problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Find the smallest natural number that is greater than the sum of its digits by 1755. | 1770 | 0.875 |
Through the midpoints $M$ and $N$ of the edges $AD$ and $CC_1$ of the parallelepiped $ABCD A_1 B_1 C_1 D_1$, a plane is drawn parallel to the diagonal $DB_1$. Construct the section of the parallelepiped by this plane. In what ratio does it divide the edge $BB_1$? | 5:1 | 0.625 |
Compute the number of ordered pairs of positive integers \((a, b)\) satisfying the equation \(\operatorname{gcd}(a, b) \cdot a + b^2 = 10000\). | 99 | 0.625 |
The sets \( A \) and \( B \) form a partition of positive integers if \( A \cap B = \emptyset \) and \( A \cup B = \mathbb{N} \). The set \( S \) is called prohibited for the partition if \( k + l \neq s \) for any \( k, l \in A, s \in S \) and any \( k, l \in B, s \in S \).
a) Define Fibonacci numbers \( f_{i} \) by letting \( f_{1} = 1, f_{2} = 2 \) and \( f_{i+1} = f_{i} + f_{i-1} \), so that \( f_{3} = 3, f_{4} = 5 \), etc. How many partitions for which the set \( F \) of all Fibonacci numbers is prohibited are there? (We count \( A, B \) and \( B, A \) as the same partition.)
b) How many partitions for which the set \( P \) of all powers of 2 is prohibited are there? What if we require in addition that \( P \subseteq A \)? | 1 | 0.75 |
I1.1 Find \( a \) if \( 2t+1 \) is a factor of \( 4t^{2}+12t+a \).
I1.2 \( \sqrt{K} \) denotes the nonnegative square root of \( K \), where \( K \geq 0 \). If \( b \) is the root of the equation \( \sqrt{a-x} = x-3 \), find \( b \).
I1.3 If \( c \) is the greatest value of \( \frac{20}{b + 2 \cos \theta} \), find \( c \).
I1.4 A man drives a car at \( 3c \ \text{km/h} \) for 3 hours and then \( 4c \ \text{km/h} \) for 2 hours. If his average speed for the whole journey is \( d \ \text{km/h} \), find \( d \). | d = 34 | 0.625 |
One way to pack a 100 by 100 square with 10000 circles, each of diameter 1, is to put them in 100 rows with 100 circles in each row. If the circles are repacked so that the centers of any three tangent circles form an equilateral triangle, what is the maximum number of additional circles that can be packed? | 1443 | 0.125 |
The non-negative numbers \(a, b, c, d, e, f, g\) have a sum of 1. Select the largest value among the sums \(a+b+c, b+c+d, c+d+e, d+e+f, e+f+g\). What is the minimum value of this largest sum obtained? | \frac{1}{3} | 0.625 |
Given a triangle \( \triangle ABC \) with interior angles \( \angle A, \angle B, \angle C \) and opposite sides \( a, b, c \) respectively, where \( \angle A - \angle C = \frac{\pi}{2} \) and \( a, b, c \) are in arithmetic progression, find the value of \( \cos B \). | \frac{3}{4} | 0.75 |
For \( x, y, z > 0 \), find the maximum value of the expression
\[
A = \frac{(x-y) \sqrt{x^{2}+y^{2}}+(y-z) \sqrt{y^{2}+z^{2}}+(z-x) \sqrt{z^{2}+x^{2}}+\sqrt{2}}{(x-y)^{2}+(y-z)^{2}+(z-x)^{2}+2}
\] | \frac{1}{\sqrt{2}} | 0.125 |
In trapezoid $ABCD$, points $E$ and $F$ are marked on the bases $AD = 17$ and $BC = 9$ respectively, such that $MENF$ is a rectangle, where $M$ and $N$ are the midpoints of the diagonals of the trapezoid. Find the length of segment $EF$. | 4 | 0.25 |
As shown in the figure, point $D$ is the midpoint of side $BC$ of $\triangle ABC$. Points $E$ and $F$ lie on $AB$, with $AE = \frac{1}{3} AB$ and $BF = \frac{1}{4} AB$. If the area of $\triangle ABC$ is $2018$, what is the area of $\triangle DEF$? | \frac{5045}{12} | 0.75 |
Calculate the area of the figure bounded by the curves given by the equations:
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=8 \cos ^{3} t \\
y=8 \sin ^{3} t
\end{array}\right. \\
& x=1(x \geq 1)
\end{aligned}
$$ | 8\pi | 0.125 |
What is the maximum number of colors that can be used to color the cells of an 8x8 chessboard such that each cell shares a side with at least two cells of the same color? | 16 | 0.375 |
\( N \) is an integer whose representation in base \( b \) is 777. Find the smallest positive integer \( b \) such that \( N \) is the fourth power of a decimal integer. | 18 | 0.875 |
Two overlapping triangles $P O R$ and $Q O T$ are such that points $P, Q, R,$ and $T$ lie on the arc of a semicircle of center $O$ and diameter $P Q$. Lines $Q T$ and $P R$ intersect at the point $S$. Angle $T O P$ is $3 x^{\circ}$ and angle $R O Q$ is $5 x^{\circ}$. Show that angle $R S Q$ is $4 x^{\circ}$. | 4x^\circ | 0.5 |
Let \( a \) and \( b \) be two real numbers. We set \( s = a + b \) and \( p = ab \). Express \( a^3 + b^3 \) in terms of \( s \) and \( p \) only. | s^3 - 3sp | 0.25 |
At the whistle of the physical education teacher, all 10 boys and 7 girls line up in a row in a random order as they manage.
