problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01?
|
11475
| 0.625 |
Given \( n \in \mathbb{N}^{*} \). Find the number of \( n \)-digit numbers whose digits belong to \(\{2, 3, 7, 9\}\) and are divisible by 3.
|
\frac{4^n + 2}{3}
| 0.625 |
In an arithmetic sequence \(\{a_{n}\}\), if \(\frac{a_{11}}{a_{10}} < -1\) and the sum of the first \(n\) terms \(S_{n}\) has a maximum value, then the value of \(n\) when \(S_{n}\) attains its smallest positive value is \(\qquad\).
|
19
| 0.875 |
There are two circles: one with center at point \( A \) and radius 6, and the other with center at point \( B \) and radius 3. Their common internal tangent touches the circles respectively at points \( C \) and \( D \). The lines \( AB \) and \( CD \) intersect at point \( E \). Find the length of \( CD \), given that \( AE = 10 \).
|
12
| 0.25 |
\(\frac{\sin 22^{\circ} \cos 8^{\circ}+\cos 158^{\circ} \cos 98^{\circ}}{\sin 23^{\circ} \cos 7^{\circ}+\cos 157^{\circ} \cos 97^{\circ}}\).
|
1
| 0.75 |
On a line \( r \), points \( A \) and \( B \) are marked, and on a line \( s \), parallel to \( r \), points \( C \) and \( D \) are marked so that \( A B C D \) forms a square. Point \( E \) is also marked on the segment \( C D \).
a) What is the ratio between the areas of triangles \( A B E \) and \( B C D \), if \( E \) is the midpoint of \( C D \)?
b) What is the ratio \( \frac{D E}{E C} \), for the area of triangle \( B F E \) to be twice the area of \( D F E \), where \( F \) is the intersection of segments \( A E \) and \( B D \)?
|
1
| 0.75 |
Find the number that becomes a perfect square either by adding 5 or by subtracting 11.
|
20
| 0.25 |
Determine all real values of \( A \) for which there exist distinct complex numbers \( x_{1} \) and \( x_{2} \) such that the following three equations hold:
\[
\begin{aligned}
x_{1}(x_{1}+1) &= A, \\
x_{2}(x_{2}+1) &= A, \\
x_{1}^{4} + 3x_{1}^{3} + 5x_{1} &= x_{2}^{4} + 3x_{2}^{3} + 5x_{2}.
\end{aligned}
\]
|
-7
| 0.875 |
For all \( x \in \left(0, \frac{\pi}{2}\right) \), find the largest positive integer \( n \) such that the inequality \( \sin^{n} x + \cos^{n} x > \frac{1}{2} \) holds.
|
3
| 0.875 |
If \( a \) and \( b \) are prime numbers greater than 7, then the expression
$$
\left(a^{2}-1\right)\left(b^{2}-1\right)\left(a^{6}-b^{6}\right)
$$
is divisible by 290304.
|
290304
| 0.5 |
A trapezoid with side lengths \( a \) and \( b \) is circumscribed around a circle. Find the sum of the squares of the distances from the center of the circle to the vertices of the trapezoid.
|
a^2 + b^2
| 0.125 |
Show that as \( t \rightarrow \infty \), the limit of the variable \( x = \frac{6t^3 - 9t + 1}{2t^3 - 3t} \) is 3.
|
3
| 0.75 |
How many real solutions are there to the equation
\[ |||| x|-2|-2|-2|=|||| x|-3|-3|-3| ? \]
|
6
| 0.375 |
Let \(ABCD\) be a quadrilateral inscribed in a circle with diameter \(\overline{AD}\). If \(AB = 5\), \(AC = 6\), and \(BD = 7\), find \(CD\).
|
\sqrt{38}
| 0.75 |
Let \( n \) be a natural number. For any real numbers \( x, y, z \), if the inequality \(\left(x^{2}+y^{2}+z^{2}\right) \leqslant n\left(x^{4}+y^{4}+z^{4}\right)\) always holds, then the smallest value of \( n \) is _____.
|
3
| 0.625 |
If \( 2^{200} \cdot 2^{203} + 2^{163} \cdot 2^{241} + 2^{126} \cdot 2^{277} = 32^{n} \), what is the value of \( n \)?
|
81
| 0.875 |
In how many ways can the numbers \(1, 2, 3, 4, 5, 6\) be arranged in a row so that for any three consecutive numbers \(a, b, c\), the expression \(ac - b^2\) is divisible by 7? Answer: 12.
