task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | G5.3 If $R^{2000}<5^{3000}$, where $R$ is a positive integer, find the largest value of $R$. | 11 | 38 | 2 |
math | 3A. Find all positive numbers $a$ for which both roots of the equation $a^{2} x^{2}+a x+1-7 a^{2}=0$ are integers. | \in{1,\frac{1}{2},\frac{1}{3}} | 42 | 18 |
math | G10.1 The sum of 3 consecutive odd integers (the smallest being $k$ ) is 51 . Find $k$. G10.2 If $x^{2}+6 x+k \equiv(x+a)^{2}+C$, where $a, C$ are constants, find $C$.
G10.3 If $\frac{p}{q}=\frac{q}{r}=\frac{r}{s}=2$ and $R=\frac{p}{s}$, find $R$.
$$
\begin{aligned}
R & =\frac{p}{s} \\
& =\frac{p}{q} \times \frac{q}{r}... | 15 | 213 | 2 |
math | Example 7 (20th Irish Mathematical Olympiad) Find all prime numbers $p, q$ such that $p \mid (q+6)$ and $q \mid (p+7)$. | q=13,p=19 | 43 | 8 |
math | 5. (COL 1) Consider the polynomial $p(x)=x^{n}+n x^{n-1}+a_{2} x^{n-2}+\cdots+a_{n}$ having all real roots. If $r_{1}^{16}+r_{2}^{16}+\cdots+r_{n}^{16}=n$, where the $r_{j}$ are the roots of $p(x)$, find all such roots. | r_{1}=r_{2}=\ldots=r_{n}=-1 | 102 | 17 |
math | Provide an example of a quadratic trinomial $P(x)$ such that for any $x$ the equality $P(x)+P(x+1)+\cdots+P(x+10)=x^{2}$ holds. | \frac{x^{2}-10x+15}{11} | 47 | 16 |
math | In a quadrilateral $ABCD$, we have $\angle DAB = 110^{\circ} , \angle ABC = 50^{\circ}$ and $\angle BCD = 70^{\circ}$ . Let $ M, N$ be the mid-points of $AB$ and $CD$ respectively. Suppose $P$ is a point on the segment $M N$ such that $\frac{AM}{CN} = \frac{MP}{PN}$ and $AP = CP$ . Find $\angle AP C$. | 120^\circ | 113 | 5 |
math | 2B. An alloy of zinc and silver with a mass of 3.5 kilograms contains $75\%$ silver. When this alloy is melted and mixed with another alloy of the same elements, a third alloy with a mass of 10.5 kilograms is obtained, which has $84\%$ silver. What is the percentage of silver in the second alloy? | 88 | 79 | 2 |
math | 3. Let $M=\frac{8}{\sqrt{2008}-44}$, $a$ is the integer part of $M$, and $b$ is the fractional part of $M$. Then
$$
a^{2}+3(\sqrt{2008}+37) a b+10=
$$ | 2008 | 74 | 4 |
math | ## Task 6A - 291246A
In two urns $A$ and $B$, there are exactly $m$ red and exactly $n$ blue balls in total. The total number of balls is greater than 2; at least one of the balls is red.
At the beginning, $A$ contains all the red and $B$ all the blue balls.
By alternately taking one randomly selected ball from $A$ ... | (4,2) | 265 | 5 |
math | 9. To what power must the root $x_{0}$ of the equation $x^{11} + x^{7} + x^{3} = 1$ be raised to obtain the number $x_{0}^{4} + x_{0}^{3} - 1 ?$ | 15 | 63 | 2 |
math | 1. Given the parabola
$$
y=x^{2}+(k+1) x+1
$$
intersects the $x$-axis at two points $A$ and $B$, not both on the left side of the origin. The vertex of the parabola is $C$. To make $\triangle A B C$ an equilateral triangle, the value of $k$ is $\qquad$ | -5 | 89 | 2 |
math | Example. Find the solution to the Cauchy problem for the equation
$$
y^{\prime}-\frac{1}{x} y=-\frac{2}{x^{2}}
$$
with the initial condition
$$
y(1)=1 \text {. }
$$ | \frac{1}{x} | 59 | 7 |
math | 6. $P$ is the midpoint of the height $V O$ of the regular quadrilateral pyramid $V-A B C D$. If the distance from point $P$ to the side face is 3, and the distance from point $P$ to the base is 5, then the volume of the regular quadrilateral pyramid is | 750 | 69 | 3 |
math | [ [ Linear Recurrence Relations ]
Calculate the sum: $S_{n}=C_{n}^{0}-C_{n-1}^{1}+C_{n-2}^{2}-\ldots$.
