task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 10 1.39k | answer_tokens int64 1 98 |
|---|---|---|---|---|
math | 6. For $0<x<1$, if the complex number
$$
z=\sqrt{x}+\mathrm{i} \sqrt{\sin x}
$$
corresponds to a point, then the number of such points inside the unit circle is $n=$ | 1 | 54 | 1 |
math | For any natural number $n$, we denote $\mathbf{S}(n)$ as the sum of the digits of $n$. Calculate $\mathbf{S}^{5}\left(2018^{2018^{2018}}\right)$. | 7 | 58 | 1 |
math | Let \(Q\) be a set of permutations of \(1,2,...,100\) such that for all \(1\leq a,b \leq 100\), \(a\) can be found to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Find the largest possible number of elements in \(Q\). | 100 | 81 | 3 |
math | 4. A group of 17 middle school students went to several places for a summer social survey, with a budget for accommodation not exceeding $x$ yuan per person per day. One day, they arrived at a place with two hostels, $A$ and $B$. $A$ has 8 first-class beds and 11 second-class beds; $B$ has 10 first-class beds, 4 second... | 10 | 165 | 2 |
math | Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$. | 505 | 51 | 3 |
math | 1. $\left(12\right.$ points) Solve the equation $\left(x^{3}-3\right)\left(2^{\operatorname{ctg} x}-1\right)+\left(5^{x^{3}}-125\right) \operatorname{ctg} x=0$. | \sqrt[3]{3};\frac{\pi}{2}+\pin,\quadn\inZ | 69 | 22 |
math | The Fibonacci sequence is defined recursively by $F_{n+2}=F_{n+1}+F_{n}$ for $n \in \mathbb{Z}$ and $F_{1}=F_{2}=1$. Determine the value of:
$$
\left(1-\frac{F_{2}^{2}}{F_{3}^{2}}\right)\left(1-\frac{F_{3}^{2}}{F_{4}^{2}}\right) \cdot \ldots \cdot\left(1-\frac{F_{2019}^{2}}{F_{2020}^{2}}\right)
$$ | \frac{F_{2021}}{2F_{2019}F_{2020}} | 143 | 26 |
math | An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping? | 4.5 \text{ cm}^2 | 28 | 10 |
math | The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation. | 12 | 85 | 2 |
math | $7 \cdot 2$ When two numbers are drawn without replacement from the set $\left\{-3,-\frac{5}{4},-\frac{1}{2}, 0, \frac{1}{3}, 1, \frac{4}{5}, 2\right\}$, find the probability that the two numbers are the slopes of a pair of perpendicular lines. | \frac{3}{28} | 81 | 8 |
math | ## Task B-1.1.
Determine all natural numbers $x, y, z$ for which
$$
4 x^{2}+45 y^{2}+9 z^{2}-12 x y-36 y z=25
$$
where $x<y<z$. | (1,2,3),(1,2,5),(2,3,6) | 65 | 19 |
math | Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC =60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$.
[i]Proposed by Kaan Dokmeci[/i] | 408 | 89 | 3 |
math | A trapez has parallel sides $A B$ and $C D$, and the intersection point of its diagonals is $M$. The area of triangle $A B M$ is 2, and the area of triangle $C D M$ is 8. What is the area of the trapezoid? | 18 | 66 | 2 |
math | Let $A=\{1,2,\ldots, 2006\}$. Find the maximal number of subsets of $A$ that can be chosen such that the intersection of any 2 such distinct subsets has 2004 elements. | 2006 | 53 | 4 |
math | There is a $40\%$ chance of rain on Saturday and a $30\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive... | 107 | 92 | 3 |
math | 1. $2019^{\ln \ln 2019}-(\ln 2019)^{\ln 2019}$ $=$ . $\qquad$
Fill in the blank (8 points per question, total 64 points)
1. $2019^{\ln \ln 2019}-(\ln 2019)^{\ln 2019}$ $=$ . $\qquad$ | 0 | 100 | 1 |
math | Example 5. Solve the integral equation
$$
\varphi(x)=x+\int_{x}^{\infty} \mathrm{e}^{2(x-t)} \varphi(t) d t
$$ | \varphi(x)=2x+1+Ce^{x} | 46 | 14 |
math | ## Task 1 - 180811
The FDJ members Arnim, Bertram, Christian, Dieter, Ernst, and Fritz participated in a 400-meter race. No two of them finished at the same time.
