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 | 12.060. The base of the pyramid is an isosceles triangle, with the lateral side equal to ' $a$, and the angle at the vertex equal to $\alpha$. All lateral edges are inclined to the plane of the base at an angle $\beta$. Find the volume of the pyramid. | \frac{1}{6}^3\sin\frac{\alpha}{2}\tan\beta | 65 | 21 |
math | 5. Find the sum of all numbers of the form $x+y$, where $x$ and $y$ are natural number solutions to the equation $5 x+17 y=307$. | 164 | 42 | 3 |
math | One evening a theater sold 300 tickets for a concert. Each ticket sold for \$40, and all tickets were purchased using \$5, \$10, and \$20 bills. At the end of the evening the theater had received twice as many \$10 bills as \$20 bills, and 20 more \$5 bills than \$10 bills. How many bills did the theater receive altoge... | 1210 | 87 | 4 |
math | 14.22. How many pairs of integers $x, y$, lying between 1 and 1000, are there such that $x^{2}+y^{2}$ is divisible by $7 ?$ | 20164 | 48 | 5 |
math | Problem 8. In 28 examination tickets, each includes two theoretical questions and one problem. The student has prepared 50 theoretical questions and can solve the problems in 22 tickets. What is the probability that, by randomly picking one ticket, the student will be able to answer all the questions in the ticket? | 0.625 | 66 | 5 |
math | ### 3.19. Calculate
$$
\int_{0}^{i} z \sin z d z
$$ | -\frac{i}{e} | 27 | 6 |
math | 9. Let $[x]$ be the greatest integer not exceeding the real number $x$. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+3 a_{n}+1, \quad n \in N^{*}$, find $\left[\sum_{k=1}^{2017} \frac{a_{k}}{a_{k}+2}\right]$.
| 2015 | 113 | 4 |
math | Example 10 Let $k$ be a positive integer, find all polynomials $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}$, where $a_{i}$ are real numbers, satisfying the equation $P(P(x))=[P(x)]^{k}$.
(1975 Canadian Olympiad Problem) | P(x)=x^korP(x)=where0,\1dependingonk | 86 | 16 |
math | 1. Solve, in the set of complex numbers, the system of equations $z^{19} w^{25}=1$, $z^{5} w^{7}=1, z^{4}+w^{4}=2$. | (1,1),(-1,-1),(i,i),(-i,-i) | 50 | 18 |
math | 1. Problem: The cost of 3 hamburgers, 5 milk shakes, and 1 order of fries at a certain fast food restaurant is $\$ 23.50$. At the same restaurant, the cost of 5 hamburgers, 9 milk shakes, and 1 order of fries is $\$ 39.50$. What is the cost of 2 hamburgers, 2 milk shakes ,and 2 orders of fries at this restaurant? | 15 | 98 | 2 |
math | 6. Let $A$ and $B$ be non-empty subsets of the set $\{1,2, \cdots, 10\}$, and the smallest element in set $A$ is not less than the largest element in set $B$. Then the number of such pairs $(A, B)$ is. | 9217 | 67 | 4 |
math | 6. Find all real-coefficient polynomials $P$ such that for all non-zero real numbers $x, y, z$ satisfying $2xyz = x + y + z$, we have
$$
\begin{array}{l}
\frac{P(x)}{yz} + \frac{P(y)}{zx} + \frac{P(z)}{xy} \\
= P(x-y) + P(y-z) + P(z-x).