# | S_{n}=0,ifn\equiv2,5(\bmod6);S_{n}=1,ifn\equiv0,1(\bmod6);S_{n}=-1,ifn\equiv3,4(\bmod6) | 48 | 55 |
math | $15 \cdot 21$ Can 2 be written in the form
$$
2=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots+\frac{1}{n_{1974}} .
$$
where $n_{1}, n_{2}, \cdots, n_{1974}$ are distinct natural numbers.
(Kiev Mathematical Olympiad, 1974) | 1974 | 95 | 4 |
math | 5. Let $\sin \theta+\cos \theta=\frac{\sqrt{2}}{3}, \frac{\pi}{2}<\theta<\pi$, then the value of $\tan \theta-\cot \theta$ is $\qquad$ | -\frac{8}{7}\sqrt{2} | 52 | 11 |
math | Determine all quadruples $(p, q, r, n)$ of prime numbers $p, q, r$ and positive integers $n$ for which
$$
p^{2}=q^{2}+r^{n}
$$
is satisfied.
(Walther Janous)
Answer. There are two such quadruples $(p, q, r, n)$, namely $(3,2,5,1)$ and $(5,3,2,4)$. | (3,2,5,1)(5,3,2,4) | 98 | 17 |
math | Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that, for all $a, b \in \mathbb{N}^{*}$, we have
$$
f(y)+2 x \mid 2 f(x)+y
$$ | f(x)=x | 67 | 4 |
math | 2. Connecting the intersection points of $x^{2}+y^{2}=10$ and $y=\frac{4}{x}$ in sequence, a convex quadrilateral is formed. The area of this quadrilateral is $\qquad$ | 12 | 51 | 2 |
math | 【Question 1】Calculate: $5 \times 13 \times 31 \times 73 \times 137=$ | 20152015 | 31 | 8 |
math | Example 2 The sequence $\left\{a_{n}\right\}$:
$1,1,2,1,1,2,3,1,1,2,1,1,2,3,4, \cdots$.
Its construction method is:
First, give $a_{1}=1$, then copy this item 1 and add its successor number 2, to get $a_{2}=1, a_{3}=2$;
Next, copy all the previous items $1,1,2$, and add the successor number 3 of 2, to get
$$
a_{4}=1, ... | 3952 | 275 | 4 |
math | Let $E = \{1, 2, \dots , 16\}$ and let $M$ be the collection of all $4 \times 4$ matrices whose entries are distinct members of $E$. If a matrix $A = (a_{ij} )_{4\times4}$ is chosen randomly from $M$, compute the probability $p(k)$ of $\max_i \min_j a_{ij} = k$ for $k \in E$. Furthermore, determine $l \in E$ such that ... | l = 5 | 130 | 5 |
math | Find all function $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for every three real positive number $x,y,z$ :
$$ x+f(y) , f(f(y)) + z , f(f(z))+f(x) $$
are length of three sides of a triangle and for every postive number $p$ , there is a triangle with these sides and perimeter $p$.
[i]Proposed by Amirhossein Zo... | f(x) = x | 108 | 6 |
math | Task 1. Find all pairs $(x, y)$ of integers that satisfy
$$
x^{2}+y^{2}+3^{3}=456 \sqrt{x-y} .
$$ | (30,21),(-21,-30) | 43 | 14 |
math | 9. (16 points) Let positive real numbers $a, b$ satisfy $a+b=1$.
