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 | 3B. Solve the inequality
$$
\left(\frac{1}{2}\right)^{-|x-3|-\frac{1}{|x-3|}}=4
$$ | 4or2 | 41 | 3 |
math | . Find all triplets $(x, y, z)$, $x > y > z$ of positive integers such that $\frac{1}{x}+\frac{2}{y}+\frac{3}{z}= 1$ | (x, y, z) = \{(36, 9, 4), (20, 10, 4), (15, 6, 5)\} | 49 | 42 |
math | Example 2 Find the maximum constant $k$, such that for any $x, y, z \in R^{+}$, we have
$$
\frac{x}{\sqrt{y+z}}+\frac{y}{\sqrt{z+x}}+\frac{z}{\sqrt{x+y}} \geqslant k \sqrt{x+y+z} .
$$ | \sqrt{\frac{3}{2}} | 77 | 9 |
math | 17. (10 points) In $\triangle A B C$, the sides opposite to $\angle A, \angle B, \angle C$ are $a, b, c$ respectively,
$$
\tan C=\frac{\sin A+\sin B}{\cos A+\cos B}, \sin (B-A)=\cos C \text {. }
$$
(1) Find $\angle A, \angle C$;
(2) If $S_{\triangle A B C}=3+\sqrt{3}$, find $a, c$. | a=2\sqrt{2}, c=2\sqrt{3} | 116 | 16 |
math | 1. A fruit has a water content by weight of $m \%$. When left to dry in the sun, it loses $(m-5) \%$ of this water, leaving it with a water content by weight of $50 \%$. What is the value of $m$ ? | 80 | 59 | 2 |
math | 27. [12] Cyclic pentagon $A B C D E$ has a right angle $\angle A B C=90^{\circ}$ and side lengths $A B=15$ and $B C=20$. Supposing that $A B=D E=E A$, find $C D$. | 7 | 68 | 1 |
math | Let $p$ be some prime number.
a) Prove that there exist positive integers $a$ and $b$ such that $a^2 + b^2 + 2018$ is multiple of $p$.
b) Find all $p$ for which the $a$ and $b$ from a) can be chosen in such way that both these numbers aren’t multiples of $p$. | p \neq 3 | 90 | 7 |
math | 11.5. In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, two diagonals of the lateral faces $A_{1} C_{1}$ and $C_{1} D$ are drawn. Find the angle between them, given that the diagonal $A_{1} C_{1}$ is equal in length to one of the edges of the parallelepiped, and the diagonal $C_{1} D$ forms an angle of... | \arccos\frac{\sqrt{3}}{6} | 117 | 14 |
math | ## Subject (3).
b) To reward students who achieved good results in competitions, a school bought 60 dictionaries of the following types: Romanian Explanatory Dictionary (price 80 RON), English - Romanian / Romanian - English Dictionary (price 31 RON), and Spanish - Romanian / Romanian - Spanish Dictionary (price 30 RO... | DEX=4,E/R=13,S/R=43 | 109 | 13 |
math | Fuzzy draws a segment of positive length in a plane. How many locations can Fuzzy place another point in the same plane to form a non-degenerate isosceles right triangle with vertices consisting of his new point and the endpoints of the segment?
[i]Proposed by Timothy Qian[/i] | 6 | 62 | 1 |
math | ## Task 3 - 090723
A tourist was on the road for exactly the same amount of time on three consecutive days.
