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 | $\underline{\text { Frankin B.R. }}$
Given a polynomial $P(x)$ with real coefficients. An infinite sequence of distinct natural numbers $a_{1}, a_{2}, a_{3}, \ldots$ is such that
$P\left(a_{1}\right)=0, P\left(a_{2}\right)=a_{1}, P\left(a_{3}\right)=a_{2}$, and so on. What degree can $P(x)$ have? | 1 | 103 | 1 |
math | Example 4. How many different six-digit numbers can be written using the digits $1 ; 1 ; 1 ; 2 ; 2 ; 2$? | 20 | 35 | 2 |
math | 43rd Swedish 2003 Problem 3 Which reals x satisfy [x 2 - 2x] + 2[x] = [x] 2 ? | allnegativeintegers,0,allintervals[1,1+\sqrt{2}),[3,\sqrt{5}+1),\ldots,[n,\sqrt{n^2-2n+2}+1),\ldots | 38 | 51 |
math | 1. Five consecutive natural numbers were written on the board, and then one number was erased. It turned out that the sum of the remaining four numbers is 2015. Find the smallest of these four numbers. | 502 | 45 | 3 |
math | Mr. Stein is ordering a two-course dessert at a restaurant. For each course, he can choose to eat pie, cake, rødgrød, and crème brûlée, but he doesn't want to have the same dessert twice. In how many ways can Mr. Stein order his meal? (Order matters.) | 12 | 66 | 2 |
math | 8. Given that the number of integer points (points with integer coordinates) on the closed region (including the boundary) enclosed by the circle $x^{2}+y^{2}=8$ is one-fifth of the number of integer points on the closed region (including the boundary) enclosed by the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1$, th... | 22\leq23 | 103 | 7 |
math | Determine the least integer $n$ such that for any set of $n$ lines in the 2D plane, there exists either a subset of $1001$ lines that are all parallel, or a subset of $1001$ lines that are pairwise nonparallel.
[i]Proposed by Samuel Wang[/i]
[hide=Solution][i]Solution.[/i] $\boxed{1000001}$
Since being parallel is a... | 1000001 | 178 | 7 |
math | 12. The point on the ellipse $7 x^{2}+4 y^{2}=28$ that is closest to the line $l: 3 x-2 y-16=0$ has the coordinates | (\frac{3}{2},-\frac{7}{4}) | 47 | 14 |
math | Example 2. Find the domain of the function
$$
y=\frac{\sqrt{(4-x)(x+3)}\left(\sqrt{x^{2}-1}\right)^{0}}{(x-3) \lg (x+2)}
$$
The domain of this example is selected from the book "Comprehensive Application of Middle School Mathematics" (edited by Wei Jingnong and Zhang Wenxiong, Zhejiang Science and Technology Press). O... | \{x \mid -2<x<-1 \text{ or } 1<x \leqslant 4 \text{ and } x \neq 3\} | 97 | 38 |
math | 1. For which real parameters $a$ does there exist a complex number $z$ such that
$$
|z+\sqrt{2}|=\sqrt{a^{2}-3 a+2} \quad \text { and } \quad|z+i \sqrt{2}|<a ?
$$ | >2 | 63 | 2 |
math | 1. Given the set $M=\{2,0,11\}$. If $A \varsubsetneqq M$, and $A$ contains at least one even number, then the number of sets $A$ that satisfy the condition is $\qquad$ . | 5 | 57 | 1 |
math | Let $ n>1$ and for $ 1 \leq k \leq n$ let $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$ be the sum of the products of all possible combinations of k of the numbers $ a_1,a_2,...,a_n$. Furthermore let $ P \equal{} P(a_1, a_2, . . . , a_n)$ be the sum of all $ p_k$ with odd values of $ k$ less than or equal to $ n$.
