that $A D=B C$. The segment $B D$ is constructed as shown. Determine $\angle A B D$ in degrees. | 10 |
1,081 | Luis wrote the sequence of natural numbers starting from 1:
$$
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, \cdots
$$
When did he write the digit 3 for the 25th time? | 131 |
1,090 | 3. Solve the equation $\mathrm{p}^{3}-\mathrm{q}^{3}=2716$ in prime numbers. In your answer, specify the pair $(p ; q)$.
If there are several such pairs, specify the one for which $p-q$ is maximal. | (17; 13) |
1,113 | Find a natural number \( N \) that is divisible by 5 and 49, and has exactly 10 divisors, including 1 and \( N \). | 12005 |
1,120 | 10. (5 points) Write down the natural numbers from 1 to 2015 in sequence, to get a large number $123456789 \cdots 20142015$. When this large number is divided by 9, the remainder is | 0 |
1,154 | 15. (6 points) There are 4 different digits that can form a total of 18 different four-digit numbers, which are arranged in ascending order. The first number is a perfect square, and the second-to-last four-digit number is also a perfect square. What is the sum of these two numbers? $\qquad$ | 10890 |
1,163 | (10) The number of ordered integer tuples $\left(k_{1}, k_{2}, k_{3}, k_{4}\right)$ that satisfy $0 \leqslant k_{i} \leqslant 20, i=1,2,3,4$, and $k_{1}+k_{3}=k_{2}+k_{4}$ is $\qquad$. | 6181 |
1,218 | 9.2. On the board in the laboratory, two numbers are written. Every day, the senior researcher Pyotr Ivanovich erases both numbers from the board and writes down their arithmetic mean and harmonic mean ${ }^{2}$. In the morning of the first day, the numbers 1 and 2 were written on the board.
Find the product of the numbers written on the board in the evening of the 2016th day. | 2 |
1,223 | 194. Find the derivative of the function $y=x^{3}$.
翻译完成，保留了原文的格式和换行。 | 3x^{2} |
1,245 | In a certain city, license plate numbers consist of 6 digits (ranging from 0 to 9), but it is required that any two plates differ by at least 2 digits (for example, license plates 038471 and 030471 cannot both be used). Determine the maximum number of different license plates that can be issued and provide a proof. | 100000 |
1,254 | From the sequence of natural numbers \(1, 2, 3, 4, 5, \cdots\), remove every number that is a multiple of 3 or 4, but keep any number that is a multiple of 5 (for example, 15 and 120 are retained). After performing these steps, the remaining numbers form a new sequence: \(1, 2, 5, 7, 10, \cdots\). Find the 2010th term in this new sequence. | 3349 |
1,270 | On the complex plane, the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$ has area $\frac{35}{37}.$ If the real part of $z$ is positive, let $d$ be the smallest possible value of $\left| z + \frac{1}{z} \right|.$ Compute $d^2.$ | \dfrac{50}{37} |
1,294 | A magician has a set of \(12^2\) different cards. Each card has one red side and one blue side; on each card, a natural number from 1 to 12 is written on both sides. We call a card a duplicate if the numbers on both sides of the card match. The magician wants to draw two cards such that at least one of them is a duplicate and no number appears simultaneously on both drawn cards. In how many ways can he achieve this? | 1386 |
1,323 | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt[3]{n^{2}+2}-5 n^{2}}{n-\sqrt{n^{4}-n+1}}$ | 5 |
1,326 | Perform the following operation on a natural number: if it is even, divide it by 2; if it is odd, add 1. Continue this process until the result is 1. How many numbers require exactly 8 operations to reach the result of 1? | 21 |
1,334 | 1. Let $A$ be a set of three distinct real numbers, and let the set $B=\{x+y \mid x 、 y \in A, x \neq y\}$. If
$$
B=\left\{\log _{2} 6, \log _{2} 10, \log _{2} 15\right\} \text {, }
$$
then the set $A=$ $\qquad$ | \{1, \log_{2} 3, \log_{2} 5\} |
1,387 | Given vectors $\vec{a} = \left(x^2, x + 1\right)$ and $\vec{b} = (1 - x, t)$, find the range of values for $t$ such that the function $f(x) = \vec{a} \cdot \vec{b}$ is monotonically increasing on the interval $(-1, 1)$. | [5, \infty) |
1,398 | Example 8. Solve the Bernoulli equation
$$
x y^{\prime}+y=y^{2} \ln x
$$ | y = \dfrac{1}{\ln x + 1 + C x} |
1,423 | 4. Let's fix a point $O$ on the plane. Any other point $A$ is uniquely determined by its radius vector $\overrightarrow{O A}$, which we will simply denote as $\vec{A}$. We will use this concept and notation repeatedly in the following. The fixed point $O$ is conventionally called the initial point. We will also introduce the symbol $(A B C)$ - the simple ratio of three points $A, B, C$ lying on the same line; points $A$ and $B$ are called the base points, and point $C$ is the dividing point. The symbol $(A B C)$ is a number determined by the formula:
$$
\overrightarrow{C A}=(A B C) \cdot \overrightarrow{C B}
$$
Furthermore, for two collinear vectors $\vec{a}$ and $\vec{b}$, there is a unique number $\lambda$ such that $\vec{a}=\lambda \vec{b}$. We introduce the notation $\lambda=\frac{\vec{a}}{\vec{b}}$ (the ratio of collinear vectors). In this case, we can write:
$$
(A B C)=\frac{\overrightarrow{C A}}{\overrightarrow{C B}}
$$
Remark. The notation for the ratio of vectors is not related to the scalar product of vectors! (For example, the relation $\frac{\vec{a}}{\vec{b}}=\frac{\vec{c}}{\vec{d}}$ implies $\vec{a}=\lambda \vec{b}$ and $\vec{c}=\lambda \vec{d}$, from which $\vec{a} \cdot \vec{d}=(\lambda \vec{b}) \cdot \left(\frac{1}{\lambda} \vec{c}\right)=\vec{b} \cdot \vec{c}$, but the converse is not true - from the relation $\vec{a} \cdot \vec{d}=\vec{b} \cdot \vec{c}$ it does not follow that $\frac{\vec{a}}{\vec{b}}=\frac{\vec{c}}{\vec{d}}$ : vectors $\vec{a}$ and $\vec{b}$ do not have to be collinear. Provide a corresponding example.)