On the first day, he walked with an average speed of $6 \mathrm{~km} / \mathrm{h}$. On the second day, he used a moped with an average speed of $30 \mathrm{~km} / \mathrm{h}$. On the third day, he used a car wi... | 325 | 170 | 3 |
math | 4. [7] Find $\prod_{n=2}^{\infty}\left(1-\frac{1}{n^{2}}\right)$. | \frac{1}{2} | 34 | 7 |
math | Example 11. Let $a_{n}=6^{n}-8^{n}$. Find the remainder when $a_{94}$ is divided by 49. (Adapted from the first American Mathematical Invitational Competition) | 7 | 50 | 1 |
math | Example 7.1 Let $A=\{1,2, \cdots, 9\}$, and let the permutation $f$ on $A$ be
$$
f=\left(\begin{array}{lllllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
3 & 4 & 7 & 6 & 9 & 2 & 1 & 8 & 5
\end{array}\right)
$$
Express $f$ as a product of disjoint cycles. | (137)(246)(59)(8) | 124 | 14 |
math | 2.286. For what value of $k$ can the polynomial $x^{2}+2(k-9) x+$ $+\left(k^{2}+3 k+4\right)$ be represented as a perfect square? | \frac{11}{3} | 51 | 8 |
math | [ equations in integers ]
The number 1047, when divided by $A$, gives a remainder of 23, and when divided by $A+1$, it gives a remainder of 7. Find $A$.
# | 64 | 50 | 2 |
math | ## Task B-2.4.
The teacher has several candies that she wants to distribute to students who come for additional math lessons so that each student gets an equal number of candies and no candy is left over.
- If two students come, each student cannot get the same number of candies.
- If three students come, each of the... | 15 | 152 | 2 |
math | Suppose that for the positive numbers $x, y, z$,
$$
x^{2}+x y+y^{2}=9, \quad y^{2}+y z+z^{2}=16, \quad z^{2}+z x+x^{2}=25 .
$$
Determine the value of $x y+y z+z x$. | 8\sqrt{3} | 77 | 6 |
math | Example 4 Given a positive integer $n(n \geqslant 2)$. Find the largest real number $\lambda$, such that the inequality
$$
a_{n}^{2} \geqslant \lambda\left(a_{1}+a_{2}+\cdots+a_{n-1}\right)+2 a_{n},
$$
holds for any positive integers $a_{1}$, $a_{2}, \cdots, a_{n}$ satisfying $a_{1}<a_{2}<\cdots<a_{n}$.
(2003, Girls' ... | \frac{2(n-2)}{n-1} | 131 | 13 |
math | In a deck of cards consisting only of red and black cards, there are 2 times as many black cards as red cards. If 4 black cards are added, there are then 3 times as many black cards as red cards. How many cards were in the deck before adding the 4 black cards?
Only a numerical answer is expected here. | 12 | 71 | 2 |
math | ## Task 1 - 050721
At the Rostock public transport companies, you can buy tram tickets for adults at the following prices:
(1) A ticket from the ticket machine for 0.20 MDN
(2) A card with 6 fare sections for 1.00 MDN
(3) A block of 50 tickets for 7.50 MDN (The validity period is unlimited)
(4) A monthly pass for ... | 67 | 139 | 2 |
math | 5. 145 Let the polynomial $R(x)$ have a degree less than 4, and there exists a polynomial $P(x)$ such that
$$\begin{array}{l}
7 \sin ^{31} t + 8 \sin ^{13} t - 5 \sin ^{5} t \cos ^{4} t - 10 \sin ^{7} t + 5 \sin ^{5} t - 2 \\
\equiv P(\sin t)\left[\sin ^{4} t - (1 + \sin t)\left(\cos ^{2} t - 2\right)\right] + R(\sin t... | 13 x^{3} + 5 x^{2} + 12 x + 3 | 177 | 21 |
math | 8.4. Petya wrote down a line of three positive numbers, below it - a line of their pairwise sums, and below that - a line of the pairwise products of the numbers in the second line. The numbers in the third line coincided (in some order) with the numbers in the first line. Find these numbers. | \frac{1}{4};\frac{1}{4};\frac{1}{4} | 69 | 21 |
math | # 4.3. Condition:
In front of the elevator stand people weighing 150, 62, 63, 66, 70, 75, 79, 84, 95, 96, and 99 kg. The elevator's load capacity is 190 kg. What is the minimum number of trips needed to get everyone up? | 6 | 88 | 1 |
math | Let $ABC$ be an equilateral triangle. Find all positive integers $n$, for which the function $f$, defined on all points $M$ from the circle $S$ circumscribed to triangle $ABC$, defined by the formula $f:S \rightarrow R, f(M)=MA^n+MB^n+MC^n$, is a constant function. | n = 2, 4 | 73 | 7 |
math | 23. In $\triangle \mathrm{ABC}, \angle \mathrm{CAB}=30^{\circ}$ and $\angle \mathrm{ABC}=80^{\circ}$. The point $\mathrm{M}$ lies inside the triangle such that $\angle \mathrm{MAC}=10^{\circ}$ and $\angle \mathrm{MCA}=30^{\circ}$. Find $\angle \mathrm{BMC}$ in degrees. | 110 | 94 | 3 |
math | 5. Given that $\frac{\cos ^{4} \alpha}{\cos ^{2} \beta}+\frac{\sin ^{4} \alpha}{\sin ^{2} \beta}=1$, evaluate $\frac{\cos ^{4} \beta}{\cos ^{2} \alpha}+\frac{\sin ^{4} \beta}{\sin ^{2} \alpha}$. | 1 | 88 | 1 |
math | 【Example 1】There are $m$ people standing in a line, with the rule that person A does not stand at the left end, and person B does not stand at the right end. How many different arrangements are there? | A_{}^{}-2A_{-1}^{-1}+A_{-2}^{-2} | 48 | 24 |
math | 6. 2017 numbers are written. It is known that the sum of the squares of any 7 of them is 7, the sum of any 11 of them is positive, and the sum of all 2017 numbers is divisible by 9. Find these numbers. | Five\\\equal\to\-1,\the\rest\\equal\to\1 | 63 | 17 |
math | Example 5 If the line $y=m x+b$ intersects the hyperbola $(x-1)^{2}-a^{2} y^{2}=a^{2}$ for any real number $m$, find the conditions that the real numbers $a$ and $b$ must satisfy. | a^{2} b^{2}+a^{2}-1 \leqslant 0 | 61 | 21 |
math | The sequence $ (x_n)$, $ n\in\mathbb{N}^*$ is defined by $ |x_1|<1$, and for all $ n \ge 1$, \[ x_{n\plus{}1} \equal{}\frac{\minus{}x_n \plus{}\sqrt{3\minus{}3x_n^2}}{2}\] (a) Find the necessary and sufficient condition for $ x_1$ so that each $ x_n > 0$.
(b) Is this sequence periodic? And why? | x_1 \in \left(0, \frac{\sqrt{3}}{2}\right) | 116 | 23 |
math | (2) (20 points) In the Cartesian coordinate system, a circle with center $C\left(t, \frac{2}{t}\right)$ passes through the origin $O$, and intersects the $x$-axis and $y$-axis at points $A$ and $B$ (different from the origin $O$).
(1) Prove that the area $S$ of $\triangle A O B$ is a constant;
(2) Suppose the line $l: ... | (x-2)^{2}+(y-1)^{2}=5 | 148 | 16 |
math | 1.31. Find the eigenvalues and eigenvectors of the linear operator $\hat{A}$ with the matrix
$$
A=\left(\begin{array}{ll}
1 & 2 \\
3 & 2
\end{array}\right)
$$
Will the eigenvectors be orthogonal? | x^{(1)}\cdotx^{(2)}=-0.5^2 | 65 | 18 |
math | 3. Let $n$ be a given positive integer. Find the smallest positive integer $u_{n}$, satisfying: for every positive integer $d$, the number of integers divisible by $d$ in any $u_{n}$ consecutive positive odd numbers is not less than the number of integers divisible by $d$ in the odd numbers $1,3,5, \cdots$, $2 n-1$. | u_{n}=2 n-1 | 86 | 8 |
math | 3.2. Two identical cylindrical vessels are connected at the bottom by a small-section pipe with a valve. While the valve was closed, water was poured into the first vessel, and oil into the second, so that the level of the liquids was the same and equal to \( h = 40 \, \text{cm} \). At what level will the water stabili... | 34 | 147 | 2 |
math | In $\triangle A B C$,
$$
