task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | Let's rationalize the denominators of the following fractions:
$$
1^{\circ} . \quad \frac{\sqrt{3}+\sqrt{2}}{\sqrt{5-2 \sqrt{6}}}, \quad 2^{\circ} . \quad \frac{\sqrt{3+\sqrt{5}}}{\sqrt[3]{(4 \sqrt{2}-2 \sqrt{10})^{2}}}
$$ | 5+2\sqrt{6} | 93 | 8 |
math | Let's determine all integer values of $x$ for which $x^{2}+19 x+95$ is a perfect square. | -14,-5 | 30 | 5 |
math | The radius of the circumscribed circle $R$ is known.
a) Find $A P^{2}+B P^{2}+C P^{2}+D P^{2}$.
b) Find the sum of the squares of the sides of the quadrilateral $A B C D$. | 4R^2 | 63 | 4 |
math | 7. The sum of 100 positive integers is 101101. What is the maximum possible value of their greatest common divisor? Prove your conclusion. | 1001 | 37 | 4 |
math | 12.425 The ratio of the volume of a truncated cone to the volume of a sphere inscribed in it is $\boldsymbol{k}$. Find the angle between the slant height of the cone and the plane of its base and the permissible values of $k$. | \operatorname{arctg}\frac{2}{\sqrt{2k-3}} | 57 | 20 |
math | 74. Let's solve the inequality
$$
\sqrt{x+3}<\sqrt{x-1}+\sqrt{x-2}
$$ | (\frac{2}{3}\sqrt{21};+\infty) | 30 | 16 |
math | Find all positive integer $n$ such that the equation $x^3+y^3+z^3=n \cdot x^2 \cdot y^2 \cdot z^2$ has positive integer solutions. | n = 1, 3 | 43 | 7 |
math | 14. (6 points) Pleasant Goat opens a book and finds that the product of the page numbers on the left and right pages is 420. Then the sum of these two page numbers is $\qquad$ | 41 | 46 | 2 |
math | 26. In a competition of fun and ingenuity, 9 points were awarded for each correctly completed task, and 5 points were deducted for each uncompleted or incorrectly completed task. It is known that the team was offered no more than 15 tasks and scored 57 points. How many tasks did the team complete correctly? | 8 | 69 | 1 |
math | 7.2. There are broken scales that are off by no more than 500g in their readings. When Alexei weighed the melon, they showed 4kg. When he weighed the watermelon - 3 kg. When he weighed both the melon and the watermelon together - 8.5 kg. What is the actual weight of the melon and the watermelon separately? The scales c... | The\melon\weighs\4.5\,\\the\watermelon\weighs\3.5\ | 91 | 25 |
math | 2. Compare the fractions:
$$
\frac{1+2+3+\ldots+50}{2018} \text { and } \frac{1275}{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 7 \cdot 9}
$$ | \frac{1275}{2018}<\frac{1275}{2016} | 68 | 26 |
math | 2. A function $f(x)$ satisfies
$$
(2-x) f(x)-2 f(3-x)=-x^{3}+5 x-18
$$
for all real numbers $x$. Solve for $f(0)$. | 7 | 54 | 1 |
math | 40th Putnam 1979 Problem B3 F is a finite field with n elements. n is odd. x 2 + bx + c is an irreducible polynomial over F. For how many elements d ∈ F is x 2 + bx + c + d irreducible? | \frac{n-1}{2} | 61 | 8 |
math | 1. Find all pairs of real numbers (x, y) for which the equality $\sqrt{x^{2}+y^{2}-1}=x+y-1$ holds. | (1,),(,1) | 37 | 7 |
math | 5. There are 100 different cards with numbers $2,5,2^{2}, 5^{2}, \ldots, 2^{50}, 5^{50}$ (each card has exactly one number, and each number appears exactly once). In how many ways can 2 cards be chosen so that the product of the numbers on the chosen cards is a cube of an integer? | 1074 | 85 | 4 |
