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 | II. (25 points) Several boxes are unloaded from a cargo ship, with a total weight of 10 tons, and the weight of each box does not exceed 1 ton. To ensure that these boxes can be transported away in one go, the question is: what is the minimum number of trucks with a carrying capacity of 3 tons needed? | 5 | 73 | 1 |
math | 7. (1994 Shanghai Senior High School Competition Question) Two two-digit numbers, their difference is 52, and the last two digits of their squares are the same. Then these two numbers are | 76,24 | 43 | 5 |
math | (Small generating sets of $\mathbb{Z} \times \mathbb{Z}$) Find all generating sets of cardinality at most 2 of the group $\left(\mathbb{Z}^{2},+\right)$.
## Group Homomorphisms
Definition 71. (Group Homomorphism) Let $(G, *)$ and $(H, \circledast)$ be two groups, and $\varphi: G \rightarrow H$ be a function such that... | a_{1}b_{2}-a_{2}b_{1}=\1 | 179 | 18 |
math | Example 1 Let $m, n$ be positive integers, and $p$ be a prime. Find all triples $(m, n, p)$ that satisfy $\mathrm{C}_{m}^{3}-4=p^{n}$. | (,n,p)=(7,1,31),(6,4,2) | 49 | 18 |
math | A positive integer is called [i]oneic[/i] if it consists of only $1$'s. For example, the smallest three oneic numbers are $1$, $11$, and $111$. Find the number of $1$'s in the smallest oneic number that is divisible by $63$. | 18 | 70 | 2 |
math | 7. Find $(x+1)\left(x^{2}+1\right)\left(x^{4}+1\right)\left(x^{8}+1\right) \cdots$, where $|x|<1$. | \frac{1}{1-x} | 50 | 8 |
math | Given are positive integers $ n>1$ and $ a$ so that $ a>n^2$, and among the integers $ a\plus{}1, a\plus{}2, \ldots, a\plus{}n$ one can find a multiple of each of the numbers $ n^2\plus{}1, n^2\plus{}2, \ldots, n^2\plus{}n$. Prove that $ a>n^4\minus{}n^3$. | a > n^4 - n^3 | 102 | 10 |
math | 3. Solve in the set of integers the equation $x^{2}+16 x+55=3^{y^{2}-2 y}$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
Note: The note itself is not part of the translation but is provided to clarify the instruction. Here ... | (x,y)\in{(-2,3),(-2,-1),(-14,3),(-14,-1)} | 118 | 27 |
math | At a math competition, a team of $8$ students has $2$ hours to solve $30$ problems. If each problem needs to be solved by $2$ students, on average how many minutes can a student spend on a problem? | 16 | 51 | 2 |
math | 12. (15 points) 10 white balls numbered from 1 to 10 are arranged in a row. Now, the balls are painted red according to the following method:
（1）Paint 2 balls;
（2）The difference in the numbers of the 2 painted balls is greater than 2. How many different painting methods are there? | 28 | 76 | 2 |
math | Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that:
[list]
[*] No two numbers are equal.
[*] If a number $m$ is placed on square $C$, then at least $k$ of the squares orthogonally adjacent to $C$ have a multiple of $m$ written on... | k = 2 | 103 | 5 |
math | 5. (19th Nordic Mathematical Contest) Find all positive integers $k$, such that in decimal notation, the product of the digits of $k$ equals $\frac{25}{8} k - 211$. | 72or88 | 48 | 5 |
math | Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{2}+f(y)\right)=f(f(x))+f\left(y^{2}\right)+2 f(x y)
$$
for all real number $x$ and $y$.