\frac{\overrightarrow{A B} \cdot \overrightarrow{B C}}{3}=\frac{\overrightarrow{B C} \cdot \overrightarrow{C A}}{2}=\frac{\overrightarrow{C A} \cdot \overrightarrow{A B}}{1} \text {. }
$$
Then $\tan A=$ $\qquad$ . | \sqrt{11} | 89 | 6 |
math | Problem 4. Let the matrix
$$
A=\left(\begin{array}{rcc}
2011 & 2012 & 2013 \\
2013 & 2011 & 0 \\
-2012 & 0 & 2011
\end{array}\right)
$$
Calculate $A^{n}$, where $n \in \mathbb{N}$. | A^{n}=2011^{n}\cdotI_{3}+n\cdot2011^{n-1}B+\frac{n(n-1)}{2}\cdot2011^{n-2}B^{2} | 95 | 54 |
math | 5. (10 points) In another 12 days, it will be 2016, HaoHao sighs: I have only experienced 2 leap years so far, and the year I was born is a multiple of 9. So, how old will HaoHao be in 2016? $\qquad$ years old. | 9 | 76 | 1 |
math | 3. (6 points) $A, B, C, D$ four people stay in four rooms numbered $1, 2, 3, 4$, with one person per room; so that $B$ does not stay in room 2, and $B, C$ two people require to stay in adjacent numbered rooms. The number of ways to arrange this is. $\qquad$ | 8 | 82 | 1 |
math | 4. We call a set of professors and committees on which they serve a university if
(1) given two distinct professors there is one and only one committee on which they both serve,
(2) given any committee, $C$, and any professor, $P$, not on that committee, there is exactly one committee on which $P$ serves and no profess... | 6 | 115 | 1 |
math | The three positive integers $a, b$, and $c$ satisfy
$$
4^{a} \cdot 5^{b} \cdot 6^{c}=8^{8} \cdot 9^{9} \cdot 10^{10}
$$
Determine the value of $a+b+c$. | 36 | 67 | 2 |
math | 6. If $a>0, b>0$, and $\arcsin a+\arcsin b=\frac{\pi}{2}, m=\log _{2} \frac{1}{a}+\log _{2} \frac{1}{b}$, then the range of $m$ is $\qquad$ . | [1,+\infty) | 71 | 7 |
math | 8. (10 points) A frog starts climbing from the bottom of a 12-meter deep well at 8:00. It climbs up 3 meters and then slides down 1 meter due to the slippery well wall. The time it takes to slide down 1 meter is one-third of the time it takes to climb up 3 meters. At 8:17, the frog reaches 3 meters below the well's mou... | 22 | 122 | 2 |
math | 315. The numbers $2 \overline{a c}+1$ and $3 \overline{a c}+1$ are perfect squares. Find $\overline{a c}$. | 40 | 44 | 2 |
math | ## Task 4 - 170834
A pioneer group collected waste paper; the entire proceeds were transferred to the solidarity account. The pioneers formed two brigades, with each pioneer in the group belonging to exactly one of these brigades. The following is known about the collection results:
(1) Each pioneer in Brigade $A$ co... | 52.50\mathrm{M} | 235 | 10 |
math | 11.022. Determine the volume of a regular quadrilateral prism if its diagonal forms an angle of $30^{\circ}$ with the plane of a lateral face, and the side of the base is $a$. | ^{3}\sqrt{2} | 48 | 7 |
math | 13.211. The digits of a certain three-digit number form a geometric progression. If in this number the digits of the hundreds and units are swapped, the new three-digit number will be 594 less than the desired one. If, however, in the desired number the digit of the hundreds is erased and the digits of the resulting tw... | 842 | 108 | 3 |
math | \section*{Problem 15}
What is the minimal value of \(\mathrm{b} /(\mathrm{c}+\mathrm{d})+\mathrm{c} /(\mathrm{a}+\mathrm{b})\) for positive real numbers \(\mathrm{b}\) and \(\mathrm{c}\) and nonnegative real numbers a and \(\mathrm{d}\) such that \(\mathrm{b}+\mathrm{c} \geq \mathrm{a}+\mathrm{d}\) ?
| \sqrt{2}-\frac{1}{2} | 110 | 12 |
math | Example 5 "Tossing a coin continuously until two consecutive heads appear" occurs in all possible ways on the $n(n \geqslant 2)$-th toss.
