task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math |
NT12 Solve the equation $\frac{p}{q}-\frac{4}{r+1}=1$ in prime numbers.
| (p,q,r)=(3,2,7), | 29 | 10 |
math | 9.3. Find the smallest natural number $n$ such that in any set of $n$ distinct natural numbers, not exceeding 1000, it is always possible to select two numbers, the larger of which does not divide the smaller one. | 11 | 53 | 2 |
math | Based on a spherical segment with a base radius $r=5 \mathrm{~cm}$ and height $m=2 \mathrm{~cm}$; inscribed in this segment and around this segment, we inscribe straight cones. How do the volumes of these three bodies relate to each other? | 1659:1050:2500 | 61 | 14 |
math | ## Task 1 - 180811
The FDJ members Arnim, Bertram, Christian, Dieter, Ernst, and Fritz participated in a 400-meter race. No two of them finished at the same time.
Before the race, the following three predictions were made about the results (each participant is represented by the first letter of their first name):
| Place | 1. | 2. | 3. | 4. | 5. | 6. |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| 1st Prediction | A | B | C | D | E | F |
| 2nd Prediction | A | C | B | F | E | D |
| 3rd Prediction | C | E | F | A | D | B |
After the race, it turned out that in the first prediction, the places of exactly three runners were correctly given. No two of these three places were adjacent to each other. In the second prediction, no runner's place was correctly given. In the third prediction, the place of one runner was correctly given.
Give all possible sequences of the places achieved by the runners under these conditions! | C,B,E,D,A,F | 265 | 6 |
math | 1. (17 points) Solve the equation $12 x=\sqrt{36+x^{2}}\left(6+x-\sqrt{36+x^{2}}\right)$. | -6,0 | 41 | 4 |
math | 4. Given $\sin \alpha+\cos \alpha=-\frac{1}{5}(0<\alpha<\pi)$, then $\tan \alpha=$ $\qquad$ | -\frac{3}{4} | 38 | 7 |
math | $\begin{array}{l}\text { 1・147 Simplify } \\ \quad(2+1)\left(2^{2}+1\right)\left(2^{4}+1\right)\left(2^{8}+1\right) \cdots \cdots\left(2^{256}+1\right) .\end{array}$ | 2^{512}-1 | 85 | 7 |
math | 13.366. On a river with a current speed of 5 km/h, there are piers $A, B$, and $C$ in the direction of the current, with $B$ located halfway between $A$ and $C$. From pier $B$, a raft and a boat depart simultaneously in the direction of the current towards pier $C$, and the boat heads towards pier $A$, with the boat's speed in still water being $V$ km/h. Upon reaching pier $A$, the boat turns around and heads towards pier $C$. Find all values of $v$ for which the boat arrives at $C$ later than the raft. | 5<V<15 | 139 | 5 |
math | BELG * Find the value of the algebraic expression
$$
\arcsin \frac{1}{\sqrt{10}}+\arccos \frac{7}{\sqrt{50}}+\arctan \frac{7}{31}+\operatorname{arccot} 10
$$ | \frac{\pi}{4} | 69 | 7 |
math | Example 6. In the sequence $\left\{a_{n}\right\}$, for any natural number $n(n \geqslant 2)$, we have $a_{n}=3 a_{n-1}-2 a_{n-2}$, and $a_{0}=2, a_{1}=3$, find the general term formula of this sequence. | a_n = 2^n + 1 | 80 | 9 |
math | Volienkov S.G.
