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 | From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$. The point $ M \in (AE$ is such that $ M$ external to $ ABC$, $ \angle AMB \equal{} 20 ^\circ$ and $ \angle AMC \equal{} 30 ^ \circ$. What is the measure of the angle $ \angle MAB$? | 20^\circ | 97 | 4 |
math | $1 \cdot 146$ Simplify $\left(\frac{1 \cdot 2 \cdot 4+2 \cdot 4 \cdot 8+\cdots+n \cdot 2 n \cdot 4 n}{1 \cdot 3 \cdot 9+2 \cdot 6 \cdot 18+\cdots+n \cdot 3 n \cdot 9 n}\right)^{\frac{1}{3}}$. | \frac{2}{3} | 95 | 7 |
math | 9.1. From points A and B towards each other with constant speeds, a motorcyclist and a cyclist started simultaneously from A and B, respectively. After 20 minutes from the start, the motorcyclist was 2 km closer to B than the midpoint of AB, and after 30 minutes, the cyclist was 3 km closer to B than the midpoint of AB... | 24 | 95 | 2 |
math | A student read the book with $480$ pages two times. If he in the second time for every day read $16$ pages more than in the first time and he finished it $5$ days earlier than in the first time. For how many days did he read the book in the first time? | 15 | 65 | 2 |
math | 7. In trapezoid $A B C D, A D$ is parallel to $B C$. If $A D=52, B C=65, A B=20$, and $C D=11$, find the area of the trapezoid. | 594 | 61 | 3 |
math | a) A championship will have 7 competitors. Each of them will play exactly one game against all the others. What is the total number of games in the championship?
b) A championship will have $n$ competitors and each of them will play exactly one game against all the others. Verify that the total number of games is $\fr... | \frac{n(n-1)}{2} | 214 | 10 |
math | The surface area of the circumscribed sphere of a cube $K_{1}$ is twice as large as the surface area of the inscribed sphere of a cube $K_{2}$. Let $V_{1}$ denote the volume of the inscribed sphere of cube $K_{1}$, and $V_{2}$ the volume of the circumscribed sphere of cube $K_{2}$. What is the ratio $\frac{V_{1}}{V_{2}... | \frac{2\sqrt{2}}{27} | 101 | 13 |
math | Example 2 $x$ is a positive real number, find the minimum value of the function $y=x^{2}+x+\frac{3}{x}$. | 5 | 35 | 1 |
math | 7. Given two moving points $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ on the parabola $x^{2}=4 y$ with $y_{1}+y_{2}=2$ and $y_{1} \neq y_{2}$. If the perpendicular bisector of segment $A B$ intersects the $y$-axis at point $C$, then the maximum value of the area of $\triangle A B C$ is $\qquad$ | \frac{16\sqrt{6}}{9} | 117 | 13 |
math | Three. (50 points) On a line $l$, there are $n+1$ points numbered $1, 2, \cdots, n+1$ arranged from left to right. In the plane above $l$, connect these points with $n$ continuous curves, satisfying:
(1) Any curve connects only two different points, and at most one curve connects any two points;
(2) If point $i$ is con... | A_{n}=\frac{1}{n+1} \mathrm{C}_{2 n}^{n} | 206 | 24 |
math | 5. Find the maximum value of the quantity $x^{2}+y^{2}$, given that
$$
x^{2}+y^{2}=3 x+8 y
$$ | 73 | 41 | 2 |
math | 3. (10 points) A water pool has two drainage taps, $A$ and $B$. When both taps are opened simultaneously, it takes 30 minutes to drain the full pool; if both taps are opened for 10 minutes, then tap $A$ is closed and tap $B$ continues to drain, it also takes 30 more minutes to drain the full pool. How many minutes does... | 45 | 102 | 2 |
math | Example 6 Let $x_{1}=1, x_{n+1}=\frac{x_{n}^{2}}{\sqrt{3 x_{n}^{4}+6 x_{n}^{2}+2}}$ $(n \geqslant 1)$. Find $x_{n}$. | x_{n}=\frac{\sqrt{2}}{\sqrt{5^{2^{n-1}}-3}} | 68 | 25 |
math | Example 1. Find the function $w=f(z)$ that conformally maps the upper half-plane $\operatorname{Im} z>0$ onto the region
$$
0<\arg w<\alpha \pi, \quad \text { where } \quad 0<\alpha<2
$$
of the $w$-plane. | z^{\alpha} | 74 | 5 |
math | ## Task Condition
Find the derivative.
