problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Several island inhabitants gather in a hut, with some belonging to the Ah tribe and the rest to the Uh tribe. Ah tribe members always tell the truth, while Uh tribe members always lie. One inhabitant said, "There are no more than 16 of us in the hut," and then added, "All of us are from the Uh tribe." Another said, "There are no more than 17 of us in the hut," and then noted, "Some of us are from the Ah tribe." A third person said, "There are five of us in the hut," and looking around, added, "There are at least three Uh tribe members among us." How many Ah tribe members are in the hut? | 15 | 0.875 |
What is the number of subsets of the set $\{1, 2, 3, \ldots, n\}$ that do not contain two consecutive numbers? | F_{n+2} | 0.5 |
In a regular hexagon \( ABCDEF \), the diagonals \( AC \) and \( CE \) are divided by interior points \( M \) and \( N \) in the following ratio: \( AM : AC = CN : CE = r \). If the points \( B, M, N \) are collinear, find the ratio \( r \). | \frac{\sqrt{3}}{3} | 0.875 |
Xiaoming formed a sequence using the digits 2, 0, 1, and 6 in that order: 2, 20, 201, 2016, 20162, 201620, 2016201, 20162016, 201620162, ..., adding these digits continuously at the end of the numbers. In this sequence, there are how many prime numbers? | 1 | 0.375 |
In $\triangle ABC$, if $a + c = 2b$, then find the value of $\tan \frac{A}{2} \cdot \tan \frac{C}{2}$. | \frac{1}{3} | 0.5 |
In quadrilateral \(ABCD\), the lengths \( |AB| = a \) and \( |AD| = b \) are given. The sides \(BC\), \(CD\), and \(AD\) are tangent to a circle whose center is at the midpoint of \(AB\). Find the length of side \( |BC| \). | \frac{a^2}{4b} | 0.25 |
With square tiles of a side length of an exact number of units, a room with a surface area of 18,144 square units has been tiled in the following manner: on the first day one tile was placed, the second day two tiles, the third day three tiles, and so on. How many tiles were necessary? | 2016 | 0.25 |
In a geometric sequence $\left\{a_{n}\right\}$ with all positive terms, there exist two terms $a_{m}$ and $a_{n}$ such that $\sqrt{a_{m} a_{n}}=8 a_{1}$, and it is known that $a_{9}=a_{8}+2 a_{7}$. Find the minimum value of $\frac{1}{m}+\frac{4}{n}$. | \frac{17}{15} | 0.875 |
Determine the digits $A, B, C, D$ if
$$
\overline{A B^{2}}=\overline{A C D B}, \quad \text{and} \quad \overline{C D^{3}}=\overline{A C B D}
$$ | A=9, B=6, C=2, D=1 | 0.75 |
Find the least positive integer \( n \) for which \( \frac{n-10}{9n+11} \) is a non-zero reducible fraction. | 111 | 0.875 |
What is the sum of all natural numbers between 1 and 200 that are not multiples of 2 or 5? | 8000 | 0.75 |
The numbers $a, b, c, d$ belong to the interval $[-8.5,8.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$. | 306 | 0.375 |
Find all \( a_{0} \in \mathbb{R} \) such that the sequence \( a_{0}, a_{1}, \cdots \) determined by \( a_{n+1}=2^{n}-3a_{n}, n \in \mathbb{Z}^{+} \) is increasing. | a_0 = \frac{1}{5} | 0.5 |
Given a sequence \( a_{0}, a_{1}, a_{2}, \cdots \) satisfying \( a_{0} = a_{1} = 11 \) and \( a_{m+n} = \frac{1}{2}\left(a_{2m}+a_{2n}\right) - (m-n)^2 \) for \( m, n \geq 0 \), find \( a_{45} \). | 1991 | 0.875 |
Two ships shuttle back and forth between the ports Mumraj and Zmatek on the same route. They spend negligible time in the ports, immediately turning around and continuing their journey. In the morning, at the same moment, the blue ship sets off from the port Mumraj and the green ship sets off from the port Zmatek. The ships first meet $20 \mathrm{~km}$ from the port Mumraj, and after some time, they meet again directly at this port. By then, the blue ship has managed to travel the route between the ports four times, while the green ship has only done so three times.