Find all infinite bounded sequences of natural numbers $a_{1}, a_{2}, a_{3}, \ldots$, for all members of which, starting from the third, the following holds:
$$
a_{n}=\frac{a_{n-1}+a_{n-2}}{\left(a_{n-1}, a_{n-2}\right)}
$$ | a_{1}=a_{2}=\ldots=2 | 85 | 13 |
math | For any positive integer $k, S(k)$ is the sum of the digits of $k$ in its decimal representation. Find all integer-coefficient polynomials $P(x)$ such that for any positive integer $n \geqslant 2$016, $P(n)$ is a positive integer, and
$$
S(P(n))=P(S(n)) \text {. }
$$ | P(x)=(where1\leqslant\leqslant9)orP(x)=x | 83 | 22 |
math | 3. Given that $a$ and $b$ are real numbers. If the quadratic function
$$
f(x)=x^{2}+a x+b
$$
satisfies $f(f(0))=f(f(1))=0$, and $f(0) \neq f(1)$, then the value of $f(2)$ is $\qquad$. | 3 | 82 | 1 |
math | 1. Triangle $G R T$ has $G R=5, R T=12$, and $G T=13$. The perpendicular bisector of $G T$ intersects the extension of $G R$ at $O$. Find $T O$. | \frac{169}{10} | 55 | 10 |
math | 13. 3 (CMO 2000) Let $a, b, c$ be the three sides of $\triangle ABC$, $a \leqslant b \leqslant c$, and $R$ and $r$ be the circumradius and inradius of $\triangle ABC$, respectively. Let $f=a+b-2R-2r$, try to determine the sign of $f$ using the size of angle $C$.
| f>0\LeftrightarrowC<\pi/2 | 98 | 13 |
math | 6. [5] We say " $s$ grows to $r$ " if there exists some integer $n>0$ such that $s^{n}=r$. Call a real number $r$ "sparse" if there are only finitely many real numbers $s$ that grow to $r$. Find all real numbers that are sparse. | -1,0,1 | 73 | 6 |
math | In a rectangular $57\times 57$ grid of cells, $k$ of the cells are colored black. What is the smallest positive integer $k$ such that there must exist a rectangle, with sides parallel to the edges of the grid, that has its four vertices at the center of distinct black cells?
[i]Proposed by James Lin | 457 | 73 | 3 |
math | 6. (Mathematics of Intermediate Level, Issue 1, 2005 Olympiad Training Problem) If the inequality $x|x-a|+b<0$ (where $b$ is a constant) holds for any value of $x$ in $[0,1]$, find the range of real numbers $a$.
---
The inequality $x|x-a|+b<0$ (where $b$ is a constant) must hold for all $x$ in the interval $[0,1]$. Determine the range of real numbers $a$. | \in(1+b,2\sqrt{-b})when-1\leqslant2\sqrt{2}-3;\\in(1+b,1-b)when-1 | 119 | 40 |
math | 379. Form the equation of the tangent to the parabola $y=x^{2}$ $-4 x$ at the point with abscissa $x_{0}=1$. | -2x-1 | 40 | 5 |
math | 3 [ Quadrilateral: calculations, metric relations.]
Find the angles of the convex quadrilateral $A B C D$, in which $\angle B A C=30^{\circ}, \angle A C D=40^{\circ}, \angle A D B=50^{\circ}, \angle C B D$ $=60^{\circ}$ and $\angle A B C+\angle A D C=180^{\circ}$. | \angleABC=100,\angleADC=80,\angleBAD=\angleBCD=90 | 96 | 23 |
math | 一,(20 points) Given that $a, b, c, d$ are all non-zero real numbers. Substituting $x=a$ and $b$ into $y=x^{2}+c x$, the value of $y$ is 1 in both cases; substituting $x=c$ and $d$ into $y=x^{2}+a x$, the value of $y$ is 3 in both cases. Try to find the value of $6 a+2 b+3 c+2 d$. | \pm \frac{1}{2} | 112 | 9 |
math | Example 7. Calculate $(-1+i \sqrt{3})^{60}$.
---
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 2^{60} | 45 | 5 |
math | 2. (HUN) Find all real numbers $x$ for which $$ \sqrt{3-x}-\sqrt{x+1}>\frac{1}{2} . $$ | -1 \leq x < 1-\sqrt{31} / 8 | 38 | 18 |
math | [ Quadratic equations and systems of equations ] [Completing the square. Sums of squares] Solve the system: $\left\{\begin{array}{c}x-y \geq z \\ x^{2}+4 y^{2}+5=4 z\end{array}\right.$. | (2;-0.5;2.5) | 63 | 11 |
math | Example 12. Let $M, x, y$ be positive integers, and $\sqrt{M-\sqrt{28}}=\sqrt{x}-\sqrt{y}$. Then the value of $x+y+M$ is ( ).