$y=\frac{7^{x}(3 \sin 3 x+\cos 3 x \cdot \ln 7)}{9+\ln ^{2} 7}$ | 7^{x}\cdot\cos3x | 46 | 9 |
math | 1. Solve the equation
$$
\sqrt{x-9}+\sqrt{x+12}-\sqrt{x-3}-\sqrt{x-4}=0
$$ | \frac{793}{88} | 36 | 10 |
math | Example 1. Let's find pairs of real numbers $x$ and $y$ that satisfy the equation
$$
x^{2}+4 x \cos x y+4=0
$$ | -2 | 42 | 2 |
math | ## Aufgabe 18/80
Gesucht sind alle reellen Zahlen $x$, die der Gleichung für beliebige reelle Zahlen $a$ genügen:
$$
\log _{a^{2}+4}\left(2 a^{4} x^{2}-60 a^{4} x+\sqrt{9 x^{2}-27 x-7289}\right)=\log _{a^{2}+16}\left(a x^{2}+6 x-7379\right)
$$
| x_{1}=30 | 122 | 6 |
math | 1.- Find two positive integers $a$ and $b$ given their sum and their least common multiple. Apply this in the case where the sum is 3972 and the least common multiple is 985928.
## SOLUTION: | =1964,b=2008 | 53 | 11 |
math | 23 *. There are $k$ colors. In how many ways can the sides of a given regular $n$-gon be painted so that adjacent sides are painted in different colors (the polygon cannot be rotated)?
## $10-11$ grades | x_{n}=(k-1)^{n}+(k-1)(-1)^{n} | 53 | 23 |
math | 3. (CUB 3$)^{\mathrm{LMO1}}$ Let $n>m \geq 1$ be natural numbers such that the groups of the last three digits in the decimal representation of $1978^{m}, 1978^{n}$ coincide. Find the ordered pair $(m, n)$ of such $m, n$ for which $m+n$ is minimal. | (3,103) | 87 | 7 |
math | 13.357. A cyclist traveled 96 km 2 hours faster than he had planned. During this time, for each hour, he traveled 1 km more than he had planned to travel in 1 hour and 15 minutes. At what speed did he travel? | 16 | 60 | 2 |
math | Three. (25 points) Find all pairs of positive integers $(a, b)$ such that $a^{3}=49 \times 3^{b}+8$.
---
Please note that the translation retains the original formatting and structure of the text. | (11,3) | 54 | 6 |
math | 8. 1b.(TUR 5) Find the smallest positive integer $n$ such that
(i) $n$ has exactly 144 distinct positive divisors, and
(ii) there are ten consecutive integers among the positive divisors of $n$. | 110880 | 56 | 6 |
math | 7.244. $3 \log _{2}^{2} \sin x+\log _{2}(1-\cos 2 x)=2$. | (-1)^{n}\frac{\pi}{6}+\pin,n\inZ | 35 | 18 |
math | Paulinho is training for his arithmetic test. To become faster, he keeps performing various additions. Paulinho asks his father to help by choosing five integers from 1 to 20. Then, Paulinho adds them up. After several attempts, his father noticed that he was getting very bored with the additions and decided to make hi... | 15to90 | 138 | 5 |
math | 8. The domain of the function $f(x)$ is $\mathbf{R}$, and for any $x_{1}, x_{2} \in \mathbf{R}$, it holds that $f\left(x_{1}+x_{2}\right)=f\left(x_{1}\right)+f\left(x_{2}\right)$. When $x \geqslant 0$, the function $f(x)$ is increasing.
(1) Prove that the function $y=-(f(x))^{2}$ is increasing on $(-\infty, 0]$ and dec... | \in(-2,1) | 191 | 7 |
math | 13. Given $f(x)=\frac{x}{1+x}$. Find the value of the following expression:
$$
\begin{array}{l}
f\left(\frac{1}{2004}\right)+f\left(\frac{1}{2003}\right)+\cdots+f\left(\frac{1}{2}\right)+f(1)+ \\
f(0)+f(1)+f(2)+\cdots+f(2003)+f(2004) .