Answer: The possible functions are $f(x)=0$ for all $x$ and $f(x)=x^{2}$ for all $x$. | f(x)=0 \text{ or } f(x)=x^2 | 103 | 15 |
math | 185. Reduce the equations of the circles to their standard form:
1) $x^{2}+y^{2}-4 x+6 y-3=0$
2) $4 x^{2}+4 y^{2}-8 x+4 y-11=0$ | (x-2)^{2}+(y+3)^{2}=16(x-1)^{2}+(y+\frac{1}{2})^{2}=4 | 62 | 37 |
math | In the following equation, determine $p$ so that -1 is one of the roots of the equation:
$$
(2 p-1) x^{2}+2(1-p) x+3 p=0
$$ | \frac{3}{7} | 48 | 7 |
math | ## 9. Distribution of Euros
Ana, Janko, and Tara have certain amounts of euros and want to redistribute them among themselves. First, Ana gives Janko and Tara a portion of her money so that after this, both Janko and Tara have twice as much money as they had before. Then, Janko gives Ana and Tara a portion of his mone... | 511 | 165 | 3 |
math | 7. [15] Find a real, irreducible quartic polynomial with leading coefficient 1 whose roots are all twelfth roots of unity. | x^{4}-x^{2}+1 | 30 | 10 |
math | Problem 3. Vasya solved problems for 15 days - at least one problem each day. Each day (except the first), if the weather was cloudy, he solved one more problem than the previous day, and if it was sunny, one less problem. In the first 9 days, Vasya solved 13 problems.
a) What was the weather like on the 10th day? How... | 7 | 137 | 1 |
math | Example 4.4.4 A die has six faces marked with $1,2, \cdots, 6$, how many different dice are there? | 30 | 33 | 2 |
math | Example 3 Find all $f: \mathbf{R} \rightarrow \mathbf{R}$, satisfying the following inequality: $f(x+y)+f(y+z)+f(z+x) \geqslant 3 f(x+2 y+3 z)$, for all $x, y, z \in \mathbf{R}$. | f()=f(0) | 75 | 7 |
math | 4. In triangle $ABC$, the angle bisectors $\mathrm{AK}$ and $\mathrm{CL}$ and the median $\mathrm{BM}$ are drawn. It turns out that $\mathrm{ML}$ is the bisector of angle AMB, and $\mathrm{MK}$ is the bisector of angle CMB. Find the angles of triangle ABC. | 30,30,120 | 73 | 9 |
math | 19. (ROM 1) Consider the sequences $\left(a_{n}\right),\left(b_{n}\right)$ defined by $$ a_{1}=3, \quad b_{1}=100, \quad a_{n+1}=3^{a_{n}}, \quad b_{n+1}=100^{b_{n}} . $$ Find the smallest integer $m$ for which $b_{m}>a_{100}$. | 99 | 101 | 2 |
math | 9. In tetrahedron $ABCD$, face $ABC$ and face $BCD$ form a dihedral angle of $60^{\circ}$, the projection of vertex $A$ onto face $BCD$ is the orthocenter $H$ of $\triangle BCD$, and $G$ is the centroid of $\triangle ABC$. If $AH=4$ and $AB=AC$, then $GH=$ $\qquad$ | \frac{4 \sqrt{21}}{9} | 95 | 13 |
math | 1. A smaller square was cut out from a larger square, one of its sides lying on the side of the original square. The perimeter of the resulting octagon is $40 \%$ greater than the perimeter of the original square. By what percentage is its area less than the area of the original square? | 64 | 63 | 2 |
math | Let it be known that all roots of a certain equation $x^{3} + p x^{2} + q x + r = 0$ are positive. What additional condition must the coefficients $p, q$, and $r$ satisfy so that a triangle can be formed from segments whose lengths are equal to these roots? | p^{3}-4pq+8r>0 | 67 | 11 |
math | # Problem 1. (2 points)
Let $p_{1}, p_{2}, \ldots, p_{97}$ be prime numbers (not necessarily distinct). What is the largest integer value that the expression
$$
\sum_{i=1}^{97} \frac{p_{i}}{p_{i}^{2}+1}=\frac{p_{1}}{p_{1}^{2}+1}+\frac{p_{2}}{p_{2}^{2}+1}+\ldots+\frac{p_{97}}{p_{97}^{2}+1}
$$
can take? | 38 | 141 | 2 |
math | 4. Given a regular quadrilateral pyramid $\Gamma$ with height $h=3$, and the angle $\theta=\frac{\pi}{3}$ between the side face and the base. First, place an inscribed sphere $O_{1}$ inside the regular quadrilateral pyramid $\Gamma$, then place spheres $O_{2}, O_{3}, \cdots$ in sequence, such that each subsequent spher... | \frac{18\pi}{13} | 119 | 11 |
math | $8 \cdot 38$ Find the largest integer $x$ such that $4^{27}+4^{1000}+4^{x}$ is a perfect square.
