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 | $2 \cdot 35$ subset $X$ of the set $\{00,01, \cdots, 98,99\}$ satisfies: in any infinite sequence of digits, there are two adjacent digits that form an element of $X$. What is the minimum number of elements that $X$ should contain? | 55 | 71 | 2 |
math | For any natural number, let $S(n)$ denote sum of digits of $n$. Find the number of $3$ digit numbers for which $S(S(n)) = 2$. | 100 | 38 | 3 |
math | Let's find all non-negative real numbers $A, B$ such that for every $x > 1$, the following holds:
$$
\left[\frac{1}{A x+\frac{B}{x}}\right]=\frac{1}{A \cdot[x]+\frac{B}{[x]}}
$$ | A=0,B=1 | 67 | 6 |
math | 4. Let $n$ be a natural number. In the plane, there are $2n+2$ points, no three of which lie on the same line. A line in the plane is a separator if it passes through two of the given points and on each side of this line there are exactly $n$ points. Determine the largest number $m$ such that there are always at least ... | n+1 | 125 | 3 |
math | 3. In the set of real numbers, solve the system
$$
\left\{\begin{array}{l}
x+\sqrt{y}=1 \\
y+\sqrt{z}=1 \\
z+\sqrt{x}=1
\end{array}\right.
$$ | \frac{3-\sqrt{5}}{2} | 56 | 12 |
math | 1. In $\triangle A B C$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. If the sizes of angles $A, B, C$ form a geometric sequence, and $b^{2}-a^{2}=a c$, then the radian measure of angle $B$ is $\qquad$ | \frac{2\pi}{7} | 76 | 9 |
math | 3. (12 points) There are four weights of different masses. Katya weighs the weights in pairs. As a result, she got 1800, 1970, 2110, 2330, and 2500 grams. How many grams does the sixth weighing variant weigh? | 2190 | 71 | 4 |
math | ## Task Condition
Find the derivative.
$y=\frac{e^{2 x}(2-\sin 2 x-\cos 2 x)}{8}$ | e^{2x}\cdot\sin^{2}x | 33 | 12 |
math | 6. In $\triangle A B C$, the lengths of the three sides are $a, b, c$, and $b>\max \{a, c\}$. There are 3 non-zero real numbers $x_{0}, y_{0}, z_{0}$, satisfying that the line $a x+b y+c=0$ passes through the point $\left(\frac{z_{0}}{x_{0}}, \frac{2 y_{0}}{x_{0}}\right)$, and the point $\left(\frac{z_{0}}{y_{0}}, \fra... | \frac{5}{3} | 177 | 7 |
math | 6. On the Cartesian plane, the number of integer points (i.e., points with both coordinates as integers) on the circumference of a circle centered at $(199,0)$ with a radius of 199 is $\qquad$ | 4 | 51 | 1 |
math | 3. Real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=5$ Let $s$ $=x^{2}+y^{2}$, then $\frac{1}{s_{\text {max }}}+\frac{1}{s_{\text {min }}}$ is
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | \frac{8}{5} | 96 | 7 |
math | 4.