\end{array}
$$ | P(x)=(x^2+3) | 101 | 9 |
math | Find all functions $f$ from the set $\mathbf{R}$ of real numbers to itself such that $f(x y+1)=x f(y)+2$ for all $x, y \in \mathbf{R}$. | f(x)=2x | 50 | 5 |
math | 15. Find the minimum value of
$$
\left|\sin x+\cos x+\frac{\cos x-\sin x}{\cos 2 x}\right|
$$ | 2 | 37 | 1 |
math | ## Task 6 - 341236
Determine for each odd natural number $n \geq 3$ the number
$$
\left[\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\ldots+\frac{1}{\sqrt{n^{2}-4}+\sqrt{n^{2}-3}}+\frac{1}{\sqrt{n^{2}-2}+\sqrt{n^{2}-1}}\right]
$$
Hint: If $z$ is a real number, then [ $z]$ denot... | \frac{n-1}{2} | 169 | 8 |
math | W6 Find all natural numbers $x$ that satisfy the following conditions: the product of the digits of $x$ equals $44x - 86868$, and the sum of the digits is a cube number. | 1989 | 48 | 4 |
math | Three. (This question is worth 16 points)
Let $F$ be the set of all ordered $n$-tuples $\left(A_{1}, A_{2}, \cdots, A_{n}\right)$, where $A_{i}(1 \leqslant i \leqslant n)$ are subsets of the set $\{1,2,3, \cdots, 2002\}$. Let $|A|$ denote the number of elements in the set $A$. For all elements $\left(A_{1}, A_{2}, \cdo... | 2002\left(2^{2002 n}-2^{2001 n}\right) | 227 | 25 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{e^{2 x}-e^{3 x}}{\operatorname{arctg} x-x^{2}}$ | -1 | 48 | 2 |
math | 1. The sum of six consecutive natural numbers, none of which is divisible by 7, is divisible by 21, but not by 42. Prove it!
Determine six such numbers, such that their sum is a four-digit number and is the square of some natural number. | 659,660,661,662,663,664 | 61 | 23 |
math | 5. (10 points) There are some natural numbers written on the blackboard, with an average of 30; after writing down 100, the average becomes 40; if another number is written last, the average becomes 50, then what is the last number written?
Translate the above text into English, please keep the original text's line ... | 120 | 88 | 3 |
math | Let's determine those two-digit numbers $\overline{a b}$ in the decimal system for which the greatest common divisor of $\overline{a b}$ and $\overline{b a}$ is $a^{2}-b^{2}$. | (10,1),(21,12),(54,45) | 51 | 18 |
math | 19.1.11 * Find all positive integers $n$ such that $(n-36)(n-144)-4964$ is a perfect square. | 2061,1077,489,297 | 39 | 17 |
math | 4. Given that the complex number $z$ satisfies $z^{3}+z=2|z|^{2}$. Then all possible values of $z$ are $\qquad$ | 0, 1, -1 \pm 2i | 40 | 12 |
math | # Problem 3.
In triangle $A B C$, the bisector $B E$ and the median $A D$ are equal and perpendicular. Find the area of triangle $A B C$, if $A B=\sqrt{13}$. | 12 | 52 | 2 |
math | [ Geometry (other) ]
A sphere with radius $3 / 2$ has its center at point $N$. From point $K$, located at a distance of $3 \sqrt{5} / 2$ from the center of the sphere, two lines $K L$ and $K M$ are drawn, touching the sphere at points $L$ and $M$ respectively. Find the volume of the pyramid $K L M N$, given that $M L=... | 1 | 100 | 1 |
math | In five years, Tom will be twice as old as Cindy. Thirteen years ago, Tom was three times as old as Cindy. How many years ago was Tom four times as old as Cindy? | 19 | 40 | 2 |
math | For her daughter’s $12\text{th}$ birthday, Ingrid decides to bake a dodecagon pie in celebration. Unfortunately, the store does not sell dodecagon shaped pie pans, so Ingrid bakes a circular pie first and then trims off the sides in a way such that she gets the largest regular dodecagon possible. If the original pie wa... | 64 | 128 | 2 |
math | ## 98. Math Puzzle $7 / 73$
A plastic pipe has a density of $1.218 \mathrm{~g} / \mathrm{cm}^{3}$. The wall thickness is $2.0 \mathrm{~mm}$, the outer diameter $32 \mathrm{~mm}$, and the mass $0.8 \mathrm{~kg}$.