(6th All-Soviet Union Mathematical Olympiad, 1972) | 1972 | 59 | 4 |
math | Three, (50 points) Find the smallest positive integer $n$, such that any $n$-element set $A$ of positive integers contains 15 elements whose sum is divisible by 15. | 29 | 44 | 2 |
math | Task 4. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy
$$
f(x+y f(x))=f(x f(y))-x+f(y+f(x))
$$
for all $x, y \in \mathbb{R}$. | f(x)=1-x | 64 | 5 |
math | 11. Given the circle $P: x^{2}+y^{2}=2 x$ and the parabola $S: y^{2}=4 x$, a line $l$ is drawn through the center of circle $P$. This line intersects the two curves at four points, which are sequentially labeled as $A, B, C, D$ from top to bottom. If the lengths of segments $A B, B C, C D$ form an arithmetic sequence i... | \\frac{\sqrt{2}}{2}y+1 | 111 | 13 |
math | One. (20 points) Let the constant $a \geqslant 0$. Solve the equation $x^{4}+6 x^{3}+(9-2 a) x^{2}-6 a x+a^{2}-4=0$ for $x$.
| x_{1,2}=\frac{-3 \pm \sqrt{4 a+1}}{2}, x_{3,4}=\frac{-3 \pm \sqrt{4 a+17}}{2} | 60 | 47 |
math | 1. The range of the function $y=\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1-x^{2}}+1}$ is | [1,\sqrt{2}] | 35 | 7 |
math | Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that
$$
f(x)+f(t)=f(y)+f(z)
$$
for all rational numbers $x<y<z<t$ that form an arithmetic progression. | f(q)=f(0)+q(f(1)-f(0)) | 56 | 16 |
math | Example 17. Solve the equation
$$
\frac{2}{2-\sqrt{x}}+\frac{1}{2}=\frac{4}{2 \sqrt{x}-x}
$$ | 16 | 41 | 2 |
math | ## Task Condition
Find the derivative.
$$
y=\arcsin \sqrt{\frac{x}{x+1}}+\operatorname{arctg} \sqrt{x}
$$ | \frac{1}{\sqrt{x}(x+1)} | 38 | 13 |
math | 376. Given a triangle $A B C$, the angles of which are $\alpha, \beta$ and $\gamma$. Triangle $D E F$ is circumscribed around triangle $A B C$ such that vertices $A, B$ and $C$ are located on sides $E F$, $F D$ and $D E$ respectively, and $\angle E C A = \angle D B C = \angle F A B = \varphi$. Determine the value of th... | \operatorname{tg}\varphi_{0}=\operatorname{ctg}\alpha+\operatorname{ctg}\beta+\operatorname{ctg}\gamma | 124 | 35 |
math | Find the minimum natural number $n$ with the following property: between any collection of $n$ distinct natural numbers in the set $\{1,2, \dots,999\}$ it is possible to choose four different $a,\ b,\ c,\ d$ such that: $a + 2b + 3c = d$. | 835 | 72 | 3 |
math | 9. A person picks $n$ different prime numbers less than 150 and finds that they form an arithmetic sequence. What is the greatest possible value of $n$ ?
