task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find... | 507 | 107 | 3 |
math | $10 \cdot 27$ Find the smallest natural number $n$ that satisfies the following conditions:
(1) The last digit of $n$ is 6;
(2) If the last digit 6 of $n$ is moved to the front of the other digits, the resulting new number is 4 times $n$.
(4th International Mathematical Olympiad, 1962) | 153846 | 86 | 6 |
math | Let $n$ be a natural number. Find all real numbers $x$ satisfying the equation $$\sum^n_{k=1}\frac{kx^k}{1+x^{2k}}=\frac{n(n+1)}4.$$ | x = 1 | 49 | 4 |
math | (5) Given the sequence $\left\{a_{n}\right\}$ with the general term formula $a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}$ $\left(n \in \mathbf{N}^{*}\right)$, and its first $n$ terms sum as $S_{n}$, then in the sequence $S_{1}, S_{2}, \cdots, S_{2009}$, the number of rational terms is $\qquad$ terms. | 43 | 119 | 2 |
math | Which are those positive integers $k$ for which $2 \cdot 3^{k}$ is a perfect number? | 1 | 24 | 1 |
math | 6.51 Find all positive integers $n>1$, such that $\frac{2^{n}+1}{n^{2}}$ is an integer.
(31st International Mathematical Olympiad, 1990) | 3 | 49 | 1 |
math | 8. The range of $y=\sqrt{x}+\sqrt{1-x}$ is
The translation preserves the original text's line breaks and format. | [1,\sqrt{2}] | 31 | 7 |
math | (12) Let $[x]$ denote the greatest integer not exceeding $x$, $a_{k}=\left[\frac{2009}{k}\right]$, $k=1$,
$2, \cdots, 100$, then the number of different integers among these 100 integers is $\qquad$ | 69 | 74 | 2 |
math | 4. The maximum value of the function $y=\frac{4-\sin x}{3-\cos x}$ is
保留了源文本的换行和格式。 | \frac{6+\sqrt{6}}{4} | 35 | 12 |
math | 1.4.7 ** Summation
$$
S=[\lg 2]+[\lg 3]+\cdots+[\lg 2008]+\left[\lg \frac{1}{2}\right]+\left[\lg \frac{1}{3}\right]+\cdots+\left[\lg \frac{1}{2008}\right]
$$
Find the value of the sum. Here $[x]$ denotes the greatest integer not exceeding $x$. | -2004 | 100 | 5 |
math | Source: 2018 Canadian Open Math Challenge Part A Problem 3
-----
Points $(0,0)$ and $(3\sqrt7,7\sqrt3)$ are the endpoints of a diameter of circle $\Gamma.$ Determine the other $x$ intercept of $\Gamma.$ | (3\sqrt{7}, 0) | 58 | 10 |
math | 1. Solve the equation
$$
x^{3}-\left(m^{2}+3\right) x^{2}+\left(m^{2}+3\right) x+m^{4}-1=0
$$
for the unknown $x$ by factoring the polynomial $P(m)$ on the left side of this equation. | x_{1}=^{2}+1,\quadx_{2}=+1,\quadx_{3}=-+1 | 72 | 26 |
math | ## Task 13/84
Calculate the sum of all those natural numbers that contain each of the five digits $1 ; 2 ; 3 ; 4 ; 5$ exactly once in their decimal representation. | 3999960 | 46 | 7 |
math | 10. (5 points) Write down the natural numbers from 1 to 2015 in sequence, to get a large number $123456789 \cdots 20142015$. When this large number is divided by 9, the remainder is | 0 | 65 | 1 |
math | 39. 15 volleyball teams played a round-robin tournament, where each team won exactly seven matches. How many triplets of teams are there in this tournament where each team has exactly one win against the other two? | 140 | 46 | 3 |
math | 4. The vertices of a cube with edge length 1 are alternately colored red and blue, so that the two vertices on each edge are of different colors. The volume of the common part of the tetrahedron formed by the red vertices and the tetrahedron formed by the blue vertices is $\qquad$ | \frac{1}{6} | 66 | 7 |
math | 15. Given the family of curves $2(2 \sin \theta-\cos \theta+3) x^{2}-(8 \sin \theta+\cos \theta+1) y=0, \theta$ being the parameter. Find the maximum value of the length of the chord intercepted by the line $y=2 x$ on this family of curves. | 8\sqrt{5} | 77 | 6 |
math | Let $BCDE$ be a trapezoid with $BE\parallel CD$, $BE = 20$, $BC = 2\sqrt{34}$, $CD = 8$, $DE = 2\sqrt{10}$. Draw a line through $E$ parallel to $BD$ and a line through $B$ perpendicular to $BE$, and let $A$ be the intersection of these two lines. Let $M$ be the intersection of diagonals $BD$ and $CE$, and let $X$ be th... | 203 | 159 | 3 |
math | Find all positive integers $N$ having only prime divisors $2,5$ such that $N+25$ is a perfect square. | 200, 2000 | 30 | 9 |
math | We color all vertexs of a convex polygon with $10$ vertexs by $2$ colors: red and blue $($each vertex is colored by $1$ color$).$
How many ways to color all the vertexs such that there are no $2$ adjacent vertex that are both colored red? | 123 | 63 | 3 |
math | 9. If $x_{1}, x_{2}$ are the real roots of the equation $x^{2}+a x+a-\frac{1}{2}=0$.