How long is the pipe? | 3.49\, | 91 | 6 |
math | 3. Let $C$ be the unit circle. Four distinct, smaller congruent circles $C_{1}, C_{2}, C_{3}, C_{4}$ are internally tangent to $C$ such that $C_{i}$ is externally tangent to $C_{i-1}$ and $C_{i+1}$ for $i=$ $1, \ldots, 4$ where $C_{5}$ denotes $C_{1}$ and $C_{0}$ represents $C_{4}$. Compute the radius of $C_{1}$. | \sqrt{2}-1 | 117 | 6 |
math | Peshnin A.
What is the minimum number of colors needed to color the natural numbers so that any two numbers differing by 2 or by a factor of two are colored differently?
# | 3 | 38 | 1 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$,
$$
(x-y) f(x+y)=x f(x)-y f(y) .
$$ | f()=+b | 56 | 5 |
math | A fair coin is flipped nine times. Which is more likely, having exactly four heads or having exactly five heads? | \frac{126}{512} | 23 | 11 |
math | 7. If the sequence of positive numbers $\left\{a_{n}\right\}$ is a geometric sequence, and $a_{2} \cdot a_{6}+2 a_{4} a_{5}+a_{1} a_{9}=25$, then the value of $a_{4}+a_{5}$ is $\qquad$ . | 5 | 78 | 1 |
math | Let $a,b\in \mathbb{R}$ and $z\in \mathbb{C}\backslash \mathbb{R}$ so that $\left| a-b \right|=\left| a+b-2z \right|$.
a) Prove that the equation ${{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}={{\left| a-b \right|}^{x}}$, with the unknown number $x\in \mathbb{R}$, has a unique solution.
b) Solve the follo... | x \geq 2 | 202 | 7 |
math | $15 \cdot 2$ In the set $\{1000,1001, \cdots, 2000\}$, how many pairs of consecutive integers can be added without carrying?
(10th American Invitational Mathematics Examination, 1992) | 156 | 63 | 3 |
math | Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$. Prove that the sequence $\{x_{n}\}$ converges and find its limit. | 1 | 77 | 3 |
math | ## Problem Statement
Find the angle between the planes:
$$
\begin{aligned}
& 3 x-y-5=0 \\
& 2 x+y-3=0
\end{aligned}
$$ | \frac{\pi}{4} | 44 | 7 |
math | \section*{Problem 6}
The difference between the longest and shortest diagonals of the regular n-gon equals its side. Find all possible \(n\).
| 9 | 34 | 1 |
math | Example 4 Given $u+v=96$, and the quadratic equation $x^{2} + u x + v = 0$ has integer roots, then its largest root is $\qquad$
(1996, Anhui Province Partial Areas Junior High School Mathematics League) | 98 | 60 | 2 |
math | 6. Given a positive integer $k$ that satisfies for any positive integer $n$, the smallest prime factor of $n^{2}+n-k$ is no less than 11. Then, $k_{\text {min }}$ $=$ . $\qquad$ | 43 | 58 | 2 |
math | 5. Find all natural numbers $n \geq 2$ such that $20^{n}+19^{n}$ is divisible by $20^{n-2}+19^{n-2}$. | 3 | 48 | 1 |
math | 8. 5 (CMO 13) In a non-obtuse $\triangle ABC$, $AB > AC$, $\angle B = 45^{\circ}$, $O$ and $I$ are the circumcenter and incenter of $\triangle ABC$ respectively, and $\sqrt{2} OI = AB - AC$. Find $\sin A$. | \sinA=\frac{\sqrt{2}}{2} | 77 | 13 |
math | Task 2. Every student in the Netherlands receives a finite number of cards. Each card has a real number in the interval $[0,1]$ written on it. (The numbers on different cards do not have to be different.) Find the smallest real number $c>0$ for which the following holds, regardless of the numbers on the cards that ever... | 11-\frac{1}{91} | 120 | 10 |
math | 9. (16 points) Given that the line $l$ is tangent to the parabola $y=\frac{1}{2} x^{2}$, and intersects the circle $x^{2}+y^{2}=1$ at points $A$ and $B$. Let $M$ be the midpoint of $A B$, and $E\left(0,-\frac{1}{2}\right)$. Find the range of $|E M|$. | [\frac{\sqrt{2\sqrt{3}-3}}{2},\frac{2\sqrt{2}-1}{2}) | 100 | 29 |
math | ## Task $7 / 71$
Let $p_{n}$ be the n-th prime number. Find all $i$ such that $p_{i}=2 i+1$. | i=5i=6 | 39 | 6 |
math | Yashchenko I.V.
