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 | Let's calculate the value of the following expression:
$$
\frac{1+x}{1+\sqrt{1+x}}+\frac{1-x}{1-\sqrt{1-x}} \quad \text { if } \quad x=\frac{\sqrt{3}}{2}
$$ | 1 | 59 | 1 |
math | 4. Let $f(x)=\cos (\cos (\cos (\cos (\cos (\cos (\cos (\cos x)))))))$, and suppose that the number $a$ satisfies the equation $a=\cos a$. Express $f^{\prime}(a)$ as a polynomial in $a$. | ^{8}-4^{6}+6^{4}-4^{2}+1 | 60 | 18 |
math | Let $p$ be a prime number. Determine the remainder of the Euclidean division of $1^{k}+\cdots+(p-1)^{k}$ for any integer $k \geq 0$ by $p$ (without primitive root). | p-1 | 54 | 3 |
math | $\left[\begin{array}{l}\text { Tangent Circles } \\ \text { [Pythagorean Theorem (direct and inverse).] }\end{array}\right]$
Two circles with radii $\sqrt{19}$ and $\sqrt{76}$, touching each other externally, are inscribed in a semicircle (i.e., each circle touches this semicircle and its diameter). Find the radius of... | 4\sqrt{19} | 96 | 7 |
math | A square with sides of length $1$ cm is given. There are many different ways to cut the square into four rectangles.
Let $S$ be the sum of the four rectangles’ perimeters. Describe all possible values of $S$ with justification. | 8 \text{ cm} \text{ or } 10 \text{ cm} | 52 | 19 |
math | 9. (16 points) Given the sequence $\left\{a_{n}\right\}$ with the general term $a_{n}=1+2+\cdots+n\left(n \in \mathbf{N}^{*}\right)$, extract all multiples of 3 from this sequence to form a new sequence $b_{1}, b_{2}, \cdots, b_{m}, \cdots$. Find the sum of the first $2 m$ terms of the sequence $\left\{b_{m}\right\}$, ... | \frac{3}{2}(+1)(2+1) | 125 | 14 |
math | In a society of 30 people, any two people are either friends or enemies, and everyone has exactly six enemies. How many ways are there to choose three people from the society so that any two of them are either all friends or all enemies? | 1990 | 51 | 4 |
math | 10. For some positive integer $n$, the fraction $\frac{n+2}{3 n^{2}+7}$ is not in its simplest form, then the minimum value of $n$ is $\qquad$ . | 17 | 47 | 2 |
math | 3. In the tetrahedron $S-ABC$, $BC=3, AC=4$, the distance from point $C$ to the midpoint of $AB$ is $\frac{5}{2}$, and the dihedral angles between the lateral faces $SAB$, $SBC$, and $SAC$ and the base are all $60^{\circ}$. Then the volume of the tetrahedron is $\qquad$ | 2 \sqrt{3} | 95 | 6 |
math | Example 7 Find a rational number $x$, such that $1+5 \times 2^{x}$ is the square of a rational number.
| -2 \text{ or } 4 | 31 | 9 |
math | Let $\mathbb{N}$ denote the set of positive integers. Let $\varphi: \mathbb{N} \rightarrow \mathbb{N}$ be a bijective function and assume that there exists a finite limit
$$
\lim _{n \rightarrow \infty} \frac{\varphi(n)}{n}=L
$$
What are the possible values of $L$? | L=1 | 84 | 3 |
math | $(x_{n})_{-\infty<n<\infty}$ is a sequence of real numbers which satisfies $x_{n+1}=\frac{x_{n}^2+10}{7}$ for every $n \in \mathbb{Z}$. If there exist a real upperbound for this sequence, find all the values $x_{0}$ can take. | [2, 5] | 80 | 7 |
math | ## Task 3 - 310623
How many natural numbers are there in total that are
a) divisors of 256,
b) divisors of $2 \cdot 256$,
c) divisors of $256 \cdot 256$?