(1) Find the set of real values $A$ for $a$;
(2) Is there a real number $m$ such that the inequality $m^{2}+t m+4 \sqrt{2}+6 \geqslant\left(x_{1}-3 x_{2}\right)\left(x_{2}-3 x_{1}\right)$ holds for any $a \in A$ and $t \in[-1,1]$? If i... | \geqslant1or\leqslant-1or=0 | 156 | 17 |
math | Task 1. (5 points) Find $\frac{a^{12}+729^{2}}{729 a^{6}}$, if $\frac{a}{3}-\frac{3}{a}=2$. | 198 | 50 | 3 |
math | 10.1. Summation
$$
\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\left[\frac{2^{3}}{3}\right]+\cdots+\left[\frac{2^{1000}}{3}\right]
$$ | \frac{1}{3}\left(2^{1001}-2\right)-500 | 78 | 23 |
math | 135. Find $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+4}}{x}$. | 1 | 31 | 1 |
math | 8. (10 points) A certain exam consists of 7 questions, each of which only concerns the answers to these 7 questions, and the answers can only be one of $1, 2, 3, 4$. It is known that the questions are as follows:
(1) How many questions have the answer 4?
(2) How many questions do not have the answer 2 or 3?
(3) What is... | 16 | 191 | 2 |
math | 30th IMO 1989 shortlist Problem 20 b > 0 and a are fixed real numbers and n is a fixed positive integer. The real numbers x 0 , x 1 , ... , x n satisfy x 0 + x 1 + ... + x n = a and x 0 2 + ... + x n 2 = b. Find the range of x 0 . | (n+1)x_0^2-2ax_0+^2-nb\ | 87 | 19 |
math | ## Task 2 - 180612
A number $z$ is to be written in the form $z=\star 3 \star 60$, where each star $(\star)$ is to be replaced by one of the digits 0 to 9 so that $z$ has the following two properties:
(1) $60000<z<100000$,
(2) $z$ is divisible by 9.
Determine all numbers $z$ that satisfy these conditions! | 63360,73260,83160,93060,93960 | 113 | 29 |
math | $11 \cdot 34$ Determine the smallest natural number $n$, such that $n!$ ends with exactly 1987 zeros.
(28th International Mathematical Olympiad Candidate Question 1987) | 7960 | 49 | 4 |
math | [ Properties of bisectors, concurrency ] [ The ratio in which the bisector divides the side $]$ [ Properties of bisectors, concurrency ]
In triangle $A B C$, points $M$ and $N$ are marked on sides $A B$ and $B C$ respectively, such that $B M=B N$. A line is drawn through point $M$ perpendicular to $B C$, and a line th... | 5 | 158 | 1 |
math | 10. Let $x_{1}, x_{2}, x_{3}$ be the roots of the polynomial equation $x^{3}-10 x+11=0$.
