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 |
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math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty}\left(\frac{1}{n^{2}}+\frac{2}{n^{2}}+\frac{3}{n^{2}}+\ldots+\frac{n-1}{n^{2}}\right)$ | \frac{1}{2} | 67 | 7 |
math | 【Example 2】Eight classes of the third year of high school participate in the sports meet, to determine the first, second, and third place in the overall team score, how many different possible outcomes of the competition are there? | 336 | 47 | 3 |
math | 1. (12th "Hope Cup" Invitational Training Question) In the quadratic trinomial $a x^{2}+b x+c$, $a>100$, how many integer values of $x$ at most can make the absolute value of the quadratic trinomial not exceed 50? | 2 | 66 | 1 |
math | 7. If $x, y, k$ are positive reals such that
$$
3=k^{2}\left(\frac{x^{2}}{y^{2}}+\frac{y^{2}}{x^{2}}\right)+k\left(\frac{x}{y}+\frac{y}{x}\right) \text {, }
$$
find the maximum possible value of $k$. | \frac{-1+\sqrt{7}}{2} | 85 | 12 |
math | Four fathers wanted to sponsor a skiing trip for their children.
The first promised: "I will give 11500 CZK,"
the second promised: "I will give a third of what you others give in total,"
the third promised: "I will give a quarter of what you others give in total,"
the fourth promised: "And I will give a fifth of wh... | 7500,6000,5000 | 102 | 14 |
math | 9. Find all real polynomials $f$ and $g$ such that for all $x \in \mathbf{R}$, we have $\left(x^{2}+x+1\right) f\left(x^{2}-x+1\right)=\left(x^{2}-x+1\right) g\left(x^{2}+x+1\right)$.
(53rd Romanian Mathematical Olympiad (Final)) | f(x) = kx, g(x) = kx | 95 | 13 |
math | 14. The largest even number that cannot be expressed as the sum of two odd composite numbers is what?
Will the translated text be used for further discussion or do you need more information on this topic? | 38 | 41 | 2 |
math | Let $x_1=2021$, $x_n^2-2(x_n+1)x_{n+1}+2021=0$ ($n\geq1$). Prove that the sequence ${x_n}$ converges. Find the limit $\lim_{n \to \infty} x_n$. | \lim_{n \to \infty} x_n = \sqrt{2022} - 1 | 72 | 25 |
math | 1. In $1000 \mathrm{~kg}$ of fresh plums, there is $77 \%$ water. After the evaporation of a certain amount of water, the mass of the plums was halved. What percentage of water do the plums now contain? | 54 | 60 | 2 |
math | 1. Given non-zero real numbers $x, y$ satisfy
$$
\begin{array}{l}
(5 x+y)^{2019}+x^{2019}+30 x+5 y=0 . \\
\text { Then } \frac{x}{y}=
\end{array}
$$ | -\frac{1}{6} | 71 | 7 |
math | 2. Solve the inequality $\sqrt{\frac{x-24}{x}}-\sqrt{\frac{x}{x-24}}<\frac{24}{5}$. | x\in(-\infty;-1)\cup(24;+\infty) | 37 | 19 |
math | Calvin was asked to evaluate $37 + 31 \times a$ for some number $a$. Unfortunately, his paper was tilted 45 degrees, so he mistook multiplication for addition (and vice versa) and evaluated $37 \times 31 + a$ instead. Fortunately, Calvin still arrived at the correct answer while still following the order of operations.... | 37 | 99 | 2 |
math | Find all positive integers $n$ and prime numbers $p$ such that $$17^n \cdot 2^{n^2} - p =(2^{n^2+3}+2^{n^2}-1) \cdot n^2.$$
[i]Proposed by Nikola Velov[/i] | (p, n) = (17, 1) | 66 | 14 |
math | 5. [5 points] Around a hook with a worm, in the same plane as it, a carp and a minnow are swimming along two circles. In the specified plane, a rectangular coordinate system is introduced, in which the hook (the common center of the circles) is located at the point $(0 ; 0)$. At the initial moment of time, the carp and... | (4-\sqrt{2};4+\sqrt{2}),(4+\sqrt{2};\sqrt{2}-4),(\sqrt{2}-4;-4-\sqrt{2}),(-\sqrt{2}-4;4-\sqrt{2}) | 165 | 54 |
math | Let $ n $ be a positive integer. A sequence $(a_1,a_2,\ldots,a_n)$ of length is called $balanced$ if for every $ k $ $(1\leq k\leq n)$ the term $ a_k $ is equal with the number of distinct numbers from the subsequence $(a_1,a_2,\ldots,a_k).$
a) How many balanced sequences $(a_1,a_2,\ldots,a_n)$ of length $ n $ do exist... | 2^{n-1} | 147 | 8 |
math | A man is twice as old as his wife was when the husband was as old as his wife is now. The number of years of the husband's and wife's ages, as well as the sum of these years, are in the form of aa. How old is the husband and the wife? | 4433 | 60 | 4 |
math | 10. [6] Find the number of nonempty sets $\mathcal{F}$ of subsets of the set $\{1, \ldots, 2014\}$ such that:
(a) For any subsets $S_{1}, S_{2} \in \mathcal{F}, S_{1} \cap S_{2} \in \mathcal{F}$.