(1 mark)
某人選取了 $n$ 個不同的質數, 每個均小於 150 。他發現這些質數組成一個等差數列。求 $n$ 的最大可能值。 | 5 | 90 | 1 |
math | 4.044. Find four numbers, the first three of which form a geometric progression, and the last three form an arithmetic progression. The sum of the extreme numbers is 21, and the sum of the middle numbers is 18. | 3,6,12,18;18.75;11.25;6.75;2.25 | 52 | 31 |
math | A father and his son are skating around a circular skating rink. From time to time, the father overtakes the son. After the son starts skating in the opposite direction, they begin to meet five times more often. What is the ratio of the skating speeds of the father and the son?
(Tairova) | \frac{3}{2} | 65 | 7 |
math | 2. Let $a$, $b$, $c$ be integers, and for all real numbers $x$,
$$
(x-a)(x-8)+1=(x-b)(x-c)
$$
always holds. Then the value of $a+b+c$ is $\qquad$. | 20 \text{ or } 28 | 60 | 10 |
math | In his triangle $ABC$ Serge made some measurements and informed Ilias about the lengths of median $AD$ and side $AC$. Based on these data Ilias proved the assertion: angle $CAB$ is obtuse, while angle $DAB$ is acute. Determine a ratio $AD/AC$ and prove Ilias' assertion (for any triangle with such a ratio). | \frac{AD}{AC} = \frac{1}{2} | 78 | 16 |
math | Let $\mathbb{Q}_{+}$ be the set of positive rational numbers. Find all functions $f: \mathbb{Q}_{+} \rightarrow \mathbb{Q}_{+}$ which for all $x \in \mathbb{Q}_{+}$ fulfill
(1) $f\left(\frac{1}{x}\right)=f(x)$
(2) $\left(1+\frac{1}{x}\right) f(x)=f(x+1)$ | f\left(\frac{p}{q}\right)=a p q | 102 | 15 |
math | 1. [5] A polynomial $P$ with integer coefficients is called tricky if it has 4 as a root.
A polynomial is called teeny if it has degree at most 1 and integer coefficients between -7 and 7 , inclusive.
How many nonzero tricky teeny polynomials are there? | 2 | 62 | 1 |
math | 5. The maximum value of the function $y=|\sin x|+|\sin 2 x|$ is $\qquad$ . | \frac{\sqrt{414+66 \sqrt{33}}}{16} | 28 | 21 |
math | 15. The teacher asked the students to calculate $\overline{A B} . C+D . E$. Ma Xiaohu missed the decimal point in $D . E$, getting the wrong result of 39.6; while Ma Dahu mistook the plus sign for a multiplication sign, getting the wrong result of 36.9. What, then, should the correct calculation result be? | 26.1 | 85 | 4 |
math | 9. Given that $\left\{a_{n}\right\}$ is a geometric sequence with all terms being positive, and $a_{50}, a_{51}$ are the two distinct solutions of the equation $100 \lg ^{2} x=\lg (100 x)$. Find the value of $a_{1} a_{2} \cdots a_{100}$. | \sqrt{10} | 88 | 6 |
math | Example 8 A positive integer, if it can be expressed as the difference of squares of two positive integers, is called a "wise number". Arrange all the wise numbers in ascending order. Find the wise number at the 2009th position. | 2681 | 52 | 4 |
math | In $\triangle ABC$, $\angle A = 30^{\circ}$ and $AB = AC = 16$ in. Let $D$ lie on segment $BC$ such that $\frac{DB}{DC} = \frac23$ . Let $E$ and $F$ be the orthogonal projections of $D$ onto $AB$ and $AC$, respectively. Find $DE + DF$ in inches. | 8 | 89 | 1 |
math | Problem 2. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{2}+y+f(y)\right)=2 y+(f(x))^{2}
$$
for any $x, y \in \mathbb{R}$. | f(x)\equivx | 69 | 5 |
math | 1. (10 points) Let $[x]$ denote the greatest integer not exceeding $x$, for example $[3.14]=3$, then $\left[\frac{2017 \times 3}{11}\right]+\left[\frac{2017 \times 4}{11}\right]+\left[\frac{2017 \times 5}{11}\right]+\left[\frac{2017 \times 6}{11}\right]+\left[\frac{2017 \times 7}{11}\right]+\left[\frac{2017 \times 8}{1... | 6048 | 148 | 4 |
math | Let $a$ and $b$ be the roots of the quadratic $x^2-7x+c$. Given that $a^2+b^2=17$, compute $c$. | 16 | 40 | 2 |
math | In the $xOy$ system consider the lines $d_1\ :\ 2x-y-2=0,\ d_2\ :\ x+y-4=0,\ d_3\ :\ y=2$ and $d_4\ :\ x-4y+3=0$. Find the vertices of the triangles whom medians are $d_1,d_2,d_3$ and $d_4$ is one of their altitudes.