(1994, Hope Cup Mathematics Competition) | 16 | 62 | 2 |
math | 5. Given an isosceles triangle with a base length of 3, the range of the length of one of its base angle bisectors is $\qquad$ | (2,3\sqrt{2}) | 35 | 9 |
math | 2. Given that the first $n$ terms sum of the arithmetic sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ are $S_{n}$ and $T_{n}$ respectively, and $\frac{S_{n}}{T_{n}}=\frac{3 n+2}{2 n+1}$. Then $\frac{a_{7}}{b_{5}}=$ $\qquad$ . | \frac{41}{19} | 98 | 9 |
math | Example 4 If
$$
\cos ^{5} \theta-\sin ^{5} \theta<7\left(\sin ^{3} \theta-\cos ^{3} \theta\right)(\theta \in[0,2 \pi)),
$$
then the range of values for $\theta$ is $\qquad$ [3]
(2011, National High School Mathematics Joint Competition) | \left(\frac{\pi}{4}, \frac{5 \pi}{4}\right) | 89 | 20 |
math | 8.4. Masha and Katya wove wreaths from wildflowers: dandelions, cornflowers, and daisies. The total number of flowers in the wreaths of both girls turned out to be 70, with dandelions making up $5 / 9$ of Masha's wreath, and daisies making up 7/17 of Katya's wreath. How many cornflowers are in each girl's wreath, if the number of dandelions and daisies is the same for both? | Masha\has\2\cornflowers,\Katya\has\0\cornflowers | 115 | 19 |
math | 1. Petya's watch gains 5 minutes per hour, while Masha's watch loses 8 minutes per hour. At 12:00, they set their watches to the school clock (which is accurate) and agreed to go to the rink together at half past six. How long will Petya wait for Masha if each arrives at the rink exactly at $18-30$ by their own watches | 1.5 | 91 | 3 |
math | 5. Let the sequence of positive numbers $\left\{a_{n}\right\}$ satisfy
$$
\begin{array}{l}
a_{1}=\sqrt{2}-1, \\
a_{n+1}=\frac{2 n+1}{S_{n}+S_{n+1}+2}(n=1,2, \cdots),
\end{array}
$$
where $S_{n}$ is the sum of the first $n$ terms of $a_{n}$. Then the general term formula of the sequence is $\qquad$ . | a_{n}=\sqrt{n^{2}+1}-\sqrt{(n-1)^{2}+1} | 124 | 26 |
math | 3.270. $\frac{\sin 2 \alpha+\cos 2 \alpha-\cos 6 \alpha-\sin 6 \alpha}{\sin 4 \alpha+2 \sin ^{2} 2 \alpha-1}$. | 2\sin2\alpha | 55 | 6 |
math | 7. (10 points) For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "Six-Union Number". The smallest "Six-Union Number" greater than 2000 is $\qquad$ . | 2016 | 70 | 4 |
math | Example 1. Two students took four tests, their average scores were different, but both were below 90 and were integers. They then took a fifth test, after which their average scores both increased to 90. What were the scores of the two students on the fifth test? | 98 \text{ and } 94 | 59 | 10 |
math | 7. If $n$ is a positive integer such that $n^{6}+206$ is divisible by $n^{2}+2$, find the sum of all possible values of $n$.
(1 mark)
若 $n$ 是正整數, 且 $n^{6}+206$ 可被 $n^{2}+2$ 整除, 求 $n$ 所有可能值之和。
(1 分) | 32 | 102 | 2 |
math | Starting from a number written in the decimal system, let's calculate the sum of the different numbers that can be obtained by the permutations of its digits. For example, starting from 110, the sum $110+101+11=222$ is obtained. What is the smallest number from which, starting, the obtained sum is 4933284? | 47899 | 84 | 5 |
math | 17. (10 points) In $\triangle A B C$, let the sides opposite to $\angle A$, $\angle B$, and $\angle C$ be $a$, $b$, and $c$, respectively, satisfying
$a \cos C = (2 b - c) \cos A$.