math | 9. The minimum value of the function $y=\sqrt{x^{2}+1}+\sqrt{x^{2}-4 x+8}$ is | \sqrt{13} | 31 | 6 |
math | 14. Given the quadratic function $f(x)$ satisfies: (1) $f(-1)=0$; (2) for all values of $x$, $x \leqslant f(x) \leqslant \frac{1+x^{2}}{2}$, find the analytical expression of $f(x)$. | f(x)=\frac{1}{4}x^{2}+\frac{1}{2}x+\frac{1}{4} | 71 | 29 |
math | 1. Write down the natural numbers from 360 to 630 that have an odd number of divisors. | 361,400,441,484,529,576,625 | 26 | 27 |
math | 9. Let $a_{1}=1, a_{n+1}=\frac{1}{16}\left(1+4 a_{n}+\sqrt{1+24 a_{n}}\right)$, find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{2^{2n-1}+3\cdot2^{n-1}+1}{3\cdot2^{2n-1}}(n\geqslant1) | 68 | 46 |
math | Let $n$ be a positive integer. There are $n$ soldiers stationed on the $n$th root of unity in the complex plane. Each round, you pick a point, and all the soldiers shoot in a straight line towards that point; if their shot hits another soldier, the hit soldier dies and no longer shoots during the next round. What is th... | \lceil \log_2(n) \rceil | 100 | 12 |
math | 1. Given that $x=-2$ is a root of the equation $x^{2}+a x+2 b=0$. Then the minimum value of $a^{2}+b^{2}$ is $\qquad$ | 2 | 49 | 1 |
math | 129. A section of maximum area is made through the vertex of a right circular cone. It is known that the area of this section is twice the area of the axial section. Find the angle at the vertex of the axial section of the cone. | \frac{5}{6}\pi | 52 | 8 |
math | 13. Let $\left\{a_{n}\right\}$ be an arithmetic sequence, $\left\{b_{n}\right\}$ be a geometric sequence, and $b$ $=a_{1}^{2}, b_{2}=a_{2}^{2}, b_{3}=a_{3}^{2}\left(a_{1}<a_{2}\right)$, and $\lim _{n \rightarrow+\infty}\left(b_{1}+b_{2}+\right.$ $\left.\cdots+b_{n}\right)=\sqrt{2}+1$. Find the first term and common dif... | a_{1}=-\sqrt{2}, d=2\sqrt{2}-2 | 147 | 19 |
math | Solve the following equation:
$$
x^{\lg \tan x}+x^{\lg \cot x}=2
$$ | x_1=1,\quadx_2=\frac{\pi}{4}\k\pi,\quadk=0,1,2,\ldots | 28 | 32 |
math | Task 4. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ that satisfy
$$
f(m+n)+f(m n-1)=f(m) f(n)+2
$$
for all $m, n \in \mathbb{Z}$. | f(n)=n^2+1 | 64 | 8 |
math | 10. [20] A positive real number $x$ is such that
$$
\sqrt[3]{1-x^{3}}+\sqrt[3]{1+x^{3}}=1
$$
Find $x^{2}$. | x^{2}=\frac{\sqrt[3]{28}}{3} | 52 | 17 |
math | 176. The textbook is published in a run of 100,000 copies. The probability that a textbook is improperly bound is 0.0001. Find the probability that the run contains exactly five defective books. | 0.0375 | 51 | 6 |
math | 1.83 Mark 10 points on a circle. How many different convex polygons can be constructed using some of these points as vertices? (Polygons are considered the same only if all their vertices coincide)
| 968 | 43 | 3 |
math | 5. Positive integers $a, b$ and $c$ satisfy the equation $\frac{31}{72}=\frac{a}{8}+\frac{b}{9}-c$. What is the smallest possible value of $b$ ? | 5 | 52 | 1 |
math | ## Task A-2.7. (20 points)
Determine the natural numbers $a, b$ and $c$ such that the equality $(a+b i)^{3}-107 i=c$ holds. ( $i$ is the imaginary unit.) | 198 | 56 | 3 |
math | 3.1. The decreasing sequence $a, b, c$ is a geometric progression, and the sequence $577 a, \frac{2020 b}{7}, \frac{c}{7}$ is an arithmetic progression. Find the common ratio of the geometric progression. | 4039 | 60 | 4 |