(b) If $S \in \mathcal{F}, T \subseteq\{1, \ldots, 2014\}$, and $S \subseteq T$, then $T \in \mathcal{F}$. | 2^{2014} | 134 | 7 |
math | The sequence $1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ... $ is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to $1994$. | 1993 | 88 | 6 |
math | 4. Let $f_{1}(x)=\frac{2}{1+x}$. Define $f_{n+1}(x)=f_{1}\left[f_{n}(x)\right]$. And let $a_{n}=\frac{f_{n}(0)-1}{f_{n}(0)+2}$. Then $a_{100}=$ $\qquad$ . | -\frac{1}{2^{101}} | 84 | 11 |
math | Find all pair of integer numbers $(n,k)$ such that $n$ is not negative and $k$ is greater than $1$, and satisfying that the number:
\[ A=17^{2006n}+4.17^{2n}+7.19^{5n} \]
can be represented as the product of $k$ consecutive positive integers. | (n, k) = (0, 2) | 81 | 13 |
math | ## 126. Math Puzzle 11/75
A gardener sold half of all his apples and half an apple to the first buyer, half of the remaining apples and another half an apple to the second buyer, half of the remaining apples and half an apple to the third buyer, and so on.
To the seventh buyer, he sold half of the remaining apples an... | 127 | 102 | 3 |
math | 7. An editor uses the digits $0 \sim 9$ to number the pages of a book. If a total of 636 digits were written, then the book has $\qquad$ pages. | 248 | 44 | 3 |
math | Let $n \in \mathbb{N}^{*}$. Find the number of $n$-digit numbers whose digits are among $\{2,3,7,9\}$ and which are divisible by 3. | \frac{4^{n}+2}{3} | 48 | 12 |
math | Compute the number of ways to erase 24 letters from the string ``OMOMO$\cdots$OMO'' (with length 27), such that the three remaining letters are O, M and O in that order. Note that the order in which they are erased does not matter.
[i]Proposed by Yannick Yao | 455 | 70 | 3 |
math | 7. Put 48 chess pieces into 9 boxes, with at least 1 piece in each box, and the number of pieces in each box is different. The box with the most pieces can contain $\qquad$ pieces at most.
Put 48 chess pieces into 9 boxes, with at least 1 piece in each box, and the number of pieces in each box is different. The box wi... | 12 | 98 | 2 |
math | 3. Let $[x]$ denote the greatest integer not exceeding the real number $x$. If $\left[x-\frac{1}{2}\right]\left[x+\frac{1}{2}\right]$ is a prime number, then the range of values for the real number $x$ is | -\frac{3}{2} \leqslant x<-\frac{1}{2} \text{ or } \frac{3}{2} \leqslant x<\frac{5}{2} | 61 | 47 |
math | 47. Find all positive integers $p(\geqslant 2)$, such that there exists $k \in \mathbf{N}^{*}$, satisfying: the number $4 k^{2}$ in base $p$ representation consists only of the digit 1. | p \in \{3,7\} | 60 | 10 |
math | 4B. Solve the equation
$$
\log _{6}\left(3 \cdot 4^{-\frac{1}{x}}+2 \cdot 9^{-\frac{1}{x}}\right)+\frac{1}{x}=\log _{6} 5
$$ | -1 | 64 | 2 |
math | There is a safe that can be opened by entering a secret code consisting of $n$ digits, each of them is $0$ or $1$. Initially, $n$ zeros were entered, and the safe is closed (so, all zeros is not the secret code).