In the test, there are 4 sections, each containing the same number of questions. Andrey answered 20 questions correctly. The percentage of his correct answers was more than 60 but less than 70. How many questions were in the test? | 32 | 57 | 2 |
math | 7.2. Do there exist three integers (which may be the same) such that if the product of any two of them minus the third one equals $2018?$ | -1,-1,-2017 | 37 | 9 |
math | Problem 10.1. The entire surface of a cube $13 \times 13 \times 13$ was painted red, and then this cube was sawn into smaller cubes $1 \times 1 \times 1$. All the faces of the smaller cubes $1 \times 1 \times 1$ that were not painted red were painted blue. By what factor is the total area of the blue faces greater than... | 12 | 100 | 2 |
math | 20. 10 people go to the bookstore to buy books. If it is known that: (1) each person bought three books; (2) any two people have at least one book in common. How many people at least bought the most popular book (the one bought by the most people)? Why? | 5 | 65 | 1 |
math | A trapez has parallel sides $A B$ and $C D$, and the intersection point of its diagonals is $M$. The area of triangle $A B M$ is 2, and the area of triangle $C D M$ is 8. What is the area of the trapezoid? | 18 | 66 | 2 |
math | Example 2 The number of prime pairs \((a, b)\) that satisfy the equation
$$
a^{b} b^{a}=(2 a+b+1)(2 b+a+1)
$$
is \qquad (2]
(2011, I Love Mathematics Junior High School Summer Camp Mathematics Competition) | 2 | 69 | 1 |
math | 4. Let $n \in \mathbf{Z}_{+}$. A volleyball team has $n$ male players and $n$ female players. Initially, each player is assigned to one of the positions numbered $1,2, \cdots, 2 n$, with only positions 1 and $n+1$ being outside the court. During the game, position swaps occur, and each swap involves moving the player i... | 2^{n}(n!)^{2} | 163 | 9 |
math | 5. In a two-digit number, the number of tens is two times less than the number of units. If you subtract the sum of its digits from this two-digit number, you get 18. Find this number. | 24 | 46 | 2 |
math | The year 2015, which is about to end, has 7 consecutive days whose date numbers sum up to 100. What are the date numbers of these 7 days? $\qquad$
$\qquad$
$\qquad$
$\qquad$
$\qquad$
$\qquad$ - | 29,30,31,1,2,3,4 | 66 | 16 |
math | $2 \cdot 63$ Let the set $M=\{1,2, \cdots, 1000\}$. For any non-empty subset $X$ of $M$, let $\alpha_{X}$ denote the sum of the largest and smallest numbers in $X$. Find the arithmetic mean of all such $\alpha_{Y}$. | 1001 | 75 | 4 |
math | 2. Find the sum of all four-digit numbers in which only the digits $1,2,3,4,5$ appear, and each digit appears no more than once. (8 points) | 399960 | 41 | 6 |
math | Example 2. Given the circle $x^{2}+y^{2}=4$ and a point $A(2, 0)$ on it, the moving chord $BC$ of this circle always satisfies the condition $\angle BAC = \frac{\pi}{3}$. Try to find the equation of the locus of the centroid of $\triangle ABC$. | \left(x-\frac{2}{3}\right)^{2}+y^{2}=\left(\frac{2}{3}\right)^{2} | 75 | 34 |
math | [Mathematical Logic (Miscellaneous).]
An investigation is underway regarding a stolen mustang. There are three suspects - Bill, Joe, and Sam. In court, Sam stated that Joe stole the mustang. Bill and Joe also gave testimonies, but no one remembered what they said, and all the records were lost. During the trial, it wa... | Bill | 100 | 1 |
math | (Exercise 3 of submission 4 - Combinatorics (2020-2021))
42 students are lined up. Paul gives each student a certain positive number of pebbles. It is assumed that each student has strictly more pebbles than their right neighbor (except for the student all the way to the right of the line). How many pebbles has Paul d... | 903 | 90 | 3 |
math | 5. On a long stick, there are three types of graduation lines. The first type divides the stick into 10 equal parts;
The second type divides the stick into 12 equal parts; The third type divides the stick into 15 equal parts. If each graduation line cuts the stick, how many pieces will the stick be cut into? | 28 | 72 | 2 |
math | VIII.1. The road from Gostivar to Kicevo consists of horizontal sections, ascents, and descents. A cyclist travels this road, with a speed of $10 \mathrm{~km}$ per hour on the horizontal sections, $8 \mathrm{~km}$ per hour on the ascents, and $16 \mathrm{~km}$ per hour on the descents. The cyclist takes 6 hours to trav... | 57.5\mathrm{~} | 141 | 9 |
math | ## Task B-3.5.
The lengths of two sides of a triangle are $7 \mathrm{~cm}$ and $4 \mathrm{~cm}$. The angle opposite the longer side is twice as large as the angle opposite the shorter side. What is the length of the third side of the triangle? | 8.25 | 65 | 4 |
math | The product of the ages of Mr. Multiplier's children is 1408. The age of the youngest child is equal to half the age of the oldest child.