Find the minimum value of the function
$$
f(a, b)=\left(\frac{1}{a^{5}}+a^{5}-2\right)\left(\frac{1}{b^{5}}+b^{5}-2\right)
$$ | \frac{31^{4}}{32^{2}} | 79 | 14 |
math | 9. Solve the equation
$$
\sqrt[3]{x^{2}+2 x}+\sqrt[3]{3 x^{2}+6 x-4}=\sqrt[3]{x^{2}+2 x-4}
$$
Solution: Let's make the substitution $t=x^{2}+2 x$, as a result of which the equation takes the form $\sqrt[3]{t}+\sqrt[3]{3 t-4}=\sqrt[3]{t-4}$. Further, we have
$$
\begin{aligned}
& \sqrt[3]{t}+\sqrt[3]{3 t-4}=\sqrt[3]{t... | {-2;0} | 420 | 5 |
math | 5.1. Solve the inequality
$$
8 \cdot \frac{|x+1|-|x-7|}{|2 x-3|-|2 x-9|}+3 \cdot \frac{|x+1|+|x-7|}{|2 x-3|+|2 x-9|} \leqslant 8
$$
In the answer, write the sum of its integer solutions that satisfy the condition $|x|<120$. | 6 | 105 | 1 |
math | 14. Let $a_{1}=2006$, and for $n \geq 2$,
$$
a_{1}+a_{2}+\cdots+a_{n}=n^{2} a_{n} .
$$
What is the value of $2005 a_{2005}$ ? | 2 | 72 | 1 |
math | We know the following about the digits of a four-digit number:
I. The sum of the first and second (thousands and hundreds place) digits is equal to the sum of the last two digits.
II. The sum of the second and fourth digits is equal to twice the sum of the first and third digits.
III. Adding the first and fourth dig... | 1854 | 112 | 4 |
math | 4. In $\triangle A B C$, $3 A B=2 A C, E 、 F$ are the midpoints of $A C 、 A B$ respectively. If $B E<t C F$ always holds, then the minimum value of $t$ is $\qquad$ . | \frac{7}{8} | 63 | 7 |
math | 6) Let $\left\{a_{n}\right\}$ be a sequence with the sum of the first $n$ terms denoted as $S_{n}$. Let $T_{n}=\frac{S_{1}+S_{2}+\cdots+S_{n}}{n}$, which is called the "average" of the sequence $a_{1}, a_{2}, \cdots, a_{n}$. Given that the "average" of the sequence $a_{1}, a_{2}, \cdots, a_{1005}$ is 2012, what is the ... | 2009 | 167 | 4 |
math | 1. ( $5 \mathrm{p}$) a) Compare the natural numbers $a$ and $b$, knowing that:
$$
\begin{aligned}
& a=\left[\left(12^{2}-10^{2}\right): 11-1^{2014}\right]: 3-2014^{0} \\
& b=\left[\left(3^{4}-2^{16}: 2^{11}\right): 7-2 \cdot 3\right]^{2014}
\end{aligned}
$$
( 2p ) b) Using the numbers $a$ and $b$ determined in part a... | 89 | 179 | 2 |
math | Problem 4. Vanya thought of a two-digit number, then swapped its digits and multiplied the resulting number by itself. The result turned out to be four times larger than the number he thought of. What number did Vanya think of? | 81 | 49 | 2 |
math | 2. Given is the set $A=\{1,2,3, \ldots, 62,63\}$. A subset $X$ of the set $A$ will be called good if for every element $x \in X$ it holds that $x$ does not divide the sum of the remaining elements of the set $X$. Determine the maximum number of elements that a good subset can have. | 61 | 88 | 2 |
math | In triangle $ABC$ we have $|AB| \ne |AC|$. The bisectors of $\angle ABC$ and $\angle ACB$ meet $AC$ and $AB$ at $E$ and $F$, respectively, and intersect at I. If $|EI| = |FI|$ find the measure of $\angle BAC$. | 60^\circ | 74 | 4 |
math | 4. (7 points) On the board, 38 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 38 minutes? | 703 | 69 | 3 |
math | 4. The numbers $x$ and $y$ are such that the equalities $\operatorname{ctg} x - \operatorname{ctg} y = 2$ and $5 \sin (2 x - 2 y) = \sin 2 x \sin 2 y$ hold. Find $\operatorname{tg} x \operatorname{tg} y$. | -\frac{6}{5} | 81 | 7 |
math | Davi has a very original calculator that performs only two operations, the usual addition $(+)$ and another operation, denoted by $*$, which satisfies
(i) $a * a=a$,
(ii) $a * 0=2a$ and
(iii) $(a * b) + (c * d) = (a + c) * (b + d)$,
for any integers $a$ and $b$. What are the results of the operations $(2 * 3) + (0 ... | -2,2000 | 123 | 7 |
math | Example 3 If the function $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfies for all real numbers $x$,
$$
\sqrt{2 f(x)}-\sqrt{2 f(x)-f(2 x)} \geqslant 2
$$
then the function $f$ is said to have property $P$. Find the largest real number $\lambda$ such that if the function $f$ has property $P$, then for all real numbers ... | 12+8 \sqrt{2} | 121 | 9 |
math | Example 4 Let $\lambda>0$, find the largest constant $c=c(\lambda)$, such that for all non-negative real numbers $x, y$, we have
$$
x^{2}+y^{2}+\lambda x y \geqslant c(x+y)^{2} .
$$ | c(\lambda)=\left\{\begin{array}{ll}
1, & \lambda \geqslant 2, \\
\frac{2+\lambda}{4}, & 0<\lambda<2 .