In Mexico, ecologists have succeeded in passing a law according to which each car must not be driven at least one day a week (the owner reports to the police the car's number and the "day off" for the car). In a certain family, all adults wish to drive daily (each for their own business!). How many car... | 6 | 100 | 1 |
math | I3.1 Given that $a$ is a positive real root of the equation $2^{x+1}=8^{\frac{1}{x}-\frac{1}{3}}$. Find the value of $a$.
I3.2 The largest area of the rectangle with perimeter $a$ meter is $b$ square meter, find the value of $b$.
I3.3 If $c=\left(1234^{3}-1232 \times\left(1234^{2}+2472\right)\right) \times b$, find the... | 6 | 201 | 1 |
math | 5. Three military trains passed through the railway station. The first had 462 soldiers, the second had 546, and the third had 630. How many cars were in each train if it is known that each car had the same number of soldiers and that this number was the largest possible? | 11,13,15 | 66 | 8 |
math | Let $ABC$ be a triangle with incenter $I$. Let $M_b$ and $M_a$ be the midpoints of $AC$ and $BC$, respectively. Let $B'$ be the point of intersection of lines $M_bI$ and $BC$, and let $A'$ be the point of intersection of lines $M_aI$ and $AC$.
If triangles $ABC$ and $A'B'C$ have the same area, what are the possible va... | 60^\circ | 107 | 4 |
math | 4-8. In a chess club, 90 children attend. During the session, they divided into 30 groups of 3 people, and in each group, everyone played one game with each other. No other games were played. In total, there were 30 games of "boy+boy" and 14 games of "girl+girl". How many "mixed" groups were there, that is, groups wher... | 23 | 99 | 2 |
math | 2. In $\triangle A B C$, $A B=A C, \angle A=20^{\circ}$, point $M$ is on $A C$ and satisfies $A M=B C$. Find the degree measure of $\angle B M C$. | 30^{\circ} | 55 | 6 |
math | Let $a$ and $b$ be two non-zero digits not necessarily different. The two-digit number $\overline{a b}$ is called curious if it is a divisor of the number $\overline{b a}$, which is formed by swapping the order of the digits of $\overline{a b}$. Find all curious numbers.
Note: The bar over the numbers serves to distin... | {11,22,33,44,55,66,77,88,99} | 102 | 28 |
math | There are 2016 customers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.)
Find the maximal $k$ such that the following holds:
There are $k$ customers such that either all ... | 45 | 97 | 2 |
math | 9.4. Find the smallest natural number in which each digit occurs exactly once and which is divisible by 990. | 1234758690 | 26 | 10 |
math | Task B-4.7. Determine all natural numbers $n$ such that the value of the following expression
$$
\left(\frac{1+i \operatorname{tan} \frac{\pi}{38 n}}{1-i \operatorname{tan} \frac{\pi}{38 n}}\right)^{2014}
$$
is a real number. | n\in{1,2,53,106} | 81 | 15 |
math | In trapezium $ABCD$, $BC \perp AB$, $BC\perp CD$, and $AC\perp BD$. Given $AB=\sqrt{11}$ and $AD=\sqrt{1001}$. Find $BC$ | \sqrt{110} | 56 | 7 |
math | 2. Solve the equation $\log _{a}(1+\sqrt{x})=\log _{b} x$, where $a>1, b>1, a^{2}=b+1$. | b^{2} | 42 | 4 |
math |
Problem 4. For a positive integer $n$, two players A and B play the following game: Given a pile of $s$ stones, the players take turn alternatively with A going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a multiple of $n$ stones. The winner is the one who takes ... | n-1 | 111 | 3 |
math | Problem 9.6. Find all pairs $(x, y)$ of natural numbers that satisfy the equation
$$
x^{2}-6 x y+8 y^{2}+5 y-5=0
$$ | {(2,1),(4,1),(11,4),(13,4)} | 46 | 19 |
math | 5. Find all integers n for which the number
$$
2^{n}+n^{2}
$$
is a perfect square of some integer.