\end{array}
$$ | 2004 | 119 | 4 |
math | 1. Two cyclists started from places $A$ and $B$ towards each other. When they met, the first cyclist had traveled $\frac{4}{7}$ of the distance and an additional $\frac{24}{10} \mathrm{~km}$, while the second cyclist had traveled half the distance of the first. Find the distance from $A$ to $B$. | 25.2 | 80 | 4 |
math | 5. An astronomer discovered that the intervals between the appearances of comet $2011 Y$ near planet $12 I V 1961$ are consecutive terms of a decreasing geometric progression. The three most recent intervals (in years) are the roots of the cubic equation $t^{3}-c t^{2}+350 t-1000=0$, where $c-$ is some constant. What w... | 2.5 | 106 | 3 |
math | 11. Sides $A B$ and $C D$ of quadrilateral $A B C D$ are perpendicular and are diameters of two equal tangent circles with radius $r$. Find the area of quadrilateral $A B C D$, if $\frac{|B C|}{|A D|}=k$. | 3r^{2}|\frac{1-k^{2}}{1+k^{2}}| | 66 | 20 |
math | 3.338. a) $\sin 15^{\circ}=\frac{\sqrt{6}-\sqrt{2}}{4} ;$ b) $\cos 15^{\circ}=\frac{\sqrt{6}+\sqrt{2}}{4}$. | \sin15=\frac{\sqrt{6}-\sqrt{2}}{4};\cos15=\frac{\sqrt{6}+\sqrt{2}}{4} | 61 | 38 |
math | 4. Let $n=\overline{a b c}$ be a three-digit number, where $a, b, c$ are the lengths of the sides that can form an isosceles (including equilateral) triangle. The number of such three-digit numbers $n$ is $\qquad$.
untranslated part: $\qquad$ | 165 | 71 | 3 |
math | 7. Given positive integers $x, y, z$ satisfying $x y z=(14-x)(14-y)(14-z)$, and $x+y+z<28$. Then the maximum value of $x^{2}+y^{2}+z^{2}$ is . $\qquad$ | 219 | 66 | 3 |
math | 7. Let $f(m)$ be the product of the digits of the positive integer $m$. Find the positive integer solutions to the equation $f(m)=m^{2}-10 m-36$. | 13 | 43 | 2 |
math | 2. A polynomial $p(x)$, when divided by the polynomial $x-1$, gives a remainder of 2, and when divided by the polynomial $x-2$, gives a remainder of 1. Find the remainder obtained when the polynomial $p(x)$ is divided by $(x-1)(x-2)$. | -x+3 | 68 | 3 |
math | 1. Determine the real number $a$ such that the polynomials $x^{2}+a x+1$ and $x^{2}+x+a$ have at least one common root.
(3rd Canadian Olympiad) | =1or=-2 | 49 | 5 |
math | Example 10 Find the minimum value of the function $f(x, y)=\frac{3}{2 \cos ^{2} x \sin ^{2} x \cos ^{2} y}+$ $\frac{5}{4 \sin ^{2} y}\left(x, y \neq \frac{k \pi}{2}, k \in Z\right)$. (Example 3 in [4]) | \frac{29+4 \sqrt{30}}{4} | 91 | 16 |
math | 2. (17 points) Hooligan Vasily tore out a whole chapter from a book, the first page of which was numbered 241, and the number of the last page consisted of the same digits. How many sheets did Vasily tear out of the book?