How ... | 2 | 155 | 3 |
math | 4B. Three weary travelers arrived at an inn and asked for food. The innkeeper had nothing else to offer them except baked potatoes. While the potatoes were baking, the travelers fell asleep. After some time, once the potatoes were ready, the first traveler woke up, took $\frac{1}{3}$ of the potatoes, and continued to s... | 27 | 165 | 2 |
math | 11.6. Solve the equation
$$
\cos 2x - \sin 2x + 2 \cos x + 1 = 0
$$ | x_{0}+2k\pi,k\in{Z},x_{0}\in{-\frac{\pi}{2},\frac{\pi}{2},\pi} | 36 | 38 |
math | 5. A bag contains $n$ red balls and 5 white balls. It costs 1 yuan to draw once, and each time two balls are drawn from the bag at once. If the two balls are of different colors, it is considered a win, and the prize is 2 yuan. Then when $n=$ $\qquad$, the expected value is maximized. | 4or5 | 77 | 3 |
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 | 5. There are 10 people with distinct heights standing in a row, and each person is either taller or shorter than the person in front of them. The number of arrangements that meet this condition is $\qquad$ (answer with a number).
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The last sentence is a repetition of the instr... | 512 | 152 | 3 |
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 | ## Task B-1.3.
Solve the equation in the set of prime numbers
$$
2 p^{3}-q^{2}=2(p+q)^{2}
$$ | (3,2) | 39 | 5 |
math | 3.62. Find the sine of the angle at the vertex of an isosceles triangle, given that the perimeter of any rectangle inscribed in it, with two vertices lying on the base, is a constant value. | \frac{4}{5} | 47 | 7 |
math | 4. Find all triples of natural numbers $x, y$ and $z$ such that $(x+1)(y+1)(z+1)=3 x y z$ | (x,y,z)\in{(2,2,3),(2,3,2),(3,2,2),(5,1,4),(5,4,1),(4,1,5),(4,5,1),(1,4,5),(1,5,4),(8,1,3),(8,3,1),(3,1,8),(3,8,1),(1,3,8),(1,8} | 37 | 94 |
math | 1. Calculate: $603 \cdot 38 + 225 \cdot (514 - 476) + (15 + 23) \cdot 172$ | 38000 | 46 | 5 |
math | 5, 50 different positive integers, their sum is 2012, how many of these numbers can be odd at most?
There are 50 different positive integers, their sum is 2012, how many of these numbers can be odd at most? | 44 | 58 | 2 |
math | Let $d(n)$ denote the number of positive divisors of the positive integer $n$. Determine those numbers $n$ for which
$$
d\left(n^{3}\right)=5 \cdot d(n)
$$ | p^{3}\cdotq | 46 | 6 |
math | 28(1104). Two motorcyclists set off simultaneously from $A$ to $B$ and from $B$ to $A$. The first arrived in $B$ 2.5 hours after the meeting, while the second arrived in $A$ 1.6 hours after the meeting. How many hours was each motorcyclist on the road? | 4.5 | 77 | 3 |
math | 6.207. $\left\{\begin{array}{l}x+y+z=6, \\ x(y+z)=5, \\ y(x+z)=8 .\end{array}\right.$ | (5;2;-1),(5;4;-3),(1;2;3),(1;4;1) | 43 | 25 |
math | Find the smallest positive integer $n$ having the property: for any set of $n$ distinct integers $a_{1}, a_{2}, \ldots, a_{n}$ the product of all differences $a_{i}-a_{j}, i<j$ is divisible by 1991. | 182 | 64 | 3 |
math | Let $T = TNFTPP$. As $n$ ranges over the integers, the expression $n^4 - 898n^2 + T - 2160$ evaluates to just one prime number. Find this prime.
[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question. For those who wanted to try the problem separ... | 1801 | 129 | 4 |
math | 1. Can the number 8 be represented as the sum of eight integers such that the product of these numbers is also equal to 8? | Yes | 29 | 1 |
math | ## Task Condition
Find the derivative.
$y=(2+3 x) \sqrt{x-1}-\frac{3}{2} \operatorname{arctg} \sqrt{x-1}$ | \frac{18x^{2}-8x-3}{4x\sqrt{x-1}} | 43 | 22 |
math | 【Question 11】
Mom goes to the supermarket to sell fruits, buying 5 kilograms of apples and 4 kilograms of pears costs 48 yuan, buying 2 kilograms of apples and 3 kilograms of mangoes costs 33 yuan. It is known that each kilogram of mangoes is 2.5 yuan more expensive than pears. If 3 kilograms of apples and 3 kilograms ... | 31.5 | 103 | 4 |
math | Let $a$ be the largest positive integer so that $a^{3}$ is less than 999.