math | 14. (15 points) A non-zero natural number $n$ is both the sum of 2010 natural numbers with the same digit sum, the sum of 2012 natural numbers with the same digit sum, and the sum of 2013 natural numbers with the same digit sum. What is the smallest possible value of $n$? | 6036 | 78 | 4 |
math | 35. What is the smallest positive integer $n$, where $n \neq 11$, such that the highest common factor of $n-11$ and $3 n+20$ is greater than 1 ? | 64 | 49 | 2 |
math | 2. Given three positive numbers: $a, b, c$. Petya wrote on the board the numbers $\frac{1}{a}+b c, \frac{1}{b}+a c, \frac{1}{c}+a b, a$ and Vasya wrote the numbers $2 a^{2}, 2 b^{2}, 2 c^{2}$. It turned out that both wrote the same three numbers (possibly in a different order). What is the product $a b c?$ (N. Agakhano... | 1 | 118 | 1 |
math | A natural number $n$ is at least two digits long. If we write a certain digit between the tens digit and the units digit of this number, we obtain six times the number $n$. Find all numbers $n$ with this property. | 18 | 50 | 2 |
math | Determine all triples of natural numbers $(a,b, c)$ with $b> 1$ such that $2^c + 2^{2016} = a^b$. | (a, b, c) = (3 \cdot 2^{1008}, 2, 2019) | 40 | 30 |
math | Solve the following system of equations for real $x,y$ and $z$:
\begin{eqnarray*}
x &=& \sqrt{2y+3}\\
y &=& \sqrt{2z+3}\\
z &=& \sqrt{2x+3}.
\end{eqnarray*} | x = y = z = 3 | 69 | 8 |
math | Example 25 (2007 Bulgarian National Team Selection Test) Find all positive integers $x, y$ such that $\left(x^{2}+\right.$ $y)\left(y^{2}+x\right)$ is a fifth power of a prime. | (2,5)(5,2) | 56 | 9 |
math | 1. For which values of the parameters $p$ and $q$ does the equation
$$
\sqrt{x^{2} + p x + q} + x = 2017
$$
have more than 2017 different real solutions? | p=-4034,q=2017^{2} | 56 | 15 |
math | 4. Variant 1. An ant, starting from point A, goes $1+\frac{1}{10}$ cm north, then $2+\frac{2}{10}$ cm west, then $3+\frac{3}{10}$ cm south, then $4+\frac{4}{10}$ cm east, then $5+\frac{5}{10}$ cm north, then $6+\frac{6}{10}$ cm west, and so on. After 1000 steps, the ant is at point B. Find the distance between points A... | 605000 | 145 | 6 |
math | Some blue and red circular disks of identical size are packed together to form a triangle. The top level has one disk and each level has 1 more disk than the level above it. Each disk not at the bottom level touches two disks below it and its colour is blue if these two disks are of the same colour. Otherwise its colou... | \text{blue} | 103 | 5 |
math | There are exactly three real numbers $x$ for which $\left(x-\frac{5}{x}\right)$ is the reciprocal of $(x-4)$. What is the sum of these three real numbers? | 4 | 43 | 1 |
math | 2.245. $\frac{\left((x+2)^{-1 / 2}+(x-2)^{-1 / 2}\right)^{-1}+\left((x+2)^{-1 / 2}-(x-2)^{-1 / 2}\right)^{-1}}{\left((x+2)^{-1 / 2}+(x-2)^{-1 / 2}\right)^{-1}-\left((x+2)^{-1 / 2}-(x-2)^{-1 / 2}\right)^{-1}}$. | -\sqrt{\frac{x-2}{x+2}} | 125 | 12 |
math | 5. Find all real numbers $x$ and $y$ that satisfy the equation
$$
(x+y)^{2}=(x+3)(y-3)
$$
Solve the problem independently. You have 150 minutes for solving.
The use of notes, literature, or a pocket calculator is not allowed.