In one attempt, you can enter an arbitrary sequence of $n$ digits, each of them is $0$ or $1$. If the ente... | n | 153 | 2 |
math | 127. A truncated cone is described around a sphere. The total surface area of this cone is $S$. A second sphere touches the lateral surface of the cone along the circumference of the cone's base. Find the volume of the truncated cone, given that the part of the surface of the second sphere that is inside the first has ... | \frac{1}{3}S\sqrt{\frac{Q}{\pi}} | 74 | 18 |
math | 6. (10 points) A team currently has a win rate of $45 \%$. If they win 6 out of the next 8 games, their win rate will increase to 50\%. How many games has this team won so far? $\qquad$ | 18 | 58 | 2 |
math | Example 3. Find the complex Fourier transform of the function
$$
f(x)=\frac{1}{x^{2}+2 x+2},
$$
using the formula
$$
\frac{1}{x^{2}+1} \longmapsto \sqrt{\frac{\pi}{2}} e^{-|p|}
$$ | F(p)=\sqrt{\frac{\pi}{2}}e^{-|p|-ip} | 72 | 19 |
math | 6. Let the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ have its left and right foci as $F_{1}$ and $F_{2}$, respectively, and let $A$ be a point on the asymptote of the hyperbola such that $A F_{2} \perp F_{1} F_{2}$. The distance from the origin $O$ to the line $A F_{1}$ is $\frac{1}{3}\left|O F_{1... | \frac{\sqrt{6}}{2} | 144 | 10 |
math | ## Task 19/73
Determine all prime numbers $p$ for which $P=20 p^{2}+1$ is a prime number. | 3 | 36 | 1 |
math | Example 6 Solve the equation $\frac{13 x-x^{2}}{x+1}\left(x+\frac{13-x}{x+1}\right)=42$. (1998, Changchun City, Jilin Province Mathematics Competition (Grade 9)) | x_{1}=1, x_{2}=6, x_{3}=3+\sqrt{2}, x_{4}=3-\sqrt{2} | 61 | 32 |
math | 1. Solve the equation $\left(2^{x}-4\right)^{3}+\left(4^{x}-2\right)^{3}=\left(4^{x}+2^{x}-6\right)^{3}$. | 1,2,\frac{1}{2} | 53 | 10 |
math | 6. B. Given that the lengths of the two legs are integers $a$ and $b$ $(b<2011)$. Then the number of right triangles with the hypotenuse length $b+1$ is | 31 | 48 | 2 |
math | # Problem 1.
A rule is given by which each pair of integers $X$ and $Y$ is assigned a number $X \nabla Y$. (The symbol «»» means applying the rule to the numbers $X$ and $Y$.) It is known that for any integers $X, Y$ the following properties hold:
1) $X \nabla 0=X$
2) $X \nabla(Y-1)=(X \nabla Y)-2$
3) $X \nabla(Y+1)=... | -673 | 158 | 4 |
math | Test Question D: The sum of four positive numbers is 4, and the sum of their squares is 8. Determine the maximum value of the largest of these four numbers.
Restate it as: Let \( a \geqslant b \geqslant c \geqslant d > 0 \), and satisfy
\[
a+b+c+d=4, \quad a^{2}+b^{2}+c^{2}+d^{2}=8,
\]
Find \(\max a=\) ? | 1+\sqrt{3} | 112 | 6 |
math | Three cowboys walked into a saloon. One bought 4 sandwiches, a cup of coffee, and 10 donuts for a total of $1.69. The second bought 3 sandwiches, a cup of coffee, and 7 donuts for $1.26. How much did the third cowboy pay for a sandwich, a cup of coffee, and a donut? | 40 | 81 | 2 |
math | Nathan has discovered a new way to construct chocolate bars, but it’s expensive! He starts with a single $1\times1$ square of chocolate and then adds more rows and columns from there. If his current bar has dimensions $w\times h$ ($w$ columns and $h$ rows), then it costs $w^2$ dollars to add another row and $h^2$ dolla... | 5339 | 109 | 4 |
math | G2.4 Let $f(x)=41 x^{2}-4 x+4$ and $g(x)=-2 x^{2}+x$. If $d$ is the smallest value of $k$ such that $f(x)+k g(x)=0$ has a single root, find $d$. | -40 | 67 | 3 |
math | The integer $n$ has exactly six positive divisors, and they are: $1<a<b<c<d<n$. Let $k=a-1$. If the $k$-th divisor (according to above ordering) of $n$ is equal to $(1+a+b)b$, find the highest possible value of $n$. | 2009 | 67 | 4 |
math | There are 2023 cups numbered from 1 through 2023. Red, green, and blue balls are placed in the cups according to the following rules.