Let $b$ be the smallest positive integer so that $b^{5}$ is greater than 99 .
What is the value of $a-b$ ? | 6 | 57 | 1 |
math | 2. Solve the equation in the set of prime numbers
$$
3 p^{2}+3 p=166+q
$$ | 7 | 30 | 1 |
math | 9. If $a \in A$, and $a-1 \notin A, a+1 \notin A$, then $a$ is called an isolated element of set $A$. Therefore, the number of four-element subsets of set $M=\{1,2, \cdots, 9\}$ without isolated elements is $\qquad$ . | 21 | 74 | 2 |
math | 7. Connect the right focus $F_{2}$ of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ with a moving point $A$ on the ellipse, and construct a square $F_{2} A B C\left(F_{2} 、 A 、 B 、 C\right.$ are arranged in a clockwise direction). Then, as point $A$ moves around the ellipse once, the trajectory equation of the moving... | \frac{(x-\sqrt{5})^{2}}{4}+\frac{(y-\sqrt{5})^{2}}{9}=1 | 111 | 31 |
math | B3. Three circles with the same centre $O$ and areas $2 \pi, 3 \pi$ and $4 \pi$ are drawn. From a point $A$ on the largest circle, tangent lines are drawn to points $B$ on the middle circle and $C$ on the smallest circle. If $B, C$ are on the same side of $O A$, find the exact value of $\angle B A C$. | 15 | 93 | 2 |
math | Find all triangular numbers that are perfect squares (recall that the $n$-th triangular number is $1+2+\ldots+n$). | \frac{x_{k}^{2}-1}{8} | 30 | 13 |
math | 5. Find all solutions to the equation $2017^{x}-2016^{x}=1$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 1 | 51 | 1 |
math | A rectangle, $HOMF$, has sides $HO=11$ and $OM=5$. A triangle $\Delta ABC$ has $H$ as orthocentre, $O$ as circumcentre, $M$ be the midpoint of $BC$, $F$ is the feet of altitude from $A$. What is the length of $BC$ ?
[asy]
unitsize(0.3 cm);
pair F, H, M, O;
F = (0,0);
H = (0,5);
O = (11,5);
M = (11,0);
draw(H--O--M--F... | 28 | 172 | 2 |
math | 3. Find the relationship between the coefficients $a, b, c$ such that the system of equations
$$
\left\{\begin{array}{l}
a x^{2}+b x+c=0, \\
b x^{2}+c x+a=0, \\
c x^{2}+a x+b=0
\end{array}\right.
$$
has a solution. | a+b+c=0 | 85 | 5 |
math | [ Equations in integers ]
Find all integer solutions of the equation $21 x+48 y=6$.
# | (-2+16k,1-7k),k\in{Z} | 26 | 18 |
math | A positive integer $n$ is called $\textit{un-two}$ if there does not exist an ordered triple of integers $(a,b,c)$ such that exactly two of$$\dfrac{7a+b}{n},\;\dfrac{7b+c}{n},\;\dfrac{7c+a}{n}$$are integers. Find the sum of all un-two positive integers.
[i]Proposed by [b]stayhomedomath[/b][/i] | 660 | 99 | 3 |
math | 10.1. Find the sum of all such integers $a \in[0 ; 400]$, for each of which the equation $x^{4}-6 x^{2}+4=\sin \frac{\pi a}{200}-2\left[x^{2}\right]$ has exactly six roots. Here the standard notation is used: $[t]$ - the integer part of the number $t$ (the greatest integer not exceeding $t$). | 60100 | 100 | 5 |
math | 1. Given $n>2$ natural numbers, among which there are no three equal, and the sum of any two of them is a prime number. What is the largest possible value of $n$? | 3 | 43 | 1 |
math | $\triangle ABC$ is isosceles $AB = AC$. $P$ is a point inside $\triangle ABC$ such that
$\angle BCP = 30$ and $\angle APB = 150$ and $\angle CAP = 39$. Find $\angle BAP$.
| 13^\circ | 64 | 4 |
math | (10) The smallest positive integer that can be simultaneously expressed as the sum of 9 consecutive integers, the sum of 10 consecutive integers, and the sum of 11 consecutive integers is $\qquad$ . | 495 | 46 | 3 |
math | 10. find all pairs $(\alpha, \beta)$ of positive real numbers with the following properties:
(a) For all positive real numbers $x, y, z, w$ applies
$$
x+y^{2}+z^{3}+w^{6} \geq \alpha(x y z w)^{\beta}
$$
(b) There is a quadruple $(x, y, z, w)$ of positive real numbers such that equality holds in (a).