[i]Lucian Dragomir[/i] | A(1, 0), B(0, 4), C(5, 2) | 107 | 21 |
math | In order to save money on gas and use up less fuel, Hannah has a special battery installed in the family van. Before the installation, the van averaged $18$ miles per gallon of gas. After the conversion, the van got $24$ miles per gallong of gas.
Michael notes, "The amount of money we will save on gas over any time ... | 108 | 191 | 3 |
math | 8. A freight train leaves the town of Jenkinsville at 1:00 PM traveling due east at constant speed. Jim, a hobo, sneaks onto the train and falls asleep. At the same time, Julie leaves Jenkinsville on her bicycle, traveling along a straight road in a northeasterly direction (but not due northeast) at 10 miles per hour. ... | 62.5\mathrm{} | 161 | 7 |
math | 1. There are 4 kg of a copper-tin alloy, in which $40\%$ is copper, and 6 kg of another copper-tin alloy, in which $30\%$ is copper. What masses of these alloys need to be taken to obtain 8 kg of an alloy containing $p\%$ copper after melting? Find all $p$ for which the problem has a solution. | 32.5\leqslantp\leqslant35 | 87 | 17 |
math | Let's rationalize the denominators of the following fractions:
a) $\frac{1}{\sqrt{2}+\sqrt{3}-\sqrt{5}}$,
b) $\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}}$ | \frac{2\sqrt{3}+3\sqrt{2}+\sqrt{30}}{12},\frac{+\sqrt{^{2}-x^{2}}}{x} | 65 | 42 |
math | The sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are extended to meet at $E$. Let $H$ and $G$ be the midpoints of $BD$ and $AC$, respectively. Find the ratio of the area of the triangle $EHG$ to that of the quadrilateral $ABCD$. | \frac{1}{4} | 73 | 7 |
math | 5. Let $a_{1}, a_{2}, \cdots, a_{n}$ be a permutation of the natural numbers $1,2, \cdots, n$. For all permutations of $1,2, \cdots, n$, find the maximum value of the sum $s=\left|a_{1}-1\right|+$ $\left|a_{2}-2\right|+\cdots+\left|a_{n}-n\right|$. | \frac{1}{2}n^ | 100 | 9 |
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 | $7 \cdot 56$ If there are $n$ integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying the following equation
$$
a_{1}+a_{2}+\cdots+a_{n}=a_{1} a_{2} \cdots a_{n}=1990 \text {. }
$$
Find the smallest natural number $n(n \geqslant 2)$.
(16th All-Russian Mathematical Olympiad, 1990) | 5 | 113 | 1 |
math | Before the math competition, Dmytro overheard Olena and Mykola talking about their birthdays.
[b]О[/b]: "The day and month of my birthday are half as large as the day and month of Mykola's birthday."