(1) Find the size of $\angle A$;
(2) If $a = \sqrt{3}$, and $D$ is the midpoint of side $B C$, find the range of $A D$. | \left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right] | 110 | 21 |
math | 4. Solve the equation: $3 x^{2}+14 y^{2}-12 x y+6 x-20 y+11=0$. | 3,2 | 36 | 3 |
math | 45th Putnam 1984 Problem B1 Define f(n) = 1! + 2! + ... + n! . Find a recurrence relation f(n + 2) = a(n) f(n + 1) + b(n) f(n), where a(x) and b(x) are polynomials. | f(n+2)=(n+3)f(n+1)-(n+2)f(n) | 70 | 19 |
math | 8.5. The faces of a cube are labeled with six different numbers from 6 to 11. The cube was rolled twice. The first time, the sum of the numbers on the four side faces was 36, the second time - 33. What number is written on the face opposite the one where the digit $10$ is written? | 8 | 76 | 1 |
math | ## Task 6
Draw a line segment $\overline{A B}$ that is $7 \mathrm{~cm}$ long, and a line segment $\overline{C D}$ that is $3 \mathrm{~cm}$ shorter. | 4\mathrm{~} | 51 | 6 |
math | ## Problem Statement
Find the point $M^{\prime}$ symmetric to the point $M$ with respect to the plane.
$M(1 ; 2 ; 3)$
$2 x+10 y+10 z-1=0$ | M^{\}(0;-3;-2) | 54 | 10 |
math | 8. Let one edge of a tetrahedron be 6, and the remaining edges
all be 5. Then the radius of the circumscribed sphere of this tetrahedron is $\qquad$ . | \frac{20\sqrt{39}}{39} | 46 | 15 |
math | 49(971). A two-digit number, when added to the number written with the same digits but in reverse order, gives the square of a natural number. Find all such two-digit numbers. | 29,38,47,56,65,74,83,92 | 42 | 23 |
math | 2. What is the greatest whole number of liters of water that can be heated to boiling temperature using the amount of heat obtained from the combustion of solid fuel, if in the first 5 minutes of combustion, 480 kJ is obtained from the fuel, and for each subsequent five-minute period, 25% less than the previous one. The initial temperature of the water is $20^{\circ} \mathrm{C}$, the boiling temperature is $100^{\circ} \mathrm{C}$, and the specific heat capacity of water is 4.2 kJ. | 5 | 125 | 1 |
math | 4. Let $a_{1}, a_{2}, \cdots, a_{100}$ be
$$
\{1901,1902, \cdots, 2000\}
$$
an arbitrary permutation, and the partial sum sequence
$$
\begin{array}{l}
S_{1}=a_{1}, S_{2}=a_{1}+a_{2}, \\
S_{3}=a_{1}+a_{2}+a_{3}, \\
\cdots \cdots \\
S_{100}=a_{1}+a_{2}+\cdots+a_{100} .
\end{array}
$$
If every term in the sequence $\left\{S_{j}\right\}(1 \leqslant j \leqslant 100)$ cannot be divisible by 3, then the number of such permutations is $\qquad$ kinds. | \frac{99!\times 33!\times 34!}{66!} | 204 | 21 |
math | Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations
\[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \] | (4, 4, 4) | 62 | 10 |
math | 4. (8 points) There is a sequence, the first term is 12, the second term is 19, starting from the third term, if the sum of its previous two terms is odd, then this term equals the sum of the previous two terms, if the sum of the previous two terms is even, this term equals the difference of the previous two terms (the larger number minus the smaller number). Then, in this sequence, the $\qquad$ term first exceeds 2016. | 252 | 106 | 3 |
math | Consider the set of all rectangles with a given area $S$.
Find the largest value o $ M = \frac{16-p}{p^2+2p}$ where $p$ is the perimeter of the rectangle. | \frac{4 - \sqrt{S}}{4S + 2\sqrt{S}} | 48 | 22 |
math | 11. (7 points) A refrigerator factory needs to produce 1590 refrigerators. It has already been producing for 12 days, with 80 units produced each day. The remaining units will be produced at a rate of 90 units per day. How many more days are needed to complete the task? | 7 | 69 | 1 |
math | ## Task B-4.7.