## Solution | (2^{\frac{4}{3}}3^{\frac{1}{4}},\frac{1}{2}) | 102 | 26 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{4}}(\sin x+\cos x)^{\frac{1}{\tan x}}$ | \sqrt{2} | 42 | 5 |
math | ## Task 4 - 060514
We are looking for a natural number with the following properties:
If you divide 100 by this number, the remainder is 4, and if you divide 90 by this number, the remainder is 18.
What is the number we are looking for? | 24 | 69 | 2 |
math | 158*. Using the digits from 1 to 9 once each, form the smallest nine-digit number that is divisible by 11. | 123475869 | 30 | 9 |
math | Example 5 Find the integer solutions of the equation $x^{2}+x=y^{4}+y^{3}+y^{2}+y$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | (x, y)=(0,-1),(-1,-1),(0,0),(-1,0),(-6,2),(5,2) | 60 | 31 |
math | ## 168. Math Puzzle $5 / 79$
A poultry farm delivers 1320 eggs, a second farm delivers a third less.
a) How many eggs do both poultry farms deliver in total?
b) How many chickens does each farm have if each chicken lays 4 eggs? | 2200,330,220 | 64 | 12 |
math | Given $n(n \geqslant 2)$ real numbers $\theta_{1}, \theta_{2}, \cdots, \theta_{n}\left(\theta_{i} \in\left(0, \frac{\pi}{2}\right], 1 \leqslant i \leqslant n\right)$ satisfying $\sum_{i=1}^{n} \sin \theta_{i} \leqslant 1$. Try to find the maximum value of $\prod_{i=1}^{n} \tan \theta_{i}$. | (n^{2}-1)^{-\frac{n}{2}} | 123 | 13 |
math | \section*{Problem 2 - 171222}
Four students, A, B, C, and D, each make three statements about a natural number \(x\). Student A makes exactly two true statements, while students B, C, and D each make at least one and at most two true statements.
Determine all natural numbers \(x\) that satisfy these conditions:
(A1)... | 395935 | 322 | 6 |
math | Example 2.66. The velocity $v$ of a point moving in a straight line changes with time $t$ according to the law $v(t)=$ $=t \sin 2 t$. Determine the path $s$ traveled by the point from the start of the motion to the moment of time $t=\frac{\pi}{4}$ units of time. | 0.25 | 78 | 4 |
math | 11. Let $x, y, z \in \mathbf{R}^{+}$, and satisfy $x y z(x+y+z)=1$, find the minimum value of $(x+y)(x+z)$. | 2 | 47 | 1 |
math | Let $n$ be a fixed integer, $n \geqslant 2$.
a) Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$
holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n} \geqsla... | \frac{1}{8} | 270 | 7 |
math | ## Task 3 - 170513
Fritz wants to write down a subtraction problem where the difference between two natural numbers is to be formed.
The result should be a three-digit number whose three digits are all the same. The minuend should be a number that ends in zero. If this zero is removed, the subtrahend should result.
... | \begin{pmatrix}370\\-37\\\hline333\end{pmatrix}\quad\begin{pmatrix}740\\-\quad74\\\hline666\end{pmatrix}\quad\begin{pmatrix}1110\\-111\\\hline999\end{pmatrix} | 87 | 80 |
math | 12. The maximum value of the function $f(x)=\sqrt{x^{4}-3 x^{2}-6 x+13}-\sqrt{x^{4}-x^{2}+1}$ is | \sqrt{10} | 44 | 6 |
math | ## SUBJECT I
The sequence of numbers is given: $1,4,7,10,13,16,19$
a) Determine whether the 22nd term of the sequence is a perfect square.