(1) Given that $x_{1}, x_{2}, x_{3}$ all lie in the interval $(-5,5)$, find the integer parts of these three roots;
(2) Prove: $\arctan x_{1}+\arctan x_{2}+\arctan x_{3}=\frac{\pi}{4}$. | \arctanx_{1}+\arctanx_{2}+\arctanx_{3}=\frac{\pi}{4} | 116 | 31 |
math | ## Task 4 - 250814
A cuboid has edge lengths $a, 2a$, and $\frac{a}{2}$, where $a$ is given. From this cuboid, a straight prism is separated. The height of this prism is $\frac{a}{2}$, and its base is an isosceles right triangle with leg length $a$. The remaining body is a prism with a trapezoidal base.
a) Draw the r... | V_{R}:V_{Q}=3:4 | 155 | 11 |
math | 5. Solve the equation $\sin \frac{\pi n}{12} \cdot \sin \frac{\pi k}{12} \cdot \sin \frac{\pi m}{12}=\frac{1}{8}$. Here $k, m, n-$ are natural numbers not exceeding 5. | (2;2;2),(1;2;5),(1;5;2),(2;1;5),(2;5;1),(5;1;2),(5;2;1) | 66 | 43 |
math | 3. If numbers $a_{1}, a_{2}, a_{3}$ are taken in increasing order from the set $1, 2, \cdots, 14$, such that the following conditions are satisfied:
$$
a_{2}-a_{1} \geqslant 3 \text { and } a_{3}-a_{2} \geqslant 3 \text {. }
$$
Then the number of different ways to choose such numbers is $\qquad$ kinds. | 120 | 107 | 3 |
math | ## Task 4 - 100524
A group of young mathematicians went on an excursion. Each participant paid 1.50 marks for travel expenses. When paying for the collective ticket, a remainder of 1.10 marks was left.
If each participant had paid 1.40 marks, there would have been a shortage of 1.10 marks to cover the cost of the col... | 22 | 121 | 2 |
math | 14. For each integer $n \geqslant 2$, determine the maximum value of the sum $a_{0}+a_{1}+a_{2}+\cdots+a_{n}$ satisfying the conditions $a_{0}=1, a_{i} \geqslant a_{i+1}+a_{i+2}, i=0$. $1, \cdots, n-2$ (where $a_{0}, a_{1}, \cdots a_{n}$ are non-negative). | \frac{F_{n+1}}{F_{n-1}} | 114 | 16 |
math | (Canada, 1996). Let $r_{1}, \ldots, r_{m}$ be positive rationals such that $r_{1}+\cdots+r_{m}=1$. We define $f(n)=n-\sum\left\lfloor r_{i} n\right\rfloor$. What are the minimal and maximal values taken by $f(n)$? | 0-1 | 83 | 3 |
math | ## Task B-4.2.
On the planet "Anything is Possible," children play a game in which each of them has to choose numbers from the set of odd natural numbers less than 1000 within a certain time. In this game, the sum of no pair of different numbers chosen by any child is 1002. The winner is the one who selects the most s... | 2^{249} | 139 | 6 |
math | 1. 50 students from fifth to ninth grade published a total of 60 photos on Instagram, each not less than one. All students of the same grade (same parallel) published an equal number of photos, while students of different grades (different parallels) published a different number. How many students published only one ph... | 46 | 67 | 2 |
math | 12. In the triangle $A B C, A B=14, B C=16, A C=26, M$ is the midpoint of $B C$ and $D$ is the point on $B C$ such that $A D$ bisects $\angle B A C$. Let $P$ be the foot of the perpendicular from $B$ onto $A D$. Determine the length of $P M$. | 6 | 93 | 1 |
math | 24. Let $a, b, c, d$ be 4 distinct nonzero integers such that $a+b+c+d=0$ and the number $M=(b c-a d)(a c-b d)(a b-c d)$ lies strictly between 96100 and 98000 . Determine the value of $M$. | 97344 | 74 | 5 |
math | Exercise 4. Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of numbers such that $a_{1}=2$, $a_{2}=3$, and $a_{n}=\frac{a_{n-1}}{a_{n-2}}$ for all integers $n \geqslant 3$. For example, $a_{3}=\frac{a_{2}}{a_{1}}=\frac{3}{2}$.