Before the race, the following three predictions were made about the results (each participant is represented by the first letter of their first name):
| ... | C,B,E,D,A,F | 265 | 6 |
math | An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black. We call this oper... | 32 | 134 | 2 |
math | Let $a$ and $b$ be positive integers that satisfy $ab-7a-11b+13=0$. What is the minimum possible value of $a+b$? | 34 | 40 | 2 |
math | 2. A natural number minus 69 is a perfect square, and this natural number plus 20 is still a perfect square. Then this natural number is $\qquad$ . | 2005 | 38 | 4 |
math | A rectangle has an area of $16$ and a perimeter of $18$; determine the length of the diagonal of the rectangle.
[i]2015 CCA Math Bonanza Individual Round #8[/i] | 7 | 47 | 1 |
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 | [ Volume of a parallelepiped ]
The base of an oblique prism is a parallelogram with sides 3 and 6 and an acute angle of $45^{\circ}$. The lateral edge of the prism is 4 and is inclined to the base plane at an angle of $30^{\circ}$. Find the volume of the prism. | 18\sqrt{2} | 75 | 7 |
math | Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 647 | 69 | 3 |
math | 2. (5 points) The number of different divisors of 60 (excluding 1) is | 11 | 22 | 2 |
math | Find the sum of the two smallest odd primes $p$ such that for some integers $a$ and $b$, $p$ does not divide $b$, $b$ is even, and $p^2=a^3+b^2$.
[i]2021 CCA Math Bonanza Individual Round #13[/i] | 122 | 71 | 3 |
math | 7th Putnam 1947 Problem B3 Let O be the origin (0, 0) and C the line segment { (x, y) : x ∈ [1, 3], y = 1 }. Let K be the curve { P : for some Q ∈ C, P lies on OQ and PQ = 0.01 }. Let k be the length of the curve K. Is k greater or less than 2? Solution | k<2 | 96 | 3 |
math | 1. Two-headed and seven-headed dragons came to a meeting. At the very beginning of the meeting, one of the heads of one of the seven-headed dragons counted all the other heads. There were 25 of them. How many dragons in total came to the meeting? | 8 | 56 | 1 |
math | Let's calculate the sum $S_{n}=1 \cdot 2^{2}+2 \cdot 3^{2}+3 \cdot 4^{2}+\ldots+n(n+1)^{2}$. | \frac{n(n+1)(n+2)(3n+5)}{12} | 48 | 20 |
math | Problem 4. Place digits in the positions of $a$ and $b$, in the four-digit numbers, so that the sum $\overline{323 a}+\overline{b 410}$ is divisible by 9. Determine all possible solutions. | (0,5),(1,4),(4,1),(2,3),(3,2),(5,9),(9,5),(6,8),(8,6),(7,7) | 57 | 41 |
math | Find all positive integers $w$, $x$, $y$ and $z$ which satisfy $w! = x! + y! + z!$. | (w, x, y, z) = (3, 2, 2, 2) | 32 | 23 |
math | B2. Find all natural numbers $n$ and prime numbers $p$ for which $\sqrt[3]{n}+\frac{p}{\sqrt[3]{n}}$ is a square of a natural number. | n=1,p=3n=27,p=3 | 46 | 13 |
math | 6. (40 points) Let $f(x)$ be a polynomial of degree 2010 such that $f(k)=-\frac{2}{k}$ for $k=1,2, \cdots, 2011$.