b) Find the 2015th term of the sequence. | 6043 | 63 | 4 |
math | For example, $5 n$ is a positive integer, its base $b$ representation is 777, find the smallest positive integer $b$ such that $n$ is a fourth power of an integer. | 18 | 45 | 2 |
math | 4. On the board, there is a four-digit number that has exactly six positive divisors, of which exactly two are single-digit and exactly two are two-digit. The larger of the two-digit divisors is a square of a natural number. Determine all numbers that could be written on the board. | 1127,1421,1519,1813,2009,2107,2303 | 61 | 34 |
math | Find all the three digit numbers $\overline{a b c}$ such that
$$
\overline{a b c}=a b c(a+b+c)
$$ | 135 \text{ and } 144 | 35 | 12 |
math | Is the multiplication:
$$
79133 \times 111107=8794230231
$$
exact? (Without a calculator) | 8792230231 | 41 | 10 |
math | 7.3. Yesterday, Sasha cooked soup and put in too little salt, so he had to add more salt. Today, he put in twice as much salt, but still had to add more salt, though only half the amount he added yesterday. By what factor does Sasha need to increase today's portion of salt so that he doesn't have to add more salt tomor... | 1.5 | 87 | 3 |
math | 8. 4 people pass the ball to each other, with the requirement that each person passes the ball to someone else immediately after receiving it. Starting with person A, and counting this as the first pass, find the total number of different ways the ball can be passed such that after 10 passes, the ball is back in the ha... | 14763 | 75 | 5 |
math | A group of $n$ people play a board game with the following rules:
1) In each round of the game exactly $3$ people play
2) The game ends after exactly $n$ rounds
3) Every pair of players has played together at least at one round
Find the largest possible value of $n$ | 7 | 67 | 1 |
math | Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}$ and $g: \mathbb{R}^{+} \rightarrow \mathbb{R}$ such that
$$f(x^2 + y^2) = g(xy)$$
holds for all $x, y \in \mathbb{R}^{+}$.
| f(x) = c | 82 | 6 |
math | 10. [25] Let $P$ be a regular $k$-gon inscribed in a circle of radius 1 . Find the sum of the squares of the lengths of all the sides and diagonals of $P$. | k^2 | 49 | 3 |
math | 11. Let $n$ be a given natural number, $n \geqslant 3$, and for $n$ given real numbers $a_{1}, a_{2}$, $\cdots, a_{n}$, denote $\left|a_{i}-a_{j}\right|(1 \leqslant i<j \leqslant n)$ by $m$, the minimum value. Find the maximum value of $m$ under the condition $a_{1}^{2}+a_{2}^{2}+\cdots+$ $a_{n}^{2}=1$. | \sqrt{\frac{12}{n(n^{2}-1)}} | 128 | 15 |
math | 2. Let $\triangle A B C$ have internal angles $A, B, C$ with opposite sides $a, b, c$ respectively, and satisfy $a \cos B - b \cos A = \frac{3}{5} c$. Then the value of $\frac{\tan A}{\tan B}$ is . $\qquad$ | 4 | 73 | 1 |
math | 2. If $\sin x+\cos x=\frac{\sqrt{2}}{2}$, then $\sin ^{3} x+\cos ^{3} x=$ $\qquad$ | \frac{5\sqrt{2}}{8} | 40 | 12 |
math | Shapovalov A.V.
Little kids were eating candies. Each one ate 7 candies less than all the others together, but still more than one candy.
How many candies were eaten in total? | 21 | 41 | 2 |
math | Task 3 - 330713 Anke reports that she has drawn an isosceles triangle with a perimeter of $14 \mathrm{~cm}$, in which one of the three sides is exactly three times as long as a second of the three sides.
Beate thinks that the lengths of all three sides are uniquely determined by these specifications.
Christin, on the... | =2\mathrm{~},=b=6\mathrm{~} | 121 | 16 |
math | Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$.