(Tomáš Jurík) | n=0n=6 | 40 | 6 |
math | 25. Fresh mushrooms contain $90 \%$ water, and when they were dried, they became lighter by 15 kg with a moisture content of $60 \%$. How much did the fresh mushrooms weigh? | 20 | 45 | 2 |
math | 7.057. $5^{x+6}-3^{x+7}=43 \cdot 5^{x+4}-19 \cdot 3^{x+5}$. | -3 | 42 | 2 |
math | 6. (5 points) A fairy tale book has two volumes, upper and lower, and a total of 999 digits were used. The upper volume has 9 more pages than the lower volume. The upper volume has a total of pages. | 207 | 52 | 3 |
math | Consider a circle of radius $4$ with center $O_1$, a circle of radius $2$ with center $O_2$ that lies on the circumference of circle $O_1$, and a circle of radius $1$ with center $O_3$ that lies on the circumference of circle $O_2$. The centers of the circle are collinear in the order $O_1$, $O_2$, $O_3$. Let $A$ be a ... | \sqrt{6} | 167 | 5 |
math | Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have
\[
(z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]
Compute the number of distinct possible values of $c$. | 4 | 82 | 1 |
math | 136. Solve the equation $y^{\prime \prime}+2 y^{\prime}-8 y=0$. | C_{1}e^{-4x}+C_{2}e^{2x} | 27 | 19 |
math | Let's play heads or tails in the following way: We toss the coin four times and then as many times as there were heads in the first four tosses. What is the probability that we will get at least 5 heads in all our tosses? | \frac{47}{256} | 52 | 10 |
math | Which is the smallest positive integer that, when divided by 5, leaves a remainder of 2, when divided by 7, leaves a remainder of 3, and when divided by 9, leaves a remainder of 4? | 157 | 48 | 3 |
math | Exercise 2. Find all pairs of digits $(a, b)$ such that the integer whose four digits are $a b 32$ is divisible by 99. | 6,7 | 36 | 3 |
math | 3. Given $\triangle A B C, \tan A, \tan B, \tan C$ are all integers, and $\angle A>\angle B>\angle C$. Then $\tan B=$ $\qquad$ | 2 | 45 | 1 |
math | For how many integers $x$ is the expression $\frac{\sqrt{75-x}}{\sqrt{x-25}}$ equal to an integer? | 5 | 32 | 1 |
math | Five integers form an arithmetic sequence. Whether we take the sum of the cubes of the first four terms, or the sum of the cubes of the last four terms, in both cases we get 16 times the square of the sum of the considered terms. Determine the numbers. | 0,0,0,0,0;64,64,64,64,64;0,16,32,48,64;64,48,32,16,0 | 56 | 52 |
math | 6. Determine all solutions of the equation
(20)
$$
\left(x^{2}-5 x+5\right)^{2 \cdot 4^{x}-9 \cdot 2^{x}+4}=1
$$ | -1,1,2,3,4 | 51 | 10 |
math | 14. Use 6 white beads, 8 black beads, and 1 red bead to string into a necklace. How many different ways are there to do this? | 1519 | 35 | 4 |
math | ## 9. Diagonal Squares
Vlado covered the diagonal of a large square with a side length of $2020 \mathrm{~cm}$ using a row of smaller squares with a side length of $4 \mathrm{~cm}$ cut from green collage paper. The diagonals of the green squares align with the diagonal of the large square, and the intersection of any t... | 101 | 134 | 3 |
math | 2. A natural number is multiplied by two adjacent odd numbers respectively, and the difference between the two products is 100. What is this number?