How many children does Mr. Multiplier have and how old are they?
(L. Hozová)
Hint. How can you systematically navigate the divisors of a given number? | 3 | 73 | 1 |
math | For positive numbers $x$, $y$, and $z$ satisfying $x^{2}+y^{2}+z^{2}=1$, find
$$\frac{x}{1-x^{2}}+\frac{y}{1-y^{2}}+\frac{z}{1-z^{2}}$$
the minimum value. | \frac{3 \sqrt{3}}{2} | 69 | 12 |
math |
3. A finite non-empty set $S$ of integers is called 3 -good if the the sum of the elements of $S$ is divisble by 3 . Find the number of 3 -good non-empty subsets of $\{0,1,2, \ldots, 9\}$.
| 351 | 66 | 3 |
math | Problem 9.5. For real non-zero numbers $a, b, c$ it is known that
$$
a^{2}-b c=b^{2}-a c=c^{2}-a b
$$
(a) (1 point) What positive values can the expression
$$
\frac{a}{b+c}+\frac{2 b}{a+c}+\frac{4 c}{a+b} ?
$$
take? List all possible options. If the expression cannot take positive values, write 0 as the answer.
(b... | \frac{7}{2},-7 | 178 | 9 |
math | ## Task 2 - 060912
If we form the cross sum of a natural number and then (if possible) the cross sum of this number again, etc., we finally obtain a single-digit number, which we will call the "final cross sum." By definition, the cross sum of a single-digit number is set to be the number itself.
Calculate how many n... | 111 | 100 | 3 |
math | 1st APMO 1989 Problem 5 f is a strictly increasing real-valued function on the reals. It has inverse f -1 . Find all possible f such that f(x) + f -1 (x) = 2x for all x. Solution | f(x)=x+b | 59 | 5 |
math | Task B-1.2. Determine all integers $a$ for which the equation
$$
3 \cdot|x-2|+a \cdot|3-x|=a+4-x
$$
has integer solutions. | \in{-10,-7,-6,-5,-1,0,2} | 46 | 18 |
math | Consider two drums of sufficiently large capacity, one of them empty and the other full of liquid.
a) Determine if it is possible to place exactly one liter of liquid from the full drum into the empty one, using two buckets, one with a capacity of 5 liters and the other with a capacity of 7 liters.
b) Determine if it... | 1 | 118 | 1 |
math | Problem 4.6.2 A rectangle with a perimeter of 100 cm is divided into 70 identical smaller rectangles by six vertical and nine horizontal cuts. What is the perimeter of each of them, if the total length of all the cuts is 405 cm $?$ | 13 | 60 | 2 |
math | 1. $[\mathbf{3}]$ If $4^{4^{4}}=\sqrt[128]{2^{2^{2^{n}}}}$, find $n$. | 4 | 39 | 1 |
math | # Task 2. 15 points
Solve the equation
$$
\sqrt{\frac{x^{3}+5}{1+\sqrt{5}}}=x
$$
# | \frac{1+\sqrt{1+4\sqrt{5}}}{2},\sqrt{5} | 40 | 23 |
math | Example 1.3 Divide 4 people into two groups, with at least 1 person in each group, and find the number of different grouping methods. | 7 | 32 | 1 |
math | 16.2.25 ** Arrange natural numbers in sequence:
$$
123456789101112131415161718192021 \cdots \cdots
$$
If the $10^{n}$-th digit is in an $m$-digit number, then define $f(n)=m$. For example, the $10^{1}$-th digit 1 is in the two-digit number 10, and the $10^{2}$-th digit 5 is in the two-digit number 55, so $f(1)=2, f(2)... | 1988 | 166 | 4 |
math | 10. Question: Among $1,2,3, \cdots, 1999,2000,2001$, what is the maximum number of numbers that can be chosen such that the sum of any three chosen numbers is divisible by 21? | 95 | 60 | 2 |