Find the expected value of the quantity "Number of girls standing to the left of all boys". | \frac{7}{11} | 0.875 |
How many orderings \(\left(a_{1}, \ldots, a_{8}\right)\) of \((1, 2, \ldots, 8)\) exist such that \(a_{1} - a_{2} + a_{3} - a_{4} + a_{5} - a_{6} + a_{7} - a_{8} = 0\)? | 4608 | 0.25 |
Gru and the Minions plan to make money through cryptocurrency mining. They chose Ethereum as one of the most stable and promising currencies. They bought a system unit for 9499 rubles and two graphics cards for 31431 rubles each. The power consumption of the system unit is 120 W, and for each graphics card, it is 125 W. The mining speed for one graphics card is 32 million hashes per second, allowing it to earn 0.00877 Ethereum per day. 1 Ethereum equals 27790.37 rubles. How many days will it take for the team's investment to pay off, considering electricity costs of 5.38 rubles per kWh? (20 points) | 165 | 0.625 |
Two girls knit at constant, but different speeds. The first girl takes a tea break every 5 minutes, and the second girl every 7 minutes. Each tea break lasts exactly 1 minute. When the girls went for a tea break together, it turned out that they had knitted the same amount. By what percentage is the first girl's productivity higher if they started knitting at the same time? | 5\% | 0.125 |
Given a natural number \( x = 8^n - 1 \), where \( n \) is a natural number. It is known that \( x \) has exactly three distinct prime divisors, one of which is 31. Find \( x \). | 32767 | 0.75 |
Given a rhombus \(ABCD\), \(\Gamma_{B}\) and \(\Gamma_{C}\) are circles centered at \(B\) and \(C\) passing through \(C\) and \(B\) respectively. \(E\) is an intersection point of circles \(\Gamma_{B}\) and \(\Gamma_{C}\). The line \(ED\) intersects circle \(\Gamma_{B}\) at a second point \(F\). Find the measure of \(\angle AFB\). | 60^\circ | 0.125 |
Of the integers from 1 to \(8 \cdot 10^{20}\) (inclusive), which are more numerous, and by how much: those containing only even digits in their representation or those containing only odd digits? | \frac{5^{21}-5}{4} | 0.125 |
In triangle \(ABC\), the median \(AD\) and the angle bisector \(BE\) are perpendicular and intersect at point \(F\). It is known that \(S_{DEF} = 5\). Find \(S_{ABC}\). | 60 | 0.5 |
Given a positive geometric sequence $\left\{a_{n}\right\}$ satisfying $a_{7}=a_{6}+2a_{5}$, if there exist two terms $a_{m}$ and $a_{n}$ such that $\sqrt{a_{m} \cdot a_{n}}=4a_{1}$, find the minimum value of $\frac{1}{m}+\frac{4}{n}$. | \frac{3}{2} | 0.625 |
Petya has four cards with the digits $1, 2, 3, 4$. Each digit appears exactly once. How many natural numbers greater than 2222 can Petya form using these cards? | 16 | 0.75 |
The side of the base of a regular quadrilateral pyramid is equal to \( a \). A lateral face forms a 45° angle with the plane of the base. Find the lateral surface area of the pyramid. | a^2 \sqrt{2} | 0.75 |
How many quadruples (i, j, k, h) of ordered integers satisfy the condition \(1 \leqslant i < j \leqslant k < h \leqslant n+1\)? | \binom{n+2}{4} | 0.375 |
Given that the cube root of \( m \) is a number in the form \( n + r \), where \( n \) is a positive integer and \( r \) is a positive real number less than \(\frac{1}{1000}\). When \( m \) is the smallest positive integer satisfying the above condition, find the value of \( n \). | 19 | 0.875 |
Let \( n \) be a natural number. We call a sequence consisting of \( 3n \) letters Romanian if the letters \( I \), \( M \), and \( O \) all occur exactly \( n \) times. A swap is an exchange of two neighboring letters. Show that for every Romanian sequence \( X \) there exists a Romanian sequence \( Y \), so that at least \( \frac{3n^2}{2} \) swaps are necessary to obtain the sequence \( Y \) from the sequence \( X \). | \frac{3n^2}{2} | 0.5 |