[b]М[/b]: "Also, the day of Olena's birth and the month of my birth are consecutive positive integers."
[b]О[/b]: "And ... | (11, 6, 22, 12) | 135 | 16 |
math | Find all possible values of $a\in \mathbb{R}$ and $n\in \mathbb{N^*}$ such that $f(x)=(x-1)^n+(x-2)^{2n+1}+(1-x^2)^{2n+1}+a$
is divisible by $\phi (x)=x^2-x+1$ | (n, a) = (3c, -1) | 81 | 14 |
math | 3. Solve the inequality $\log _{9} 4+\left(16-\log _{3}^{2} 2\right) \log _{162} 3 \leqslant 64^{\log _{4}^{2} x}-15 \cdot x^{\log _{4} x}$. | x\in(0;\frac{1}{4}]\cup[4;+\infty) | 77 | 21 |
math | 3. Given an isosceles right triangle with legs equal to $\sqrt{2}$. Find the straight-line cut of the smallest length that divides the area of the triangle in half.
In the answer, specify the length of the cut, rounded to 2 decimal places after the decimal point. Answer: $0.91$. | 0.91 | 69 | 4 |
math | Positive real numbers $a$, $b$, $c$ satisfy $a+b+c=1$. Find the smallest possible value of
$$E(a,b,c)=\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}.$$ | \frac{1}{8} | 70 | 7 |
math | $7.8 \quad 4^{\sqrt{x}}-9 \cdot 2^{\sqrt{x}-1}+2=0$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
$7.8 \quad 4^{\sqrt{x}}-9 \cdot 2^{\sqrt{x}-1}+2=0$. | 4 | 88 | 1 |
math | 11. Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}$ related to $a_{n}$ by $S_{n}=-b a_{n}+1-\frac{1}{(1+b)^{n}}$, where $b$ is a constant independent of $n$, and $b \neq-1$.
(1) Find the expression for $a_{n}$ in terms of $b$ and $n$;
(2) When $b=3$, from which term does the sequence... | 4 | 155 | 1 |
math | 7. Four points are randomly chosen from the vertices of a regular 12-sided polygon. Find the probability that the four chosen points form a rectangle (including square).從一個正十二邊形的頂點中隨意選四點, 求該四點成一長方形（包括正方形）的概率。 | \frac{1}{33} | 67 | 8 |
math | 4. All natural numbers, the sum of the digits in the representation of which is divisible by 5, are listed in ascending order: $5,14,19,23,28,32, \ldots$ What is the smallest positive difference between consecutive numbers in this sequence? Provide an example and explain why it cannot be smaller.
---
The smallest pos... | 1 | 323 | 1 |
math | 121. Solve the Cauchy problem for the equation $y^{\prime \prime}=1+x+x^{2}+$ $+x^{3}$, if $y=1$ and $y^{\prime}=1$ when $x=0$. | \frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{12}+\frac{x^{5}}{20}+x+1 | 56 | 42 |
math | 8. If the complex number $x$ satisfies $x+\frac{1}{x}=-1$, then $x^{2013}+\frac{1}{x^{2013}}=$ $\qquad$ . | 2 | 49 | 1 |
math | Let $0 \leq k < n$ be integers and $A=\{a \: : \: a \equiv k \pmod n \}.$ Find the smallest value of $n$ for which the expression
\[ \frac{a^m+3^m}{a^2-3a+1} \]
does not take any integer values for $(a,m) \in A \times \mathbb{Z^+}.$ | n = 11 | 96 | 6 |
math | Players A and B play a game. They are given a box with $n=>1$ candies. A starts first. On a move, if in the box there are $k$ candies, the player chooses positive integer $l$ so that $l<=k$ and $(l, k) =1$, and eats $l$ candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of... | n \equiv 1 \pmod{2} | 94 | 12 |