|
12
| 0.875 |
Let \( X = \{1, 2, 3, \ldots, 17\} \). Find the number of subsets \( Y \) of \( X \) with odd cardinalities.
|
65536
| 0.75 |
Jack Sparrow needed to distribute 150 piastres across 10 purses. After placing a certain amount of piastres in the first purse, he placed more piastres in each subsequent purse than in the previous one. As a result, the number of piastres in the first purse was not less than half the number of piastres in the last purse. How many piastres are in the 6th purse?
|
16
| 0.375 |
Formulate the equation of the plane \( Q \), which intercepts a segment \( OA = 3 \) on the \( OX \) axis and is perpendicular to the vector \( \bar{n} \{2, -3, 1\} \).
|
2 x - 3 y + z = 6
| 0.125 |
In the equation, $\overline{\mathrm{ABCD}}+\overline{\mathrm{EFG}}=2020$, different letters represent different digits. What is $A+B+C+D+E+F+G=$ $\qquad$?
|
31
| 0.625 |
Given \( n \in \mathbb{N} \) and \( a \in [0, n] \), under the condition
$$
\sum_{i=1}^{n} \sin ^{2} x_{i}=a
$$
find the maximum value of \( \left| \sum_{i=1}^{n} \sin 2 x_{i} \right| \).
|
2 \sqrt{a(n - a)}
| 0.75 |
In an isosceles trapezoid with bases \(a = 21\), \(b = 9\) and height \(h = 8\), find the radius of the circumscribed circle.
|
\frac{85}{8}
| 0.625 |
Petrov booked an apartment in a newly built house, which has five identical entrances. Initially, the entrances were numbered from left to right, and Petrov's apartment number was 636. Later, the developer changed the numbering to the opposite direction (right to left, as shown in the diagram). Then, Petrov's apartment number became 242. How many apartments are in the building? (The numbering of apartments within each entrance has not changed.)
|
985
| 0.75 |
It is required to make a box with a square base for placing boxes that are 9 cm wide and 21 cm long. What should be the minimum length of the side of the square base so that the boxes fit in the box snugly?
|
63 \text{ cm}
| 0.875 |
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 |
Find all positive integers \( n \) that satisfy
$$
n=2^{2x-1}-5x-3=\left(2^{x-1}-1\right)\left(2^x+1\right)
$$
for some positive integer \( x \).
|
2015
| 0.375 |
Let $K$ be the incenter of $\triangle ABC$. Points $C_{1}$ and $B_{1}$ are the midpoints of sides $AB$ and $AC$, respectively. The line $AC$ intersects $C_{1}K$ at point $B_{2}$, and the line $AB$ intersects $B_{1}K$ at point $C_{2}$. If the area of $\triangle AB_{2}C_{2}$ equals the area of $\triangle ABC$, find $\angle CAB$.
|
60^\circ
| 0.75 |
Three numbers are given. If each of them is increased by 1, their product also increases by 1. If all the original numbers are increased by 2, their product also increases by 2. Find these numbers.
|
-1, -1, -1
| 0.625 |
Find the smallest natural number that is simultaneously twice an exact square and three times an exact cube.
|
648
| 0.875 |
How many digits are both prime and represented by a prime number of illuminated bars?
A 0
B 1
C 2
D 3
E 4
|
4
| 0.375 |
The sequence \( \{a_n\} \) satisfies: \( a_1 = 1 \), and for each \( n \in \mathbf{N}^{*} \), \( a_n \) and \( a_{n+1} \) are the roots of the equation \( x^2 + 3n x + b_n = 0 \). Find the value of \( \sum_{k=1}^{20} b_k \).
|
6385
| 0.375 |
Let \(ABCD\) be a convex quadrilateral inscribed in a circle with center \(O\). Let \(P\) be the intersection point of the diagonals and \(Q\) the second intersection point of the circumcircles of triangles \(APD\) and \(BPC\).