Explain how you found each of your three answers! | 9,10,17 | 77 | 7 |
math | ## Task Condition
Find the derivative of the specified order.
$y=\left(4 x^{3}+5\right) e^{2 x+1}, y^{V}=?$ | 32(4x^{3}+30x^{2}+60x+35)e^{2x+1} | 40 | 29 |
math | 31. The number of poplars planted by the students is expressed as a three-digit number, and the number of lindens as a two-digit number. In total, they planted 144 trees. If the extreme digits in these two numbers are swapped and the resulting numbers are added, the sum is 603. How many poplars and how many lindens wer... | 115 | 84 | 3 |
math | 3. Find all positive real numbers $t$ such that there exists an infinite set $X$ of real numbers, for which, for any $x, y, z \in X$ (where $x, y, z$ can be the same), and any real number $a$ and positive real number $d$, we have
$$
\max \{|x-(a-d)|,|y-a|,|z-(a+d)|\}>t d
$$ | 0<t<\frac{1}{2} | 99 | 10 |
math | 3. (10 points) A barrel of oil, the oil it contains is $\frac{3}{5}$ of the barrel's full capacity. After selling 18 kilograms, 60% of the original oil remains. Therefore, this barrel can hold $\qquad$ kilograms of oil. | 75 | 62 | 2 |
math | Problem 5. Martha is playing a computer game called "Destroy the Balloons". In each move, Martha can destroy exactly 1, 11, 15, or 27 balloons. At the same time, if she destroys 1 balloon, exactly 10 new balloons appear on the screen; if she destroys 11 balloons, exactly 8 new balloons appear; if she destroys 15 balloo... | 15,15,15,15,15,15,11,1,11,15 | 142 | 28 |
math | $\begin{array}{l}\text { 1. Let } p, q \in \mathbf{R}_{+} \text {, and satisfy } \\ \log _{9} p=\log _{12} q=\log _{16}(p+q) \text {. } \\ \text { Then } \frac{q}{p}=\end{array}$ | \frac{1+\sqrt{5}}{2} | 83 | 12 |
math | 9. Let $m$ be a positive integer, and let $T$ denote the set of all subsets of $\{1,2, \ldots, m\}$. Call a subset $S$ of $T \delta$-good if for all $s_{1}, s_{2} \in S, s_{1} \neq s_{2},\left|\Delta\left(s_{1}, s_{2}\right)\right| \geq \delta m$, where $\Delta$ denotes symmetric difference (the symmetric difference of... | 2048 | 172 | 4 |
math | 3. (1991 Japan Mathematical Olympiad) $n$ is a non-negative integer, determined by $f(0)=0, f(1)=1, f(n)=$ $f\left(\left[\frac{n}{2}\right]\right)+n-2\left[\frac{n}{2}\right]$. Find the maximum value of $f(n)$ when $0 \leqslant n \leqslant 1991$. (Here $[x]$ denotes the greatest integer not exceeding $x$) | 10 | 115 | 2 |
math | 9. Given: $c>0$, let $p$ : the function $y=c^{x}$ is monotonically decreasing on $\mathbf{R}$, $q$ : the solution set of the inequality $x+|x-2 c|>1$ is
R. If $p$ and $q$ are such that exactly one of them is true, find the range of values for $c$. | (0,\frac{1}{2}]\cup[1,+\infty) | 88 | 18 |
math | 7. If a natural number can be written as the sum of two consecutive natural numbers and also as the sum of three consecutive natural numbers, it is called a "good number." Please find the "good number" among 2007, 2008, 2009, 2010, 1011. | 2007,1011 | 74 | 9 |