How many such ways are there? | 233 | 45 | 3 |
math | 6. In an accident, a maternity hospital lost all the labels of 8 babies born on that day. Without any markings, the probability that exactly 4 families take the right babies is $\qquad$ . | \frac{1}{64} | 43 | 8 |
math | 1. Solve the inequality $\sqrt{x^{2}-4} \cdot \sqrt{2 x-1} \leq x^{2}-4$. | x\in{2}\cup[3;+\infty) | 32 | 14 |
math | Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where both Ga... | 19152 | 94 | 5 |
math | 9.1. Petya wrote 10 integers on the board (not necessarily distinct).
Then he calculated the pairwise products (that is, he multiplied each of the written numbers by each other). Among them, there were exactly 15 negative products. How many zeros were written on the board? | 2 | 62 | 1 |
math | ## 194. Math Puzzle $7 / 81$
From a textbook by Adam Ries, who lived from 1492 to 1559, this problem was taken:
A son asks his father how old he is. The father answers: "If you were as old as I am, and half as old, and a quarter as old, and one year more, you would be 134 years old."
How old is the father? | 76 | 99 | 2 |
math | 1. One sixth of the total quantity of a certain commodity is sold at a profit of $20 \%$, and half of the total quantity of the same commodity is sold at a loss of $10 \%$. By what percentage profit should the remainder of the commodity be sold to cover the loss? | 5 | 61 | 1 |
math | 8,9
The height of a regular triangular pyramid is $6 \sqrt{6}$, and the lateral edge forms an angle of $45^{\circ}$ with the base plane. Find the distance from the center of the base of the pyramid to a lateral face. | \frac{36}{\sqrt{30}} | 57 | 12 |
math | 8.398. $\left\{\begin{array}{l}x-y=-\frac{1}{3}, \\ \cos ^{2} \pi x-\sin ^{2} \pi y=\frac{1}{2} .\end{array}\right.$ | k-\frac{1}{6},k+\frac{1}{6},k\in\mathbb{Z} | 60 | 25 |
math | 112(981). A pedestrian left point $A$ for point $B$. After 1 hour and 24 minutes, a cyclist left point $A$ in the same direction. After one hour, the cyclist was 1 km behind the pedestrian, and another hour later, the distance remaining for the cyclist to $B$ was half the distance remaining for the pedestrian. Find the... | v_{1}=5,v_{2}=11 | 104 | 11 |
math | 2. Points $A, B, C, D$ are chosen in the plane such that segments $A B, B C, C D, D A$ have lengths $2,7,5$, 12 , respectively. Let $m$ be the minimum possible value of the length of segment $A C$ and let $M$ be the maximum possible value of the length of segment $A C$. What is the ordered pair $(m, M)$ ? | (7,9) | 96 | 5 |
math | 8. (10 points) If the expression $\frac{1}{1 \times 2}-\frac{1}{3 \times 4}+\frac{1}{5 \times 6}-\frac{1}{7 \times 8}+\cdots+\frac{1}{2007 \times 2008}$ is converted to a decimal, then the first digit after the decimal point is $\qquad$ . | 4 | 94 | 1 |
math | $6 \cdot 31$ Try to find the domain of the function: $y=\lg \left(\sin \frac{x}{2}-\frac{1}{2}\right)+\sqrt{\frac{1}{x-2}}$. (Shanghai Mathematical Competition, China, 1962) | 2<x<\frac{5\pi}{3},\frac{\pi}{3}+4k\pi<x<\frac{5\pi}{3}+4k\pi\cdot(k=1,2,\cdots) | 66 | 51 |
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 | 11.4. It is known that the lengths of the sides of a triangle are consecutive natural numbers, and the radius of its inscribed circle is 4. Find the radius of the circumscribed circle of this triangle. | \frac{65}{8} | 47 | 8 |
math | 5. Can you use the previous question to propose a method for solving a linear congruence equation? Use the method you propose to solve (i) $6 x \equiv 7(\bmod 23)$; (ii) $5 x \equiv 1(\bmod 12)$. Point out what to pay attention to when applying this method. | x \equiv 5(\bmod 23) | 76 | 12 |
math | 1- 17 Let $A$ be the sum of the digits of the decimal number $4444^{4444}$, and let $B$ be the sum of the digits of $A$. Find the sum of the digits of $B$ (all numbers here are in decimal). | 7 | 64 | 1 |
math | 13 (14 points) Let the equation of the ellipse $C_{1}$ be $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, and the equation of the curve $C_{2}$ be $y=\frac{1}{x}$. Suppose $C_{1}$ and $C_{2}$ have only one common point $P$ in the first quadrant.