How long is the route between the ports Mumraj and Zmatek? | 35 \ \text{km} | 0.875 |
Point $A$ lies on the line $y = \frac{12}{5} x - 3$, and point $B$ lies on the parabola $y = x^2$. What is the minimum length of the segment $AB$? | \frac{3}{5} | 0.125 |
Find all natural numbers \( a \) and \( b \) such that
\[ a^3 - b^3 = 633 \cdot p \]
where \( p \) is some prime number. | a = 16, b = 13 | 0.125 |
The elements $x_n$ of the sequence $\mathrm{Az}\left(x_{n}\right)$ are positive real numbers, and for every positive integer $n$, the following equation holds:
$$
2\left(x_{1}+x_{2}+\ldots+x_{n}\right)^{4}=\left(x_{1}^{5}+x_{2}^{5}+\ldots+x_{n}^{5}\right)+\left(x_{1}^{7}+x_{2}^{7}+\ldots+x_{n}^{7}\right)
$$
Determine the elements of the sequence. | x_{n} = n | 0.5 |
There are seven red cubes, three blue cubes, and nine green cubes. Ten cubes were placed into a gift bag. In how many different ways could this have been done? | 31 | 0.75 |
Find the number of roots of the equation
\[ z^{2} - a e^{z} = 0, \quad \text{where} \quad 0 < a < e^{-1} \]
inside the unit disk \( |z| < 1 \). | 2 | 0.875 |
In triangle \( MNK \), \( MN = NK \). From point \( A \) on side \( MN \), a perpendicular \( AP \) is dropped to side \( NK \). It turns out that \( MA = AP \). Find the angle \( \angle PMK \). | 45^{\circ} | 0.5 |
If $2 - \sin^{2}(x + 2y - 1) = \frac{x^{2} + y^{2} - 2(x + 1)(y - 1)}{x - y + 1}$, then the minimum value of the product $xy$ is $\qquad$ . | \frac{1}{9} | 0.875 |
G9.1 Find \( A \) from the sequence: \( 0,3,8, A, 24,35, \ldots \)
G9.2 The roots of the equation \( x^{2}-Ax+B=0 \) are 7 and \( C \). Find \( B \) and \( C \).
G9.3 If \( \log _{7} B=\log _{7} C+7^{X} \); find \( X \). | X = 0 | 0.625 |
Replace the larger of two distinct natural numbers with their difference. Continue this operation until the two numbers are the same. For example, for the pair (20, 26), the process is as follows:
\[ (20, 26) \rightarrow (20, 6) \rightarrow (14, 6) \rightarrow (8, 6) \rightarrow (2, 6) \rightarrow (2, 4) \rightarrow (2, 2) \]
1. Perform the above operation on the pair (45, 80).
2. If the operation is performed on two four-digit numbers and the final same number obtained is 17, find the maximum possible sum of these two four-digit numbers. | 19975 | 0.75 |
Find the minimum value of the expression \(\left(a^{2}+x^{2}\right) / x\), where \(a>0\) is a constant and \(x>0\) is a variable. | 2a | 0.875 |
Triangle \( ABC \) is isosceles, and \( \angle ABC = x^\circ \). If the sum of the possible measures of \( \angle BAC \) is \( 240^\circ \), find \( x \). | 20 | 0.625 |
For a $k$-element subset $T$ of the set $\{1,2,\cdots,242\}$, every pair of elements (which may be the same) in $T$ has a sum that is not an integer power of 3. Find the maximum value of $k$. | 121 | 0.375 |
Alice chooses three primes \( p, q, r \) independently and uniformly at random from the set of primes of at most 30. She then calculates the roots of \( p x^{2} + q x + r \). What is the probability that at least one of her roots is an integer? | \frac{3}{200} | 0.125 |
On the \(xy\)-plane, let \(S\) denote the region consisting of all points \((x, y)\) for which
\[
\left|x+\frac{1}{2} y\right| \leq 10, \quad |x| \leq 10, \quad \text{and} \quad |y| \leq 10.