math | 3. (15 points). In a hat box, there are 21 hats and 18 caps. They were distributed on two shelves: the first should accommodate 20 items, and the second - 19. After the distribution, the percentage of hats on each shelf was calculated and the resulting numbers were added together. What should be the distribution of hat... | Ontheshelf-19hats,onthefirstshelf-2hats18caps | 87 | 21 |
math | 3. Let $O$ be the circumcenter of acute $\triangle A B C$, with $A B=6, A C=10$. If $\overrightarrow{A O}=x \overrightarrow{A B}+y \overrightarrow{A C}$, and $2 x+10 y=5$, then $\cos \angle B A C=$ $\qquad$ . | \frac{1}{3} | 82 | 7 |
math | 11.18. The height of a regular quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is half the length of the side of the base. Find the maximum value of the angle $A_{1} M C_{1}$, where $M$ is a point on the edge $A B$. | 90 | 78 | 2 |
math | 7. In square $ABCD$ with side length $4$, $E$ is the midpoint of $AB$, and $F$ is a moving point on side $BC$. If $\triangle ADE$ and $\triangle DCF$ are folded along $DE$ and $DF$ respectively, so that $A$ and $C$ coincide at $A'$, then the maximum distance from point $A'$ to the plane $DEF$ is $\qquad$. | \frac{4\sqrt{5}}{5} | 96 | 12 |
math | ## Task 3 - 020723
Emil tells: "My brother Heinz is only half as old as I am. If you multiply the number of his years by itself, you get my father's age. My mother is 5 years younger than my father. All together, we are 85 years old."
How old is Emil? Describe how you found the solution! | 12 | 83 | 2 |
math | Three. (20 points) Given $\sin (2 \alpha+\beta)=3 \sin \beta$. Let $\tan \alpha=x, \tan \beta=y$, and denote $y=f(x)$.
(1) Find the expression for $f(x)$;
(2) Define the sequence of positive numbers $\left\{a_{n}\right\}: a_{1}=\frac{1}{2}, a_{n+1}^{2}=$ $2 a_{n} f\left(a_{n}\right)\left(n \in \mathbf{N}_{+}\right)$. F... | a_{n}=\sqrt{\frac{2^{n-2}}{2^{n-1}+1}} | 147 | 25 |
math | Example 6 Let $n$ be a positive integer of the form $a^{2}+b^{2}$, where $a, b$ are two coprime positive integers. It satisfies: if $p$ is a prime and $p \leqslant \sqrt{n}$, then $p \mid a b$. Find all positive integers $n$ that meet the requirements. | 2, 5, 13 | 82 | 8 |
math | 5. Tanya was sequentially writing down numbers of the form $n^{7}-1$ for natural numbers $n=2,3, \ldots$ and noticed that for $n=8$ the resulting number is divisible by 337. For what smallest $n>1$ will she get a number divisible by $2022?$ | 79 | 74 | 2 |
math | How many $8$-digit numbers in base $4$ formed of the digits $1,2, 3$ are divisible by $3$? | 2187 | 32 | 4 |
math | Example 2 Given $\frac{\cos ^{4} A}{\cos ^{2} B}+\frac{\sin ^{4} A}{\sin ^{2} B}=1$, find the value of $\frac{\cos ^{4} B}{\cos ^{2} A}+\frac{\sin ^{4} B}{\sin ^{2} A}$. | 1 | 83 | 1 |
math | Problem 3. Let $n$ be a non-zero natural number and $A=\{1,2, \ldots, n\}$. Determine the number of increasing functions $f: A \rightarrow A$ with the property that $|f(x)-f(y)| \leq|x-y|$ for any $x, y \in A$. | (n+1)2^{n-2} | 73 | 10 |
math | Calculate the following indefinite integrals.