Let \( x, y, z \) be positive real numbers such that:
\[ \begin{aligned}
& x^2 + xy + y^2 = 2 \\
& y^2 + yz + z^2 = 5 \\
& z^2 + xz + x^2 = 3
\end{aligned} \]
Determine the value of \( xy + yz + xz \). | 2 \sqrt{2} | 0.375 |
\[
\frac{\log_{a} b - \log_{\sqrt{a} / b^{3}} \sqrt{b}}{\log_{a / b^{4}} b - \log_{a / b^{6}} b} : \log_{b}\left(a^{3} b^{-12}\right)
\] | \log_{a} b | 0.875 |
Let the set \(I = \{0, 1, 2, \ldots, 22\}\). Define \(A = \{(a, b, c, d) \mid a, b, c, d \in I, a + d \equiv 1 \pmod{23}, \text{ and } a d - b c \equiv 0 \pmod{23}\}\). Determine the number of elements in the set \(A\). | 552 | 0.25 |
Given that bus types $A$, $B$, and $C$ all depart at 6:00 AM from a certain station, with departure intervals of 10 minutes, 12 minutes, and 15 minutes, respectively, if Xiaoming arrives at the station at some time between 8:00 AM and 12:00 PM to catch one of these buses, what is Xiaoming's average waiting time in minutes? | \frac{19}{6} | 0.375 |
What is the maximum number of L-shaped figures, consisting of 5 squares of size \(1 \times 1\), that can be placed in a \(7 \times 7\) square? (The L-shaped figures can be rotated and flipped, but cannot overlap with each other.) | 9 | 0.25 |
In a warehouse, there are 8 cabinets, each containing 4 boxes, and each box contains 10 mobile phones. The warehouse, each cabinet, and each box are locked with a key. The manager is tasked with retrieving 52 mobile phones. What is the minimum number of keys the manager must take with him? | 9 | 0.375 |
A positive integer \( n \) is said to be increasing if, by reversing the digits of \( n \), we get an integer larger than \( n \). For example, 2003 is increasing because, by reversing the digits of 2003, we get 3002, which is larger than 2003. How many four-digit positive integers are increasing? | 4005 | 0.375 |
Vojta bought 24 identical square tiles. Each tile had a side length of $40 \mathrm{~cm}$. Vojta wanted to arrange them in front of a cabin to form a rectangular platform with the smallest possible perimeter. What was the perimeter of the tiled rectangle in meters, given that no tile was leftover and Vojta did not cut or break any tiles? | 8 | 0.875 |
Find the product of two approximate numbers: $0.3862 \times 0.85$. | 0.33 | 0.625 |
Determine all pairs of positive integers \((m, n)\) such that \(2^m + 1 = n^2\). | (3, 3) | 0.875 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow -2} \frac{\operatorname{tg}\left(e^{x+2}-e^{x^{2}-4}\right)}{\operatorname{tg} x + \operatorname{tg} 2}
\] | 5 \cos^2 2 | 0.125 |
The transgalactic ship encountered an amazing meteor shower. One part of the meteors flies along a straight line with equal speeds, one after another, at equal distances from each other. Another part flies similarly but along another straight line, parallel to the first one, with the same speeds but in the opposite direction, at the same distance from each other. The ship flies parallel to these lines. Astronaut Gavrila recorded that the ship encounters meteors flying towards it every 7 seconds, and those flying in the same direction as the ship every 13 seconds. He wondered how often the meteors would pass by if the ship were stationary. He thought it necessary to take the arithmetic mean of the two given times. Is Gavrila right? If so, write this arithmetic mean as the answer. If not, indicate the correct time in seconds, rounded to one decimal place. | 9.1 | 0.25 |
In the set of the first ten thousand positive integers $\{1, 2, \cdots, 10000\}$, how many elements leave a remainder of 2 when divided by 3, a remainder of 3 when divided by 5, and a remainder of 4 when divided by 7? | 95 | 0.75 |