\end{array}\right.} | 65 | 54 |
math | 4. Determine all complex numbers $z$ and $w$ such that
$$
|z|^{2}+z w+\bar{w}=2+6 i \quad \text { and } \quad|w|^{2}+\overline{z w}+z=2-4 i .
$$ | (z,w)=(-2,2-2i)(z,w)=(-\frac{7+i}{4},\frac{3-9i}{4}) | 67 | 33 |
math | 7. If 653 divides $\overline{a b 2347}$, then $a+b=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 11 | 52 | 2 |
math | Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that the sequence $S_n=a_1+a_2+\ldots+a_n$ is upperbounded and lowerbounded and find its limit as $n\to\infty$.
| \lim_{n \to \infty} S_n = \frac{1}{2} | 69 | 20 |
math | 10.151. A circle is divided into two segments by a chord equal to the side of an inscribed regular triangle. Determine the ratio of the areas of these segments. | \frac{4\pi-3\sqrt{3}}{8\pi+3\sqrt{3}} | 38 | 24 |
math | 20. Pauli and Bor are playing the following game. There is a pile of 99! molecules. In one move, a player is allowed to take no more than 1% of the remaining molecules. The player who cannot make a move loses. They take turns, with Pauli starting. Who among them can win, regardless of how the opponent plays? | Pauli | 76 | 2 |
math | Example 12 Let $m>n \geqslant 1$, find the smallest $m+n$ such that
$$
1000 \mid 1978^{m}-1978^{n} .
$$ | 106 | 51 | 3 |
math | Example $\mathbf{5}$ equation
$$
\begin{array}{l}
\left(x^{3}-3 x^{2}+x-2\right)\left(x^{3}-x^{2}-4 x+7\right)+ \\
6 x^{2}-15 x+18=0
\end{array}
$$
all distinct real roots are $\qquad$ | 1, \pm 2, 1 \pm \sqrt{2} | 83 | 16 |
math | 8. Given that the sum of the distance from any point on curve $C$ to point $A(0,0)$ and the line $x=4$ is equal to 5, for a given point $B(b, 0)$, there are exactly three pairs of distinct points on the curve that are symmetric with respect to point $B$, then the range of $b$ is $\qquad$ | 8.2<b<4 | 85 | 6 |
math | 1. (10 points) $\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{20}\right)+\left(\frac{2}{3}+\cdots+\frac{2}{20}\right)+\cdots+\left(\frac{18}{19}+\frac{18}{20}\right)+\frac{19}{20}=$ $\qquad$ | 95 | 97 | 2 |
math | 4. A three-digit number, all digits of which are different and non-zero, will be called balanced if it is equal to the sum of all possible two-digit numbers formed from the different digits of this number. Find the smallest balanced number. | 132 | 49 | 3 |
math | In a tetrahedron, at each vertex, edges of lengths $5$, $\sqrt{41}$, and $\sqrt{34}$ meet. What are the lengths of the segments connecting the midpoints of the opposite edges of the tetrahedron? | 3,4,5 | 55 | 5 |
math | ## Example. Solve the integral equation
$$
\int_{0}^{x} \cos (x-t) \varphi(t) d t=x
$$ | \varphi(x)=1+\frac{x^{2}}{2} | 34 | 15 |
math | 12. $\left(1+\tan 1^{\circ}\right)\left(1+\tan 2^{\circ}\right)\left(1+\tan 3^{\circ}\right) \cdots\left(1+\tan 43^{\circ}\right)\left(1+\tan 44^{\circ}\right)=$ | 2^{22} | 76 | 5 |
math | Find the integers $n \geq 1$ and $p$ primes such that $n p+n^{2}$ is a perfect square. | ((\frac{p-1}{2})^{2},p) | 30 | 15 |
math | Given 4. $m, n$ are positive integers, and in the polynomial $f(x)=(1+x)^{m}+(1+x)^{n}$, the coefficient of $x$ is 19.