[1] $\int \sin x\cos ^ 3 x dx$
[2] $\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx$
[3] $\int x^2 \sqrt{x^3+1}dx$
[4] $\int \frac{e^{2x}-3e^{x}}{e^x}dx$
[5] $\int (1-x^2)e^x dx$ | -(x - 1)^2 e^x + C | 107 | 12 |
math | Example 6 Real numbers $x_{1}, x_{2}, \cdots, x_{2001}$ satisfy $\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001$, let $y_{k}$ $=\frac{1}{k}\left(x_{1}+x_{2}+\cdots+x_{k}\right), k=1,2, \cdots, 2001$, find the maximum possible value of $\sum_{k=1}^{2000} \mid y_{k}-y_{k+1}$. (2001 Shanghai High School Mathematics Comp... | 2000 | 150 | 4 |
math | 6. (10 points) A class plants trees on Tree Planting Day, divided into 3 groups. The first group plants 5 trees per person, the second group plants 4 trees per person, and the third group plants 3 trees per person. It is known that the number of people in the second group is one-third of the sum of the number of people... | 32 | 128 | 2 |
math | Find all integer solutions $(p, q, r)$ of the equation $r + p ^ 4 = q ^ 4$ with the following conditions:
$\bullet$ $r$ is a positive integer with exactly $8$ positive divisors.
$\bullet$ $p$ and $q$ are prime numbers. | (2, 5, 609) | 68 | 12 |
math | 2. Let $m$ be a non-zero integer, and the quadratic equation in $x$, $m x^{2}-(m-1) x+1=0$, has rational roots. Find the value of $m$.
| 6 | 49 | 1 |
math | ## Problem 4.
An encyclopedia consists of 2000 numbered volumes. The volumes are stacked in order with number 1 on top and 2000 in the bottom. One may perform two operations with the stack:
(i) For $n$ even, one may take the top $n$ volumes and put them in the bottom of the stack without changing the order.
(ii) For... | (1000!)^2 | 128 | 8 |
math | Problem 10.4. (15 points) Once, 45 friends living in different parts of the world decided to exchange news with each other. To do this, they plan to arrange $k$ video meetings, at each of which every person will share their news, as well as all the news from other people they have learned previously.
For the video mee... | 5 | 192 | 1 |
math | 8. Define the sequence $\left\{a_{n}\right\}: a_{n}=n^{3}+4, n \in \mathbf{N}_{+}$, and let $d_{n}=\left(a_{n}, a_{n+1}\right)$, i.e., $d_{n}$ is the greatest common divisor of $a_{n}$ and $a_{n+1}$, then the maximum value of $d_{n}$ is $\qquad$. | 433 | 104 | 3 |
math | 8. $[\mathbf{6}]$ Let $A:=\mathbb{Q} \backslash\{0,1\}$ denote the set of all rationals other than 0 and 1. A function $f: A \rightarrow \mathbb{R}$ has the property that for all $x \in A$,
$$
f(x)+f\left(1-\frac{1}{x}\right)=\log |x| .
$$
Compute the value of $f(2007)$. | \log(2007/2006) | 112 | 13 |
math | 1. Given that the intercepts of line $l$ on the two coordinate axes are opposite numbers, and the distance from point $M(1,-1)$ to $l$ is $\sqrt{2}$. Then the number of lines $l$ that meet the conditions is $\qquad$ lines. | 2 | 63 | 1 |
math | 144. Extract the cube root $\sqrt[3]{2+\sqrt{5}}$. | \frac{1+\sqrt{5}}{2} | 20 | 12 |
math | Example 9. Simplify $1-\frac{1}{4} \sin ^{2} 2 \alpha-\sin ^{2} \beta-\cos ^{4} \alpha$ into a product form of trigonometric functions. (84 College Entrance Examination for Liberal Arts)
(84年高考文科试题)
Note: The last line is kept in Chinese as it is a reference to the source of the problem and does not need to be trans... | \sin (\alpha+\beta) \sin (\alpha-\beta) | 120 | 14 |
math | Example 6: There are three types of goods, A, B, and C. If you buy 3 pieces of A, 7 pieces of B, and 1 piece of C, it costs 3.15 yuan; if you buy 4 pieces of A, 10 pieces of B, and 1 piece of C, it costs 4.20 yuan. Now, if you buy 1 piece each of A, B, and C, it will cost $\qquad$ yuan.