Mathematical Competition for High School Students in Slovenia
## Invitational Competition... | -3,3 | 102 | 4 |
math | 7. (10 points) The calculation result of the expression $1007 \times \frac{1 \frac{3}{4} \div \frac{3}{4}+3 \div 2 \frac{1}{4}+\frac{1}{3}}{(1+2+3+4+5) \times 5-22} \div 19$ is | 4 | 86 | 1 |
math | Example 14. There are 10 pieces of paper, on each of which the positive integers from 1 to 10 are written. Then they are all folded and placed in a hat. Five people are then asked to each draw two pieces of paper (the pieces drawn by each person are not returned to the hat). Unfortunately, an error occurred in recordin... | A=4+7, B=1+3, C=2+5, D=10+6, E=8+9 | 148 | 30 |
math | 5. Given the equation of the ellipse $\Gamma$ as $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$, a line passing through the left focus $F(-2,0)$ with a slope of $k_{1}\left(k_{1} \neq 0\right)$ intersects the ellipse at points $A$ and $B$. Let $R(1,0)$, and extend $A R$ and $B R$ to intersect the ellipse at points $C$ and $D$, re... | \frac{4}{7} | 149 | 7 |
math | In how many ways can $1000 \mathrm{Ft}$ be made using only 1, 2, and $5 \mathrm{Ft}$ coins? | 50401 | 37 | 5 |
math | Example 5 Find all odd prime numbers $p$ such that $p \mid \sum_{k=1}^{103} k^{p-1}$.
untranslated text remains the same as requested. | 3 | 44 | 1 |
math | Given a semicircle with radius $R$ and diameter $AB$, the point $C$ on $AB$ is to be determined such that if the projection of a point $D$ on the semicircle arc is $C$, then $AC + AD = l$, where $l$ is a given length. For what values of $l$ does a solution exist? | \leq4R | 78 | 5 |
math | 9. (16 points) Let the inequality $\left|2^{x}-a\right|<\left|5-2^{x}\right|$ hold for all $x \in[1,2]$. Find the range of real number $a$.
| 3<<5 | 56 | 3 |
math | 2. The numbers 1 through 25 are each colored blue or red. Determine all possible colorings that satisfy the following rules:
- The number 5 is red.
- If the numbers $x$ and $y$ have different colors and $x+y \leqslant 25$, then $x+y$ is blue.
- If the numbers $x$ and $y$ have different colors and $x \cdot y \leqslant ... | 3 | 110 | 1 |
math | 13. Given vector $\vec{a}=\{2,-3\}$, vector $\overrightarrow{A B}$ is perpendicular to $\vec{a}$, and $|\overrightarrow{A B}|=3 \sqrt{13}$, point $A$ has coordinates $(-3,2)$, find the coordinates of the position vector $\overrightarrow{O B}$. | {\begin{pmatrix}6\\8\end{pmatrix}. | 81 | 15 |
math | 12. Xiao Qian, Xiao Lu, and Xiao Dai are guessing a natural number between 1-99, and the result is:
Xiao Qian says: “It is a perfect square, and it is less than 5.”
Xiao Lu says: “It is less than 7, and it is a two-digit number.”
Xiao Dai says: “The first half of what Xiao Qian said is true, but the second half is fal... | 9 | 198 | 1 |
math | 7.280. $\left\{\begin{array}{l}2\left(\log _{1 / y} x-2 \log _{x^{2}} y\right)+5=0, \\ x y^{2}=32 .\end{array}\right.$ | (2;4),(4\sqrt{2};2\sqrt[4]{2}) | 62 | 19 |
math | 25. Let
$$
\begin{array}{c}
A=\left(\binom{2010}{0}-\binom{2010}{-1}\right)^{2}+\left(\binom{2010}{1}-\binom{2010}{0}\right)^{2}+\left(\binom{2010}{2}-\binom{2010}{1}\right)^{2} \\
+\cdots+\left(\binom{2010}{1005}-\binom{2010}{1004}\right)^{2}
\end{array}
$$
Determine the minimum integer $s$ such that
$$
s A \geq\bino... | 2011 | 250 | 4 |
math | 9.1 The sum of 100 numbers is 1000. The largest of these numbers was doubled, and some other number was decreased by 10. After these actions, the sum of all the numbers did not change. Find the smallest of the original numbers. | 10 | 59 | 2 |
math | 3. Solve the inequality:
$$
x^{2}-2 x+3 \leqslant \sqrt{4-x^{2}}
$$ | nosolution | 30 | 2 |
math | 4. If the function $f(x)=\frac{1}{3} x^{2}+\frac{2}{3} x-\frac{11}{3}$ has both its domain and range as the closed interval $M$, then $M=$ | M=[-4,-1]orM=[-4,\frac{1+3\sqrt{5}}{2}] | 53 | 26 |
math | ## Task 4 - 060514
We are looking for a natural number with the following properties:
If you divide 100 by this number, the remainder is 4, and if you divide 90 by this number, the remainder is 18.