(1) Try to express the coordinates of point $P$ in terms of $a$;
(2) L... | \{f(),()}={\begin{pmatrix}^{2}-\frac{4}{^{2}},&\sqrt{2}\sqrt[4]{6}0\end{pmatrix}.} | 224 | 43 |
math | Four, (50 points) In a round-robin tournament with $2 n+1$ teams, each team plays exactly one match against every other team, and there are no ties. If three teams $A, B, C$ satisfy: $A$ beats $B, B$ beats $C, C$ beats $A$, then they form a “cyclic triplet”. Find:
(1) the minimum possible number of cyclic triplets;
(2)... | \frac{1}{6} n(n+1)(2 n+1) | 105 | 17 |
math | A circle with center $O$ has a diameter $A B$. A point $C$ on the circumference of the circle, different from $A$ and $B$, draws a perpendicular to $A B$, intersecting $A B$ at $D$. A perpendicular from $O$ to $B C$ intersects $B C$ at $M$. Determine the measure of angle $A B C$, given that $D B = 3 \cdot O M$. | 30 | 96 | 2 |
math | 6. $12 f(n)$ is a function defined on the set of positive integers, taking non-negative integer values, and for all $m, n$ we have:
$$\begin{array}{l}
f(m+n)-f(m)-f(n)=0 \text { or } 1 ; \\
f(2)=0, f(3)>0, f(9999)=3333 .
\end{array}$$
Try to find: $f(1982)$. | 660 | 108 | 3 |
math | 5. What remainder do we get when the number $2002^{2001}$ is divided by 2003? Justify your answer.
Mathematical Competition for High School Students in Slovenia
## Optional Competition
March 28, 2003
## Problems for 3rd Year Students | 2002 | 69 | 4 |
math | 7. In the expansion of $(2+\sqrt{x})^{2 n+1}$, the sum of the coefficients of the terms where the power of $x$ is an integer is | \frac{3^{2n+1}+1}{2} | 38 | 15 |
math | Given the equation of a circle as $x^{2}+y^{2}=4$. Try to find two points $A(s, t), B(m, n)$ on the coordinate plane, such that the following two conditions are satisfied:
(1) The ratio of the distance from any point on the circle to point $A$ and to point $B$ is a constant $k ;$.