1) Try to find the minimum value of the coefficient of $x^{2}$ in $f(x)$;
2) For the $m, n$ that make the coefficient of $x^{2}$ in $f(x)$ the smallest, find the term containing $x^{7}$ at this time. | 156 | 109 | 3 |
math | 10.339. The center of an equilateral triangle with a side length of 6 cm coincides with the center of a circle with a radius of 2 cm. Determine the area of the part of the triangle that lies outside this circle. | 2(3\sqrt{3}-\pi)^2 | 53 | 12 |
math | 1. A certain 4-digit number was added to the number written with the same digits but in reverse order, and the result was 4983. What numbers were added | 19922991 | 37 | 8 |
math | 3. If $P$ is a point inside the cube $A B C D-E F G H$, and satisfies $P A=P B=\frac{3 \sqrt{3}}{2}$, $P F=P C=\frac{\sqrt{107}}{2}$, then the edge length of the cube is | 5 | 68 | 1 |
math | Task B-4.5. In the race, 100 people participated, and no two people finished the race with the same time. At the end of the race, each participant was asked what place they finished in, and everyone answered with a number between 1 and 100.
The sum of all the answers is 4000. What is the smallest number of incorrect a... | 12 | 95 | 2 |
math | 6. In her fourth year at Hogwarts, Hermione was outraged by the infringement of house-elf rights and founded the Association for the Restoration of Elven Independence. Of course, even the brightest and noblest idea requires funding for promotion, so Hermione decided to finance her campaign by producing merchandise, sta... | 7.682 | 263 | 5 |
math | 4. Calculate $\sqrt{6+\sqrt{32}}-\sqrt{6-\sqrt{32}}$. | 2\sqrt{2} | 24 | 6 |
math | For example, $73 \leqslant n \in \mathbf{N}$, let $S$ be the set of all non-empty subsets of the set $\{2,3, \cdots, n\}$. For each $S_{i} \subseteq S, i=1,2, \cdots, 2^{n-1}-1$, let $p_{i}$ be the product of all elements in $S_{i}$. Find $p_{1}+p_{2}+\cdots+p_{2^{n-1}-1}$. | \frac{(n+1)!}{2}-1 | 124 | 11 |
math | ## Task A-4.3.
A die is rolled three times in a row. Determine the probability that each subsequent roll (after the first) results in a number that is not less than the previous one. | \frac{7}{27} | 43 | 8 |
math | What are all pairs of integers $(r, p)$ for which $r^{2}-r(p+6)+p^{2}+5 p+6=0$ ? | (3,1),(4,1),(0,-2),(4,-2),(0,-3),(3,-3) | 36 | 25 |
math | Let $m$ and $n$ be relatively prime positive integers. Determine all possible values of \[\gcd(2^m - 2^n, 2^{m^2+mn+n^2}- 1).\] | 1 \text{ and } 7 | 47 | 8 |
math | 3. On a line, consider the points $A_{0}, A_{1}, A_{2}, \ldots, A_{10}$, in this order, such that $A_{1}$ is the midpoint of $\left[A_{0} A_{2}\right]$, $A_{2}$ is the midpoint of $\left[A_{0} A_{3}\right]$, and so on, up to $A_{9}$ being the midpoint of $\left[A_{0} A_{10}\right]$. Knowing that $A_{0} A_{1}=2 \text{~c... | 341 | 207 | 3 |
math | 16. $[7]$ Let $\mathbb{R}$ be the set of real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that for all real numbers $x$ and $y$, we have
$$
f\left(x^{2}\right)+f\left(y^{2}\right)=f(x+y)^{2}-2 x y \text {. }
$$
Let $S=\sum_{n=-2019}^{2019} f(n)$. Determine the number of possible values of $S... | 2039191 | 127 | 7 |
math | 5. Let $n$ be an arbitrary number written with 2000 digits and divisible by 9. Let the sum of the digits of the number $n$ be denoted by $a$, the sum of the digits of $a$ be denoted by $b$, and the sum of the digits of $b$ be denoted by $c$. Determine the number $c$. | 9 | 82 | 1 |
math | 2. Let the sequence $\left\{a_{n}\right\}$ satisfy, $a_{1}=1, a_{n+1}=5 a_{n}+1(n=1,2, \cdots)$, then $\sum_{n=1}^{2018} a_{n}=$ | \frac{5^{2019}}{16}-\frac{8077}{16} | 68 | 25 |
math | 10. (ROM 4) Consider two segments of length $a, b(a>b)$ and a segment of length $c=\sqrt{a b}$.