# | 86 | 59 | 2 |
math | 4. Calculate the value of the expression
$$
\left(\frac{1+i}{\sqrt{2}}\right)^{n+4}+\left(\frac{1-i}{\sqrt{2}}\right)^{n+4}+\left(\frac{1+i}{\sqrt{2}}\right)^{n}+\left(\frac{1-i}{\sqrt{2}}\right)^{n},
$$
( i is the imaginary unit). | 0 | 100 | 1 |
math | 3. If the real numbers $x, \alpha, \beta$ satisfy
$$
x=\log _{3} \tan \alpha=-\log _{3} \tan \beta \text {, and } \alpha-\beta=\frac{\pi}{6} \text {, }
$$
then the value of $x$ is $\qquad$ . | \frac{1}{2} | 78 | 7 |
math | What is the last digit of $2019^{2020^{2021}}$? | 1 | 24 | 1 |
math | "Finding all positive integer solutions to the equation $x^{y}=y^{x}+1$ is a problem I studied over twenty years ago. At that time, I provided a solution, but there was actually an issue with one step in the process. It was pointed out by readers after I included this problem in the 2015 Special Issue of *High School M... | (2,1),(3,2) | 164 | 9 |
math | For every positive integer $n$, let $s(n)$ be the sum of the exponents of $71$ and $97$ in the prime factorization of $n$; for example, $s(2021) = s(43 \cdot 47) = 0$ and $s(488977) = s(71^2 \cdot 97) = 3$. If we define $f(n)=(-1)^{s(n)}$, prove that the limit
\[ \lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n... | \frac{20}{21} | 156 | 10 |
math | For how many integer values of $m$,
(i) $1\le m \le 5000$
(ii) $[\sqrt{m}] =[\sqrt{m+125}]$
Note: $[x]$ is the greatest integer function | 72 | 55 | 2 |
math | 6. Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from eleven consecutive natural numbers to some number $a$ is 902, and the sum of the distances from these same eleven numbers to some number $b$ is 374. Find all possible values of $a$, given th... | =107,=-9,=25 | 82 | 11 |
math | 4.1. Find the maximum of the expression $a / b + b / c$ given $0 < a \leqslant b \leqslant a + b \leqslant c$. | \frac{3}{2} | 45 | 7 |
math | If $x, y$ are real numbers, and $1 \leqslant x^{2}+4 y^{2} \leqslant 2$. Find the range of $x^{2}-2 x y+4 y^{2}$. | \left[\frac{1}{2}, 3\right] | 55 | 14 |
math | Example 3 Let $S_{n}=1+2+3+\cdots+n, n \in \mathbf{N}$, find the maximum value of $f(n)=\frac{S_{n}}{(n+32) S_{n+1}}$. | \frac{1}{50} | 58 | 8 |
math | Éveriste listed all of the positive integers from 1 to 90 . He then crossed out all of the multiples of 3 from the list. Of the remaining numbers, he then crossed out all of the multiples of 5 . How many numbers were not crossed out? | 48 | 57 | 2 |
math | 1. Find all functions $f: \mathbf{Q}_{+} \rightarrow \mathbf{Q}_{+}$ such that for all $x, y \in \mathbf{Q}_{+}$, we have
$$
f\left(x^{2} f^{2}(y)\right)=f^{2}(x) f(y) .
$$ | f(x)=1 | 76 | 4 |
math | 6. In $\triangle A B C, A B=A C=13$ and $B C=10 . P$ is a point on $B C$ with $B P<P C$, and $H, K$ are the orthocentres of $\triangle A P B$ and $\triangle A P C$ respectively. If $H K=2$, find $P C$.
(1 mark)
在 $\triangle A B C$ 中, $A B=A C=13$ 而 $B C=10 \circ P$ 是 $B C$ 上滿足 $B P<P C$ 的一點, 而 $H 、 K$ 分別是 $\triangle A ... | 7.4 | 178 | 3 |
math | Let $\overline{AB}$ be a line segment with length $10$. Let $P$ be a point on this segment with $AP = 2$. Let $\omega_1$ and $\omega_2$ be the circles with diameters $\overline{AP}$ and $\overline{P B}$, respectively. Let $XY$ be a line externally tangent to $\omega_1$ and $\omega_2$ at distinct points $X$ and $Y$ , re... | \frac{16}{5} | 114 | 8 |
math | 4. Determine the largest natural number less than the value of the expression $\underbrace{\sqrt{6+\sqrt{6+\cdots \sqrt{6}}}}_{2001 \text { roots }}+\underbrace{\sqrt[3]{6+\sqrt[3]{6+\cdots \sqrt[3]{6}}}}_{2001 \text { roots }}$, given that $\sqrt[3]{6}>1.8$. | 4 | 94 | 1 |
math | 4. The smallest positive integer $a$ that makes the inequality $\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n+1}<a-2007 \frac{1}{3}$ hold for all positive integers $n$ is $\qquad$. | 2009 | 70 | 4 |
math | 10. The function $f(x)$ satisfies: for any real numbers $x, y$, we have
$$
\frac{f(x) f(y)-f(x y)}{3}=x+y+2 .