Derive a formula expressing the vector $\vec{C}$ in terms of the vectors $\vec{A}$ and $\vec{B}$, given that $(A B C)=k$. | \dfrac{\vec{A} - k\vec{B}}{1 - k} |
1,453 | Find the radius of a circle in which a segment corresponding to a chord of length 6 cm contains a square with a side length of 2 cm. | \sqrt{10} |
1,467 | Show that if there exists a prime number \( p \) such that \( n \equiv 0 \pmod{p^2} \), then there are at most \( \frac{n-1}{p} \) dishonest numbers. | \frac{n-1}{p} |
1,475 | A circle with a radius of 6 is inscribed around the trapezoid \(ABCD\). The center of this circle lies on the base \(AD\), and \(BC = 4\). Find the area of the trapezoid. | 32\sqrt{2} |
1,508 | There are two ingots of different alloys of copper and tin weighing 6 kg and 12 kg, respectively. Identical pieces were cut from each of them, and the first piece was fused with the remnants of the second ingot, while the second piece was fused with the remnants of the first ingot. As a result, the ratio of copper to tin in the two new ingots was the same. Find the weight of each of the cut pieces. | 4 |
1,520 | ## Task 1 - 180521
The tracks of the BAM will have a total length of $3200 \mathrm{~km}$ upon completion. Every $1 \mathrm{~m}$ of track corresponds to $2 \mathrm{~m}$ of rail.
How many tons of steel will be needed for the rails of the BAM in total, if $65 \mathrm{~kg}$ of steel is required for every $1 \mathrm{~m}$ of rail? | 416000 |
1,527 |
The bases of a truncated pyramid are two regular octagons. The side of the lower base of the pyramid is 0.4 m, and the side of the upper base is 0.3 m; the height of the truncated pyramid is 0.5 m. The truncated pyramid is extended to form a complete pyramid. Determine the volume of the complete pyramid. | \dfrac{16(1 + \sqrt{2})}{75} |
1,533 | From a checkered square, a smaller square was cut out along the lines of $1 \times 1$ squares. Could there be 250 squares left after this? | No |
1,538 | A cube is circumscribed around a sphere of radius 1. From one of the centers of the cube's faces, vectors are drawn to all other face centers and vertices. The dot products of each pair of these vectors are calculated, totaling 78. What is the sum of these dot products? | 76 |
1,559 | Let \(a, b, n\) be natural numbers where \(a>1, \quad b>1, \quad n>1\). \(A_{n-1}\) and \(A_{n}\) are numbers in base \(a\), and \(B_{n-1}\) and \(B_{n}\) are numbers in base \(b\). These numbers can be represented as follows:
$$
\begin{array}{l}
A_{n-1}=\overline{x_{n-1} x_{n-2} \cdots x_{0}}, \quad A_{n}=\overline{x_{n} x_{n-1} \cdots x_{0}}, \\
B_{n-1}=\overline{x_{n-1} x_{n-2} \cdots x_{0}}, \quad B_{n}=\overline{x_{n} x_{n-1} \cdots x_{0}},
\end{array}
$$
where \(x_{n} \neq 0\) and \(x_{n-1} \neq 0\). Prove that when \(a > b\),
$$
\frac{A_{n-1}}{A_{n}} < \frac{B_{n-1}}{B_{n}}.
$$
(IMO 1970) | \frac{A_{n-1}}{A_{n}} < \frac{B_{n-1}}{B_{n}} |
1,567 | Example 9 Simplify: $\arctan \frac{1+|x|-\sqrt{1-x^{2}}}{1+|x|+\sqrt{1-x^{2}}}+\frac{1}{2} \arccos |x|(-1 \leqslant x \leqslant 1)$. | \dfrac{\pi}{4} |
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