Show that \(\widehat{OQP} = 90^\circ\).
|
90^\circ
| 0.875 |
At 9 a.m., ships "Anin" and "Vanin" departed from port O to port E. At the same moment, the ship "Sanin" set off from port E to port O. All three vessels are traveling along the same course (with "Sanin" heading towards "Anin" and "Vanin") at constant but different speeds. At 10 a.m., "Vanin" was equidistant from both "Anin" and "Sanin." At 10:30 a.m., "Sanin" was equidistant from both "Anin" and "Vanin." At what moment will "Anin" be exactly in the middle between "Vanin" and "Sanin"?
|
12:00
| 0.125 |
As shown in the figure, in a square \(ABCD\), \(AB=1\). Let the midpoints of \(AB\) and \(BC\) be \(E\) and \(F\) respectively. If line \(CE\) intersects line \(AF\) at point \(G\), find the area of quadrilateral \(GEBF\).
|
\frac{1}{6}
| 0.75 |
An experienced sawmiller, Harik, can make cuts. In one day of continuous work, he cuts 600 nine-meter logs into equal three-meter logs (the only difference from the original logs is their length). How much time will it take for the experienced sawmiller Harik to cut 400 twelve-meter logs (which only differ in length from the nine-meter logs) into the same three-meter logs?
|
1 \text{ day}
| 0.875 |
Given a set $M$ of $n$ points on a plane, if every three points in $M$ form the vertices of an equilateral triangle, find the maximum value of $n$.
|
3
| 0.75 |
Find \( x \) and \( y \), if
\[
\sqrt[5]{119287 - 48682 \sqrt{6}} = x + y \sqrt{6}
\]
|
x = 7, \; y = -2
| 0.75 |
If the side length of an equilateral triangle \( \triangle ABC \) is 6 and the distances from its three vertices to the plane \( \alpha \) are 1, 2, and 3 respectively, find the distance from the centroid \( G \) of \( \triangle ABC \) to the plane \( \alpha \).
|
2
| 0.75 |
There are 20 numbers arranged in a circle. It is known that the sum of any six consecutive numbers is 24. What number is in the 12th position if the number in the 1st position is 1?
|
7
| 0.125 |
A prime number \( p \) is such that the number \( p + 25 \) is the seventh power of a prime number. What can \( p \) be? List all possible options.
|
103
| 0.75 |
Find the product of all divisors of the number \( N \), given that there are \( n \) divisors.
|
N^{\frac{n}{2}}
| 0.625 |
Find the sum of the first 10 elements that are present both in the arithmetic progression $\{5, 8, 11, 14, \ldots\}$ and in the geometric progression $\{10, 20, 40, 80, \ldots\}$.
|
6990500
| 0.25 |
Given three points \( A, B, C \) on a plane such that \( |\overrightarrow{AB}| = 3 \), \( |\overrightarrow{BC}| = 4 \), \( |\overrightarrow{CA}| = 5 \), find the value of \( \overrightarrow{AB} \cdot \overrightarrow{BC} + \overrightarrow{BC} \cdot \overrightarrow{CA} + \overrightarrow{CA} \cdot \overrightarrow{AB} \).
|
-25
| 0.75 |
The villages "Verkhnie Vasyuki" and "Nizhnie Vasyuki" are located on the riverbank. A steamboat travels the distance from Verkhnie Vasyuki to Nizhnie Vasyuki in one hour, while a motorboat covers the same distance in 45 minutes. It is known that the speed of the motorboat in still water is twice the speed of the steamboat (also in still water). Determine the time (in minutes) required for a raft to drift from Verkhnie Vasyuki to Nizhnie Vasyuki.
|
90 \text{ minutes}
| 0.875 |
Given \(a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n}\) with \(a_{n} \neq a_{n}\), \(\sum_{i=1}^{n} x_{i}=0\), \(\sum_{i=1}^{n}|x_{i}|=1\), find the minimum value of \(\lambda\) such that the inequality \(\left|\sum_{i=1}^{n} a_{i} x_{i}\right| \leqslant \lambda(a_{1}-a_{n})\) always holds.
|
\frac{1}{2}
| 0.875 |
Consider a 2 × 2 grid of squares. Each of the squares will be colored with one of 10 colors, and two colorings are considered equivalent if one can be rotated to form the other. How many distinct colorings are there?
|
2530
| 0.5 |
Find the range of the function \( f(x) = 2 \sin \left( \left( \frac{\pi}{4} \right) \sin (\sqrt{x-2} + x + 2) - \frac{5\pi}{2} \right) \).
|
[-2, -\sqrt{2}]
| 0.25 |
The numbers \( p, q, r, \) and \( t \) satisfy \( p < q < r < t \). When these numbers are paired, each pair has a different sum and the four largest sums are 19, 22, 25, and 28. What is the sum of the possible values for \( p \)?