Find $f(2012)$. | -\frac{1}{503} | 65 | 9 |
math | 5. (5 points) $A$, $B$, and $C$ are three fractions, where both the numerators and denominators are natural numbers. The ratio of the numerators is $3: 2: 1$, and the ratio of the denominators is $2: 3: 4$. The sum of the three fractions is $\frac{29}{60}$. Then, $A-B-C=$ $\qquad$ . | \frac{7}{60} | 94 | 8 |
math | ## Task $3 / 83$
For which natural numbers $n$ does the area $A_{2 n}$ of the regular $2 n$-gon equal twice the area $A_{n}$ of the regular $n$-gon with the same circumradius? | 3 | 57 | 1 |
math | ## Task 5 - V00505
How many matches (5 cm long, 2 mm wide, and 2 mm high) can fit into a cube with a side length of 1 m? | 5000000 | 45 | 7 |
math | The class teacher calculated the class average grades for each subject, and Kati helped her by recalculating the grades based on how many students received each consecutive grade. When comparing the results of the first subject, it turned out that Kati had used the data for consecutive fives in reverse order, taking th... | 6 | 107 | 1 |
math | ## Task B-2.4.
Solve the equation in the set of integers
$$
x^{8}+y^{2016}=32 x^{4}-256
$$ | (2,0)(-2,0) | 43 | 10 |
math | For example, $1 k$ is a natural number, and $\frac{1001 \cdot 1002 \cdot \cdots \cdot 2005 \cdot 2006}{11^{k}}$ is an integer, what is the maximum value of $k$? | 101 | 67 | 3 |
math | 2ag * Find all natural numbers $x$, such that $x^{2}$ is a twelve-digit number of the form: $2525 * * * * * * 89$ (the six “*”s represent six unknown digits, which are not necessarily the same). | 502517,502533,502567,502583 | 60 | 27 |
math | 4. The integer values of $n$ that satisfy the equation $\left(n^{2}-5 n+5\right)^{n+1}=1$ are $\qquad$ .
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 1,4,3,-1 | 64 | 7 |
math | Example 6 Given 2014 real numbers $x_{1}, x_{2}, \cdots, x_{2014}$ satisfy the system of equations
$$
\sum_{k=1}^{2014} \frac{x_{k}}{n+k}=\frac{1}{2 n+1}(n=1,2, \cdots, 2014) .
$$
Try to calculate the value of $\sum_{k=1}^{2014} \frac{x_{k}}{2 k+1}$. | \frac{1}{4}(1-\frac{1}{4029^{2}}) | 123 | 21 |
math | 2. 10 people go to the bookstore to buy books, it is known that: (1) each person bought three types of books; (2) any two people have at least one book in common.
How many people at most could have bought the book that was purchased by the fewest people? | 5 | 63 | 1 |
math | 5. Given two points $A(0,1), B(6,9)$. If there is an integer point $C$ (Note: A point with both coordinates as integers is called an integer point), such that the area of $\triangle A B C$ is minimized. Then the minimum value of the area of $\triangle A B C$ is $\qquad$ | 1 | 77 | 1 |
math | ## Task 24/72
Determine all solutions in the domain of natural numbers for the system of equations
$$
\begin{aligned}
\binom{y+1}{x+1}-\binom{y}{x+1} & =6 \\
\binom{x}{x-2} \cdot \frac{2}{x-1}+\binom{y}{y-1} \cdot \frac{1}{y} & =3
\end{aligned}
$$ | 2,4 | 107 | 3 |
math | Three, (25 points) Given that $m, n, p, q$ satisfy $m n p q = 6(m-1)(n-1)(p-1)(q-1)$.
(1) If $m, n, p, q$ are all positive integers, find the values of $m, n, p, q$;
(2) If $m, n, p, q$ are all greater than 1, find the minimum value of $m+n+p+q$. | (9,4,2,2),(6,5,2,2),(4,3,3,2) | 108 | 25 |
math | 3. Find all natural numbers $n$ for which the number $2^{10}+2^{13}+2^{14}+3 \cdot 2^{n}$ is a square of a natural number.