$$
Then the value of $f(36)$ is | 39 | 59 | 2 |
math | 7.3. In 60 chandeliers (each chandelier has 4 lampshades), lampshades need to be replaced. Each electrician spends 5 minutes replacing one lampshade. A total of 48 electricians will be working. Two lampshades in a chandelier cannot be replaced simultaneously. What is the minimum time required to replace all the lampsha... | 25 | 87 | 2 |
math | 2. A natural number ends with zero, and the greatest of its divisors, not equal to itself, is a power of a prime number. Find the second-to-last digit of this number. | 1or5 | 40 | 3 |
math | Let $n>1$ be an integer and $S_n$ the set of all permutations $\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \}$ where $\pi$ is bijective function. For every permutation $\pi \in S_n$ we define:
\[ F(\pi)= \sum_{k=1}^n |k-\pi(k)| \ \ \text{and} \ \ M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi) \]
where $M_n$ is taken wi... | \frac{n^2 - 1}{3} | 151 | 11 |
math | 13. Let \( f(x, y) = \frac{a x^{2} + x y + y^{2}}{x^{2} + y^{2}} \), satisfying
\[
\max _{x^{2}+y^{2}+0} f(x, y) - \min _{x^{2}+y^{2}+0} f(x, y) = 2 \text{. }
\]
Find \( a \). | a=1 \pm \sqrt{3} | 101 | 10 |
math | Example 12 Find $\sum_{k=1}^{8} \frac{1}{k!}$ equals what. | 1.71827877 \cdots | 26 | 13 |
math | (2) Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=2,(n+1) a_{n+1}=a_{n}+n\left(n \in \mathbf{N}^{*}\right)$, then $a_{n}=$ $\qquad$. | a_{n}=\frac{1}{n!}+1 | 69 | 14 |
math | Once, a king called all his pages and lined them up. He gave the first page a certain number of ducats, the second page two ducats less, the third page again two ducats less, and so on. When he reached the last page, he gave him the appropriate number of ducats, turned around, and proceeded in the same manner back to t... | 16 | 184 | 2 |
math | Linguistics
$[\quad$ Case enumeration $\quad]$
A three-digit number consists of different digits in ascending order, and in its name, all words start with the same letter. Another three-digit number, on the contrary, consists of the same digits, but in its name, all words start with different letters. What are these ... | 147111 | 71 | 6 |
math | Mom brought 56 strawberries and 39 raspberries to the bowl and took them to Ema, who was reading. Ema clarified her reading by snacking, and she did so by taking two random pieces of fruit at a time:
- If she drew two raspberries, she traded them with Mom for one strawberry and returned it to the bowl.
- If she drew t... | raspberry | 174 | 3 |
math | Five. (15 points) Given a function $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for any real numbers $x, y, z$ we have
$$
\begin{array}{l}
\frac{1}{2} f(x y)+\frac{1}{2} f(x z)-f(x) f(y z) \geqslant \frac{1}{4} . \\
\text { Find }[1 \times f(1)]+[2 f(2)]+\cdots+[2011 f(2011)]
\end{array}
$$
where $[a]$ denotes th... | 1011030 | 154 | 7 |
math | Amy has divided a square up into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates \( x \), the sum... | 2.5 | 99 | 3 |
math | Inscribe a rectangle of base $b$ and height $h$ in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of $h$ do the rectangle and triangle have the same area? | h = \frac{2}{5} | 72 | 10 |
math | Three. (25 points) A total of no more than 30 football teams from the East and West participate in the Super League. The East has 1 more team than the West. Any two teams play exactly one match. Each team earns 3 points for a win, 1 point for a draw, and 0 points for a loss. After the league ends, the statistics show t... | 7 | 159 | 1 |
math | 1. A complete number is a 9-digit number that contains each of the digits 1 through 9 exactly once. Furthermore, the difference number of a number $N$ is the number you get when you take the difference of each pair of adjacent digits in $N$ and concatenate all these differences. For example, the difference number of 25... | 4,5 | 179 | 3 |
math | $6 \cdot 25$ Find all positive integers $k$ such that the polynomial $x^{2 k+1}+x+1$ is divisible by $x^{k}+x+1$. For each $k$ that satisfies this condition, find all positive integers $n$ such that $x^{n}+x+1$ is divisible by $x^{k}+x+1$.