\]
The largest circle centered at \((0,0)\) that can be fitted in the region \(S\) has area \(k \pi\). Find the value of \(k\). | 80 | 0.75 |
Two natural numbers \(a\) and \(b\) have a sum of 100. Additionally, when \(a\) is divided by 5, the remainder is 2; when \(b\) is divided by 6, the remainder is 3. Find the maximum product of \(a\) and \(b\). | 2331 | 0.875 |
The edge length of cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \) is 1. What is the distance between the lines \( A_{1}C_{1} \) and \( BD_{1} \)? | \frac{\sqrt{6}}{6} | 0.375 |
In a square $\mathrm{ABCD}$, point $\mathrm{E}$ is on $\mathrm{BC}$ with $\mathrm{BE} = 2$ and $\mathrm{CE} = 1$. Point $\mathrm{P}$ moves along $\mathrm{BD}$. What is the minimum value of $\mathrm{PE} + \mathrm{PC}$? | \sqrt{13} | 0.5 |
Points \( A \) and \( C \) lie on the circumference of a circle with radius \(\sqrt{50}\). \( B \) is a point inside the circle such that \(\angle A B C = 90^{\circ}\). If \( A B = 6 \) and \( B C = 2\), find the distance from \( B \) to the centre of the circle. | \sqrt{26} | 0.5 |
Find the smallest positive integer \( k \) such that \( z^{10} + z^{9} + z^{6} + z^{5} + z^{4} + z + 1 \) divides \( z^k - 1 \). | 84 | 0.875 |
Let the strictly increasing sequence $\left\{a_{n}\right\}$ consist of positive integers with $a_{7}=120$ and $a_{n+2}=a_{n}+a_{n+1}$ for $n \in \mathbf{Z}_{+}$. Find $a_{8}=$. | 194 | 0.75 |
Vanya received three sets of candies for New Year. Each set contains three types of candies: hard candies, chocolates, and gummy candies. The total number of hard candies in all three sets is equal to the total number of chocolates in all three sets, and also to the total number of gummy candies in all three sets. In the first set, there are equal numbers of chocolates and gummy candies, and 7 more hard candies than chocolates. In the second set, there are equal numbers of hard candies and chocolates, and 15 fewer gummy candies than hard candies. How many candies are in the third set if it is known that there are no hard candies in it? | 29 | 0.875 |
From the similarity of the right triangles \(A K C\) and \(B L C\), formed by the segments of the tangent, line \(l\), and the radii \(A K\) and \(B L\) drawn to the points of tangency at time \(T\), we obtain
$$
\frac{a T}{a T - R} = \frac{L + x}{x}
$$
where \(x\) represents the distance \(B C\).
From the given equation, we find \( x = a T \frac{L}{R} - L \). Hence, the speed of the moving point \(C\) at the intersection of the lines is \(a \frac{L}{R}\). | a \frac{L}{R} | 0.75 |
Given the function \( f(x) = \frac{1}{2} + \log_{2} \frac{x}{1-x} \), define \( S_{n} = f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + f\left(\frac{3}{n}\right) + \cdots + f\left(\frac{n-1}{n}\right) \), where \( n \in \mathbb{N}^* (n \geq 2) \). Find \( S_{n} \). | \frac{n-1}{2} | 0.375 |
Two circles with radii $\sqrt{5}$ and $\sqrt{2}$ intersect at point $A$. The distance between the centers of the circles is 3. A line through point $A$ intersects the circles at points $B$ and $C$ such that $A B = A C$ (point $B$ does not coincide with $C$). Find $A B$. | \frac{6\sqrt{5}}{5} | 0.75 |
How many times does 5 occur as a divisor in the numbers from 1 to 50000? | 12499 | 0.5 |
Increasing sequence of positive integers \( a_1, a_2, a_3, \ldots \) satisfies \( a_{n+2} = a_n + a_{n+1} \) (for \( n \geq 1 \)). If \( a_7 = 120 \), what is \( a_8 \) equal to? | 194 | 0.75 |
In trapezoid \(ABCD\), side \(AB\) is perpendicular to the bases \(AD\) and \(BC\). Point \(E\) is the midpoint of side \(CD\).