$(16$ points) | 13,15 | 54 | 5 |
math | 12. Let the function $f(x)$ be a differentiable function defined on the interval $(-\infty, 0)$, with its derivative being $f^{\prime}(x)$, and $2 f(x) + x f^{\prime}(x) > x^{2}$. Then
$$
(x+2017)^{2} f(x+2017)-f(-1)>0
$$
The solution set is $\qquad$ . | (-\infty,-2018) | 102 | 10 |
math | 5. For which $n$ can a square grid $n \times n$ be divided into one $2 \times 2$ square and some number of strips of five cells, such that the square is adjacent to the side of the board? | 5k+2 | 51 | 4 |
math | 1. Let the sum of the digits of the natural number $x$ be $S(x)$, then the solution set of the equation $x+S(x)+S(S(x))+S(S(S(x)))=2016$ is | 1980 | 49 | 4 |
math | Example 6 The number of non-negative integer solutions to the equation $2 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}=3$ is?
(1985 National High School League Question) | 174 | 80 | 3 |
math | On the lateral sides $A B$ and $C D$ of trapezoid $A B C D$, points $M$ and $N$ are taken such that segment $M N$ is parallel to the bases and divides the area of the trapezoid in half. Find the length of $M N$, if $B C=a$ and $A D=b$. | x^2=\frac{^2+b^2}{2} | 79 | 14 |
math | 10.1. Ivan was walking from Pakhomovo to Vorobyevo. At noon, when he had covered $4 / 9$ of the entire distance, Foma set off after him from Pakhomovo on a bicycle, and Erema set off towards him from Vorobyevo. Foma overtook Ivan at 1 PM, and met Erema at 1:30 PM. When will Ivan and Erema meet? | 14:30 | 93 | 5 |
math | $4 \cdot 94$ Solve the equation $\operatorname{arctg} x+\operatorname{arctg}(1-x)=2 \operatorname{arctg} \sqrt{x-x^{2}}$.
(Kyiv Mathematical Olympiad, 1936) | \frac{1}{2} | 62 | 7 |
math | Given $a_n = (n^2 + 1) 3^n,$ find a recurrence relation $a_n + p a_{n+1} + q a_{n+2} + r a_{n+3} = 0.$ Hence evaluate $\sum_{n\geq0} a_n x^n.$ | \frac{1 - 3x + 18x^2}{1 - 9x + 27x^2 - 27x^3} | 69 | 36 |
math | 10. Let $f(x)=\frac{1}{x^{3}-x}$, find the smallest positive integer $n$ that satisfies the inequality $f(2)+f(3)+\cdots+f(n)>\frac{499}{2020}$. | 13 | 60 | 2 |
math | Given an integer $n\ge\ 3$, find the least positive integer $k$, such that there exists a set $A$ with $k$ elements, and $n$ distinct reals $x_{1},x_{2},\ldots,x_{n}$ such that $x_{1}+x_{2}, x_{2}+x_{3},\ldots, x_{n-1}+x_{n}, x_{n}+x_{1}$ all belong to $A$. | k = 3 | 108 | 5 |
math | 10. Let $x_{1}, x_{2}, x_{3}$ be non-negative real numbers, satisfying $x_{1}+x_{2}+x_{3}=1$, find the minimum and maximum values of $\left(x_{1}+3 x_{2}+5 x_{3}\right)\left(x_{1}+\frac{x_{2}}{3}+\frac{x_{3}}{5}\right)$. | \frac{9}{5} | 94 | 7 |
math | ## Aufgabe $3 / 72$
Gegeben seien die vier Scheitelpunkte einer Ellipse. Man konstruiere unter ausschließlicher Verwendung von Zirkel und Lineal das der Ellipse umbeschriebene Quadrat.