A \(101 \times 101\) grid is given, where all cells are initially colored white. You are allowed to choose several rows and paint all the cells in those rows black. Then, choose exactly the same number of columns and invert the color of all cells in those columns (i.e., change white cells to black and black cells to white). What is the maximum number of black cells that the grid can contain after this operation? | 5100 | 0.75 |
Given that complex numbers \( z_{1}, z_{2}, z_{3} \) satisfy \( \left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1 \) and \( \left|z_{1}+z_{2}+z_{3}\right|=r \), where \( r \) is a given real number, express the real part of \( \frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}} \) in terms of \( r \). | \frac{r^{2}-3}{2} | 0.125 |
What is the minimum number of colors required to paint all the vertices, edges, and diagonals of a convex \( n \)-gon, given that two conditions must be met:
1) Every two segments emanating from the same vertex must be of different colors;
2) The color of any vertex must differ from the color of any segment emanating from it? | n | 0.375 |
A school uses a sum of money to buy balls. If they only buy volleyballs, they can buy exactly 15 of them. If they only buy basketballs, they can buy exactly 12 of them. Now, with the same amount of money, they buy a total of 14 volleyballs and basketballs. How many more volleyballs than basketballs were bought? | 6 | 0.75 |
In the Cartesian coordinate system \(xOy\), points \(A\) and \(B\) lie on the parabola \(y^2 = 2x\) and satisfy \(\overrightarrow{OA} \cdot \overrightarrow{OB} = -1\). Let \(F\) be the focus of the parabola. Find the minimum value of \(S_{\triangle OFA} + S_{\triangle OFB} \). | \frac{\sqrt{2}}{2} | 0.875 |
There are three types of people, A, B, and C, with a total of 25 people. Type A people always tell the truth, Type B people always lie, and Type C people alternate between telling the truth and lying (for example, if a Type C person tells the truth this time, their next statement will certainly be a lie, and the statement after that will be true again).
The priest asks each person, "Are you a Type A person?" 17 people answer "yes."
The priest then asks each person, "Are you a Type C person?" 12 people answer "yes."
The priest finally asks each person, "Are you a Type B person?" 8 people answer "yes."
How many of the 25 people are Type C people? | 16 | 0.125 |
Arrange six positive integers \(a, b, c, d, e, f\) in a sequence in alphabetical order such that \(a=1\). If any positive integer is greater than 1, then the number that is one less than this integer must appear to its left. For example, \(1,1,2,1,3,2\) meets the requirement; \(1,2,3,1,4,1\) meets the requirement; \(1,2,2,4,3,2\) does not meet the requirement. Find the number of different sequences that meet the requirement. | 203 | 0.125 |
The monkeys - Masha, Dasha, Glasha, and Natasha - ate 16 bowls of semolina porridge for lunch. Each monkey had some portion of it. Glasha and Natasha together ate 9 portions. Masha ate more than Dasha, more than Glasha, and more than Natasha. How many bowls of porridge did Dasha get? | 1 | 0.875 |
a) Determine the number of digits in the product \(111111 \cdot 1111111111\), where the first factor has 6 digits and the second has 10 digits.
b) The numbers \(2^{2016}\) and \(5^{2016}\) are written side by side to form a single number \(N\) which has a number of digits that is the sum of the number of digits of the two numbers. For example, if we did this with \(2^{3}\) and \(5^{3}\), we would obtain the number 8125, which has 4 digits. Determine the number of digits of \(N\). | 2017 | 0.5 |
The sum of two nonzero natural numbers is 210, and their least common multiple is 1547. What is their product? $\qquad$ | 10829 | 0.5 |
It is given that \( x = \frac{1}{2 - \sqrt{3}} \). Find the value of
\[
x^{6} - 2 \sqrt{3} x^{5} - x^{4} + x^{3} - 4 x^{2} + 2 x - \sqrt{3}.