(British Mathematical Olympiad, 1991) | k=2,n=3m+2 | 101 | 9 |
math | 1. Find the complex number \( z \) that satisfies the equations
$$
|z+2i|=|z-4i| \quad \text{and} \quad |z-4|=1.
$$ | 4+i | 46 | 2 |
math | Example 12 Let $x, y, z$ be real numbers, not all zero, find the maximum value of $\frac{x y+2 y z}{x^{2}+y^{2}+z^{2}}$. | \frac{\sqrt{5}}{2} | 49 | 10 |
math | A rhombus $\mathcal R$ has short diagonal of length $1$ and long diagonal of length $2023$. Let $\mathcal R'$ be the rotation of $\mathcal R$ by $90^\circ$ about its center. If $\mathcal U$ is the set of all points contained in either $\mathcal R$ or $\mathcal R'$ (or both; this is known as the [i]union[/i] of $\mathca... | \frac{1}{2023} | 193 | 10 |
math | 6. Given a triangle with three interior angles in arithmetic progression, the difference between the longest and shortest sides is 4 times the height on the third side. Then the largest interior angle is larger than the smallest interior angle by $\qquad$ (express using inverse trigonometric functions).
| \pi-\arccos\frac{1}{8} | 58 | 13 |
math | 3. The number of positive integers $m$ that make $m^{2}+m+7$ a perfect square is $\qquad$ . | 2 | 31 | 1 |
math | ## Task 2 - 070832
The cross sum of a natural number is understood to be the sum of its digits: for example, 1967 has the cross sum $1+9+6+7=23$.
Determine the sum of all cross sums of the natural numbers from 1 to 1000 inclusive! | 13501 | 78 | 5 |
math | 16. How many 10-digit numbers are there whose digits are all 1,2 or 3 and in which adjacent digits differ by 1 ? | 64 | 33 | 2 |
math | 2. Philatelist Andrey decided to distribute all his stamps equally into 3 envelopes, but it turned out that one stamp was extra. When he distributed them equally into 5 envelopes, 3 stamps were extra; finally, when he distributed them equally into 7 envelopes, 5 stamps remained. How many stamps does Andrey have in tota... | 208 | 114 | 3 |
math | A net for hexagonal pyramid is constructed by placing a triangle with side lengths $x$, $x$, and $y$ on each side of a regular hexagon with side length $y$. What is the maximum volume of the pyramid formed by the net if $x+y=20$? | 128\sqrt{15} | 60 | 9 |
math | 154. a) Find the number if the product of all its divisors is 108.
b) Find the number if the product of its divisors is 5832. | 18 | 42 | 2 |
math | ## problem statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(1 ; 1 ; 2)$
$M_{2}(-1 ; 1 ; 3)$
$M_{3}(2 ;-2 ; 4)$
$M_{0}(2 ; 3 ; 8)$ | 7\sqrt{\frac{7}{10}} | 90 | 11 |
math | 11、(20 points) Given the hyperbola
$$
C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)
$$
$F_{1}, F_{2}$ are the left and right foci of the hyperbola $C$, respectively, and $P$ is a point on the right branch of the hyperbola $C$ such that $\angle F_{1} P F_{2}=\frac{\pi}{3}$. The area of $\triangle F_{1} P F_{2}$... | e=2,\lambda=2 | 258 | 7 |
math | 3 [ Perpendicularity of a line and a plane (other).]
The base of the pyramid is an equilateral triangle with a side length of 6. One of the lateral edges is perpendicular to the base plane and equals 4. Find the radius of the sphere circumscribed around the pyramid.