math | 5. Let $x, y, z$ be the roots of the equation $t^{3}-5 t-3=0$. Find $x^{3} y^{3}+x^{3} z^{3}+y^{3} z^{3}$. | -98 | 57 | 3 |
math | 14th CanMO 1982 Problem 3 What is the smallest number of points in n-dimensional space R n such that every point of R n is an irrational distance from at least one of the points. Solution | 3 | 46 | 1 |
math | For how many positive integers $k$ do the lines with equations $9 x+4 y=600$ and $k x-4 y=24$ intersect at a point whose coordinates are positive integers? | 7 | 45 | 1 |
math | 2. In an acute triangle, the interior angles $A, B$ satisfy $\tan A-\frac{1}{\sin 2 A}=\tan B$ and $\cos ^{2} \frac{B}{2}=\frac{\sqrt{6}}{3}$, then $\sin 2 A=$ | \frac{2\sqrt{6}-3}{3} | 66 | 13 |
math | Example 1 Suppose the equation $3 x^{2}-6(m-1) x+m^{2}+1$ $=0$ has 2 roots whose magnitudes sum to 2. Find the value of the real number $m$. | m=0 \text{ or } m=\sqrt{2} | 51 | 14 |
math | $7 \cdot 76$ A highway has 10 bus stops $A_{0}, A_{1}, A_{2}, \cdots, A_{9}$. The distance between adjacent stops is $a$ kilometers. A car departs from station $A_{0}$, travels to all stations to deliver goods. The car only stops once at each station and finally returns to the starting station $A_{0}$. Due to the need ... | 50a | 180 | 3 |
math | ## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(-8 ; 0 ; 7)$
$B(-3 ; 2 ; 4)$
$C(-1 ; 4 ; 5)$ | 2x+2y+z+9=0 | 63 | 10 |
math | .
Find all positive integers $n$ for which $n^{n-1}-1$ is divisible by $2^{2015}$, but not by $2^{2016}$. | 2^{2014}v-1 | 43 | 10 |
math | We are approaching a 120-meter tall skyscraper on a horizontal road. After traveling 300 meters, we see the building at an elevation angle that is $45^{\circ}$ greater than at the start of the road. How close have we approached the skyscraper? | 60 | 60 | 2 |
math | 4. Solve the system of equations
$$
\left\{\begin{array}{l}
x^{2}+y^{2}=1 \\
x^{3}+y^{5}=1
\end{array}\right.
$$ | (0;1),(1;0) | 50 | 9 |
math | 7. Given two unequal positive integers $a$ and $b$ that satisfy $a^{2}-b^{2}=2018-2 a$.
Find the value of $a+b$. | 672 | 42 | 3 |
math | 1. Find the value of the expression $(3 x-y)(x+3 y)+(3 x+y)(x-3 y)$ given that $x^{2}+y^{2}+4 x-10 y+29=0$. | -126 | 52 | 4 |
math | 1. Let $O$ and $H$ be the circumcenter and orthocenter of $\triangle ABC$, respectively. Please derive the complex number corresponding to the orthocenter $H$ with $O$ as the origin of the complex plane. | z_{1}+z_{2}+z_{3} | 51 | 14 |
math | ## Task Condition
Find the derivative.
$$
y=\sqrt{1+x^{2}} \operatorname{arctg} x-\ln \left(x+\sqrt{1+x^{2}}\right)
$$ | \frac{x\cdot\arctanx}{\sqrt{1+x^{2}}} | 45 | 19 |
math | Problem 2. In the class, two students sit at each desk. The number of desks where two boys sit is twice the number of desks where two girls sit. And the number of desks where two girls sit is twice the number of desks where a boy and a girl sit. How many boys are there in the class if there are 10 girls?
Answer: there... | 18 | 85 | 2 |
math | Example 1 Solve the inequalities: (1) $\sqrt{x-1}<1$; (2) $\sqrt{2 x-3} \leqslant \sqrt{x-1}$. | [1,2)[\frac{3}{2},2] | 42 | 14 |
math | 18. In $\triangle A B C, A B=13, B C=14$ and $C A=15 . P$ is a point inside $\triangle A B C$ such that $\angle P A B=\angle P B C=\angle P C A$. Find $\tan \angle P A B$.