\] | 2 | 0.75 |
For which (real) values of the variable \( x \) are the following equalities valid:
a) \(\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=\sqrt{2}\),
b) \(\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=1\),
c) \(\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=2\).
(Without sign, the square root always means the non-negative square root!) | \frac{3}{2} | 0.75 |
In how many ways can two people divide 10 distinct items between themselves so that each person gets 5 items? | 252 | 0.625 |
Given the equation \(x^{2} - 402x + k = 0\), one root plus 3 is equal to 80 times the other root. Determine the value of \(k\). | 1985 | 0.875 |
Given a parallelogram \(ABCD\) with sides \(AB=2\) and \(BC=3\), find the area of this parallelogram, given that the diagonal \(AC\) is perpendicular to the segment \(BE\), where \(E\) is the midpoint of side \(AD\). | \sqrt{35} | 0.875 |
In a cube \( ABCD-A_1 B_1 C_1 D_1 \) with edge length 1, points \( X \) and \( Y \) are the centers of the squares \( AA_1 B_1 B \) and \( BB_1 C_1 C \) respectively. Point \( Z \) lies on the diagonal \( BD \) such that \( DZ = 3 ZB \). Determine the area of the cross-section cut by the plane \( XYZ \) on the circumscribed sphere of the cube. | \frac{5 \pi}{8} | 0.5 |
Triangle \( \triangle ABC \) has area 1. Points \( E \) and \( F \) are on \( AB \) and \( AC \) respectively such that \( EF \parallel BC \). If \( \triangle AEF \) and \( \triangle EBC \) have equal areas, find the area of \( \triangle EFC \). | \sqrt{5} - 2 | 0.875 |
In $\triangle ABC$, $\angle ABC = 40^\circ$, $\angle ACB = 20^\circ$. Point $N$ is inside $\triangle ABC$, $\angle NBC = 30^\circ$, and $\angle NAB = 20^\circ$. Find the measure of $\angle NCB$. | 10^\circ | 0.875 |
At the first site, higher-class equipment was used, and at the second site, first-class equipment was used. There was less higher-class equipment than first-class equipment. First, 40% of the equipment from the first site was transferred to the second site. Then, 20% of the equipment at the second site was transferred back to the first site, with half of the transferred equipment being first-class. After this, the amount of higher-class equipment at the first site was 26 units more than at the second site, and the total amount of equipment at the second site increased by more than 5% compared to the original amount. Find the total amount of first-class equipment. | 60 | 0.625 |
From the first 539 positive integers, we select some such that their sum is at least one-third of the sum of the original numbers. What is the minimum number of integers we need to select for this condition to be satisfied? | 99 | 0.375 |
Solve the problem: Octopuses with an even number of legs always lie, while octopuses with an odd number always tell the truth. Five octopuses met, each having between 7 to 9 legs.
The first said, "Together we have 36 legs";
The second said, "Together we have 37 legs";
The third said, "Together we have 38 legs";
The fourth said, "Together we have 39 legs";
The fifth said, "Together we have 40 legs".
How many legs did they actually have? | 39 | 0.875 |
Let \( a \) and \( b \) be integers such that the difference between \( a^2 \) and \( b^2 \) is 144. Determine the largest possible value of \( d = a + b \). | 72 | 0.625 |
Given the non-negative real numbers \(x, y, z\) that satisfy \(x + y + z = 1\). Find the maximum and minimum values of \(x^3 + 2y^2 + \frac{10}{3}z\). | \frac{14}{27} | 0.375 |
Find the largest integer \( n \) such that \(\frac{(n-2)^{2}(n+1)}{2n-1}\) is an integer. | 14 | 0.875 |
Given real numbers \( x, y, z, w \) satisfying \( x + y + z + w = 1 \), find the maximum value of \( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \). | \frac{3}{2} | 0.75 |
Find the measure of the angle
$$
\delta=\arccos \left(\left(\sin 2539^{\circ}+\sin 2540^{\circ}+\cdots+\sin 6139^{\circ}\right)^{\cos } 2520^{\circ}+\cos 2521^{\circ}+\cdots+\cos 6120^{\circ}\right)
$$ | 71^\circ | 0.5 |
Find all values of the parameter \(a\) for which there exists a number \(b\) such that the system
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+2 a(a-x-y)=64 \\
y=8 \sin (x-2 b)-6 \cos (x-2 b)
\end{array}\right.