(1985. National Junior High Scho... | 1.05 | 120 | 4 |
math | Find all triples $(x, n, p)$ of positive integers $x$ and $n$ and prime numbers $p$ for which
$$
x^{3}+3 x+14=2 \cdot p^{n} \text {. }
$$ | (1,2,3) \text{ and } (3,2,5) | 54 | 19 |
math | 15. The line $y=k x+1$ intersects the left branch of the hyperbola $x^{2}-y^{2}=1$ at points $A$ and $B$. The line $l$ passes through the point $(-2,0)$ and the midpoint of $A B$. Find the range of the y-intercept $b$ of the line $l$. | (-\infty,-2-\sqrt{2})\cup(2,+\infty) | 81 | 20 |
math | Find all continuous and nondecreasing functions $ f:[0,\infty)\longrightarrow\mathbb{R} $ that satisfy the inequality:
$$ \int_0^{x+y} f(t) dt\le \int_0^x f(t) dt +\int_0^y f(t) dt,\quad\forall x,y\in [0,\infty) . $$ | f(x) = c | 81 | 6 |
math | 5. If the function $f(x)=\frac{a+\sin x}{2+\cos x}+b \tan x$ has a sum of its maximum and minimum values equal to 4, then $a+b=$ $\qquad$ | 3 | 51 | 1 |
math | 2. Find all pairs of integers $(x, y)$ that satisfy the equation $x^{2}-x y-6 y^{2}-11=0$. For each pair $(x, y)$ found, calculate the product $x y$. In the answer, write the sum of these products. | 8 | 62 | 1 |
math | Find the smallest exact square with last digit not $0$, such that after deleting its last two digits we shall obtain another exact square. | 121 | 27 | 3 |
math | 5. Find all subsequences $\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ of the sequence $\{1,2, \cdots, n\}$, such that for $1 \leqslant i \leqslant n$, the property $i+1 \mid 2\left(a_{1}+a_{2}+\cdots+a_{i}\right)$ holds. | (1,2,\cdots,n)or(2,1,3,4,\cdots,n) | 96 | 23 |
math | 10. (20 points) Given the functions $f(x)=a x^{2}-2 \sqrt{4+2 b-b^{2}} x, g(x)=-\sqrt{1-(x-a)^{2}}(a, b \in \mathbf{R})$.
(1) Find all real number pairs $(a, b): a \in \mathbf{Z}$, such that there exists $x_{0}$, making $f\left(x_{0}\right)$ the maximum value of $f(x)$, and $g\left(x_{0}\right)$ the minimum value of $g... | (,b)=(-1,-1)or(-1,3) | 267 | 15 |
math | Example 2 Find all increasing functions $f: \mathbf{N} \rightarrow \mathbf{Z}$ satisfying:
$$
\begin{array}{l}
f(2)=7, \\
f(m n)=f(m)+f(n)+f(m) f(n) .
\end{array}
$$
(2014, Taiwan Mathematical Olympiad Training Camp) | f(n)=n^{3}-1 | 79 | 8 |
math | We shuffle a 52-card French deck, then draw cards one by one from the deck until we find a black ace. On which draw is it most likely for the first black ace to appear? | 1 | 41 | 1 |
math | 3.94. A cube is inscribed in a regular quadrilateral pyramid; the vertices of its upper base lie on the lateral edges, and the vertices of its lower base are in the plane of the pyramid's base. Find the ratio of the volume of the cube to the volume of the pyramid, if the lateral edge of the pyramid makes an angle $\alp... | \frac{3\sqrt{2}\operatorname{ctg}\alpha}{(1+\sqrt{2}\operatorname{ctg}\alpha)^{3}} | 80 | 35 |
math | 4.056. Find the fifth term of an increasing geometric progression, given that its first term is equal to $7-3 \sqrt{5}$ and that each of its terms, starting from the second, is equal to the difference of the two adjacent terms. | 2 | 56 | 1 |
math | Edward, the author of this test, had to escape from prison to work in the grading room today. He stopped to rest at a place $1,875$ feet from the prison and was spotted by a guard with a crossbow.