Solve the inequality
$$
\begin{aligned}
& \log _{\sqrt{5}}\left(3^{x^{2}-x-1}+2\right)+\log _{5}\left(3^{x^{2}-x-1}+2\right)+\log _{25}\left(3^{x^{2}-x-1}+2\right)+ \\
& \log _{625}\left(3^{x^{2}-x-1}+2\right)+\ldots+\log _{5^{2}}\left(3^{x^{2}-x-1}+2\right)+\ldots \leqslant 4
\end{aligned}
$$ | x\in[-1,2] | 167 | 8 |
math | Solve the following system of equations:
$$
\begin{gathered}
\log _{x}(x+y)+\log _{y}(x+y)=4 \\
(x-1)(y-1)=1 .
\end{gathered}
$$ | 2 | 53 | 1 |
math | Task 14. (8 points)
To attend the section, Mikhail needs to purchase a tennis racket and a set of tennis balls. Official store websites have product catalogs. Mikhail studied the offers and compiled a list of stores where the items of interest are available:
| Item | Store | |
| :--- | :---: | :---: |
| | Higher League | Sport-Guru |
| Tennis racket | 5600 rub. | 5700 rub. |
| Set of tennis balls (3 pcs.) | 254 rub. | 200 rub. |
Mikhail plans to use delivery for the items. The delivery cost from the store "Higher League" is 550 rub., and from the store "Sport-Guru" it is 400 rub. If the purchase amount exceeds 5000 rub., delivery from the store "Higher League" is free.
Mikhail has a discount card from the store "Sport-Guru," which provides a 5% discount on the purchase amount.
Determine the total expenses for purchasing the sports equipment, including delivery costs, assuming that Mikhail chooses the cheapest option.
In your answer, provide only the number without units of measurement! | 5854 | 255 | 4 |
math | With three different digits, all greater than $0$, six different three-digit numbers are formed. If we add these six numbers together the result is $4.218$. The sum of the three largest numbers minus the sum of the three smallest numbers equals $792$. Find the three digits. | 8, 7, 4 | 62 | 7 |
math | Maria has just won a huge chocolate bar as an Easter gift. She decides to break it into pieces to eat it gradually. On the first day, she divides it into 10 pieces and eats only one of them. On the second day, she divides one of the remaining pieces from the previous day into 10 more pieces and eats only one of them. On the third day, she does the same, that is, she divides one of the remaining pieces from the previous day into 10 more and eats only one of them. She continues repeating this process until the next Easter.
a) How many pieces will she have at the end of the third day?
b) Is it possible for her to have exactly 2014 pieces on some day? | 25 | 156 | 2 |
math | For real numbers $B,M,$ and $T,$ we have $B^2+M^2+T^2 =2022$ and $B+M+T =72.$ Compute the sum of the minimum and maximum possible values of $T.$ | 48 | 56 | 2 |
math | 8. Among the following four numbers $1307674368000$, $1307674368500$, $1307674368200$, $1307674368010$, only one is exactly the product of the integers from 1 to 15. This number is ( ). | 1307674368000 | 89 | 13 |
math | Five, divide a circle into $n(n \geqslant 2)$ sectors, sequentially denoted as $S_{1}, S_{2}, \cdots, S_{n}$. Each sector can be painted with any of the three different colors: red, white, and blue, with the requirement that adjacent sectors must have different colors. How many ways are there to color the sectors? | 2 \left[2^{n-1} - (-1)^{n-1}\right] | 82 | 21 |
math | ## Task 1 - 340821
A four-digit natural number is called "symmetric" if and only if its thousands digit is equal to its units digit and its hundreds digit is equal to its tens digit. Tanja claims that every four-digit symmetric number is divisible by 11.
a) Check this divisibility with three self-chosen examples!
b) Prove Tanja's claim in general!
c) How many four-digit symmetric numbers are there in total?