(2 marks)
在 $\triangle A B C$ 中, $A B=13 、 B C=14 、 C A=15 \circ P$ 是 $\triangle A B C$ 內的一點, 使得 $\angle P A B$ $=\angle P B C=\angle P C A$ 。求 $\ta... | \frac{168}{295} | 150 | 11 |
math | Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$ | \frac{f(0) + 4f(1) + f(2)}{3} | 40 | 22 |
math | If we double the number of sides in a polygon, then the difference between the diagonals and sides increases by 99. How many sides does the polygon have? | 9 | 34 | 1 |
math | A certain right-angled triangle has an area $t=121.5 \mathrm{~cm}^{2}$; one of its angles $\alpha=36^{\circ} 52^{\prime} 10.7^{\prime \prime}$; calculate the surface area and volume of the double cone that is formed by the triangle's rotation around its hypotenuse. $\pi=3.14159$. | 1068.75 | 94 | 7 |
math | 9. (16 points) Given the function
$$
\begin{array}{l}
f(x)=a x^{3}+b x^{2}+c x+d(a \neq 0), \\
\text { when } 0 \leqslant x \leqslant 1 \text {, }|f^{\prime}(x)| \leqslant 1 .
\end{array}
$$
Try to find the maximum value of $a$. | \frac{8}{3} | 104 | 7 |
math | Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
(You may use the fact that $\sin{n^\circ}$ is irrational for positive in... | 178 | 134 | 3 |
math | Problem 2. The distances from three points lying in a horizontal plane to the base of a television tower are 800 m, 700 m, and 500 m, respectively. From each of these three points, the tower is visible (from base to top) at a certain angle, and the sum of these three angles is $90^{\circ}$. A) Find the height of the te... | 374 | 107 | 3 |
math | I am Ivan and I am three times younger than my father. I have brothers Vincent and Jakub, who are 11 and 9 years old. My age is equal to five times the third of the age of the younger of the brothers. A peculiarity of our family is that we were all born on April 12th, so today we are celebrating our birthdays.
How man... | 5 | 114 | 1 |
math | 3. Let $a, b$ be positive real numbers, $\frac{1}{a}+\frac{1}{b} \leqslant 2 \sqrt{2},(a-b)^{2}=4(a b)^{3}$, then $\log _{a} b=$ | -1 | 63 | 2 |
math | An elephant writes a sequence of numbers on a board starting with 1. Each minute, it doubles the sum of all the numbers on the board so far, and without erasing anything, writes the result on the board. It stops after writing a number greater than one billion. How many distinct prime factors does the largest number on ... | 2 | 78 | 1 |
math | One, (20 points) Solve the equation:
$$
\sqrt{x+\sqrt{2 x-1}}+\sqrt{x-\sqrt{2 x-1}}=\sqrt{a}(a)
$$
$0)$.
Discuss the solution of this equation for the value of the positive number $a$.
| (1) \text{ When } a>2, x=\frac{2+a}{4}; (2) \text{ When } a=2, \frac{1}{2} \leqslant x \leqslant 1; (3) \text{ When } a<2, \text{ no solution} | 65 | 72 |
math | 6. (12 points) $A, B, C$ three people are guessing a natural number between $1 \sim 99$.
A: “It is an even number, less than 6.”
B: “It is less than 7, a two-digit number.”
C: “The first half of A's statement is true, the second half is false.”
If among these 3 people, 1 person tells two truths, 1 person tells two lies... | 8 | 121 | 1 |
math | In a communication system consisting of 2001 subscribers, each subscriber is connected to exactly $n$ others. Determine all possible values of $n$.
# | n=2tfor=0,1,\ldots,1000 | 34 | 17 |
math | Among the poor, 120 K was distributed. If the number of the poor was 10 less, then each would receive exactly as much more as they would receive less if the number of the poor was 20 more. How many poor were there? | 40 | 55 | 2 |
math | Mommy preserves plums in jars so that the plums from one jar are enough for either 16 cups, or 4 pies, or half a sheet of fruit slices.