# | 4 | 63 | 1 |
math | We unpack 187 books in a box, which weigh a total of $189 \mathrm{~kg}$. The average weight of large books is $2.75 \mathrm{~kg}$, medium-sized books weigh $1.5 \mathrm{~kg}$ on average, and small books weigh $\frac{1}{3} \mathrm{~kg}$ on average. How many books are there of each size, if the total weight of the large ... | 36,34,117 | 115 | 9 |
math | 3. [4 points] Solve the inequality $\left(\sqrt{x^{3}-10 x+7}+1\right) \cdot\left|x^{3}-18 x+28\right| \leqslant 0$. | -1+\sqrt{15} | 53 | 8 |
math | Let $A$ be the number of 2019-digit numbers, that is made of 2 different digits (For example $10\underbrace{1...1}_{2016}0$ is such number). Determine the highest power of 3 that divides $A$. | \nu_3(A) = 5 | 61 | 10 |
math | ## Task A-3.1.
How many natural numbers $n$ are there for which there exists a triangle with sides of length
$$
3, \quad \log _{2} n \quad \text{and} \quad \log _{4} n \quad ?
$$ | 59 | 61 | 2 |
math | Of the students attending a school athletic event, $80\%$ of the boys were dressed in the school colors, $60\%$ of the girls were dressed in the school colors, and $45\% $ of the students were girls. Find the percentage of students attending the event who were wearing the school colors. | 71\% | 69 | 4 |
math | 2. Given $a, b, c > 0$, find the maximum value of the expression
$$
A=\frac{a^{3}+b^{3}+c^{3}}{(a+b+c)^{3}-26 a b c}
$$ | 3 | 56 | 1 |
math | Points $D$ and $E$ are chosen on the exterior of $\vartriangle ABC$ such that $\angle ADC = \angle BEC = 90^o$. If $\angle ACB = 40^o$, $AD = 7$, $CD = 24$, $CE = 15$, and $BE = 20$, what is the measure of $\angle ABC $ in,degrees? | 70^\circ | 91 | 4 |
math | Let $m=999 \ldots 99$ be the number formed by 77 digits all equal to 9 and let $n=777 \ldots 77$ be the number formed by 99 digits all equal to 7. What is the number of digits of $m \cdot n$? | 176 | 72 | 3 |
math | Example 2 (2006 National High School Mathematics Competition Question) Express 2006 as the sum of 5 positive integers $x_{1}, x_{2}, x_{3}, x_{4}$, $x_{5}$. Let $S=\sum_{1 \leqslant i<j \leqslant 5} x_{i} x_{j}$. Questions:
(1) For what values of $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ does $S$ attain its maximum value;
(2... | x_{1}=x_{2}=x_{3}=402,x_{4}=x_{5}=400 | 201 | 26 |
math | ## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(7, -5, 1)$
$B(5, -1, -3)$
$C(3, 0, -4)$ | -2x+y-z+20=0 | 63 | 10 |
math | 1. Let $A \cup B=\{1,2, \cdots, 10\},|A|=|B|$. Then the number of all possible ordered pairs of sets $(A, B)$ is | 8953 | 46 | 4 |
math | ## Task A-2.2.
Solve the equation $\sqrt{9-5 x}-\sqrt{3-x}=\frac{6}{\sqrt{3-x}}$ in the set of real numbers. | -3 | 45 | 2 |
math | 8. Let $a \in \mathbf{R}$, and the complex numbers
$$
z_{1}=a+\mathrm{i}, z_{2}=2 a+2 \mathrm{i}, z_{3}=3 a+4 \mathrm{i} \text {. }
$$
If $\left|z_{1}\right|, \left|z_{2}\right|, \left|z_{3}\right|$ form a geometric sequence, then the value of the real number $a$ is . $\qquad$ | 0 | 111 | 1 |
math | 8. (10 points) In the expression $(x+y+z)^{2036}+(x-y-z)^{2036}$, the brackets were expanded and like terms were combined. How many monomials $x^{a} y^{b} z^{c}$ with a non-zero coefficient were obtained? | 1038361 | 69 | 7 |
math | 8. Arrange all positive integers whose sum of digits is 10 in ascending order to form the sequence $\left\{a_{n}\right\}$. If $a_{n}=2017$, then $n=$ . $\qquad$ | 120 | 53 | 3 |
math | Let $k$ be a positive integer and $S$ be a set of sets which have $k$ elements. For every $A,B \in S$ and $A\neq B$ we have $A \Delta B \in S$. Find all values of $k$ when $|S|=1023$ and $|S|=2023$.
Note:$A \Delta B = (A \setminus B) \cup (B \setminus A)$ | k = 2^9 \times m | 103 | 10 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.