$$
has at least one solution \((x, y)\). | a \in [-18, 18] | 0.125 |
The number \( n \) is such that \( 8d \) is a 100-digit number, and \( 81n - 102 \) is a 102-digit number. What can be the second digit from the beginning of \( n \)? | 2 | 0.375 |
Assume that \( a_{i} \in \{1, -1\} \) for all \( i=1, 2, \ldots, 2013 \). Find the least positive value of the following expression:
\[
\sum_{1 \leq i < j \leq 2013} a_{i} a_{j}
\] | 6 | 0.875 |
One side of the cards is painted in some color, and the other has a smiley face. There are four cards in front of you: the first is yellow, the second is black, the third has a happy smiley, and the fourth has a sad smiley. You need to check the statement: "If there is a happy smiley on one side of a card, then the other side is painted yellow."
What is the minimum number of cards you need to turn over to check the truth of this statement? Which ones? (7 points) | 2 | 0.75 |
A polygon is said to be friendly if it is regular and it also has angles that, when measured in degrees, are either integers or half-integers (i.e., have a decimal part of exactly 0.5). How many different friendly polygons are there? | 28 | 0.5 |
How many students are there in our city? The number expressing the quantity of students is the largest of all numbers where any two adjacent digits form a number that is divisible by 23. | 46923 | 0.625 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow 3}\left(2-\frac{x}{3}\right)^{\sin (\pi x)}
\] | 1 | 0.875 |
Given six points $A$, $B$, $C$, $D$, $E$, $F$ in space with no four points coplanar. What is the maximum number of line segments that can be drawn such that no tetrahedron is formed in the figure? | 12 | 0.875 |
In an acute-angled triangle \( ABC \), the altitude \( BH \) and the median \( AM \) are drawn. It is known that the angle \( MCA \) is twice the angle \( MAC \), and \( BC = 10 \).
Find \( AH \). | 5 | 0.875 |
Petya plans to spend all 90 days of his vacation in the village. Every second day (i.e., every other day), he goes swimming in the lake. Every third day, he goes to the store for groceries. Every fifth day, he solves math problems. (On the first day, Petya did all three activities and was very tired.) How many "pleasant" days will Petya have, when he has to go swimming but does not need to go to the store or solve problems? How many "boring" days will there be when there are no activities at all? | 24 | 0.5 |
Among all natural numbers that are multiples of 20, what is the sum of those that do not exceed 3000 and are also multiples of 14? | 32340 | 0.875 |
A semicircle is constructed over the line segment $AB$ with midpoint $M$. Let $P$ be a point on the semicircle other than $A$ and $B$, and let $Q$ be the midpoint of the arc $AP$. The intersection of the line $BP$ with the line parallel to $PQ$ through $M$ is denoted as $S$.
Show that $PM = PS$.
(Karl Czakler) | PM = PS | 0.875 |
Given that \( n \) is a positive integer and \( n! \) denotes the factorial of \( n \), find all positive integer solutions \((x, y)\) to the equation
\[
20(x!) + 2(y!) = (2x + y)!
\] | (1,2) | 0.75 |
In quadrilateral \(ABCD\), it is given that \(\angle DAB = 150^\circ\), \(\angle DAC + \angle ABD = 120^\circ\), and \(\angle DBC - \angle ABD = 60^\circ\). Find \(\angle BDC\). | 30^\circ | 0.875 |
On the radius \( AO \) of a circle with center \( O \), a point \( M \) is chosen. On the same side of \( AO \) on the circle, points \( B \) and \( C \) are chosen so that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \), given that the radius of the circle is 12 and \( \sin \alpha = \frac{\sqrt{11}}{6} \). | 20 | 0.125 |
A group of monkeys is divided into three subgroups, with each subgroup having an equal number of monkeys. They go to the orchard to pick peaches. After finishing the picking, they combine all the peaches and distribute them equally among all the monkeys. If each monkey receives 5 peaches, there are 27 peaches left over; if each monkey receives 7 peaches, then at least one monkey receives fewer than 7 peaches. What is the total number of peaches this group of monkeys has picked? | 102 | 0.5 |