Find the ratio \(AD : BC\) if \(AE = 2AB\) and \(AE \perp CD\). | 8:7 | 0.5 |
Find the following limits as \(n\) approaches infinity:
1. \(\lim_{n \to \infty} \sum_{1 \le i \le n} \frac{1}{\sqrt{n^2 + i^2}}\)
2. \(\lim_{n \to \infty} \sum_{1 \le i \le n} \frac{1}{\sqrt{n^2 + i}}\)
3. \(\lim_{n \to \infty} \sum_{1 \le i \le n^2} \frac{1}{\sqrt{n^2 + i}}\) | \infty | 0.375 |
Alice has an equilateral triangle \(ABC\) with an area of 1. Point \(D\) is on \(BC\) such that \(BD = DC\), point \(E\) is on \(CA\) such that \(CE = 2EA\), and point \(F\) is on \(AB\) such that \(2AF = FB\). Line segments \(AD\), \(BE\), and \(CF\) intersect at a single point \(M\). What is the area of triangle \(EMC\)? | \frac{1}{6} | 0.5 |
A truck and a car are moving in the same direction on adjacent lanes at speeds of 65 km/h and 85 km/h respectively. How far apart will they be 3 minutes after they are even with each other? | 1 \text{ km} | 0.75 |
Let \( a \), \( b \), and \( c \) be the side lengths of a triangle, and assume that \( a \leq b \) and \( a \leq c \). Let \( x = \frac{b + c - a}{2} \). If \( r \) and \( R \) denote the inradius and circumradius, respectively, find the minimum value of \( \frac{a x}{r R} \). | 3 | 0.625 |
Given a sequence $\left\{a_{n}\right\}$ that satisfies the conditions: $a_{1}=1$, $a_{2}=2$, and for any three consecutive terms $a_{n-1}, a_{n}, a_{n+1}$, the relation $a_{n}=\frac{(n-1) a_{n-1}+(n+1) a_{n+1}}{2 n}$ holds. Determine the general term $a_{n}$. | 3 - \frac{2}{n} | 0.375 |
A circle is divided into seven arcs such that the sum of any two adjacent arcs does not exceed $103^\circ$. Determine the largest possible value of $A$ such that, in any such division, each of the seven arcs contains at least $A^\circ$. | 51 | 0.625 |
The altitudes of an acute isosceles triangle, where \(AB = BC\), intersect at point \(H\). Find the area of triangle \(ABC\), given \(AH = 5\) and the altitude \(AD\) is 8. | 40 | 0.5 |
Vovochka approached an arcade machine which displayed the number 0 on the screen. The rules of the game stated: "The screen shows the number of points. If you insert a 1 ruble coin, the number of points increases by 1. If you insert a 2 ruble coin, the number of points doubles. If you reach 50 points, the machine gives out a prize. If you get a number greater than 50, all the points are lost." What is the minimum amount of rubles Vovochka needs to get the prize? Answer: 11 rubles. | 11 | 0.625 |
Given 6 digits: \(0, 1, 2, 3, 4, 5\). Find the sum of all four-digit even numbers that can be written using these digits (the same digit can be repeated in a number). | 1769580 | 0.25 |
Show that in any connected planar graph, denoting $s$ as the number of vertices, $f$ as the number of faces, and $a$ as the number of edges, we have: $f = a - s + 2$. | f = a - s + 2 | 0.75 |
From the set \( \{1, 2, 3, \ldots, 999, 1000\} \), select \( k \) numbers. If among the selected numbers, there are always three numbers that can form the side lengths of a triangle, what is the smallest value of \( k \)? Explain why. | 16 | 0.5 |
Fill each of the boxes and triangles in the following equation with a natural number, so that the equation holds:
\[ \square^{2} + 12 = \triangle^{2} \]
Then find: \(\square + \triangle = 6\). | 6 | 0.875 |
Oleg has four cards, each with a natural number on each side (a total of 8 numbers). He considers all possible sets of four numbers where the first number is written on the first card, the second number on the second card, the third number on the third card, and the fourth number on the fourth card. For each set of four numbers, he writes the product of the numbers in his notebook. What is the sum of the eight numbers on the cards if the sum of the sixteen numbers in Oleg’s notebook is $330? | 21 | 0.5 |
In an acute-angled triangle \( \triangle ABC \) with sides \( a \), \( b \), and \( c \) opposite the angles \( A \), \( B \), and \( C \) respectively, if \( 2a^{2} = 2b^{2} + c^{2} \), what is the minimum value of \( \tan A + \tan B + \tan C \)? | 6 | 0.5 |
Alice and Bob are independently trying to figure out a secret password to Cathy's bitcoin wallet. They know:
- It is a 4-digit number whose first digit is 5;
- It is a multiple of 9;
- The larger number is more likely to be the password than a smaller number.