In the pantry, she has 4 such jars and wants to bake one sheet of fruit slices and 6 pies. $\mathrm{On}$ how many cups will the remaining plums be enough?
(M. Petrová) | 8 | 83 | 1 |
math | 3. Some natural numbers form an increasing geometric progression with an integer common ratio. Find these numbers if their sum is 211. | {211},{1;210},{1;14;196} | 28 | 20 |
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 | 3.266. $\sin (x+2 \pi) \cos \left(2 x-\frac{7 \pi}{2}\right)+\sin \left(\frac{3 \pi}{2}-x\right) \sin \left(2 x-\frac{5 \pi}{2}\right)$. | \cos3x | 69 | 4 |
math | 1. By replacing every $*$ in the expression $1 * 2 * 3 * 4 * 5 * \cdots * 2019 * 2020$ with a plus or minus sign ( + or - ), a long arithmetic expression is formed. Place the plus and minus signs in such a way that the result is the smallest possible positive number (greater than 0).
What is this result? | 2 | 89 | 1 |
math | II. (40 points) Find all real numbers $x$ that satisfy the equation
$$
\left[x^{2}-2 x\right]+[x]=[x]^{2} \text {, }
$$
where $[a]$ denotes the greatest integer not exceeding the real number $a$. | x=0 \text{ or } u+\varepsilon, \text{ where } \varepsilon \in \left(1-u-\sqrt{u^2-u+2}, 1-u-\sqrt{u^2-u+1}\right] \text{ for } u \geqslant 2 \text{ and } u < 0 | 64 | 77 |
math | Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have:
\[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\]
Show that this sequence has a finite limit. Determine this limit. | 3 | 128 | 1 |
math | Let's find different digits for the letters $A, B, C, D$ such that the following division holds in the decimal system:
$$
A B C: B B B B=0, B C D B \quad B C D B \quad \ldots
$$
(i.e., the quotient should be a repeating decimal). | 219:1111=0.19711971\ldots | 70 | 22 |
math | 4. A fixed point of a function $f$ is a value of $x$ for which $f(x)=x$. Let $f$ be the quadratic function defined by $f(x)=x^{2}-c x+c$ where $c \in \mathbb{R}$. Find, in interval notation, the set consisting of all values of $c$ for which $f \circ f$ has four distinct fixed points. | (-\infty,-1)\cup(3,+\infty) | 90 | 15 |
math | # Problem 5. (3 points)
In trapezoid $A B C D$, the lateral side $A D$ is equal to the diagonal $B D$. On the smaller arc $C D$ of the circumscribed circle of triangle $A C D$, a point $E$ is chosen such that $A D=D E$. Find the angle $\angle B E C$. | 90 | 81 | 2 |
math | 7. Let the three interior angles $\angle A, \angle B, \angle C$ of $\triangle ABC$ satisfy $\angle A=3 \angle B=9 \angle C$. Then
$$
\begin{array}{l}
\cos A \cdot \cos B+\cos B \cdot \cos C+\cos C \cdot \cos A \\
=\quad .