What is the number we are looking for? | 24 | 69 | 2 |
math | 2. [3] Let $\ell$ be the line through $(0,0)$ and tangent to the curve $y=x^{3}+x+16$. Find the slope of $\ell$. | 13 | 42 | 2 |
math | 2. Let $k$ be a constant. If for all $x, y \in(0,1)$, we have
$$
x^{k}+y^{k}-x^{k} y^{k} \leqslant \frac{1}{x^{k}}+\frac{1}{y^{k}}-\frac{1}{x^{k} y^{k}},
$$
then the range of the real number $k$ is $\qquad$ . | (-\infty, 0] | 101 | 8 |
math | Solve the following equation if $x$ and $y$ are integers:
$$
x^{2}-2 x y+2 y^{2}-4 y^{3}=0
$$ | 0 | 39 | 1 |
math | Let $A B C$ be an isosceles triangle at $B$, and let $F$ be a point on the bisector of $\widehat{A B C}$ such that $(A F)$ is parallel to $(B C)$. Finally, let $E$ be the midpoint of $[B C]$, and let $D$ be the symmetric point of $A$ with respect to $F$. Calculate the ratio of the distances $E F / B D$. | \frac{1}{2} | 100 | 7 |
math | Example 8 Try to find all natural number triples $(A, B, C)$, such that
$$
A^{2}+B-C=100, A+B^{2}-C=124 .
$$
(1990, Leningrad (now St. Petersburg) Mathematical Olympiad Third Round, Grade 8) | A=12, B=13, C=57 | 73 | 14 |
math | Find the least positive integer $n$ such that for every prime number $p, p^2 + n$ is never prime. | 5 | 27 | 1 |
math | 4. At a point $R$ on a line, there is a robot that moves along this line to the left or right as it wishes. It is programmed to take 2 steps on the first move, 4 steps on the second move, 6 steps on the third move, and in general, $2n$ steps on the $n$-th move.
a) Describe a variant of the robot's movement such that i... | 3 | 254 | 1 |
math | 1. Given the sequence $\left\{a_{n}\right\}$ :
$$
a_{0}=1, a_{n}=2 \sum_{i=0}^{n-1} a_{i}(n \geqslant 1) \text {. }
$$
Then the maximum positive integer $n$ that satisfies $a_{n} \leqslant 2018$ is . $\qquad$ | 7 | 93 | 1 |
math | 3. If the real numbers $x, y, z, w$ satisfy
$$
\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}=1, \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}=1, \\
\frac{z^{2}}{6^{2}-5^{2}}+\frac{w^{2}}{6^{2}-7^{2}}=1, \\
\frac{z^{2}}{8^{2}-5^{2}}+\frac{w^{2}}{8^{2}-7^{2}}=1 .
\end{array}
$$
then $x^{2... | 36 | 196 | 2 |
math | 9. Given positive real numbers $x, y, z>1$ satisfy $x^{\log _{y} x} \cdot y^{\log _{z} y} \cdot z^{\log _{x} z}=10$, find the maximum value of $x y z$. | 10 | 64 | 2 |
math | 3. On the blackboard, it is written
$$
1!\times 2!\times \cdots \times 2011!\times 2012!\text {. }
$$
If one of the factorials is erased so that the remaining product equals the square of some positive integer, then the erased term is . $\qquad$ | 1006! | 74 | 5 |
math | 1. A plank of wood has one end, $A$, against a vertical wall. Its other end, $B$, is on horizontal ground. When end $A$ slips down $8 \mathrm{~cm}$, end $B$ moves $4 \mathrm{~cm}$ further away from the wall. When end $A$ slips down a further $9 \mathrm{~cm}$, end $B$ moves a further $3 \mathrm{~cm}$ away from the wall.... | 65\mathrm{~} | 110 | 7 |
math | 2B. Determine the real number $x$ such that the sequence $\sqrt{x-7}, \sqrt[4]{2 x+19}$, $\sqrt{x+3}$ is a geometric progression. | 10 | 43 | 2 |
math | Find the distance from the point $D(1 ; 3 ; 2)$ to the plane passing through the points $A(-3 ; 0 ; 1), B(2 ; 1 ;-1)$ and $C(-2 ; 2 ; 0)$.