| \sqrt{^{2}+b^{2}} | 55 | 11 |
math | 5. Solve the inequality $\log _{1+x^{2}}\left(1+8 x^{5}\right)+\log _{1-3 x^{2}+16 x^{4}}\left(1+x^{2}\right) \leqslant 1+\log _{1-3 x^{2}+16 x^{4}}\left(1+8 x^{5}\right)$. Answer: $x \in\left(-\frac{1}{\sqrt[3]{8}} ;-\frac{1}{2}\right] \cup\left(-\frac{\sqrt{3}}{4} ; 0\right) \cup\left(0 ; \frac{\sqrt{3}}{4}\right) \c... | x\in(-\frac{1}{\sqrt[3]{8}};-\frac{1}{2}]\cup(-\frac{\sqrt{3}}{4};0)\cup(0;\frac{\sqrt{3}}{4})\cup{\frac{1}{2}} | 179 | 61 |
math | Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$ | (k, \ell, m, n) = (3, 3, 3, 2) | 56 | 22 |
math | 865*. Solve the equation in natural numbers $x, y$ and $z$
$$
x y + y z + z x = 2(x + y + z).
$$ | (1;2;4),(1;4;2),(2;1;4),(2;4;1),(2;2;2),(4;1;2),(4;2;1) | 39 | 43 |
math | 5. Find the number of pairs of natural numbers $(x, y), 1 \leqslant x, y \leqslant 1000$, such that $x^{2}+y^{2}$ is divisible by 5. | 360000 | 54 | 6 |
math | 8. (10 points) Let for positive numbers $x, y, z$ the following system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=48 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=64
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 32 | 102 | 2 |
math | In how many ways can 32 knights be placed on an $8 \times 8$ chessboard so that no two attack each other? | 2 | 30 | 1 |
math | 1. The number $a_{n}$ is formed by writing down the first $n$ squares of consecutive natural numbers in sequence. For example, $a_{11}=149162536496481100$ 121. Determine how many numbers divisible by twelve are among the numbers $a_{1}, a_{2}, \ldots, a_{100000}$. | 16667 | 94 | 5 |
math | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{-1 ; 7 ;-4\}$
$p=\{-1 ; 2 ; 1\}$
$q=\{2 ; 0 ; 3\}$
$r=\{1 ; 1 ;-1\}$ | 2p-q+3r | 73 | 6 |
math | 1.51. The side of a regular triangle is equal to $a$. Determine the area of the part of the triangle that lies outside a circle of radius $\frac{a}{3}$, the center of which coincides with the center of the triangle. | \frac{^{2}(3\sqrt{3}-\pi)}{18} | 54 | 19 |
math | ## 261. Math Puzzle $2 / 87$
The pioneers of class 6a are going on a group trip over the weekend. If each participant pays 12 marks, there will be a surplus of 33 marks compared to the required total amount. If each pays 10 marks, however, 11 marks will be missing. How many people are participating in the trip? How mu... | 22 | 94 | 2 |
math | \section*{Problem 2 - 121242}
All pairs \((x, y)\) of integers are to be specified for which the equation is satisfied:
\[
x(x+1)(x+7)(x+8)=y^{2}
\] | (x,y)\in{(-9,\12),(-8,0),(-7,0),(-4,\12),(-1,0),(0,0),(1,\12)} | 59 | 41 |
math | 4. If $x, y, z$ are real numbers, satisfying
$$
x+\frac{1}{y}=2 y+\frac{2}{z}=3 z+\frac{3}{x}=k \text{, and } x y z=3 \text{, }
$$
then $k=$ | 4 | 67 | 1 |
math | Let $ABC$ be a triangle where$\angle$[b]B=55[/b] and $\angle$ [b]C = 65[/b]. [b]D[/b] is the mid-point of [b]BC[/b]. Circumcircle of [b]ACD[/b] and[b] ABD[/b] cuts [b]AB[/b] and[b] AC[/b] at point [b]F[/b] and [b]E[/b] respectively. Center of circumcircle of [b]AEF[/b] is[b] O[/b]. $\angle$[b]FDO[/b] = ? | 30^\circ | 143 | 4 |
math | We have a pile of 2013 black balls and 2014 white balls, and a reserve of as many white balls as we want. We draw two balls: if they are of the same color, we put back one white ball, otherwise we put back the black ball. We repeat this until only one ball remains. What is its color? | black | 75 | 1 |