math | A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n... | \frac{k}{m} | 133 | 7 |
math | 4. Through the focus $F$ of the parabola $E: y^{2}=2 x$, two lines $l_{1}, l_{2}$ are drawn with the product of their slopes being $-\frac{1}{2}$. Line $l_{1}$ intersects $E$ at points $A, C$, and line $l_{2}$ intersects $E$ at points $B, D$. Then the minimum value of $|A C|+2|B D|$ is $\qquad$. | 8\sqrt{2}+6 | 108 | 8 |
math | Example 2 Factorize:
$$
(x y-1)^{2}+(x+y-2)(x+y-2 x y)
$$ | (x-1)^{2}(y-1)^{2} | 30 | 14 |
math | 2. In the Cartesian coordinate system, points of the form $\left(m, n^{2}\right)$ are painted red (where $m, n$ are integers), referred to as red points, and their surrounding points are not colored. Then, the parabola $y=x^{2}-$ $196 x+9612$ has $\qquad$ red points. | 2 | 81 | 1 |
math | 3 . Find the integers $x, y$ that satisfy the equation $2 x-3 y=17$, and for which $x^{2}+y^{2}$ is minimized, as well as the minimum value of $x^{2}+y^{2}$. | 25 | 58 | 2 |
math | ## PROBLEM 2
Determine the rational number $a$, knowing that $\left(\frac{1}{\sqrt{9+4 \sqrt{5}}}+\frac{1}{\sqrt{9-4 \sqrt{5}}}\right) \cdot a+\sqrt{5}$ is a rational number. | -\frac{1}{2} | 66 | 7 |
math | $$
\text { 4.62 Solve the equation } \frac{x}{2+\frac{x}{2+\frac{x}{2+\ddots \frac{x}{2+\frac{x}{1+\sqrt{1+x}}}}}}=1
$$
(In the above expression, there are 1985 twos.) | 3 | 70 | 1 |
math | 8 Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=0, a_{n+1}=a_{n}+1+2 \sqrt{1+a_{n}}(n=1$, $2, \cdots)$, then $a_{n}=$ $\qquad$ | a_{n}=n^{2}-1 | 69 | 9 |
math | 4. Let $a, b, c$ be the sides opposite to the interior angles $A, B, C$ of $\triangle A B C$, respectively, and the area $S=\frac{1}{2} c^{2}$. If $a b=\sqrt{2}$, then the maximum value of $a^{2}+b^{2}+c^{2}$ is . $\qquad$ | 4 | 87 | 1 |
math | 451986 can be written in the form: $1986=a_{1}+a_{2} \cdot 2!+a_{3} \cdot 3!+\cdots+a_{n} \cdot n$ !, where $a_{k}$ are integers, and $0 \leqslant a_{k} \leqslant k(k=1,2,3, \cdots, n)$, find the values of $a_{k}$. | a_{1}=0,a_{2}=0,a_{3}=3,a_{4}=2,a_{5}=4,a_{6}=2 | 107 | 30 |
math | Let $A=\left(
\begin{array}{ccc}
1 & 1& 0 \\
0 & 1& 0 \\
0 &0 & 2
\end{array}
\right),\ B=\left(
\begin{array}{ccc}
a & 1& 0 \\
b & 2& c \\
0 &0 & a+1
\end{array}
\right)\ (a,\ b,\ c\in{\mathbb{C}}).$
(1) Find the condition for $a,\ b,\ c$ such that ${\text{rank} (AB-BA})\leq 1.$
(2) Under the condition of (1), fin... | b = 0, \quad c = 0, \quad a \neq 2 | 167 | 21 |
math | 11. Given a moving line $l$ that is tangent to the circle $O: x^{2}+y^{2}=1$ and intersects the ellipse $\frac{x^{2}}{9}+y^{2}=1$ at two distinct points $A, B$, find the maximum distance from the origin to the perpendicular bisector of $A B$. | \frac{4}{3} | 76 | 7 |
math | $2016$ numbers written on the board: $\frac{1}{2016}, \frac{2}{2016}, \frac{3}{2016}, ..., \frac{2016}{2016}$. In one move, it is allowed to choose any two numbers $a$ and $b$ written on the board, delete them, and write the number $3ab - 2a - 2b + 2$ instead. Determine what number will remain written on the board aft... | \frac{2}{3} | 124 | 7 |