[i]Proposed by Kevin Sun[/i] | 450 | 57 | 3 |
math | (2) If the foci and vertices of an ellipse are the vertices and foci of the hyperbola $\frac{y^{2}}{9}-\frac{x^{2}}{16}=1$, then the equation of the ellipse is $\qquad$ | \frac{x^{2}}{16}+\frac{y^{2}}{25}=1 | 56 | 22 |
math | G10.2Let $f(x)=x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$. When $f\left(x^{10}\right)$ is divided by $f(x)$, the remainder is $b$. Find the value of $b$. | 10 | 87 | 2 |
math | 4. In tetrahedron $ABCD$, $\angle ADB = \angle BDC = \angle CDA = 60^{\circ}$, the areas of $\triangle ADB$, $\triangle BDC$, and $\triangle CDA$ are $\frac{\sqrt{3}}{2}$, $2$, and $1$, respectively. Then the volume of the tetrahedron is . $\qquad$ | \frac{2\sqrt{6}}{9} | 89 | 12 |
math | 29. $\int x^{2 / 3} d x$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
29. $\int x^{2 / 3} d x$. | \frac{3}{5}x^{5/3}+C | 56 | 15 |
math | Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which satisfies the following:
[list][*] $f(m)=m$, for all $m\in\mathbb{Z}$;[*] $f(\frac{a+b}{c+d})=\frac{f(\frac{a}{c})+f(\frac{b}{d})}{2}$, for all $a, b, c, d\in\mathbb{Z}$ such that $|ad-bc|=1$, $c>0$ and $d>0$;[*] $f$ is monotonically increasing.[/list]
(a) ... | f(x) = a_0 - 2 \sum_{n \geq 1} \frac{(-1)^n}{2^{A_n}} | 170 | 34 |
math | Find all values of $ \alpha$ for which the curves $ y\equal{}\alpha x^2\plus{}\alpha x\plus{}\frac1{24}$ and $ x\equal{}\alpha y^2\plus{}\alpha y\plus{}\frac1{24}$ are tangent to each other. | \frac{13 \pm \sqrt{601}}{12}, \frac{2}{3}, \frac{3}{2} | 65 | 32 |
math | How many natural numbers are there whose representations in both base 9 and base 11 are three-digit numbers? Determine among them those numbers whose representations in these two bases consist of the same digits but in reverse order. | 608,302_{9}=203_{11}=245_{10},604_{9}=406_{11}=490_{10} | 44 | 44 |
math | 55. Find four such numbers that the sums of the numbers, taken three at a time, are $22, 24, 27, 20$. (This problem was given by the Greek mathematician Diophantus, who lived in the second century AD). | 11,\quad4,\quad7,\quad=9 | 60 | 12 |
math | Example 3 Given that $\alpha, \beta$ are the roots of the equation $x^{2}-7 x+8=0$, and $\alpha>\beta$. Without solving the equation, find the value of $\frac{2}{\alpha}+3 \beta^{2}$. | \frac{1}{8}(403-85 \sqrt{17}) | 60 | 19 |
math | Let $a$ be the sum of the numbers:
$99 \times 0.9$
$999 \times 0.9$
$9999 \times 0.9$
$\vdots$
$999\cdots 9 \times 0.9$
where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$.
Find the sum of the digits in the number $a$. | 891 | 110 | 3 |
math | Problem 5. The numbers $a, b$, and $c$ satisfy the equation $\sqrt{a}=\sqrt{b}+\sqrt{c}$. Find $a$, if $b=52-30 \sqrt{3}$ and $c=a-2$. | 27 | 59 | 2 |
math | Let $ABCD$ be a tetrahedron with $AB=CD=1300$, $BC=AD=1400$, and $CA=BD=1500$. Let $O$ and $I$ be the centers of the circumscribed sphere and inscribed sphere of $ABCD$, respectively. Compute the smallest integer greater than the length of $OI$.
[i] Proposed by Michael Ren [/i] | 1 | 93 | 1 |
math | 5. In the Empire of Westeros, there were 1000 cities and 2017 roads (each road connected some two cities). From any city, you could travel to any other. One day, an evil wizard enchanted $N$ roads, making it impossible to travel on them. As a result, 7 kingdoms formed, such that within each kingdom, you could travel fr... | 1024 | 116 | 4 |
math | Let $\Gamma_1$ be a circle with radius $\frac{5}{2}$. $A$, $B$, and $C$ are points on $\Gamma_1$ such that $\overline{AB} = 3$ and $\overline{AC} = 5$. Let $\Gamma_2$ be a circle such that $\Gamma_2$ is tangent to $AB$ and $BC$ at $Q$ and $R$, and $\Gamma_2$ is also internally tangent to $\Gamma_1$ at $P$. $\Gamma_2$ i... | 19 | 178 | 2 |
math | # 2. CONDITION
Solve the equation in natural numbers
$$
\text { LCM }(a ; b)+\text { GCD }(a ; b)=a b .
$$
(GCD - greatest common divisor, LCM - least common multiple).