- If cups $m$ and $n$ both contain a red ball, then $m-n$ is a multiple of 2 .
- If cups $m$ and $n$ both contain a green ball, then $m-n$ is a multiple of 3 .
- If cups $m$ and $n$ both... | 538 | 131 | 3 |
math | [ Arithmetic. Mental calculation, etc.]
When the barrel is empty by $30 \%$, it contains 30 liters more honey than when it is filled to $30 \%$. How many liters of honey are in a full barrel? | 75 | 49 | 2 |
math | Determine all prime numbers $p$ for which there exists a unique $a$ in $\{1, \ldots, p\}$ such that $a^{3}-3 a+1$ is divisible by $p$.
---
The translation maintains the original formatting and structure of the source text. | 3 | 62 | 1 |
math | Example 4 Find the maximum of $y=\sin ^{2} x+2 \sin x \cos x+3 \cos ^{2} x$. | y_{\max }=2+\sqrt{2} | 34 | 12 |
math | The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$
If one writes down the number $S$, how often does the digit `$5$' occur in the result? | 4022 | 67 | 4 |
math | 4. Let $\left\{a_{n}\right\}(n \geqslant 1)$ be a sequence of non-negative numbers with the first term $a$, and the sum of the first $n$ terms be $S_{n}$, and $S_{n}=\left(\sqrt{S_{n-1}}+\sqrt{a}\right)^{2}(n \geqslant 2)$. If $b_{n}=\frac{a_{n+1}}{a_{n}}+\frac{a_{n}}{a_{n+1}}$, and the sum of the first $n$ terms of th... | \frac{4n^{2}+6n}{2n+1} | 169 | 17 |
math | 1. In a certain country, the alphabet consists of three letters: "M", "G", and "U". A word is any finite sequence of these letters, in which two consonants cannot stand next to each other and two vowels cannot stand next to each other. How many 200-letter words are there in this country that contain each of the three l... | 2^{101}-4 | 79 | 7 |
math | 10,11
All dihedral angles of a trihedral angle are equal to $60^{\circ}$. Find the angles formed by the edges of this trihedral angle with the planes of the opposite faces. | \arccos\frac{\sqrt{3}}{3} | 49 | 14 |
math | Example 1 Consider the equation in $x$
$$
(1+a) x^{4}+x^{3}-(3 a+2) x^{2}-4 a=0
$$
(1) Does the equation always have a real solution for any real number $a$?
(2) Is there a real number $x_{0}$ such that, regardless of what real number $a$ is, $x_{0}$ is never a solution to this equation? | -2,x_{0}=2 | 100 | 7 |
math | Let $y(x)$ be the unique solution of the differential equation
$$
\frac{\mathrm{d} y}{\mathrm{~d} x}=\log _{e} \frac{y}{x}, \quad \text { where } x>0 \text { and } y>0,
$$
with the initial condition $y(1)=2018$.
How many positive real numbers $x$ satisfy the equation $y(x)=2000$ ? | 1 | 103 | 1 |
math | 4. Given the curve $C_{1}: y=\sqrt{-x^{2}+10 x-9}$ and point $A(1,0)$. If there exist two distinct points $B$ and $C$ on curve $C_{1}$, whose distances to the line $l: 3 x+1=0$ are $|A B|$ and $|A C|$, respectively, then $|A B|+|A C|=$ | 8 | 99 | 1 |
math | 2B. Let $a$ and $b$ be rational numbers such that $a>1, b>0$ and $a b=a^{b}$ and $\frac{a}{b}=a^{3 b}$. Determine the values of $a$? | 4 | 55 | 1 |
math | (Netherlands). Find all triples $(a, b, c)$ of real numbers such that $a b+b c+$ $c a=1$ and
$$
a^{2} b+c=b^{2} c+a=c^{2} a+b
$$ | (0,1,1), (0,-1,-1), (1,0,1), (-1,0,-1), (1,1,0), (-1,-1,0), \left(\frac{1}{3} \sqrt{3}, \frac{1}{3} \sqrt{3}, \frac{1}{3} \sqrt{3}\right), \left(-\frac{1}{3} \sqrt{3}, - | 54 | 98 |
math | 7. The length of the minor axis of the ellipse $\rho=\frac{1}{2-\cos \theta}$ is $\qquad$ . | \frac{2\sqrt{3}}{3} | 30 | 12 |
math | 7. Find all positive integers $n$ such that $n-[n\{\sqrt{n}\}]=2$ holds.