Grisha wrote 100 numbers on the board. Then he increased each number by 1 and noticed that the product of all 100 numbers did not change. He increased each number by 1 again, and again the product of all the numbers did not change, and so on. Grisha repeated this procedure $k$ times, and each of the $k$ times the product of the numbers did not change. Find the largest possible value of $k$. | 99 | 0.625 |
Given the equation
\[ x^{2} + ax + b - 3 = 0 \quad (a, b \in \mathbf{R}) \]
has a real root in the interval \([1, 2]\), find the minimum value of \( a^{2} + (b - 4)^{2} \). | 2 | 0.875 |
Given \( x = -2272 \), \( y = 10^3 + 10^2 c + 10 b + a \), and \( z = 1 \), which satisfy the equation \( a x + b y + c z = 1 \), where \( a \), \( b \), \( c \) are positive integers and \( a < b < c \). Find \( y \). | 1987 | 0.25 |
There are 4 different digits that can form 18 different four-digit numbers arranged in ascending order. The first four-digit number is a perfect square, and the second-last four-digit number is also a perfect square. What is the sum of these two numbers? | 10890 | 0.75 |
Find the minimum value of \( a^{2} + b^{2} + c^{2} + d^{2} \) given that \( a + 2b + 3c + 4d = 12 \). | \frac{24}{5} | 0.25 |
In a trapezoid, the diagonals intersect at a right angle, and one of them is equal to the midsegment. Determine the angle that this diagonal forms with the bases of the trapezoid. | 60^\circ | 0.5 |
Inside a square, point $P$ has distances $a, b, c$ from vertices $A, B, C$ respectively. What is the area of the square? What is it in the case when $a = b = c$? | 2a^2 | 0.875 |
When \((1+x)^{38}\) is expanded in ascending powers of \(x\), \(N_{1}\) of the coefficients leave a remainder of 1 when divided by 3, while \(N_{2}\) of the coefficients leave a remainder of 2 when divided by 3. Find \(N_{1} - N_{2}\). | 4 | 0.25 |
On a plate, there are various pancakes with three different fillings: 2 with meat, 3 with cottage cheese, and 5 with strawberries. Sveta consecutively ate all of them, choosing each next pancake at random. Find the probability that the first and the last pancakes she ate had the same filling. | \frac{14}{45} | 0.75 |
Given the sequence \(\left\{a_{n}\right\}\) defined by:
\[
\begin{array}{l}
a_{1}=2, a_{2}=6, \\
a_{n+1}=\frac{a_{n}^{2}-2 a_{n}}{a_{n-1}} \text{ for } n=2,3, \ldots
\end{array}
\]
Determine \(\lim _{n \rightarrow \infty}\left\{\sqrt{a_{n} + n}\right\} = \) | 1 | 0.75 |
In the parallelogram \(ABCD\), point \(K\) is the midpoint of side \(BC\), and point \(M\) is the midpoint of side \(CD\). Find \(AD\) if \(AK = 6\) cm, \(AM = 3\) cm, and \(\angle KAM = 60^\circ\). | 4 | 0.75 |
In a classroom, 34 students are seated in 5 rows of 7 chairs. The place at the center of the room is unoccupied. A teacher decides to reassign the seats such that each student will occupy a chair adjacent to his/her present one (i.e., move one desk forward, back, left, or right). In how many ways can this reassignment be made? | 0 | 0.75 |
Given a point \( P \) inside \( \triangle ABC \), perpendiculars are drawn from \( P \) to \( BC, CA, \) and \( AB \) with feet \( D, E, \) and \( F \) respectively. Semicircles are constructed externally on diameters \( AF, BF, BD, CD, CE, \) and \( AE \). These six semicircles have areas denoted \( S_1, S_2, S_3, S_4, S_5, \) and \( S_6 \). Given that \( S_5 - S_6 = 2 \) and \( S_1 - S_2 = 1 \), find \( S_4 - S_3 \). | 3 | 0.875 |
In a certain region are five towns: Freiburg, Göttingen, Hamburg, Ingolstadt, and Jena.
On a certain day, 40 trains each made a journey, leaving one of these towns and arriving at one of the other towns. Ten trains traveled either from or to Freiburg. Ten trains traveled either from or to Göttingen. Ten trains traveled either from or to Hamburg. Ten trains traveled either from or to Ingolstadt.
How many trains traveled from or to Jena?
A) 0
B) 10
C) 20
D) 30
E) 40 | 40 | 0.5 |
Find all functions \( f \) from the set of real numbers to the set of real numbers that satisfy the following conditions:
1. \( f(x) \) is strictly increasing;
2. For all real numbers \( x \), \( f(x) + g(x) = 2x \), where \( g(x) \) is the inverse function of \( f(x) \). | f(x) = x + c | 0.625 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.