(2) $s>m, t>n$, and $m, n$ are both na... | (2,2),(1,1) | 103 | 9 |
math | Calculate the three sides of a certain triangle, if we know that the measures of the sides and the measure of the area are integers and form an arithmetic progression in the said order. | 4,3,5 | 36 | 5 |
math | 13. B. If five pairwise coprime distinct integers $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ are randomly selected from $1,2, \cdots, n$, and one of these integers is always a prime number, find the maximum value of $n$. | 48 | 68 | 2 |
math | 11.1. While walking in the park, Seryozha and Misha stumbled upon a meadow surrounded by lindens. Seryozha walked around the meadow, counting the trees. Misha did the same but started from a different tree (although he went in the same direction). The tree that was the $20-\mathrm{th}$ for Seryozha was the $7-\mathrm{t... | 100 | 131 | 3 |
math | 2. To walk 4 km, ride 6 km on a bicycle, and drive 40 km by car, Uncle Vanya needs 2 hours and 12 minutes. If he needs to walk 5 km, ride 8 km on a bicycle, and drive 30 km by car, it will take him 2 hours and 24 minutes. How much time will Uncle Vanya need to walk 8 km, ride 10 km on a bicycle, and drive 160 km by car... | 5.8 | 110 | 3 |
math | 137. A body moves with a velocity $v=\left(3 t^{2}-1\right) \mathrm{m} / \mathrm{s}$. Find the law of motion $s(t)$, if at the initial moment the body was 5 cm away from the origin. | ^{3}-+0.05 | 61 | 8 |
math | Find the coefficient of $x^2$ after expansion and collecting the terms of the following expression (there are $k$ pairs of parentheses): $((... (((x - 2)^2 - 2)^2 -2)^2 -... -2)^2 - 2)^2$. | \frac{4^k - 1}{3} \cdot 4^{k-1} | 60 | 21 |
math | 15. (6 points) A car and a truck start from locations $A$ and $B$ respectively at the same time, heading towards each other. It is known that the car's speed is twice that of the truck. The car arrives at point $C$ on the way at 8:30, and the truck arrives at point $C$ at 15:00 on the same day. The two vehicles do not ... | 10:40 | 118 | 5 |
math | 7.3. Find the number of all integer solutions of the inequality $\sqrt{1+\sin \frac{\pi x}{4}-3 \cos \frac{\pi x}{2}}+\sqrt{6} \cdot \sin \frac{\pi x}{4} \geq 0$, belonging to the interval [1991; 2013]. | 9 | 78 | 1 |
math | $1 \cdot 146$ Simplify $\left(\frac{1 \cdot 2 \cdot 4+2 \cdot 4 \cdot 8+\cdots+n \cdot 2 n \cdot 4 n}{1 \cdot 3 \cdot 9+2 \cdot 6 \cdot 18+\cdots+n \cdot 3 n \cdot 9 n}\right)^{\frac{1}{3}}$. | \frac{2}{3} | 95 | 7 |
math | Ricsi, Dénes, and Attila often play ping pong against each other, with two of them standing on one side. Dénes and Attila win against Ricsi three times as often as they lose; Dénes wins against Ricsi and Attila as often as he loses; finally, Attila wins against Ricsi and Dénes twice as often as he loses. Recently, they... | \frac{15}{16} | 129 | 9 |
math | Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$. Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined. | 2 | 71 | 1 |
math | 20. Let $a_{1}, a_{2}, \ldots$ be a sequence satisfying the condition that $a_{1}=1$ and $a_{n}=10 a_{n-1}-1$ for all $n \geq 2$. Find the minimum $n$ such that $a_{n}>10^{100}$. | 102 | 78 | 3 |
math | 2. Ana sells pies at the market.
- In the first hour, she sold a quarter of the number of all the pies she brought and another quarter of the pies.
- In the second hour, she sold a fifth of the number of the remaining pies and another fifth of the pies.
- In the third hour, she sold a quarter of the number of the rema... | 399 | 173 | 3 |
math | 12th VMO 1974 Problem A2 Find all positive integers n and b with 0 < b < 10 such that if a n is the positive integer with n digits, all of them 1, then a 2n - b a n is a square. | 2foranyn,7forn=1 | 61 | 10 |
math | Jerry buys a bottle of 150 pills. Using a standard 12 hour clock, he sees that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what hour will the clock read when he takes the last pill in