Alice knows the second and third digits, and Bob knows the third and fourth digits. Initially:
Alice: "I have no idea what the number is."
Bob: "I have no idea too."
After this conversation, they both knew which number they should try first. Identify this number. | 5949 | 0.25 |
How many positive integers less than 500 have exactly 15 positive integer factors? | 3 | 0.375 |
Let \( f(x) \) be a function defined on \(\mathbf{R}\), such that \( f(0)=1008 \), and for any \( x \in \mathbf{R} \), it holds that:
\[ f(x+4) - f(x) \leq 2(x+1) \]
\[ f(x+12) - f(x) \geq 6(x+5) \]
Find \( \frac{f(2016)}{2016} \). | 504 | 0.875 |
Erin lists all three-digit primes that are 21 less than a square. What is the mean of the numbers in Erin's list? | 421 | 0.75 |
Using the digits 1 through 9 to form three three-digit numbers \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) (each digit can be used only once), what is the maximum value of \( a + b - c \)? | 1716 | 0.375 |
Given that \( f(x) \) and \( g(x) \) are two quadratic functions both with a leading coefficient of 1, where \( g(6) = 35 \) and \( \frac{f(-1)}{g(-1)} = \frac{f(1)}{g(1)} = \frac{21}{20} \), what is \( f(6) \)? | 35 | 0.75 |
\(ABCD\) is a convex quadrilateral. It is known that \(\angle CAD = \angle DBA = 40^\circ\), \(\angle CAB = 60^\circ\), and \(\angle CBD = 20^\circ\). Find the angle \(\angle CDB\). | 30^\circ | 0.375 |
Huanhuan and Lele are playing a game together. In the first round, they both gain the same amount of gold coins, and in the second round, they again gain the same amount of gold coins.
At the beginning, Huanhuan says: "The number of my gold coins is 7 times the number of your gold coins."
At the end of the first round, Lele says: "The number of your gold coins is 6 times the number of my gold coins now."
At the end of the second round, Huanhuan says: "The number of my gold coins is 5 times the number of your gold coins now."
Find the minimum number of gold coins Huanhuan had at the beginning. | 70 | 0.75 |
Six IMO competitions are hosted sequentially by two Asian countries, two European countries, and two African countries, where each country hosts once but no continent can host consecutively. How many such arrangements are possible? | 240 | 0.375 |
Two points, A and B, are 30 cm apart. Two strings, A and B, start simultaneously from A and B respectively and move towards each other. String A is 151 cm long and moves at 2 cm per second; String B is 187 cm long and moves at 3 cm per second. If both strings are ignited at their tail ends upon departure, string A burns at a rate of 1 cm per second and string B burns at a rate of 2 cm per second. How many seconds will it take for the two strings to completely pass each other from the moment they meet? | 40 \ \text{s} | 0.375 |
Find all triples of natural numbers \( n \), \( p \), and \( q \) such that \( n^p + n^q = n^{2010} \). | (2, 2009, 2009) | 0.875 |
What work is required to stretch a spring by \(0.06 \, \text{m}\), if a force of \(1 \, \text{N}\) stretches it by \(0.01 \, \text{m}\)? | 0.18 \, \text{J} | 0.25 |
A settlement is built in the shape of a square consisting of 3 blocks by 3 blocks (each block is a square with a side length $b$, for a total of 9 blocks). What is the shortest path that a paver must take to pave all the streets if they start and end their path at a corner point $A$? (The sides of the square are also streets). | 28b | 0.375 |
Solve the inequality \(\frac{\sqrt{\frac{x}{\gamma}+(\alpha+2)}-\frac{x}{\gamma}-\alpha}{x^{2}+a x+b} \geqslant 0\).
Indicate the number of integer roots of this inequality in the answer. If there are no integer roots or there are infinitely many roots, indicate the number 0 on the answer sheet.
Given:
\[
\alpha = 3, \gamma = 1, a = -15, b = 54.