(a) For what values of $a / b$ can these segments be sides of a triangle?
(b) For what values of $a / b$ is this triangle right-angled, obtuse-angled, or acute-angled? | 1<k<\frac{3+\sqrt{5}}{2},k=\frac{1+\sqrt{5}}{2},1<k<\frac{1+\sqrt{5}}{2},\frac{1+\sqrt{5}}{2}<k<\frac{3+\sqrt{5}}{2} | 82 | 69 |
math | Example 2 If the set $A=\{x \mid-2 \leqslant x \leqslant 5\}, B=\{x \mid m+1 \leqslant x \leqslant 2 m-1\}$ and $B \subseteq A$, find the set of values that $m$ can take. | \leqslant3 | 76 | 6 |
math | ## Task Condition
Find the derivative of the specified order.
$$
y=\left(2 x^{3}+1\right) \cos x, y^{V}=?
$$ | (30x^{2}-120)\cosx-(2x^{3}-120x+1)\sinx | 38 | 28 |
math | 10.085. A circle of radius $R$ is inscribed in an isosceles trapezoid. The upper base of the trapezoid is half the height of the trapezoid. Find the area of the trapezoid. | 5R^{2} | 58 | 5 |
math | (3) In $\triangle A B C$, the lengths of the sides opposite to angles $A, B, C$ are $a, b, c$ respectively, and $S_{\triangle}=a^{2}-$ $(b-c)^{2}$. Then $\tan \frac{A}{2}=$ $\qquad$. | \frac{1}{4} | 70 | 7 |
math | Find all positive integer solutions to the equation $w! = x! + y! + z!$ . (1983 Canadian Mathematical Olympiad problem) | x=2, y=2, z=2, w=3 | 33 | 15 |
math | $$
a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b) \geqslant k a b c
$$
holds for all right triangles, and determine when equality occurs. | 2+3\sqrt{2} | 47 | 8 |
math | 11.4. The base of a right prism is a quadrilateral inscribed in a circle with a radius of $25 \mathrm{~cm}$. The areas of the lateral faces are in the ratio 7:15:20:24, and the length of the diagonal of the largest lateral face is 52 cm. Calculate the surface area of the prism. (7 points) | 4512\mathrm{~}^{2} | 85 | 12 |
math | 314. Find the derivative of the function $y=\sin \left(x^{3}-3 x^{2}\right)$. | (3x^{2}-6x)\cos(x^{3}-3x^{2}) | 28 | 19 |
math | 2. Given $\sin \alpha+\cos \alpha=\frac{\sqrt{2}}{2}$. Then $\sin ^{4} \alpha+\cos ^{4} \alpha=$ $\qquad$ | \frac{7}{8} | 44 | 7 |
math | On an island, two types of people live: good and bad. The good ones always tell the truth, the bad ones always lie. Naturally, everyone is either a boy or a girl on the island. Once, two people said the following about each other:
- Ali: We are bad.
- Bali: We are boys.
Determine whether each of them is good and what... | Ali | 81 | 1 |
math | 3. The number of ordered pairs $(a, b)$ that satisfy $(a+b \mathrm{i})^{6}=a-b \mathrm{i}$ (where $\left.a, b \in \mathbf{R}, \mathrm{i}^{2}=-1\right)$ is
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 8 | 84 | 1 |
math | 8. Let $S=\left\{r_{1}, r_{2}, \cdots, r_{n}\right\} \subseteq\{1,2,3, \cdots, 50\}$, and any two numbers in $S$ do not sum to a multiple of 7, find the maximum value of $n$.