d) How many even four-digit symmetric numbers are there in total? | 40 | 115 | 2 |
math | 1. (8 points) The calculation result of the expression $2016 \times\left(\frac{8}{7 \times 9}-\frac{1}{8}\right)$ is | 4 | 42 | 1 |
math | 10. (6 points) Xiaoming went to the store to buy a total of 66 red and black pens. The red pens are priced at 5 yuan each, and the black pens are priced at 9 yuan each. Due to the large quantity purchased, the store offered a discount, with red pens paid at 85% of the listed price and black pens paid at 80% of the listed price. If the amount he paid was 18% less than the listed price, then how many red pens did he buy? $\qquad$ . | 36 | 119 | 2 |
math | 6. Let $f(x)$ be an odd function defined on $\mathbf{R}$, and when $x \geqslant 0$, $f(x)=x^{2}$. If for any $x \in[a, a+2]$, the inequality $f(x+a) \geqslant 2 f(x)$ always holds, then the range of the real number $a$ is | [\sqrt{2},+\infty) | 86 | 9 |
math | 1. Using the digits 0 to 9, we form two-digit numbers $A B, C D, E F, G H, I J$, using each digit exactly once. Determine how many different values the sum $A B+C D+E F+G H+I J$ can take and what these values are. (Expressions like 07 are not considered two-digit numbers.)
(Jaroslav Zhouf) | 180to360 | 87 | 7 |
math | Problem 8.7. Along an alley, maples and larches were planted in one row, a total of 75 trees. It is known that there are no two maples between which there are exactly 5 trees. What is the maximum number of maples that could have been planted along the alley? | 39 | 66 | 2 |
math | Find all pairs of prime numbers $p\,,q$ for which:
\[p^2 \mid q^3 + 1 \,\,\, \text{and} \,\,\, q^2 \mid p^6-1\]
[i]Proposed by P. Boyvalenkov[/i] | \{(3, 2), (2, 3)\} | 66 | 15 |
math | We can decompose the number 3 into the sum of positive integers in four different ways: $3 ; 2+1 ; 1+2$; and $1+1+1$. In the addition, the order of the terms matters, so $1+2$ and $2+1$ are considered different decompositions. How many different ways can we decompose the natural number $n$? | 2^{n-1} | 85 | 6 |
math | ## Task 1 - 260621
In the following five-digit number, two digits have become illegible and have been replaced by asterisks.
$$
27 * * 7
$$
Instead of the asterisks, two digits are to be inserted so that the number is divisible by 9. Give all five-digit numbers that can arise from such insertion! Verify that all the required numbers have been provided by you! | 27027,27117,27207,27297,27387,27477,27567,27657,27747,27837,27927 | 91 | 65 |
math | 4. An $8 \times 6$ grid is placed in the first quadrant with its edges along the axes, as shown. A total of 32 of the squares in the grid are shaded. A line is drawn through $(0,0)$ and $(8, c)$ cutting the shaded region into two equal areas. What is the value of $c$ ? | 4 | 76 | 1 |
math | 3. In the number $2016 * * * * 02 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,2,4,5,7,9$ (digits can be repeated) so that the resulting 11-digit number is divisible by 15. In how many ways can this be done? | 864 | 79 | 3 |
math | Example 3 Given the real-coefficient equation $x^{3}+2(k-1) x^{2}+9 x+5(k-1)=0$ has a complex root with a modulus of $\sqrt{5}$, find the value of $k$, and solve this equation.