Determine the value of $a_{2014}$. | \frac{1}{2} | 121 | 7 |
math | 8.82 Suppose there are 128 ones written on the blackboard. In each step, you can erase any two numbers $a$ and $b$ on the blackboard, and write $ab+1$. After 127 such steps, only one number remains. Let the maximum possible value of this remaining number be $A$. Find the last digit of $A$.
| 2 | 82 | 1 |
math | Ten birds land on a $10$-meter-long wire, each at a random point chosen uniformly along the wire. (That is, if we pick out any $x$-meter portion of the wire, there is an $\tfrac{x}{10}$ probability that a given bird will land there.) What is the probability that every bird sits more than one meter away from its closest... | \frac{1}{10^{10}} | 82 | 11 |
math | Let $n$ be a positive integer. Determine the first digit after the decimal point of the number
$$
\sum_{k=1}^{n} \frac{k(k+1)}{n}
$$ | 6 | 44 | 1 |
math | 9. (16 points) When $x \geqslant 0$, find the minimum value $g(a)$ of the function
$$
f(x)=2 x+(|x-1|-a)^{2}
$$
| g(a)=\left\{\begin{array}{ll}2 a+1, & a \geqslant 4 ; \\ a^{2}-2 a+1, & 0 \leqslant a<4 ; \\ 1-2 a, & -1<a<0 ; \\ a^{2}+2, & a \leqslant-1 .\end{array}\right.} | 50 | 90 |
math | 56 (979). After writing the test, the students Volodya, Sasha, and Petya reported at home:
Volodya: “I got a ‘5’.”
Sasha: “I got a ‘3’.”
Petya: “I did not get a ‘5’.”
After the test was checked, it turned out that one of the boys received a ‘4’, another a ‘3’, and the third a ‘5’. What grade did each receive, given... | Vovochka-5,Sasha-4,Petya-3 | 117 | 15 |
math | 4.79. Integrate the Lagrange equation
$$
y=x\left(\frac{d y}{d x}\right)^{2}-\frac{d y}{d x}
$$ | {\begin{pmatrix}\frac{p-\ln|p|+C}{(p-1)^{2}}\\xp^{2}-p\end{pmatrix}.} | 42 | 38 |
math | 4. Let $\Varangle A O D$ with $m(\Varangle A O D)=98^{\circ}$ and $(O B,(O C$ half-lines included in the interior of $\Varangle A O D$, (OC half-line included in the interior of $\Varangle B O D$ such that
$$
a \cdot m(\Varangle A O B)=c \cdot m(\Varangle B O C) \text { and } b \cdot m(\Varangle B O C)=c \cdot m(\Vara... | (\angleAOB)=18,(\angleBOC)=30,(\angleCOD)=50 | 204 | 22 |
math | Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q... | 71 | 140 | 2 |
math | 10. Let real numbers $a, b$ satisfy $a=x_{1}+x_{2}+x_{3}=x_{1} x_{2} x_{3}, a b=x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1}$, where $x_{1}, x_{2}, x_{3}>0$. Then the maximum value of $P=\frac{a^{2}+6 b+1}{a^{2}+a}$ is $\qquad$. | \frac{9+\sqrt{3}}{9} | 117 | 12 |
math | Example 4 Let $x, y, z \in \mathbf{R}^{+}$, and $x y z(x+y+z)=1$, find the minimum value of $(x+y)(x+z)$. | 2 | 46 | 1 |
math | 4. A palindrome is a positive integer which is the same when reading from the right hand side or from the left hand side, e.g. 2002. Find the largest five-digit palindrome which is divisible by 101 .
(1 mark)
若某正整數不論從左面或右面讀起皆相同（例如：2002），則該數稱為「回文數」。求可被 101 整除的最大五位回文數。 | 49894 | 107 | 5 |
math | For which values of $p$ and $q$ integers strictly positive does $pq$ divide $3(p-1)(q-1) ?$ | (6,5),(4,9),(2,3) | 31 | 13 |
math | 15.8 (League 2001) $D, E$ are points on side $BC$ of $\triangle ABC$, $F$ is a point on the extension of $BA$, $\angle DAE = \angle CAF$.
(1) Determine the positional relationship between the circumcircle of $\triangle ABD$ and the circumcircle of $\triangle AEC$, and prove your conclusion.
(2) If the circumradius of $... | \frac{8}{3} | 984 | 7 |
math | 7. Given that $f(x)$ is an odd function with a period of 5, $f(-3)=1$, $\tan \alpha=2$, $f(20 \sin \alpha \cdot \cos \alpha)=$ $\qquad$ . | -1 | 55 | 2 |
math | 18. Let $p, q, r, s$ be the four roots of the equation $2(10 x+13)^{2}(5 x+8)(x+1)=1$. If $p q+r s$ is real, find the value of this real number.