|
\frac{17}{2}
| 0.125 |
Vijay chooses three distinct integers \(a, b, c\) from the set \(\{1,2,3,4,5,6,7,8,9,10,11\}\). If \(k\) is the minimum value taken on by the polynomial \(a(x-b)(x-c)\) over all real numbers \(x\), and \(l\) is the minimum value taken on by the polynomial \(a(x-b)(x+c)\) over all real numbers \(x\), compute the maximum possible value of \(k-l\).
|
990
| 0.625 |
Márcia is in a store buying a recorder she has wanted for a long time. When the cashier registers the price, she exclaims: "It's not possible, you have recorded the number backwards, you swapped the order of two digits, I remember that last week it cost less than 50 reais!" The cashier responds: "I'm sorry, but yesterday all our items had a 20% increase." What is the new price of the recorder?
|
54 \text{ reais}
| 0.75 |
Let $\left(u_{n}\right)_{n \in \mathbb{N}}$ be a sequence such that:
(i) $u_{0}=5$
(ii) $u_{n+1}=u_{n}+\frac{1}{u_{n}}$
Show that $u_{1000}>45$.
|
u_{1000} > 45
| 0.875 |
In the 100th year of his reign, the Immortal Treasurer decided to start issuing new coins. This year, he issued an unlimited supply of coins with a denomination of \(2^{100} - 1\), next year with a denomination of \(2^{101} - 1\), and so on. As soon as the denomination of a new coin can be obtained without change using previously issued new coins, the Treasurer will be removed from office. In which year of his reign will this happen?
|
200
| 0.75 |
A train took $X$ minutes ($0 < X < 60$) to travel from platform A to platform B. Find $X$ if it's known that at both the moment of departure from A and the moment of arrival at B, the angle between the hour and minute hands of the clock was $X$ degrees.
|
48
| 0.375 |
For every real number \( x \), let \( \lfloor x \rfloor \) denote the greatest integer less than or equal to \( x \). The fractional part of \( x \), denoted as \( \langle x \rangle \), is defined as \( \langle x \rangle = x - \lfloor x \rfloor \). How many real numbers \( x \) are there such that \( 1 \leqslant x \leqslant 10 \) and \( \langle x \rangle^{2} = \langle x^{2} \rangle \)?
|
91
| 0.5 |
Place the numbers $1,2,\cdots,n$ on a circle such that the absolute difference between any two adjacent numbers is either 3, 4, or 5. Find the smallest $n$ that satisfies these conditions.
|
7
| 0.25 |
Solve the equation \(2021x = 2022 \cdot \sqrt[202 \sqrt{x^{2021}}]{ } - 1\). (10 points)
|
1
| 0.75 |
Let \( M = \{1, 2, \cdots, 17\} \). If there exist four distinct numbers \( a, b, c, d \in M \) such that \( a + b \equiv c + d \pmod{17} \), then \( \{a, b\} \) and \( \{c, d\} \) are called a balanced pair of the set \( M \). Find the number of balanced pairs in the set \( M \).
|
476
| 0.5 |
Given that \( a, b, c \) are the lengths of the sides of a right triangle, and for any natural number \( n > 2 \), the equation \(\left(a^{n} + b^{n} + c^{n}\right)^{2} = 2\left(a^{2n} + b^{2n} + c^{2n}\right)\) holds, find \( n \).
|
n = 4
| 0.875 |
Find all functions $f: \mathbf{N} \rightarrow \mathbf{N}$ such that for all $m, n \in \mathbf{N}$, the equation $f(m^{2}+n^{2}) = f^{2}(m) + f^{2}(n)$ is satisfied and $f(1) > 0$.
|
f(n) = n
| 0.875 |
A point is randomly thrown onto the segment [3, 8] and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2 k-3\right) x^{2}+(3 k-5) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$.
|
\frac{4}{15}
| 0.75 |
In a football championship, 16 teams participated. A team receives 2 points for a win; in case of a draw in regular time, both teams shoot penalty kicks, and the team that scores more goals receives one point. After 16 rounds, all teams have accumulated a total of 222 points. How many matches ended in a draw in regular time?
|
34 \text{ matches}
| 0.625 |
Carla wrote the integers from 1 to 21 on the blackboard. Diana wants to erase some of these numbers such that the product of the remaining numbers is a perfect square.
a) Show that Diana necessarily needs to erase the numbers 11, 13, 17, and 19 to achieve her goal.
b) What is the minimum number of numbers that Diana must erase to achieve her goal?