| 23 | 75 | 2 |
math | 14.27. Given 6 digits: $0,1,2,3,4,5$. Find the sum of all four-digit even numbers that can be written using these digits (the same digit can be repeated in a number). | 1769580 | 51 | 7 |
math | If $$\Omega_n=\sum \limits_{k=1}^n \left(\int \limits_{-\frac{1}{k}}^{\frac{1}{k}}(2x^{10} + 3x^8 + 1)\cos^{-1}(kx)dx\right)$$Then find $$\Omega=\lim \limits_{n\to \infty}\left(\Omega_n-\pi H_n\right)$$ | \pi \left( \frac{2}{11} \zeta(11) + \frac{3}{9} \zeta(9) \right) | 96 | 37 |
math | Example 10 Simplify:
$$
\frac{(\sqrt{x}-\sqrt{y})^{3}+2 x \sqrt{x}+y \sqrt{y}}{x \sqrt{x}+y \sqrt{y}}+\frac{3 \sqrt{x y}-3 y}{x-y} .
$$ | 3 | 68 | 1 |
math | 4. Kolya has 440 identical cubes with a side length of 1 cm. Kolya assembled a rectangular parallelepiped from them, with all edges having a length of at least 5 cm. Find the total length of all the edges of this parallelepiped | 96 | 59 | 2 |
math | Example 5 Solve the system of simultaneous equations
$$
\left\{\begin{array}{l}
x+y+z=3, \\
x^{2}+y^{2}+z^{2}=3, \\
x^{5}+y^{5}+z^{5}=3 .
\end{array}\right.
$$
Find all real or complex roots. | x=y=z=1 | 78 | 5 |
math | One degree on the Celsius scale is equal to 1.8 degrees on the Fahrenheit scale, while $0^{\circ}$ Celsius corresponds to $32^{\circ}$ Fahrenheit.
Can a temperature be expressed by the same number of degrees both in Celsius and Fahrenheit? | -40 | 55 | 3 |
math | P r o b l e m 6. Let's find a four-digit number that is a perfect square, if its first two digits and last two digits are equal. | 7744=88^{2} | 35 | 10 |
math | 2. (20 points) Let a line passing through the origin with a positive slope intersect the ellipse $\frac{x^{2}}{4}+y^{2}=1$ at points $E$ and $F$. Points $A(2,0)$ and $B(0,1)$. Find the maximum value of the area of quadrilateral $A E B F$.
| 2\sqrt{2} | 80 | 6 |
math | ## Task Condition
Find the point $M^{\prime}$ symmetric to the point $M$ with respect to the line.
$M(0; -3; -2)$
$$
\frac{x-0.5}{0}=\frac{y+1.5}{-1}=\frac{z-1.5}{1}
$$ | M^{\}(1;2;3) | 74 | 10 |
math | 2. In a convex quadrilateral $A B C D$, the diagonals $A C$ and $B D$ intersect at point $O$. If $S_{\triangle O A D}=4, S_{\triangle O B C}=9$, then the minimum value of the area of the convex quadrilateral $A B C D$ is | 25 | 72 | 2 |
math | 1B. In the set of real numbers, solve the equation
$$
\sqrt{\frac{x^{2}-2 x+3}{x^{2}+2 x+4}}+\sqrt{\frac{x^{2}+2 x+4}{x^{2}-2 x+3}}=\frac{5}{2} \text {. }
$$ | x_{1}=2,x_{2}=\frac{4}{3} | 74 | 16 |
math | For example, if $5 x, y$ are prime numbers, solve the equation $x^{y}-y^{x}=x y^{2}-19$.
(2004 Balkan Mathematical Olympiad) | 2,7;2,3 | 46 | 7 |
math | GS. 4 If the mean, mode and median of the data $30,80,50,40, d$ are all equal, find the value of $d$. | 50 | 40 | 2 |
math | Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive in... | 25 | 91 | 2 |
math | 1. Given positive real numbers $a, b (a>b)$, the product of the distances from the points $\left(\sqrt{a^{2}-b^{2}}, 0\right)$ and $\left(-\sqrt{a^{2}-b^{2}}, 0\right)$ to the line
$$
\frac{x \cos \theta}{a}+\frac{y \sin \theta}{b}=1
$$
is $\qquad$ | b^2 | 97 | 3 |
math | 1. (16 points) Solve the equation $x-5=\frac{3 \cdot|x-2|}{x-2}$. If the equation has multiple roots, write their sum in the answer.
# | 8 | 45 | 1 |
math | $12 \cdot 93$ Find the positive integer solutions of the equation $x^{2 y}+(x+1)^{2 y}=(x+2)^{2 y}$.
(21st Moscow Mathematical Olympiad, 1958) | 3,1 | 57 | 3 |
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