(2 marks)
設 $p 、 q 、 r 、 s$ 為方程 $2(10 x+13)^{2}(5 x+8)(x+1)=1$ 的四個根。若 $p q+r s$ 是實數, 求此實數的值。
(2 分) | \frac{329}{100} | 137 | 11 |
math | 1. Find all values of $p$, for each of which the numbers $4 p+5, 2 p$ and $|p-3|$ are respectively the first, second, and third terms of some geometric progression. | p=-1,p=\frac{15}{8} | 47 | 12 |
math | 4. Five cubes with edge lengths of $15,16,20,24$ and $30 \mathrm{~cm}$ respectively, need to be melted into one sphere. Calculate the diameter of the sphere and the ratio of the sum of the areas of the cubes to the area of the sphere. | 2\sqrt[3]{\frac{42221.25}{\pi}},\frac{P_{1}+P_{2}+P_{3}+P_{4}+P_{5}}{P}\approx1.9 | 66 | 55 |
math | 16. $2.29{ }^{\star \star} N$ is an integer, its base $b$ representation is 777. Find the smallest positive integer $b$, such that $N$ is a fourth power of an integer. | 18 | 55 | 2 |
math | Determine whether there exist a positive integer $n<10^9$, such that $n$ can be expressed as a sum of three squares of positive integers by more than $1000$ distinct ways? | \text{Yes} | 45 | 5 |
math | 9. The function
$$
f(x)=\sqrt{2 x-7}+\sqrt{12-x}+\sqrt{44-x}
$$
has a maximum value of $\qquad$ | 11 | 43 | 2 |
math | All natural numbers from 1 to 1000 inclusive are divided into two groups: even and odd.
In which of the groups is the sum of all digits used to write the numbers greater, and by how much?
# | 499 | 47 | 3 |
math | 20. How many different triangles with integer side lengths are there such that the sum of the lengths of any two sides exceeds the length of the third side by at least 5 units, and that the area is numerically twice the perimeter? (Two triangles are regarded to be the same if they are congruent.)
(2 marks)
有多少個不同的三角形各邊... | 8 | 133 | 1 |
math | [b]i.)[/b] Calculate $x$ if \[ x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})} \]
[b]ii.)[/b] For each positive number $x,$ let \[ k = \frac{\left( x + \frac{1}{x} \r... | 10 | 226 | 2 |
math | Find all quadruples $(a, b, c, d)$ of real numbers for which
$$
\begin{aligned}
& a b+c+d=3, \\
& b c+d+a=5, \\
& c d+a+b=2, \\
& d a+b+c=6 .
\end{aligned}
$$ | (2,0,0,3) | 67 | 9 |
math | (9) Let the function $f(x)=\frac{x^{2}+n}{x^{2}+x+1}\left(n \in \mathbf{N}^{*}\right)$ have a maximum value of $a_{n}$ and a minimum value of $b_{n}$, then $a_{n}-b_{n}=$ | \frac{4}{3}\sqrt{n^{2}-n+1} | 75 | 16 |
math | $$
\frac{\frac{1}{a}-\frac{1}{b+c}}{\frac{1}{a}+\frac{1}{b+c}}: \frac{\frac{1}{b}-\frac{1}{a+c}}{\frac{1}{b}+\frac{1}{a+c}}=?
$$
| \frac{b+-}{+-b} | 71 | 9 |
math | If $a$ and $b$ are each randomly and independently chosen in the interval $[-1, 1]$, what is the probability that $|a|+|b|<1$? | \frac{1}{2} | 42 | 7 |
math | ## Task B-1.2.
If $a \neq 0$ and $b \neq 0$ are two distinct real numbers for which
$$
\frac{a}{2017}+\frac{2017}{a}=\frac{b}{2017}+\frac{2017}{b}
$$
calculate $\sqrt{a \cdot b}$. | 2017 | 88 | 4 |
math | ## Problem Statement
Find the derivative.