\end{array}
$$ | -\frac{1}{4} | 84 | 7 |
math | 1A. Solve the equation in $\mathbb{R}$
$$
\left(x^{2}-3\right)^{3}-(4 x+6)^{3}+6^{3}=18(4 x+6)\left(3-x^{2}\right)
$$ | x_{1}=2+\sqrt{7},x_{2}=2-\sqrt{7},x_{3}=-3 | 61 | 26 |
math | 8.1. In the wagon, several kilograms of apple jam were loaded, of which $20 \%$ was good and $80 \%$ was bad. Every day, half of the existing bad jam rotted, and it was thrown away. After several days, it turned out that $20 \%$ of the jam in the wagon was bad and $80 \%$ was good. How many days have passed since the l... | 4 | 92 | 1 |
math | 30. (Training Team) Given a positive integer $n \geqslant 2$ and a positive real number $a$, positive real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy $\prod_{i=1}^{n} x_{i}=1$. Find the smallest real number $M=M(n, a)$, such that
$$\sum_{i=1}^{n} \frac{1}{a+S-x_{i}} \leqslant M$$
always holds, where $S=x_{1}+x_{2}+... | \left\{\begin{array}{ll}
\frac{n}{a-1+n} & \text { if } a \geqslant 1 , \\
\frac{1}{a} & \text { if } 0<a<1
\end{array}\right.} | 135 | 62 |
math | 5. For a set, the difference between the maximum and minimum elements is called the "capacity" of the set. Let $2 \leqslant r \leqslant n$, and let $F(n, r)$ denote the arithmetic mean of the capacities of all $r$-element subsets of the set $M=\{1,2, \cdots, n\}$. Then $F(n, r)=$ | \frac{(r-1)(n+1)}{r+1} | 90 | 16 |
math | Find the greatest positive integer $m$ with the following property:
For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference. | 3 | 95 | 3 |
math | Let $A$ be a subset of $\{1, 2, \dots , 1000000\}$ such that for any $x, y \in A$ with $x\neq y$, we have $xy\notin A$. Determine the maximum possible size of $A$. | 999001 | 66 | 6 |
math | ## Task 21/73
Determine a polynomial $P(x)$ of the smallest degree with integer coefficients that is always divisible by 8 for odd $x$. The polynomial $P(x)$ should not be divisible by 2 for every integer $x$. | P(x)=x^2-^2withodd | 55 | 11 |
math | [ Product of chord segments or secant segments ]
In triangle $A B C$, the bisector $A P$ is drawn. It is known that $B P=16, P C=20$ and that the center of the circumcircle of triangle $A B P$ lies on the segment $A C$. Find the side $A B$.
# | \frac{144\sqrt{5}}{5} | 76 | 14 |
math | Let $a \leq b \leq c$ be the sides of a triangle. What values can
$$
\frac{(a+b+c)^{2}}{b c}
$$
take?
OKTV, 1995-96 | 0.]4;9] | 54 | 6 |
math | Positive integers $x, y, z$ satisfy $(x + yi)^2 - 46i = z$. What is $x + y + z$? | 552 | 34 | 3 |
math | Problem 2. A train consists of 20 cars, numbered from 1 to 20, starting from the beginning of the train. Some of the cars are postal cars. It is known that
- the total number of postal cars is even;
- the number of the nearest postal car to the beginning of the train is equal to the total number of postal cars
- the n... | 4,5,15,16 | 119 | 9 |
math | A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all possible values of $n$.
A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all possible values of $n$. | 324 | 75 | 3 |
math | Let $ABC$ be a Poncelet triangle, $A_1$ is the reflection of $A$ about the incenter $I$, $A_2$ is isogonally conjugated to $A_1$ with respect to $ABC$. Find the locus of points $A_2$. | A_2 | 64 | 4 |
math | 19.1 .5 * Find a four-digit square number, where the first two digits are the same, and the last two digits are also the same. | 7744 | 33 | 4 |
math | Example 2 Given $f(x)=\left\{\begin{array}{cc}x^{2}-1, & 0 \leqslant x \leqslant 1, \\ x^{2}, & -1 \leqslant x<0 .\end{array}\right.$ Find $f^{-1}(x)$. | f^{-1}(x)={\begin{pmatrix}\sqrt{x+1},(-1\leqslantx\leqslant0),\\-\sqrt{x},(0<x\leqslant1)0\end{pmatrix}.} | 74 | 56 |
math | 2. In 1986, Janez will be as many years old as the sum of the digits of the year he was born. How old will Janez be in 1986? | 21 | 43 | 2 |
math | 1.63. The entire arc of a circle with radius $R$ is divided into four large and four small segments, alternating with each other. The large segment is twice as long as the small one. Determine the area of the octagon whose vertices are the points of division of the circle's arc. | R^{2}(\sqrt{3}+1) | 63 | 12 |
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