$(2002$, Bulgarian Spring Mathematical Competition) | 2,8,15 | 37 | 6 |
math | 3. $n$ is a natural number, $19 n+14$ and $10 n+3$ are both multiples of some natural number $d$ not equal to 1, then $d=$ $\qquad$ . | 83 | 51 | 2 |
math | 10. (20 points) Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms as $S_{n}$, and $S_{n}=2 n-a_{n}\left(n \in \mathbf{N}^{*}\right)$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) If the sequence $\left\{b_{n}\right\}$ satisfies $b_{n}=2^{n-1} a_{n}$, prove that... | \frac{5}{3} | 174 | 7 |
math | 18. Collinear points $A, B$, and $C$ are given in the Cartesian plane such that $A=(a, 0)$ lies along the $x$-axis, $B$ lies along the line $y=x, C$ lies along the line $y=2 x$, and $A B / B C=2$. If $D=(a, a)$, the circumcircle of triangle $A D C$ intersects $y=x$ again at $E$, and ray $A E$ intersects $y=2 x$ at $F$,... | 7 | 127 | 1 |
math | 14.47 The sequence $\left\{x_{n}\right\}$ satisfies $x_{1}=\frac{1}{2}, x_{k+1}=x_{k}^{2}+x_{k}, k \in N$. Find the integer part of the sum $\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{100}+1}$.
(China National Training Team Problem, 1990) | 1 | 116 | 1 |
math | \section*{Problem 6 - 071046}
Determine all real \(x\) that satisfy the following equation:
\[
\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}=\frac{3}{2} \sqrt{\frac{x}{x+\sqrt{x}}}
\] | \frac{25}{16} | 66 | 9 |
math | Shapovalovo $A . B$. The numbers from 1 to 37 were written in a row such that the sum of any initial segment of numbers is divisible by the next number in the sequence.
What number is in the third position if the first position is occupied by the number 37, and the second by 1? # | 2 | 71 | 1 |
math | 2. Solve the inequality $\frac{\sqrt{1-x}-12}{1-\sqrt{2-x}} \geq 1+\sqrt{2-x}$. | x\in[-8;1) | 35 | 8 |
math | Problem 2. Determine the irrational numbers $\mathrm{x}$, with the property that the numbers $\mathrm{x}^{2}+\mathrm{x}$ and
$$
x^{3}+2 x^{2} \text { are integers. }
$$
Mathematical Gazette | x\in{\frac{-1\\sqrt{5}}{2}} | 57 | 15 |
math | 6. Two Skaters
Allie and Bllie are located
at points $A$ and $B$ on
a smooth ice surface, with $A$ and $B$
being 100 meters apart. If Allie starts from $A$
and skates at 8 meters/second
along a line that forms a $60^{\circ}$
angle with $A B$, while Bllie starts from $B$ at 7 meters/second
along the shortest line to me... | 160 | 122 | 3 |
math | 2. Solve the inequality $\sqrt{x^{2}-1} \leqslant \sqrt{5 x^{2}-1-4 x-x^{3}}$. | (-\infty;-1]\cup{2} | 35 | 11 |
math | 4. Given prime numbers $p$ and $q$, such that $p^{3}-q^{5}=(p+q)^{2}$. Then $\frac{8\left(p^{2013}-p^{2010} q^{5}\right)}{p^{2011}-p^{2009} q^{2}}=$ $\qquad$. | 140 | 83 | 3 |
math | 3.1. Kolya had 10 sheets of paper. On the first step, he chooses one sheet and divides it into two parts. On the second step, he chooses one sheet from the available ones and divides it into 3 parts, on the third step, he chooses one sheet from the available ones and divides it into 4, and so on. After which step will ... | 31 | 91 | 2 |
math | Suppose that $n\ge2$ and $a_1,a_2,...,a_n$ are natural numbers that $ (a_1,a_2,...,a_n)=1$. Find all strictly increasing function $f: \mathbb{Z} \to \mathbb{R} $ that:
$$ \forall x_1,x_2,...,x_n \in \mathbb{Z} : f(\sum_{i=1}^{n} {x_ia_i}) = \sum_{i=1}^{n} {f(x_ia_i})$$
[i]Proposed by Navid Safaei and Ali Mir... | f(x) = c x | 147 | 7 |
math | Let $ n>0$ be a natural number. Determine all the polynomials of degree $ 2n$ with real coefficients in the form
$ P(X)\equal{}X^{2n}\plus{}(2n\minus{}10)X^{2n\minus{}1}\plus{}a_2X^{2n\minus{}2}\plus{}...\plus{}a_{2n\minus{}2}X^2\plus{}(2n\minus{}10)X\plus{}1$,
if it is known that all the roots of them are positive ... | P(X) = X^2 - 8X + 1 | 145 | 15 |
math | 1. Let $x, y$ be real numbers, and satisfy
$$
\left\{\begin{array}{l}
(x-1)^{3}+1997(x-1)=-1 \\
(y-1)^{3}+1997(y-1)=1 .