The guard fired an arrow with an initial velocity of $100 \dfrac{\text{ft}}{\text{s}}$. At the same time, Edward started... | 75 \, \text{ft/s} | 175 | 10 |
math | 164. Find a perfect number of the form $p^{2} q$. | 28 | 18 | 2 |
math | Example 3 Let $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfy
$$
\begin{array}{l}
f(x f(x)+f(x) f(y)+y-1) \\
=f(x f(x)+x y)+y-1 .
\end{array}
$$
Find the analytical expression of the function $f(x)$. | f(x)=x | 79 | 4 |
math | 9. When $\theta$ varies, the equation of the locus of the vertex of the parabola $y=x^{2}+2 x \sin \theta+1$ is | -x^{2}+1,-1\leqslantx\leqslant1 | 38 | 20 |
math | ## Problem Statement
Find the point of intersection of the line and the plane.
$$
\begin{aligned}
& \frac{x-1}{-1}=\frac{y+5}{4}=\frac{z-1}{2} \\
& x-3 y+7 z-24=0
\end{aligned}
$$ | (0,-1,3) | 72 | 7 |
math | Problem 9.1. Find all values of $a$ such that the equation
$$
\left(a^{2}-a-9\right) x^{2}-6 x-a=0
$$
has two distinct positive roots.
Ivan Landjev | \in(-3,(1-\sqrt{37})/2) | 55 | 15 |
math | 10. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=1, a_{n+1}=\frac{\sqrt{3} a_{n}-1}{a_{n}+\sqrt{3}}\left(n \in \mathbf{N}^{*}\right)$, then $a_{2004}=$ | 2+\sqrt{3} | 81 | 6 |
math | 6. Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f$ : $\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ satisfying for all $x, y \in \mathbb{R}^{+}$ the equation
$$
f(x f(y))=f(x y)+x .
$$
(P. Kaňovský) | f(x)=x+1 | 93 | 6 |
math | [ Combinatorics (miscellaneous). ] [Principle of the extreme (miscellaneous) ]
How many ways are there to rearrange the numbers from 1 to 100 such that adjacent numbers differ by no more than 1?
# | 2 | 52 | 1 |
math | 42nd Putnam 1981 Problem B2 What is the minimum value of (a - 1) 2 + (b/a - 1) 2 + (c/b - 1) 2 + (4/c - 1) 2 , over all real numbers a, b, c satisfying 1 ≤ a ≤ b ≤ c ≤ 4. Solution | 12-8\sqrt{2} | 81 | 9 |
math | 4. Find the largest constant $k$, such that for all real numbers $a, b, c, d$ in $[0,1]$, the inequality $a^{2} b+b^{2} c+c^{2} d+d^{2} a+4 \geqslant k\left(a^{3}+b^{3}+c^{3}+d^{3}\right)$ holds. | 2 | 88 | 1 |
math | 11.3. Solve the system of equations $\left\{\begin{array}{l}\sin x \cos y=\sin z \\ \cos x \sin y=\cos z\end{array}\right.$, if the numbers $x, y$ and $z$ lie in the interval $\left[0 ; \frac{\pi}{2}\right]$ | (\frac{\pi}{2};0;\frac{\pi}{2}),(0;\frac{\pi}{2};0) | 76 | 26 |