$y=\arcsin \left(e^{-4 x}\right)+\ln \left(e^{4 x}+\sqrt{e^{8 x}-1}\right)$ | 4\sqrt{\frac{e^{4x}-1}{e^{4x}+1}} | 44 | 21 |
math | 2. The participants of the Olympiad left 9 pens in the office. Among any four pens, at least two belong to the same owner. And among any five pens, no more than three belong to the same owner. How many students forgot their pens, and how many pens does each student have? | 3 | 62 | 1 |
math | $\mathbf{F 1 7}$ (39-6, Bulgaria) Let $\mathbf{N}^{*}$ be the set of all positive integers, and the function $f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*}$ satisfies: for any $s$ and $t$ in $\mathbf{N}^{*}$, we have
$$
f\left(t^{2} f(s)\right)=s(f(t))^{2},
$$
Determine the smallest possible value of $f(1998)$ among al... | 120 | 132 | 3 |
math | 57 Let $f(x)=2^{x}, g(x)=\log _{\sqrt{2}} 8 x$, then the value of $x$ that satisfies $f[g(x)]=g[f(x)]$ is
Let's translate the problem and solution step by step.
### Problem:
Let \( f(x) = 2^x \) and \( g(x) = \log_{\sqrt{2}} (8x) \). Find the value of \( x \) that satisfies \( f[g(x)] = g[f(x)] \).
### Solution:
1. ... | \frac{1+\sqrt{385}}{64} | 1,087 | 15 |
math | 6-152 Find all functions $f$ from the set of real numbers to the set of real numbers that satisfy the following conditions:
(1) $f(x)$ is strictly increasing;
(2) For all real numbers $x, f(x) + g(x) = 2x$, where $g(x)$ is the inverse function of $f(x)$. | f(x)=x+ | 77 | 5 |
math | 6-12 $f(n)$ is a function defined on the set of positive integers, taking non-negative integer values, and for all $m, n$ we have:
$$
\begin{array}{l}
f(m+n)-f(m)-f(n)=0 \text { or } 1 ; \\
f(2)=0, f(3)>0, f(9999)=3333 .
\end{array}
$$
Try to find: $f(1982)$. | 660 | 109 | 3 |
math | 10.060. The radii of the inscribed and circumscribed circles of a right triangle are 2 and 5 cm, respectively (Fig. 10.59). Find the legs of the triangle. | 6;8 | 49 | 3 |
math | 2. Solve the equation
$$
(1+x)^{8}+\left(1+x^{2}\right)^{4}=2 x^{4}
$$ | x_{1}=x_{2}=\frac{-1+i\sqrt{3}}{2},x_{3}=x_{4}=\frac{-1-i\sqrt{3}}{2},x_{5/6}=\frac{-1+i\sqrt{6}\\sqrt{-9-2i\sqrt{6}}}{2},x_{7/8}=\frac{-1-i\sqrt{6}\\sqrt{} | 34 | 90 |
math | The radius $r$ of a circle is increasing at a rate of $2$ meters per minute. Find the rate of change, in $\text{meters}^2/\text{minute}$, of the area when $r$ is $6$ meters. | 24\pi | 55 | 4 |
math | Given four points $O,\ A,\ B,\ C$ on a plane such that $OA=4,\ OB=3,\ OC=2,\ \overrightarrow{OB}\cdot \overrightarrow{OC}=3.$
Find the maximum area of $\triangle{ABC}$. | 2\sqrt{7} + \frac{3\sqrt{3}}{2} | 57 | 19 |
math | Problem 11.5. Determine the number of possible values of the product $a \cdot b$, where $a, b-$ are integers satisfying the inequalities
$$
2019^{2} \leqslant a \leqslant b \leqslant 2020^{2}
$$
Answer: $\mathrm{C}_{2 \cdot 2019+2}^{2}+2 \cdot 2019+1=2 \cdot 2019^{2}+5 \cdot 2019+2=8162819$. | 8162819 | 134 | 7 |
math | What is the largest possible value of $|a_1 - 1| + |a_2-2|+...+ |a_n- n|$ where $a_1, a_2,..., a_n$ is a permutation of $1,2,..., n$? | \left\lfloor \frac{n^2}{2} \right\rfloor | 60 | 18 |
math | 663. Find all natural $n$ for which the number $n^{2}+3 n$ is a perfect square. | 1 | 28 | 1 |
math | 12. As Nest:
(1) $a, b, c, d$ all belong to $\{1,2,3,4\}$;
(2) $a \neq b, b \neq c, c \neq d, d \neq a$;
(3) $a$ is the smallest value among $a, b, c, d$.