\end{array}\right.
$$
Then $x+y=$ $\qquad$ (Proposed by the Problem Group) | 2 | 90 | 1 |
math | 5. (5 points) For any two numbers $x, y$, the operation “*” is defined as: $x^{*} y=\frac{4 \times \mathrm{x} \times \mathrm{y}}{\mathrm{m} \times \mathrm{x}+3 \times \mathrm{y}}$ (where $m$ is a fixed number). If $1 * 2=1$, then $m=$ $\qquad$ , $3 * 12=$ $\qquad$ . | 2,\frac{24}{7} | 108 | 9 |
math | ## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
\operatorname{arctg}\left(\frac{3 x}{2}-x^{2} \sin \frac{1}{x}\right), x \neq 0 \\
0, x=0
\end{array}\right.
$$ | \frac{3}{2} | 90 | 7 |
math | (7) Let a fixed point $P$ outside the plane $\alpha$ be at a distance $h$ from $\alpha$, and let three moving points $A$, $B$, and $C$ on $\alpha$ be at distances $a$, $b$, and $c$ from $P$ respectively, with $\angle PBA = 90^{\circ}$. Then the maximum area of $\triangle ABC$ is $\qquad$ (expressed in terms of $a$, $b$... | \frac{1}{2}\sqrt{^{2}-b^{2}}(\sqrt{b^{2}-^{2}}+\sqrt{^{2}-^{2}}) | 113 | 36 |
math | II. Fill-in-the-blank Questions (9 points each, total 54 points)
1. Remove all perfect squares and cubes from the natural numbers, and arrange the remaining numbers in ascending order to form a sequence $\left\{a_{n}\right\}$. Then $a_{2008}=$ $\qquad$ | 2062 | 71 | 4 |
math | 4. On a line, several points were marked, including points $A$ and $B$. All possible segments with endpoints at the marked points are considered. Vasya calculated that point $A$ is inside 50 of these segments, and point $B$ is inside 56 segments. How many points were marked? (The endpoints of a segment are not consider... | 16 | 81 | 2 |
math | 3. There are 207 different cards with numbers $1,2,3,2^{2}, 3^{2}, \ldots, 2^{103}, 3^{103}$ (each card has exactly one number, and each number appears exactly once). In how many ways can 3 cards be chosen so that the product of the numbers on the chosen cards is a square of an integer divisible by 6? | 267903 | 93 | 6 |
math | 20. List the natural numbers from $1,2,3, \cdots, 99,100$ that are both odd and composite in a row, such that no two adjacent numbers are coprime (if one line is not enough, continue on the second line, and if the second line is still not enough, continue on the third line). | 25,35,55,65,85,95,15,9,21,27,33,39,45,51,57,63,69,75,81,87,93,99,77,91,49 | 77 | 73 |
math |
4. Suppose 36 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
| 5412 | 47 | 4 |
math | 3. Find all values of the parameter $a$, for which the interval ( $3 a ; 5 a-2)$ contains at least one integer.
# | (1.2;4/3)\cup(1.4;+\infty) | 33 | 19 |
math | Consider 2015 lines in the plane, no two of which are parallel and no three of which are concurrent. Let $E$ be the set of their intersection points.
We want to assign a color to each point in $E$ such that any two points on the same line, whose segment connecting them contains no other point of $E$, are of different ... | 3 | 91 | 1 |
math | Example 5 Let $n$ be a positive integer. How many solutions does $x^{2}-\left[x^{2}\right]=$ $(x-[x])^{2}$ have in $1 \leqslant x \leqslant n$?