math | Suppose $ A_1,\dots, A_6$ are six sets each with four elements and $ B_1,\dots,B_n$ are $ n$ sets each with two elements, Let $ S \equal{} A_1 \cup A_2 \cup \cdots \cup A_6 \equal{} B_1 \cup \cdots \cup B_n$. Given that each elements of $ S$ belogs to exactly four of the $ A$'s and to exactly three of the $ B$'s, find ... | 9 | 116 | 1 |
math | Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$. | 1540 | 133 | 4 |
math | 1. Find the number of points in the $x O y$ plane having natural coordinates $(x, y)$ and lying on the parabola $y=-\frac{x^{2}}{4}+9 x+19$. | 18 | 49 | 2 |
math | 9.46 Let real numbers $a, b, c, d$ satisfy $a^{2}+b^{2}+c^{2}+d^{2} \leqslant 1$, find
$$S=(a+b)^{4}+(a+c)^{4}+(a+d)^{4}+(b+c)^{4}+(b+d)^{4}+(c+d)^{4}$$
the maximum value. | 6 | 97 | 1 |
math | 35th Swedish 1995 Problem 5 A, B, C, D are points on a circle radius r in that order. AB = BC = CD = s < r and AD = s+r. Find the angles of the quadrilateral. | \angleA=\angleD=72,\angleB=\angleC=108 | 53 | 19 |
math | 5. Given $n!=n \times(n-1) \times(n-2) \times \cdots \times 2 \times 1$, then $10!\div(5!\times 2!)=$ | 15120 | 47 | 5 |
math | All integers are written in a row, starting from one. Determine which digit stands at the 206788th position.
# | 7 | 29 | 1 |
math | 5.1. Find the limit of the sequence
$$
\lim _{n \rightarrow \infty} \frac{3 n^{5}-6 n^{4}+3 n+5}{(n+1)^{6}-(n-2)^{6}}
$$ | \frac{1}{6} | 60 | 7 |
math | 3.382. Find $\sin 2 \alpha$, if it is known that $2 \operatorname{tg}^{2} \alpha-7 \operatorname{tg} \alpha+3=0$ and the number $\alpha$ satisfies the inequalities: a) $\pi<\alpha<\frac{5 \pi}{4}$; b) $\frac{5 \pi}{4}<\alpha<\frac{3 \pi}{2}$. | \frac{4}{5};\frac{3}{5} | 99 | 14 |
math | 8.3.1. (12 points) Five numbers form an increasing arithmetic progression. The sum of their cubes is zero, and the sum of their squares is 70. Find the smallest of these numbers. | -2\sqrt{7} | 45 | 7 |
math | Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$
[i]Proposed by Ray Li[/i] | 200 | 98 | 3 |
math | A high-speed train, weighing $P=150$ tons, travels at a maximum speed of $v=72 \frac{\mathrm{km}}{\text { hour }}$ on a horizontal track where $\rho=0,005$. At what speed can it travel on a track that has the same friction conditions but an elevation of $e=0,030$? [^0]
[^0]: ${ }^{1}$ In this problem, $\rho$ is th... | 10.3\frac{\mathrm{}}{\text{}} | 121 | 14 |
math | Condition of the problem
To find the equations of the tangent and normal lines to the curve at the point corresponding to the parameter value $t=t_{0}$.