Translating the above text into English, while retaining the original text's line breaks and format, the result is as follows:
Example 3 Given the real-coefficient equation $x^{3}+2(k-1) x^{2}+9 x+5(k-1)=0$ has a complex root with a modulus of $\sqrt{5}$, find the value of $k$, and solve this equation. | -1or3 | 148 | 4 |
math | 3. There are 21 different applications installed on the phone. In how many ways can six applications be selected for deletion so that among them are three applications from the following six $T V F T' V' F'$, but none of the pairs $T T', V V', F F'$ are included? | 3640 | 65 | 4 |
math | 8,9 In an isosceles triangle $ABC (AB = BC)$, the bisectors $BD$ and $AF$ intersect at point $O$. The ratio of the area of triangle $DOA$ to the area of triangle $BOF$ is $\frac{3}{8}$. Find the ratio $\frac{AC}{AB}$. | \frac{1}{2} | 74 | 7 |
math | Each entry in the list below is a positive integer:
$$
a, 8, b, c, d, e, f, g, 2
$$
If the sum of any four consecutive terms in the list is 17 , what is the value of $c+f$ ? | 7 | 61 | 1 |
math | 7.3. Kolya wants to multiply all the natural divisors of the number 1024 (including the number itself) on his calculator. Will he be able to get the result on a screen with 16 decimal places? | 2^{55}>10^{16} | 51 | 11 |
math | 1.4. In a right-angled triangle, the bisector of an acute angle divides the opposite leg into segments of 4 and 5 cm. Determine the area of the triangle. | 54\mathrm{~}^{2} | 39 | 10 |
math | Problem 6. Find the smallest positive root of the equation
$$
2 \sin (6 x)+9 \cos (6 x)=6 \sin (2 x)+7 \cos (2 x)
$$ | \frac{\alpha+\beta}{8}\approx0.1159 | 44 | 16 |
math | PROBLEM 4. Determine the values of $n \in \mathbb{N}$, for which
$$
I_{n}=\int_{0}^{1} \frac{(1+x)^{n}+(1-x)^{n}}{1+x^{2}} \mathrm{~d} x \in \mathbb{Q}
$$[^0]
## NATIONAL MATHEMATICS OLYMPIAD Local stage - 14.02. 2015 GRADING GUIDE - 12th Grade | n\in{4k+2:k\in\mathbb{N}} | 112 | 17 |
math | Solve the following equation:
$$
\frac{x-49}{50}+\frac{x-50}{49}=\frac{49}{x-50}+\frac{50}{x-49}
$$ | 0,99,\frac{4901}{99} | 51 | 15 |
math | 13. Two cleaning vehicles, A and B, are tasked with cleaning the road between East City and West City. Vehicle A alone would take 10 hours to clean the road, while Vehicle B alone would take 15 hours. The two vehicles start from East City and West City respectively, heading towards each other. When they meet, Vehicle A has cleaned 12 kilometers more than Vehicle B. How far apart are East City and West City? | 60 | 93 | 2 |
math | A prime number$ q $is called[b][i] 'Kowai' [/i][/b]number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one [b][i]'Kowai'[/i][/b] number can be found. Find the summation of all [b][i]'Kowai'[/i][/b] numbers.
| 59 | 118 | 2 |
math | $7 \cdot 117$ Let $S=\{1,2,3,4\} ; n$ terms of the sequence $a_{1}, a_{2}, \cdots a_{n}$ have the following property: for any non-empty subset $B$ of $S$ (the number of elements in set $B$ is denoted as $|B|$ ), there are adjacent $|B|$ terms in the sequence that exactly form the set $B$. Find the minimum value of the number of terms $n$. | 8 | 113 | 1 |
math | Kovács had a dinner party for four couples. After the introductions, Mr. Kovács noted that apart from himself, each of the other nine people present had shaken hands with a different number of people.
How many people did Mrs. Kovács introduce herself to? | 4 | 58 | 1 |
math | 4.1. Find the general solution of the ODE $y^{\prime}=\frac{y^{2}}{x}$. | -\frac{1}{\ln(Cx)} | 29 | 10 |
math | 1. It is known that $\alpha^{2005}+\beta^{2005}$ can be expressed as a bivariate polynomial in terms of $\alpha+\beta$ and $\alpha \beta$. Find the sum of the coefficients of this polynomial.
（Zhu Huawei provided the problem） | 1 | 62 | 1 |
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 | ## Task 16/88
Let $P(x)$ be a polynomial of degree n, which leaves a remainder of 1 when divided by $(x-1)$, a remainder of 2 when divided by $(x-2)$, and a remainder of 3 when divided by $(x-3)$.
What remainder does it leave when divided by $(x-1)(x-2)(x-3)$? | R(x)=x | 88 | 4 |
math | ## Task A-2.5.
Determine all pairs $\{a, b\}$ of distinct real numbers such that the equations
$$
x^{2} + a x + b = 0 \quad \text{and} \quad x^{2} + b x + a = 0
$$
have at least one common solution in the set of real numbers. | -1 | 79 | 2 |
math | # Task 4. (12 points)
Solve the equation $x^{2}+y^{2}+1=x y+x+y$.