math | XXV - I - Task 1
During World War I, a battle took place near a certain castle. One of the shells destroyed a statue of a knight with a spear standing at the entrance to the castle. This happened on the last day of the month. The product of the day of the month, the month number, the length of the spear expressed in f... | 1714 | 119 | 4 |
math | 3. Solve the equation
$3 \sqrt{6 x^{2}+13 x+5}-6 \sqrt{2 x+1}-\sqrt{3 x+5}+2=0$ | -1/3;-4/9 | 45 | 8 |
math | Example 5 Let $p$ be a given positive integer, $A$ is a subset of $X=\left\{1,2,3,4, \cdots, 2^{p}\right\}$, and has the property: for any $x \in A$, $2 x \notin A$. Find the maximum value of $|A|$. (1991 French Mathematical Olympiad) | \frac{2^{p+1}+(-1)^{p}}{3} | 87 | 19 |
math | 2. Simplify the fraction: $\frac{x^{14}+x^{13}+\ldots+x+1}{x^{5}+x^{4}+x^{3}+x^{2}+x}$. | \frac{x^{10}+x^{5}+1}{x} | 50 | 17 |
math | 5. The maximum value of the function $f(x)=(6-x)^{3}(x-1)^{\frac{2}{3}}(1 \leqslant x \leqslant 6)$ is
保留了源文本的换行和格式。 | \frac{3^{6} \times 5^{3}}{11^{4}} \sqrt[3]{1100} | 57 | 30 |
math | 2. In the Cartesian coordinate system, a line segment $A B$ of length 1 moves on the $x$-axis (point $A$ is to the left of $B$), point $P(0,1)$ is connected to $A$ by a line, and point $Q(1,2)$ is connected to $B$ by a line. Then the equation of the trajectory of the intersection point $R$ of lines $P A$ and $Q B$ is $... | y(x-2)=-2 | 107 | 7 |
math | Given a set of points in space, a [i]jump[/i] consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in $10$-dimensional-space, it is possible to perform a finite number of jumps so that some two points ... | 1025 | 99 | 6 |
math | Example 22. How many times do you need to roll two dice so that the probability of rolling at least one double six is greater than $1 / 2$? (This problem was first posed by the French mathematician and writer de Mere ( $1610-1684$ ), hence the problem is named after him). | 25 | 73 | 2 |
math | 4. The length and width of a rectangular prism are 20 cm and 15 cm, respectively. If the numerical value of its volume is equal to the numerical value of its surface area, then its height is $\qquad$ cm (write the answer as an improper fraction)
The length and width of a rectangular prism are 20 cm and 15 cm, respecti... | \frac{60}{23} | 116 | 9 |
math | Example 2 Let $x, y, z, w$ be real numbers, not all zero. Find the maximum value of $P=\frac{x y+2 y z+z w}{x^{2}+y^{2}+z^{2}+w^{2}}$. | \frac{\sqrt{2}+1}{2} | 59 | 12 |
math | Task B-2.6. Two cyclists started simultaneously from two places $A$ and $B$ towards each other. After one hour, the first cyclist had traveled $10 \mathrm{~km}$ more than the second cyclist. The first cyclist arrived 50 minutes earlier at place $B$ than the second cyclist at place $A$. What is the distance between plac... | 50\mathrm{~} | 86 | 7 |
math | Let $ABCD$ be a cyclic quadrilateral, and suppose that $BC = CD = 2$. Let $I$ be the incenter of triangle $ABD$. If $AI = 2$ as well, find the minimum value of the length of diagonal $BD$. | 2\sqrt{3} | 58 | 6 |