$(1982$, Swedish Mathematical Olympiad) | n^2 - n + 1 | 66 | 8 |
math | 8. A certain intelligence station has four different passwords $A, B, C, D$, and uses one of them each week, with each week's password being randomly selected with equal probability from the three passwords not used in the previous week. If the first week uses password $A$, then the probability that the seventh week al... | \frac{61}{243} | 84 | 10 |
math | 5. If the equation $x^{2}=a \mathrm{e}^{x}$ has three distinct real roots, then the range of the real number $a$ is $\qquad$ | (0,4\mathrm{e}^{-2}) | 40 | 12 |
math | Determine all solutions of the equation
$$
a^{2}=b \cdot(b+7)
$$
with integers $a \geq 0$ and $b \geq 0$.
W. Janous, WRG Ursulinen, Innsbruck | (0,0)(12,9) | 58 | 10 |
math | For any natural number $n$, expressed in base $10$, let $S(n)$ denote the sum of all digits of $n$. Find all natural numbers $n$ such that $n=2S(n)^2$. | 50, 162, 392, 648 | 48 | 17 |
math | 10.2. Find all values of parameters $a, b, c$, for which the system of equations $\left\{\begin{array}{l}a x+b y=c \\ b x+c y=a \\ c x+a y=b,\end{array}\right\}$
has at least one negative solution (when $x, y<0$). | +b+=0 | 75 | 3 |
math | ## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(-10 ; 0 ; 9)$
$B(12 ; 4 ; 11)$
$C(8 ; 5 ; 15)$ | -4x+y+4z-76=0 | 67 | 12 |
math | [ Chess coloring ] [ Examples and counterexamples. Constructions ]
a) What is the maximum number of bishops that can be placed on a 1000 by 1000 board so that they do not attack each other?
b) What is the maximum number of knights that can be placed on an $8 \times 8$ board so that they do not attack each other? | 1998 | 81 | 4 |
math | [u]Set 3[/u]
[b]3.1[/b] Annie has $24$ letter tiles in a bag; $8$ C’s, $8$ M’s, and $8$ W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw?
[b]3.2[/b] Let $T$ be the answer from the previous problem. Charlotte is initially standi... | \frac{1}{5} | 306 | 7 |
math | \section*{Problem 3 - 011043}
Let
\[
s=\sqrt[3]{20+14 \sqrt{2}}+\sqrt[3]{20-14 \sqrt{2}}
\]
Calculate \(s_{2}\) and \(s_{3}\) and try to find a rational value for \(s\)! (The root values must not be replaced by approximate values.)
Translated as requested, preserving the original formatting and line breaks. | 4 | 104 | 1 |
math | 6.52. $\lim _{x \rightarrow+\infty} \frac{\ln x}{x}$. | 0 | 25 | 1 |
math | 9. Find the sum of the areas of all different rectangles that can be formed using 9 squares (not necessarily all), if the side of each square is $1 \mathrm{~cm}$. | 72 | 41 | 2 |
math | Find all integers $x, y, z \geq 0$ such that $5^{x} 7^{y}+4=3^{z}$. | (1,0,2) | 35 | 7 |
math | 3. Solve the inequality $27^{\log _{3}^{2} x}-8 \cdot x^{\log _{3} x} \geqslant \log _{25} 4+\left(9-\log _{5}^{2} 2\right) \log _{250} 5$. | x\in(0;\frac{1}{3}]\cup[3;+\infty) | 76 | 21 |
math | 8.140. $\operatorname{tg}\left(x-15^{\circ}\right) \operatorname{ctg}\left(x+15^{\circ}\right)=\frac{1}{3}$. | 45(4k+1),k\inZ | 49 | 12 |
math | 1. Petya and Vasya competed in a 60 m race. When Petya finished, Vasya was 9 m behind him. During the second race, Petya started exactly 9 m behind Vasya. Who finished first in the second race and by how many meters did he outpace his opponent? (Assume that each boy ran at the same constant speed both times). | 1.35 | 86 | 4 |
math | For a positive integer $n$, let $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ digits of $n$. For example, $f(96)=1+2+3+6+9=21$. Find the largest positive integer $n$ such that for all positive integers $k$, there is some positive integer $a$ such that $f^k(a)=n$, where $f^... | 15 | 135 | 2 |
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