$\qquad$ _. | 50 | 37 | 2 |
math | 3. Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $C:$
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)
$$
If there exists a point $P$ on the ellipse $C$ such that $P F_{1} \perp P F_{2}$, then the range of the eccentricity $e$ of the ellipse is $\qquad$ | [\frac{\sqrt{2}}{2},1) | 111 | 12 |
math | Example 9 For $n \in \mathbf{N}$, let $S_{n}=\min \left(\sum_{\mathrm{k}=1}^{\mathrm{n}} \sqrt{(2 \mathrm{k}-1)^{2}+\mathrm{a}_{\mathrm{k}}^{2}}\right)$, where $\mathrm{a}_{1}$, $\mathrm{a}_{2}, \cdots, \mathrm{a}_{\mathrm{n}} \in \mathbf{R}^{+}, \sum_{i=1}^{n} a_{n}=17$, if there exists a unique $n$ such that $S_{n}$ ... | 12 | 149 | 2 |
math | At the camp, the scouts were weighing themselves on an old-fashioned scale. The leader warned them that the scale did not weigh correctly, but the difference between the actual and measured weight was always the same. The scale showed $30 \mathrm{~kg}$ for Misha, $28 \mathrm{~kg}$ for Emil, but when they both stood on ... | Míša'actualweightis28\mathrm{~},Emil'actualweightis26\mathrm{~} | 107 | 28 |
math | Solve the following system of equations:
$$
\begin{gathered}
\frac{10}{2 x+3 y-29}+\frac{9}{7 x-8 y+24}=8 \\
\frac{2 x+3 y-29}{2}=\frac{7 x-8 y}{3}+8
\end{gathered}
$$ | 5,\quad7 | 82 | 4 |
math | Consider a pyramid with a square base where all edges are equal. Place a right circular cylinder ($m=2r$) inside this pyramid so that its base lies on the pyramid's base and its top touches the pyramid's side faces. What is the height of the cylinder? | =2r=(\sqrt{2}-1) | 56 | 11 |
math | 12.4 Compute the limits, then verify or refute the statement
$$
\lim _{x \rightarrow 3} \frac{x^{2}+x-12}{3 x-9}=5+\lim _{x \rightarrow 2} \frac{x^{4}-16}{8-x^{3}}
$$
Find the derivatives of the functions (12.5-12.7):
| 2\frac{1}{3} | 88 | 8 |
math | 3. Solve the system of equations:
$$
\left\{\begin{array}{l}
x y+5 y z-6 x z=-2 z \\
2 x y+9 y z-9 x z=-12 z \\
y z-2 x z=6 z
\end{array}\right.
$$ | (-2,2,1/6),(x,0,0),(0,y,0) | 68 | 20 |
math | Khachaturyan A.V.
Petr was born in the 19th century, and his brother Pavel - in the 20th century. Once, the brothers met to celebrate their shared birthday. Petr said: "My age is equal to the sum of the digits of the year of my birth." - "Mine too," replied Pavel. How much younger is Pavel than Petr? | 9 | 80 | 1 |
math | Let $ABC$ be an isosceles triangle with $AB=4$, $BC=CA=6$. On the segment $AB$ consecutively lie points $X_{1},X_{2},X_{3},\ldots$ such that the lengths of the segments $AX_{1},X_{1}X_{2},X_{2}X_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{4}$. On the segment $CB... | (a, b, c) = (1, 2c, c) \text{ or } (2c, 1, c) | 307 | 32 |
math | Example 2.12. $I=\int_{0}^{4} x\left(1-x^{2}\right)^{1 / 5} d x$.