# | 1 | 33 | 1 |
math | Kuba wrote down a four-digit number, two of whose digits were even and two were odd. If he crossed out both even digits in this number, he would get a number four times smaller than if he crossed out both odd digits in the same number.
What is the largest number with these properties that Kuba could have written down?
(M. Petrová)
Hint. What is the largest number Kuba could get after crossing out the even digits? | 6817 | 92 | 4 |
math | 8. There is a square on the plane, whose sides are parallel to the horizontal and vertical directions. Draw several line segments parallel to the sides of the square such that: no two line segments belong to the same straight line, and any two intersecting line segments intersect at an endpoint of one of the line segments. It is known that the drawn line segments divide the square into several rectangles, and any vertical line intersecting the square but not containing any of the drawn line segments intersects exactly $k$ rectangles, and any horizontal line intersecting the square but not containing any of the drawn line segments intersects exactly $l$ rectangles. Find all possible values of the number of rectangles in the square. | k l | 140 | 2 |
math | 7.130. $\left\{\begin{array}{l}\lg \left(x^{2}+y^{2}\right)=2-\lg 5, \\ \lg (x+y)+\lg (x-y)=\lg 1.2+1 .\end{array}\right.$ | (4;2)(4;-2) | 65 | 9 |
math | 8. Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci $F_{1}, F_{2}$, a line $l$ passing through the right focus intersects the ellipse at points $P, Q$. Then the maximum value of the area of the incircle of $\triangle F_{1} P Q$ is $\qquad$ | \frac{9\pi}{16} | 88 | 10 |
math | Example. Solve the Dirichlet boundary value problem for Poisson's equation in a sphere
$$
\begin{gathered}
\Delta u=14 x y, \quad 0 \leqslant r<2 \\
\left.u\right|_{r=2}=14
\end{gathered}
$$ | u(x,y,z)=(x^{2}+y^{2}+z^{2}-4)xy+14 | 70 | 25 |
math | 9. Given the function $f(x)$ satisfies
$$
\begin{array}{l}
f(x+1)+f(1-x)=0, \\
f(x+2)-f(2-x)=0,
\end{array}
$$
and $f\left(\frac{2}{3}\right)=1$. Then $f\left(\frac{1000}{3}\right)=$ | -1 | 86 | 2 |
math | Solve the following equation in the set of real numbers:
$$
\frac{1}{x^{2}}+\frac{1}{x^{6}}+\frac{1}{x^{8}}+\frac{1}{x^{12}}-\frac{4}{x^{7}}=0
$$ | 1 | 64 | 1 |
math | 14. (6 points) There are two barrels of wine, A and B. If 8 kilograms of wine are added to barrel A, the two barrels will weigh the same. If 3 kilograms of wine are transferred from barrel A to barrel B, the wine in barrel B will be 3 times that in barrel A. Barrel A originally had $\qquad$ kilograms of wine, and barrel B had $\qquad$ kilograms. | 10,18 | 90 | 5 |
math | 15. Find the value of $100\left(\sin 253^{\circ} \sin 313^{\circ}+\sin 163^{\circ} \sin 223^{\circ}\right)$ | 50 | 55 | 2 |
math | Let $p$ and $q$ be different prime numbers. In how many ways can the fraction $1 / p q$ be decomposed into the sum of the reciprocals of 2 different natural numbers? | 4 | 44 | 1 |
math | 4. Let $k$ be one of the quotients of the roots of the quadratic equation $\left.p x^{2}-q x\right\lrcorner q=0$, where $p, q>0$. Express the roots of the equation $\sqrt{p} x^{2}-\sqrt{q} x+\sqrt{p}=0$ in terms of $k$ (not in terms of $p$ and $q$). | a_{1}=\sqrt{k},b_{1}=\frac{1}{\sqrt{k}} | 94 | 21 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty}\left(\frac{n+5}{n-7}\right)^{\frac{n}{6}+1}$ | e^2 | 45 | 3 |
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