Then, the number of different four-digit numbers abcd that can be formed is $\qquad$ | 28 | 103 | 2 |
math | $1 \cdot 109$ Choose 3 numbers from $0,1,2, \cdots \cdots, 9$ such that their sum is an even number not less than 10. How many different ways are there to do this? | 51 | 56 | 2 |
math | A board $64$ inches long and $4$ inches high is inclined so that the long side of the board makes a $30$ degree angle with the ground. The distance from the ground to the highest point on the board can be expressed in the form $a+b\sqrt{c}$ where $a,b,c$ are positive integers and $c$ is not divisible by the square of a... | 37 | 161 | 2 |
math | 2. Given a positive integer $n$ less than 2006, and
$$
\left[\frac{n}{3}\right]+\left[\frac{n}{6}\right]=\frac{n}{2} \text {. }
$$
Then the number of such $n$ is $\qquad$. | 334 | 65 | 3 |
math | 18 Let $a, b, c, d, m, n \in \mathbf{N}, a^{2}+b^{2}+c^{2}+d^{2}=1989, a+b+c+d=m^{2}$, and the largest of $a, b, c, d$ is $n^{2}$, determine and prove the values of $m, n$.
Determine and prove the values of $m, n$. | =9,n=6 | 100 | 5 |
math | For a fixed integer $n\geqslant2$ consider the sequence $a_k=\text{lcm}(k,k+1,\ldots,k+(n-1))$. Find all $n$ for which the sequence $a_k$ increases starting from some number. | n = 2 | 57 | 5 |
math | Find all triples of positive integers $(a,b,c)$ such that the following equations are both true:
I- $a^2+b^2=c^2$
II- $a^3+b^3+1=(c-1)^3$ | (6, 8, 10) | 52 | 11 |
math | A triangle's sides, perimeter, and area with the usual notations: $a, b, c, 2s, T$. What is the angle opposite to the side $c$ of the triangle if
$$
T+\frac{a b}{2}=s(s-c) ?
$$ | 45 | 61 | 2 |
math | Example 2. Solve the inequality
$$
\sqrt{x^{2}+6 x+10}+\sqrt{x^{2}-6 x+10}>10 .
$$ | x>\frac{5}{4} \sqrt{15} \text { or } x<-\frac{5}{4} \sqrt{15} | 39 | 34 |
math | 4. (28th Russian Mathematical Olympiad) Find the smallest positive integer that can be represented as the sum of 2002 positive integers with equal sums of digits, and also as the sum of 2003 positive integers with equal sums of digits. | 10010 | 56 | 5 |
math | We form a decimal code of $21$ digits. the code may start with $0$. Determine the probability that the fragment $0123456789$ appears in the code. | \frac{12 \cdot 10^{11} - 30}{10^{21}} | 43 | 26 |
math | $2 \cdot 56$ Find the smallest natural number, such that when its last digit is moved to the first position, the number is multiplied by 5. | 142857 | 35 | 6 |
math | 【Example 3】To distribute 11 papers to 4 experts for review, one expert should review 4 papers, one should review 3 papers, and the other two should each review 2 papers. How many different ways are there to distribute the papers? | 831600 | 55 | 6 |
math | ## Task A-3.7. (10 points)
Find all two-digit natural numbers $a$ for which the equation
$$
2^{x+y}=2^{x}+2^{y}+a
$$
has a solution $(x, y)$ in natural numbers. | 14,20,30,44,48,62,92 | 60 | 20 |
math | To 9. On a plane, there is a fixed point $P$, consider all possible equilateral triangles $ABC$, where $AP=3, BP=2$. What is the maximum length of $CP$? (1961 Autumn Competition) | 5 | 53 | 1 |
math | 10. The numbers 2.75 and 8 have the property that the product of these numbers equals the sum of their digits: $2.75 \cdot 8=2+7+5+8=22$. Find at least one more such pair of unequal numbers. | 2.6\cdot5=13;2+6+5=13 | 61 | 18 |
math | Six, Given the hyperbola \( C_{1}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{2 a^{2}}=1(a>0) \), and the parabola \( C_{2} \) with its vertex at the origin \( O \), and the focus of \( C_{2} \) is the left focus \( F_{1} \) of \( C_{1} \).
(1) Prove that \( C_{1} \) and \( C_{2} \) always have two different intersection points;... | 6^{2} | 208 | 4 |
math | Example 3 Find all integer solutions $(a, b)$ of the equation
$$
a^{4}-3 a^{2}+4 a-3=7 \times 3^{b}
$$ | (,b)=(3,2),(5,4),(-4,3) | 42 | 17 |
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