# | \frac{10}{\sqrt{11}} | 58 | 12 |
math | At point $O$ on the shore of the Dongjiang Lake (the lake shore can be considered a straight line), a rescue boat is parked. Due to the sudden breakage of the rope, the boat is blown away, with its direction forming a $15^{\circ}$ angle with the shore, at a speed of $2.5 \mathrm{~km} / \mathrm{h}$. At the same time, a ... | 2\sqrt{2}\mathrm{~}/\mathrm{} | 184 | 13 |
math | 7. The Brazilian IMO leader chooses two natural numbers $n$ and $k$ with $n>k$, and then tells these to his deputy and a participant. The leader then whispers a binary sequence of length $n$ into the deputy's ear. The deputy writes down all binary sequences of length $n$ that differ from the leader's sequence at exactl... | 2 | 174 | 1 |
math | How many rearrangements of the letters of "$HMMTHMMT$" do not contain the substring "$HMMT$"? (For instance, one such arrangement is $HMMHMTMT$.) | 361 | 43 | 3 |
math | 6. Choose 5 different numbers from 1, 2, ..., 20, the probability that at least two of them are consecutive is $\qquad$ . | \frac{232}{323} | 35 | 11 |
math | 4. Given an arithmetic sequence $\left\{a_{n}\right\}$ with a common difference of $d \neq 0$, and $a_{1} 、 a_{3} 、 a_{9}$ form a geometric sequence, the value of $\frac{a_{1}+a_{3}+a_{9}}{a_{2}+a_{4}+a_{10}}$ is $\qquad$ . | \frac{13}{16} | 96 | 9 |
math | Task 2. (10 points) A numerical sequence is given:
$x_{0}=\frac{1}{n} ; x_{k}=\frac{1}{n-k}\left(x_{0}+x_{1}+\ldots+x_{k-1}\right) ; k=1,2, \ldots, n-1$.
Find $S_{n}=x_{0}+x_{1}+\ldots+x_{n-1}$, if $n=2021$. | 1 | 110 | 1 |
math | 4. Given the function $f(x)=a+b c^{x}$. Determine the real numbers $a, b, c$, if $f(0)=5$, $f(1)=14$ and $f(2)=50$. | f(x)=2+3\cdot4^{x} | 52 | 12 |
math | 1. Determine all values of the natural number $n$ for which the system
$$
\left\{\begin{array}{l}
x+y=n^{2} \\
10 x+y=n^{3}
\end{array}\right.
$$
has a solution in the set of natural numbers. | 3,6,9 | 63 | 5 |
math | Solve the following inequalities:
a) $\frac{x(x+1)}{x-1}>3$;
b) $\frac{x(x+1)}{x-1}>6$ | 1<x<2\quador\quadx>3 | 39 | 12 |
math | Let $a$ and $b$ be two real numbers such that $a+b=7$ and $a b=3$. What is the value of $a^{3}+b^{3}$? We will not seek to express $a$ and $b$.