\] | 7 | 0.625 |
The equation \( x^{4} - 7x - 3 = 0 \) has exactly two real roots \( a \) and \( b \), where \( a > b \). Find the value of the expression \( \frac{a - b}{a^{4} - b^{4}} \). | \frac{1}{7} | 0.625 |
Find the number of 9-digit numbers in which each digit from 1 to 9 occurs exactly once, the digits 1, 2, 3, 4, 5 are in ascending order, and the digit 6 is placed before the digit 1 (for example, 916238457). | 504 | 0.25 |
The plane is covered by a grid of squares with a side length of 1. Is it possible to construct an equilateral triangle with vertices at the grid points? | \text{No} | 0.75 |
Let \([x]\) denote the greatest integer less than or equal to the real number \(x\). Define
\[ A = \left[\frac{19}{20}\right] + \left[\frac{19^2}{20}\right] + \cdots + \left[\frac{19^{2020}}{20}\right]. \]
Find the remainder when \(A\) is divided by 11. | 2 | 0.375 |
Team A and Team B each field 7 players who will compete in a Go match according to a predetermined order. Initially, the 1st player from each team competes. The loser is eliminated, and the winner competes with the next player from the losing team. This continues until all players from one team are eliminated. The other team is declared the winner, forming a complete match process. What is the total number of possible match processes? | 3432 | 0.625 |
The numbers from 1 to 37 are written in a sequence such that the sum of any initial segment of numbers is divisible by the next number following that segment.
What number is in the third position if the first number is 37 and the second number is 1? | 2 | 0.125 |
Among all possible triangles \(ABC\) such that \(BC = 2 \sqrt[4]{3}\) and \(\angle BAC = \frac{\pi}{3}\), find the one with the maximum area. What is the area of this triangle? | 3 | 0.75 |
A square and an equilateral triangle together have the property that the area of each is the perimeter of the other. Find the square's area. | 12 \sqrt[3]{4} | 0.5 |
Let \( a \) and \( b \) be real numbers such that there exist distinct real numbers \( m \), \( n \), and \( p \) satisfying the following equations:
$$
\left\{
\begin{array}{l}
m^{3} + a m + b = 0 \\
n^{3} + a n + b = 0 \\
p^{3} + a p + b = 0
\end{array}
\right.
$$
Show that \( m + n + p = 0 \). | m+n+p=0 | 0.875 |
ABCD is a convex quadrilateral with AB = BC, ∠CBD = 2 ∠ADB, and ∠ABD = 2 ∠CDB. Show that AD = DC. | AD = DC | 0.875 |
In $\triangle ABC$, point $O$ is the midpoint of $BC$. A line passing through point $O$ intersects line segments $AB$ and $AC$ at different points $M$ and $N$, respectively. If $\overrightarrow{AB} = m \overrightarrow{AM}$ and $\overrightarrow{AC} = n \overrightarrow{AN}$, find the value of $m + n$. | 2 | 0.875 |
Let \( p \) and \( q \) be integers such that \( p + q = 2010 \). If both roots of the equation \( 10x^{2} + px + q = 0 \) are positive integers, find the sum of all possible values of \( p \). | -3100 | 0.75 |
\(ABCD\) is a rectangle with \(AB = 2\) and \(BC = 1\). A point \(P\) is randomly selected on \(CD\). Find the probability that \(\angle APB\) is the largest among the three interior angles of \(\triangle PAB\). | \sqrt{3}-1 | 0.875 |
Five cards labeled A, B, C, D, and E are placed consecutively in a row. How many ways can they be re-arranged so that no card is moved more than one position away from where it started? (Not moving the cards at all counts as a valid re-arrangement.) | 8 | 0.375 |
Let \( a, b, c, d \) be any positive real numbers. Determine the range of the sum
$$
S = \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d}.