the bottle? | 1 | 67 | 1 |
math | 3. Solve the equation: $[20 x+23]=20+23 x$. Recall that $[a]$ denotes the integer part of the number, that is, the greatest integer not exceeding $a$. (S. S. Koresheva) | \frac{16}{23},\frac{17}{23},\frac{18}{23},\frac{19}{23},\frac{20}{23},\frac{21}{23},\frac{22}{23},1 | 57 | 64 |
math | ## Task 5 - 301245
Determine a polynomial
$$
f(x)=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n} \quad\left(a_{0}, a_{1}, \ldots, a_{n} \text { real } ; a_{n} \neq 0\right)
$$
that satisfies the conditions
$$
f(-4)=0, f(-3)=1, f(-2)=0, f(-1)=-1, f(0)=0, f(1)=1, f(2)=0, f(3)=-1, f(4)=0
$$
and has the lowest possible ... | -\frac{1}{630}x^{7}+\frac{1}{18}x^{5}-\frac{26}{45}x^{3}+\frac{32}{21}x | 166 | 48 |
math | 5. Find all solutions of the equation $|\sin 2 x-\cos x|=|| \sin 2 x|-| \cos x||$ on the interval $(-2 \pi ; 2 \pi]$. Answer: $(-2 \pi ;-\pi] \cup[0 ; \pi] \cup\left\{-\frac{\pi}{2} ; \frac{3 \pi}{2} ; 2 \pi\right\}$. | (-2\pi;-\pi]\cup[0;\pi]\cup{-\frac{\pi}{2};\frac{3\pi}{2};2\pi} | 98 | 36 |
math | Find the sum : $C^{n}_{1}$ - $\frac{1}{3} \cdot C^{n}_{3}$ + $\frac{1}{9} \cdot C^{n}_{5}$ - $\frac{1}{27} \cdot C^{n}_{9}$ + ... | 2^n \cdot 3^{\frac{1-n}{2}} \cdot \sin \frac{n\pi}{6} | 62 | 27 |
math | For a triangle, we know that $a b + b c + c a = 12$ (where $a, b$, and $c$ are the sides of the triangle.) What are the limits within which the perimeter of the triangle can fall? | 6\leqk\leq4\sqrt{3} | 53 | 14 |
math | 33. [15] The polynomial $a x^{2}-b x+c$ has two distinct roots $p$ and $q$, with $a, b$, and $c$ positive integers and with $0<p, q<1$. Find the minimum possible value of $a$. | 5 | 61 | 1 |
math | 4. For what values of the parameter $a$ does the system of equations
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+z^{2}+2 y=0 \\
x+a y+a z-a=0
\end{array}\right.
$$
have a unique solution? | \\frac{\sqrt{2}}{2} | 72 | 10 |
math | 5. (3 points) In triangle $A B C$, a square $K L M N$ with side length 1 is inscribed: points $K$ and $L$ lie on side $A C$, points $M$ and $N$ lie on sides $A B$ and $B C$ respectively. The area of the square is half the area of the triangle. Find the length of the height $B H$ of triangle $A B C$. | 2 | 97 | 1 |
math | 7. If $n \in \mathbf{N}^{*}$, then $\lim _{n \rightarrow \infty} \sin ^{2}\left(\pi \sqrt{n^{2}+n}\right)=$ $\qquad$ (Contributed by Jian Weifeng) | 1 | 63 | 1 |
math | 2 [ Arithmetic. Mental calculation, etc. $\quad]$
One tea bag can brew two or three cups of tea. Milla and Tanya divided a box of tea bags equally. Milla brewed 57 cups of tea, and Tanya - 83 cups. How many tea bags could have been in the box? | 56 | 68 | 2 |
math | (7) In a certain game activity, the rewards are divided into first, second, and third prizes (all participants in the game activity will receive a prize), and the corresponding winning probabilities form a geometric sequence with the first term $a$ and a common ratio of 2. The corresponding prizes form an arithmetic se... | 500 | 103 | 3 |
math | Example 6 Given a natural number $n \geqslant 2$, find the smallest positive number $\lambda$, such that for any $a_{i} \geqslant 0,0 \leqslant b_{i} \leqslant \frac{1}{2}(i=1,2, \cdots$, $n)$, and $\sum_{i=1}^{n} a_{i}=\sum_{i=1}^{n} b_{i}=1$, we have $a_{1} a_{2} \cdots a_{n} \leqslant \lambda \sum_{i=1}^{n} a_{i} b_... | \frac{1}{2(n-1)^{n-1}} | 153 | 15 |
math | Two cars start at the same time from city A heading towards city B. One travels at a constant speed of $60 \mathrm{~km} / \mathrm{h}$ and the other at a constant speed of $70 \mathrm{~km} / \mathrm{h}$. If the faster car completes the journey from A to B in 15 minutes less than the other car, what is the distance betwe... | 105\mathrm{~} | 92 | 8 |
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