# | =b=2 | 56 | 3 |
math | Let $ABC$ be a right triangle with $\angle{ACB}=90^{\circ}$. $D$ is a point on $AB$ such that $CD\perp AB$. If the area of triangle $ABC$ is $84$, what is the smallest possible value of $$AC^2+\left(3\cdot CD\right)^2+BC^2?$$
[i]2016 CCA Math Bonanza Lightning #2.3[/i] | 1008 | 102 | 4 |
math | Let $$C=\{1,22,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}$$ and let
$$S=\{4,5,9,14,23,37\}$$ Find two sets $A$ and $B$ with the properties
(a) $A\cap B=\emptyset$.
(b) $A\cup B=C$.
(c) The sum of two distinct elements of $A$ is not in $S$.
(d) The sum of two distinct elements of $B$ is not in $S$.
| A = \{1, 2, 5, 6, 10, 11, 14, 15, 16, 19, 20\} | 157 | 45 |
math | Let the lengths of the three sides of a triangle be integers $l$, $m$, $n$, and $l > m > n$. It is known that
$$
\left\{\frac{3^{l}}{10^{4}}\right\}=\left\{\frac{3^{m}}{10^{4}}\right\}=\left\{\frac{3^{n}}{10^{4}}\right\},
$$
where $\{x\}=x-[x]$, and $[x]$ represents the greatest integer not exceeding $x$. Find the min... | 3003 | 132 | 4 |
math | 1. Consider the function $f:(0, \infty) \backslash\{1\} \rightarrow \mathbb{R}$, $f(x)=\frac{1}{\log _{x} 2 \cdot \log _{x} 4}+\frac{1}{\log _{x} 4 \cdot \log _{x} 8}+\ldots .+\frac{1}{\log _{x}\left(2^{2013}\right) \cdot \log _{x}\left(2^{2014}\right)}$
a) Calculate $f\left(\frac{1}{2}\right)$.
b) Show that $f$ is n... | x\in{\frac{1}{4},4} | 181 | 12 |
math | 7. Find all values of $\alpha$ such that all terms in the sequence $\cos \alpha, \cos 2 \alpha, \cdots, \cos 2^{n} \alpha \cdots$ are negative. | \alpha=2k\pi\\frac{2\pi}{3},k\in\mathbb{Z} | 48 | 25 |
math | Problem 7. The dragon has 40 piles of gold coins, and the number of coins in any two of them differs. After the dragon plundered a neighboring city and brought back more gold, the number of coins in each pile increased by either 2, 3, or 4 times. What is the smallest number of different piles of coins that could result... | 14 | 76 | 2 |
math | 1.48 In the expression $x_{1}: x_{2}: \cdots: x_{n}$, use parentheses to indicate the order of operations, and the result can be written in fractional form:
$$\frac{x_{i_{1}} x_{i_{2}} \cdots x_{i_{k}}}{x_{j_{j_{1}}}^{x_{j_{2}}} \cdots x_{j_{n-k}}}$$
(At the same time, each letter in $x_{1}, x_{2}, \cdots, x_{n}$ may a... | 2^{n-2} | 147 | 6 |
math | Find the continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property:
$$ f\left( x+\frac{1}{n}\right) \le f(x) +\frac{1}{n},\quad\forall n\in\mathbb{Z}^* ,\quad\forall x\in\mathbb{R} . $$ | f(x) = x + a | 84 | 7 |
math | Example 2 Suppose the foci of an ellipse are the same as those of the hyperbola $4 x^{2}-5 y^{2}=$ 20, and it is tangent to the line $x-y+9=0$. Find the standard equation of this ellipse. | \frac{x^{2}}{45}+\frac{y^{2}}{36}=1 | 59 | 22 |
math | Example 1. Find $P_{n}=\prod_{k=1}^{n}\left(2 \cos 2^{k-1} \theta-1\right)$. | \frac{2 \cos 2^{n} \theta+1}{2 \cos \theta+1} | 40 | 24 |
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