math | 4. Take a random eight-digit number composed of 0s and 1s. The probability that the sum of the digits in the even positions is equal to the sum of the digits in the odd positions is $\qquad$ . | \frac{35}{128} | 48 | 10 |
math | 3. Find the maximum value of the expression
$$
\frac{a}{x}+\frac{a+b}{x+y}+\frac{a+b+c}{x+y+z}
$$
where $a, b, c \in[2,3]$, and the triplet of numbers $x$, $y$, and $z$ is some permutation of the triplet of numbers $a, b, c$. | \frac{15}{4} | 86 | 8 |
math | 19. (6 points) There is a strip of paper, on which there are three types of markings, dividing the strip into 6 equal parts, 10 equal parts, and 12 equal parts along its length. Now, if we cut the strip along all the markings with scissors, the strip will be divided into $\qquad$ parts. | 20 | 74 | 2 |
math | A 2 meter long bookshelf is filled end-to-end with 46 books. Some of the books are 3 centimeters thick while all the others are 5 centimeters thick. Find the number of books on the shelf that are 3 centimeters thick. | 15 | 55 | 2 |
math | ### 7.301 The equation $4^{x}+10^{x}=25^{x}$ has a unique root.
Find it and compare it with 0 and 1. | \log_{0.4}\frac{\sqrt{5}-1}{2},0<x<1 | 42 | 21 |
math | 3A. For the angles $\alpha, \beta$ and $\gamma$ in triangle $ABC$, the following equality holds:
$$
\frac{\sin ^{2} \beta+\sin ^{2} \gamma-\sin ^{2} \alpha}{\sin \beta \sin \gamma}=1
$$
Calculate the angle $\alpha$. | \alpha=\frac{\pi}{3} | 74 | 9 |
math | Task B-1.3. Master Zvonko cut a needle of length $30 \mathrm{~cm}$ into three parts to make a right-angled triangle from those parts. Zvonko first cut the needle into two parts, one of which was $6 \mathrm{~cm}$ shorter, and he took it as one of the legs. Determine the lengths of all three parts into which Zvonko cut t... | 5\mathrm{~},12\mathrm{~},13\mathrm{~} | 90 | 20 |
math | 3. $A B C D$ — a convex quadrilateral, where $A B=7, B C=4, A D=D C, \angle A B D=\angle D B C$. Point E is on segment $A B$ such that $\angle D E B=90^{\circ}$. Find the length of segment $A E$. (Spain, local phase, 2020-2021) | 1.5 | 92 | 3 |
math | 10.1. Solve the equation: $\sqrt{x+\sqrt{2 x-1}}+\sqrt{x-\sqrt{2 x-1}}=\sqrt{2}$. | \frac{1}{2}\leqx\leq1 | 37 | 13 |
math | GS. 4 How many positive integers $m$ are there for which the straight line passing through points $A(-m, 0)$ and $B(0,2)$ and also passes through the point $P(7, k)$, where $k$ is a positive integer? | 4 | 60 | 1 |
math | The integers $1,2, \ldots, 2018$ are written on the board. Then 2017 operations are performed as follows: choose two numbers $a$ and $b$, erase them, and write $a+b+2 a b$ in their place. At the end, only one integer remains on the board.
What are the possible values that its units digit can take? | 7 | 86 | 1 |
math | Colin has $900$ Choco Pies. He realizes that for some integer values of $n \le 900$, if he eats n pies a day, he will be able to eat the same number of pies every day until he runs out. How many possible values of $n$ are there? | 27 | 67 | 2 |
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