| \frac{5}{12}((-15)^{\frac{6}{5}}-1^{\frac{6}{5}})\approx10.33 | 38 | 36 |
math | In trapezoid $A B C E$ base $A E$ is equal to $16, C E=8 \sqrt{3}$. The circle passing through points $A, B$ and $C$ intersects line $A E$ again at point $H ; \angle A H B=60^{\circ}$. Find $A C$. | 8 | 77 | 1 |
math | Triangle $ABC$ has $AB=10$, $BC=17$, and $CA=21$. Point $P$ lies on the circle with diameter $AB$. What is the greatest possible area of $APC$? | \frac{189}{2} | 49 | 10 |
math | 149. Find the general solution of the equation $y^{\prime \prime}-4 y^{\prime}+13 y=0$. | e^{2x}(C_{1}\cos3x+C_{2}\sin3x) | 32 | 20 |
math | 6. (8 points) On the board, 25 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 25 minutes? | 300 | 69 | 3 |
math | 16. A square fits snugly between the horizontal line and two touching circles of radius 1000 , as shown. The line is tangent to the circles.
What is the side-length of the square? | 400 | 44 | 3 |
math | 4. Let the sequence $\left\{a_{n}\right\}$ satisfy
$$
a_{1}=a_{2}=1, a_{n}=\sqrt{3} a_{n-1}-a_{n-2}(n \geqslant 3) \text {. }
$$
Then $a_{2013}=$ $\qquad$ | 1-\sqrt{3} | 80 | 6 |
math | 11. (25 points) Let positive numbers $a, b$ satisfy $a+b=1$. Find
$$
M=\sqrt{1+2 a^{2}}+2 \sqrt{\left(\frac{5}{12}\right)^{2}+b^{2}}
$$
the minimum value. | \frac{5\sqrt{34}}{12} | 68 | 14 |
math | 10. (12 points) Optimus Prime, in robot form, departs from location $A$ to location $B$, and can arrive on time; if he transforms into a car from the beginning, his speed increases by $\frac{1}{4}$, and he can arrive 1 hour earlier; if he travels 150 kilometers in robot form, then transforms into a car, with his speed ... | 750 | 121 | 3 |
math | ## 133. Math Puzzle $6 / 76$
A catch device that secures the hoist basket in a shaft fails in at most one out of 1000 operational cases. Another independent safety device fails in at most one out of 100 cases where it is called upon.
What is the probability that the occupants will be saved by the safety devices if th... | 99.999 | 86 | 6 |
math | 13. To obtain phosphorus, calcium phosphate was previously treated with sulfuric acid, the resulting orthophosphoric acid was mixed with carbon and calcined. In this process, orthophosphoric acid turned into metaphosphoric acid, which, upon interaction with carbon, yielded phosphorus, hydrogen, and carbon monoxide. Ill... | 23.4 | 144 | 4 |
math | How many solutions does the equation
$$
x^{2}+y^{2}+2 x y-1988 x-1988 y=1989
$$
have in the set of positive integers? | 1988 | 50 | 4 |
math | 1. If the quadratic equation with integer coefficients
$$
x^{2}+(a+3) x+2 a+3=0
$$
has one positive root $x_{1}$ and one negative root $x_{2}$, and $\left|x_{1}\right|<\left|x_{2}\right|$, then
$$
a=
$$
$\qquad$ | -2 | 82 | 2 |
math | Is there a 6-digit number that becomes 6 times larger when its last three digits are moved to the beginning of the number, keeping the order of the digits? | 142857 | 34 | 6 |
math | Three, given $y=y_{1}-$
$y_{2}, y_{1}$ is inversely proportional to $x$,
$y_{2}$ is directly proportional to $x-2$, when
$x=1$, $y=-1$; when $x$
$=-3$, $y_{1}$ is 21 less than $y_{2}$.
Find the corresponding value of $y$ when $x=-5$. (10 points) | -28 \frac{3}{5} | 98 | 10 |
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