math | Let $P(x)=x^3+ax^2+bx+c$ be a polynomial where $a,b,c$ are integers and $c$ is odd. Let $p_{i}$ be the value of $P(x)$ at $x=i$. Given that $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$, find the value of $p_{2}+2p_{1}-3p_{0}.$ | 18 | 113 | 2 |
math | 13. In $\triangle A B C$, $\angle B=\frac{\pi}{4}, \angle C=\frac{5 \pi}{12}, A C$ $=2 \sqrt{6}, A C$'s midpoint is $D$. If a line segment $P Q$ of length 3 (point $P$ to the left of point $Q$) slides on line $B C$, then the minimum value of $A P+D Q$ is $\qquad$. | \frac{\sqrt{30}+3\sqrt{10}}{2} | 104 | 19 |
math | Four. (50 points) The International Mathematical Olympiad Chief Committee has $n$ countries participating, with each country being represented by a team leader and a deputy leader. Before the meeting, the participants shake hands with each other, but the team leader does not shake hands with their own deputy leader. Af... | =0,n=50 | 137 | 6 |
math | 42. Find the algebraic complements of the elements $a_{13}, a_{21}, a_{31}$ of the determinant
$$
D=\left|\begin{array}{rrr}
-1 & 2 & 3 \\
2 & 0 & -3 \\
3 & 2 & 5
\end{array}\right|
$$ | A_{13}=4,A_{21}=-4,A_{31}=-6 | 78 | 20 |
math | 1. A mowing crew mowed the entire meadow in two days. On the first day, half of the meadow and another 3 hectares were mowed, and on the second day, a third of the remaining area and another 6 hectares were mowed. What is the area of the meadow? | 24 | 65 | 2 |
math | 11. Let $f(x)=x^{2}+6 x+c$ for all real numbers $x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly 3 distinct real roots? | \frac{11-\sqrt{13}}{2} | 55 | 14 |
math | How many ways are there to rearrange the letters of CCAMB such that at least one C comes before the A?
[i]2019 CCA Math Bonanza Individual Round #5[/i] | 40 | 42 | 2 |
math | ## Subject IV. (20 points)
Emil is waiting in line at a ticket booth, along with other people, standing in a row. Andrei, who is right in front of Emil, says: "Behind me, there are 5 times as many people as in front of me." Mihai, who is right behind Emil, says: "Behind me, there are 3 times as many people as in front... | 25 | 157 | 2 |
math | Determine the smallest positive integer $n$ with the following property: For all positive integers $x, y$, and $z$ with $x \mid y^{3}$ and $y \mid z^{3}$ and $z \mid x^{3}$, it always holds that $x y z \mid (x+y+z)^{n}$.
(Gerhard J. Woeginger)
Answer. The smallest such number is $n=13$. | 13 | 96 | 2 |
math | 6. In $\triangle A B C$, $\angle B=\frac{\pi}{3}, A C=\sqrt{3}$, point $D$ is on side $A B$, $B D=1$, and $D A=D C$. Then $\angle D C A=$ $\qquad$ | \frac{\pi}{6} | 62 | 7 |
math | 17. Calculate $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\cdots+\frac{2001}{1999!+2000!+2001!}$ The value is $\qquad$ . | \frac{1}{2}-\frac{1}{2001!} | 69 | 18 |
math | What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive? | 6 | 40 | 1 |
math | Given four positive numbers: $a, b, c, d$. Among the products $a b, a c, a d, b c, b d, c d$, we know the values of five of them, which are 2, 3, 4, 5, and 6. What is the value of the sixth product? | \frac{12}{5} | 72 | 8 |
math | Problem 11.2. In the store, there are 9 headphones, 13 computer mice, and 5 keyboards. In addition, there are 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse". In how many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646. | 646 | 79 | 3 |
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