| 280 | 57 | 3 |
math | 7. (10 points) Calculate: $481 \frac{1}{6}+265 \frac{1}{12}+904 \frac{1}{20}-184 \frac{29}{30}-160 \frac{41}{42}-703 \frac{55}{56}=$ | 600\frac{3}{8} | 81 | 10 |
math | 5、Insert $n$ real numbers between 1 and 100, such that these $n+2$ numbers form an increasing geometric sequence. Let the product of these $n+2$ numbers be denoted as $T_{n}$, and let $a_{n}=\lg T_{n}, n \geq 1$. Define $b_{n}=\tan a_{n} \cdot \tan a_{n+1}$, then the sum of the first $n$ terms of the sequence $\left\{b... | \frac{\tan(n+3)-\tan3}{\tan1}-n | 131 | 17 |
math | 2. Petya came up with a four-digit number, in which all the digits are different. It is known that the sum of the first three digits of this number is divisible by 9 and the sum of the last three digits of this number is divisible by 9. What values can the sum of all the digits of this number take? Find all possible va... | 18 | 83 | 2 |
math | 223. Cryptarithm with multiplication. After performing regular multiplication "in column," a person replaced each even digit with the letter $E$, and each odd digit - with the letter $O^{*}$. As a result, the following expression was obtained:
| $O E E$ E EOEE |
| :--- |
| EOE |
| $O O E E$ |
Restore the given mult... | 348\cdot28=9744 | 88 | 12 |
math | 4. Through the moving point $M$, draw the tangent line $M N$ to the circle $C:(x-2)^{2}+(y-2)^{2}=1$, where $N$ is the point of tangency. If $|M N|=|M O|(O$ is the origin), then the minimum value of $|M N|$ is $\qquad$ . | \frac{7\sqrt{2}}{8} | 83 | 12 |
math | 2. Find the maximum value of the expression for $a, b>0$
$$
\frac{|4 a-10 b|+|2(a-b \sqrt{3})-5(a \sqrt{3}+b)|}{\sqrt{a^{2}+b^{2}}}
$$ | 2\sqrt{87} | 65 | 7 |
math | (11) (15 points) Divide each side of the equilateral $\triangle A B C$ with side length 3 into three equal parts, and draw lines parallel to the other two sides through each division point. The 10 points formed by the intersections of the sides of $\triangle A B C$ and these parallel lines are called grid points. If $n... | 5 | 120 | 1 |
math | For example, the rules of a "level-passing game" stipulate: on the $n$-th level, a die must be rolled $n$ times. If the sum of the points from these $n$ rolls is greater than $2^{n}$, the player passes the level. Questions:
( I ) What is the maximum number of levels a person can pass in this game?
( II ) What is the pr... | \frac{100}{243} | 157 | 11 |
math | Let $n$ be an integer greater than 1. If all digits of $97n$ are odd, find the smallest possible value of $n$. | 35 | 33 | 2 |
math | 9.1. In the morning, a dandelion blooms, it flowers yellow for three days, on the fourth morning it turns white, and by the evening of the fifth day, it withers. On Monday afternoon, there were 20 yellow and 14 white dandelions on the meadow, and on Wednesday - 15 yellow and 11 white. How many white dandelions will the... | 6 | 95 | 1 |
math | 7.6. Along the shore of a round lake, apple trees grow. Petya and Vasya start walking from point $A$ on the shore in opposite directions along the shore and count all the apple trees they encounter, as well as all the apples growing on the trees. Meeting at some point $B$, they compared their results. It turned out tha... | 3 | 277 | 1 |
math | 1. (CZS) Determine all real solutions of the equation $\sqrt{x^{2}-p}+2 \sqrt{x^{2}-1}=$ $x$, where $p$ is a real number. | \frac{4-p}{2\sqrt{4-2p}}for\frac{4}{3}<p<2 | 45 | 26 |
math | P r o b l e m 1. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ (Fig. 1), with an edge length of 1. A sphere is drawn through the vertices $A$ and $C$ and the midpoints $F$ and $E$ of the edges $B_{1} C_{1}$ and $C_{1} D_{1}$, respectively. Find the radius $R$ of this sphere. | \frac{\sqrt{41}}{8} | 108 | 11 |
math | 17. [10] How many ways are there to color every integer either red or blue such that $n$ and $n+7$ are the same color for all integers $n$, and there does not exist an integer $k$ such that $k, k+1$, and $2 k$ are all the same color? | 6 | 71 | 1 |
math | Example 14 Let $S$ be the unit circle in the complex plane (i.e., the set of complex numbers with modulus 1), and let $f$ be a mapping from $S$ to $S$, defined by $f_{1}(z)=f(z), f_{k+1}(z)=f\left(f_{k}(z)\right), k=1,2,3, \cdots$. If $c \in S$ and a positive integer $n$ exist such that $f_{n}(c)=c$, and $f_{k}(c) \neq... | ^{1989}-^{663}-^{153}-^{117}+^{51}+^{39}+^{9}-^{3} | 249 | 38 |
math | ## Task Condition
Find the $n$-th order derivative.
$y=\sqrt{e^{3 x+1}}$ | (\frac{3}{2})^{n}\cdot\sqrt{e^{3x+1}} | 27 | 21 |
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