$\left\{\begin{array}{l}x=t^{3}+1 \\ y=t^{2}+t+1\end{array}\right.$
$t_{0}=1$ | x+1-x+5 | 74 | 6 |
math | Example 2 Let
$$
\begin{aligned}
B= & \frac{a}{|a|}-\frac{b}{|b|}-\frac{c}{|c|}+\frac{d}{|d|}+\frac{a b}{|a b|}+\frac{a c}{|a c|}- \\
& \frac{a d}{|a d|}-\frac{b c}{|b c|}+\frac{b d}{|b d|}+\frac{c d}{|c d|}+\frac{a b c}{|a b c|}- \\
& \frac{a b d}{|a b d|}-\frac{a c d}{|a c d|}+\frac{b c d}{|b c d|}-\frac{a b c d}{|a ... | 1 \text{ or } -15 | 203 | 9 |
math | Four. (20 points) Given the function $f_{n}(x)=n^{2} x^{2}(1-$ $x)^{n}, x \in[0,1], n \in \mathbf{N}_{+}$. If the maximum value of $f_{n}(x)$ is denoted as $a_{n}$, try to find the minimum term of the sequence $\left\{a_{n}\right\}$. | \frac{4}{27} | 96 | 8 |
math | 9. 1 For which values of $x$ is the inequality
$$\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9$$
true. | -\frac{1}{2} \leqslant x < 0 \text{ or } 0 < x < \frac{45}{8} | 46 | 34 |
math | For $ n \in \mathbb{N}$, let $ f(n)\equal{}1^n\plus{}2^{n\minus{}1}\plus{}3^{n\minus{}2}\plus{}...\plus{}n^1$. Determine the minimum value of: $ \frac{f(n\plus{}1)}{f(n)}.$ | \frac{8}{3} | 73 | 7 |
math | 4. For a set $X$, denote $\mathscr{P}(X)=\{Y: Y \subseteq X\}$. (For example: $\mathscr{P}(\{1\})=$ $\{\varnothing,\{1\}\}$, because the subsets of the set $\{1\}$ are the sets $\varnothing$ and $\{1\} ; \mathscr{P}(\varnothing)=\{\varnothing\}$, because the set $\varnothing$ has exactly one subset, which is $\varnothi... | {\varnothing,{\varnothing}} | 227 | 8 |
math | [Numerical inequalities. Comparing numbers.]
What is greater: $2011^{2011}+2009^{2009}$ or $2011^{2009}+2009^{2011}$? | 2011^{2011}+2009^{2009}>2011^{2009}+2009^{2011} | 59 | 42 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$:
$$
f(2 f(x)+f(y))=2 x+f(y)
$$ | f(x)=x | 52 | 4 |
math | 245. A discrete random variable $X$ is given by the distribution law
$$
\begin{array}{lll}
X & 0.3 & 0.6 \\
p & 0.2 & 0.8
\end{array}
$$
Using Chebyshev's inequality, estimate the probability that $|X-M(X)|<0.2$. | 0.64 | 82 | 4 |
math | How many numbers $\overline{abcd}$ with different digits satisfy the following property:
if we replace the largest digit with the digit $1$ results in a multiple of $30$? | 162 | 40 | 3 |
math | Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is $3$ jumps east and $10$ jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at Frist, located $2$ jumps east and $4$ jumps north of Fine Hall. How many ways can he get... | 181 | 91 | 3 |
math | 5. (5 points) When 4 identical small cubes (the six faces of the small cubes are marked with the numbers $1, 2, 3, 4, 5$, and $6$) are thrown simultaneously, the number of possible sums of the 4 numbers on the top faces is $\qquad$ kinds. | 21 | 70 | 2 |
math | 3.1. Determine the total surface area of a cube if the distance between non-intersecting diagonals of two adjacent faces of this cube is 8. If the answer is not an integer, round it to the nearest integer. | 1152 | 48 | 4 |
math | Let $a,b,c$ be distinct real numbers such that $a+b+c>0$. Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$. Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value.
[i]Mircea Becheanu[/i] | 6 | 92 | 1 |
math | 10.148. Find the area of an isosceles trapezoid if its height is equal to $h$, and the lateral side is seen from the center of the circumscribed circle at an angle of $60^{\circ}$. | ^{2}\sqrt{3} | 56 | 7 |
math | [ Algebraic inequalities (miscellaneous).] [ Case analysis $]$
$x, y>0$. Let $S$ denote the smallest of the numbers $x, 1 / y, y+1 / x$. What is the maximum value that $S$ can take? | \sqrt{2} | 57 | 5 |
math | 3. If $a, b, c, x, y, z$ are real numbers, and $a^{2}+b^{2}+c^{2}=25, x^{2}+y^{2}+z^{2}=36, a x+b y+c z=$ 30, then $\frac{a+b+c}{x+y+z}=$ $\qquad$ | \frac{5}{6} | 84 | 7 |
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