$$ | (1, 2) | 0.375 |
Calculate the area of the figure bounded by the graphs of the functions:
$$
x=4-(y-1)^{2}, \quad x=y^{2}-4 y+3
$$ | 9 | 0.875 |
Given three positive numbers \(a, b, c\) satisfying \(a \leqslant b+c \leqslant 3a\), \(3b^{2} \leqslant a(a+c) \leqslant 5b^{2}\), find the minimum value of \(\frac{b-2c}{a}\). | -\frac{18}{5} | 0.625 |
How many integers from 1 to 33000 are there that are neither divisible by 3 nor by 5, but are divisible by 11? | 1600 | 0.75 |
Solve the inequality
$$
\sqrt{3 x-7}-\sqrt{3 x^{2}-13 x+13} \geqslant 3 x^{2}-16 x+20
$$
In the answer, indicate the sum of all integer values \( x \) that satisfy the inequality. | 3 | 0.875 |
Find the remainders of the following divisions:
a) $19^{10}$ by 6
b) $19^{14}$ by 70
c) $17^{9}$ by $48$
d) $14^{14^{14}}$ by 100 | 36 | 0.875 |
The fishermen caught several crucian carps and pikes. Each fisherman caught as many crucian carps as the total number of pikes caught by all the other fishermen. How many fishermen were there if the total number of crucian carps caught is 10 times the total number of pikes caught? Justify your answer. | 11 | 0.875 |
Find all prime numbers \( p \) such that both \( 4p^2 + 1 \) and \( 6p^2 + 1 \) are also prime numbers.
| 5 | 0.25 |
Anton thought of a three-digit number, and Alex is trying to guess it. Alex successively guessed the numbers 109, 704, and 124. Anton observed that each of these numbers matches the thought number exactly in one digit place. What number did Anton think of? | 729 | 0.625 |
Let \( S(n) \) represent the sum of the digits of the non-negative integer \( n \). For example, \( S(1997) = 1 + 9 + 9 + 7 = 26 \). Find the value of \( S(1) + S(2) + \cdots + S(2012) \). | 28077 | 0.125 |
A three-digit number \( X \) was composed of three different digits, \( A, B, \) and \( C \). Four students made the following statements:
- Petya: "The largest digit in the number \( X \) is \( B \)."
- Vasya: "\( C = 8 \)."
- Tolya: "The largest digit is \( C \)."
- Dima: "\( C \) is the arithmetic mean of the digits \( A \) and \( B \)."
Find the number \( X \), given that exactly one of the students was mistaken. | 798 | 0.875 |
A boy named Vasya wrote down the non-zero coefficients of a 9th degree polynomial \( P(x) \) in a notebook. Then, he calculated the derivative of the resulting polynomial and wrote down its non-zero coefficients, continuing this process until he obtained a constant, which he also wrote down.
What is the minimum number of different numbers that he could have written down?
Coefficients are recorded with their signs, and constant terms are written down as well. If there is a monomial of the form \( \pm x^n \), it is written as \( \pm 1 \). | 9 | 0.25 |
A stalker throws a small nut from the Earth's surface at an angle of $\alpha=30^{\circ}$ to the horizontal with an initial speed $v_{0}=10 \, \mathrm{m}/\mathrm{s}$. The normal acceleration due to gravity is $g=10 \, \mathrm{m}/\mathrm{s}^{2}$. At the highest point of its trajectory, the nut enters a zone of gravitational anomaly (an area where the acceleration due to gravity changes its magnitude sharply), and continues to move within it. As a result, the nut falls to the ground at a distance $S=3 \sqrt{3} \, \mathrm{m}$ from the stalker. Determine the acceleration due to gravity inside the anomaly. (15 points) | 250 \, \text{m/s}^2 | 0.125 |
Find the largest positive integer \( n \) such that the inequality \(\frac{8}{15} < \frac{n}{n+k} < \frac{7}{13}\) holds for exactly one integer \( k \). (5th American Mathematics Invitational, 1987) | 112 | 0.75 |
Given that $\left\{a_{n}\right\}$ is a geometric series and $a_{1} a_{2017}=1$. If $f(x)=\frac{2}{1+x^{2}}$, then find the value of $f\left(a_{1}\right)+f\left(a_{2}\right)+f\left(a_{3}\right)+\cdots+f\left(a_{2017}\right)$. | 2017 | 0.875 |
In a tournament, any two players play against each other. Each player earns one point for a victory, 1/2 for a draw, and 0 points for a loss. Let \( S \) be the set of the 10 lowest scores. We know that each player obtained half of their score by playing against players in \( S \).
a) What is the sum of the scores of the players in \( S \)?
b) Determine how many participants are in the tournament.
Note: Each player plays only once with each opponent. | 25 | 0.625 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.