|
5
| 0.375 |
Given one hundred numbers: \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{100}\). We compute 98 differences: \(a_{1} = 1 - \frac{1}{3}, a_{2} = \frac{1}{2} - \frac{1}{4}, \ldots, a_{98} = \frac{1}{98} - \frac{1}{100}\). What is the sum of all these differences?
|
\frac{14651}{9900}
| 0.375 |
If \(x\), \(y\), and \(z\) are distinct positive integers such that \(x^2 + y^2 = z^3\), what is the smallest possible value of \(x + y + z\)?
|
18
| 0.75 |
Let $H_{A}$, $H_{B}$, and $H_{C}$ be the feet of the altitudes from vertices $A$, $B$, and $C$ respectively in triangle $ABC$. Let $H$ be the orthocenter of triangle $ABC$, $P$ be the orthocenter of triangle $AH_{B}H_{C}$, and $Q$ be the orthocenter of triangle $CH_{A}H_{B}$. Show that $PQ = H_{C}H_{A}$.
|
PQ = H_C H_A
| 0.75 |
A smooth ball with a radius of 1 cm was dipped in red paint and then placed between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm respectively (the ball ended up outside the smaller sphere but inside the larger one). When the ball comes into contact with both spheres, it leaves a red mark. During its motion, the ball traced a closed path, resulting in a region outlined by a red contour on the smaller sphere, with an area of 47 square cm. Find the area of the region outlined by the red contour on the larger sphere. Give the answer in square centimeters, rounded to hundredths if necessary.
|
105.75 \text{ cm}^2
| 0.625 |
In an acute-angled triangle \( ABC \), the altitude \( AA_1 \) is drawn. \( H \) is the orthocenter of triangle \( ABC \). It is known that \( AH = 3 \), \( A_1H = 2 \), and the radius of the circumcircle of triangle \( ABC \) is 4. Find the distance from the center of this circumcircle to \( H \).
|
2
| 0.5 |
Buses leave Moscow for Voronezh every hour, at 00 minutes. Buses leave Voronezh for Moscow every hour, at 30 minutes. The trip between cities takes 8 hours. How many buses from Voronezh will a bus leaving Moscow meet on its way?
|
16
| 0.125 |
Two medians of a triangle, measuring 18 and 24, are mutually perpendicular. Find the length of the third median of this triangle.
|
30
| 0.5 |
The second term of an infinite decreasing geometric progression is 3. Find the smallest possible value of the sum \( A \) of this progression, given that \( A > 0 \).
|
12
| 0.875 |
For what smallest natural number \( a \) are there exactly 50 perfect squares in the numerical interval \( (a, 3a) \)?
|
4486
| 0.125 |
In an isosceles triangle $ABC$ with base $AB$, the bisector of angle $B$ is perpendicular to the median of side $BC$. Find the cosine of angle $C$.
|
\frac{7}{8}
| 0.375 |
Determine all real polynomials \(P\) such that \(P(0)=0\) and \(P\left(X^2+1\right)=P(X)^2+1\).
|
P(X) = X
| 0.25 |
A convex polygon has \( n \) sides. Each vertex is joined to a point \( P \) not in the same plane. If \( A, B, C \) are adjacent vertices of the polygon, consider the angle between the planes \( PBA \) and \( PBC \). The sum of the \( n \) such angles equals the sum of the \( n \) angles subtended at \( P \) by the sides of the polygon (such as the angle \( APB \)). Show that \( n = 3 \).
|
n = 3
| 0.875 |
If \(a\) is defined as a number that is not divisible by 4 but whose last digit is 4, then show that
\[ a\left(a^{2}-1\right)\left(a^{2}-4\right) \]
is divisible by 480.
|
480
| 0.375 |
Find the minimum value of the expression
$$
\frac{|a-3b-2| + |3a-b|}{\sqrt{a^2 + (b+1)^2}}
$$
for \(a, b \geq 0\).
|
2
| 0.875 |
A triangle has two medians of lengths 9 and 12. Find the largest possible area of the triangle. (Note: A median is a line segment joining a vertex of the triangle to the midpoint of the opposite side.)
|
72
| 0.875 |
A tram ticket is called "lucky in Leningrad style" if the sum of its first three digits is equal to the sum of its last three digits. A tram ticket is called "lucky in Moscow style" if the sum of its digits in even positions is equal to the sum of its digits in odd positions. How many tickets are there that are both lucky in Leningrad style and lucky in Moscow style, including the ticket 000000?
|
6700
| 0.375 |
Given three natural numbers 1, 2, 3, perform an operation by replacing one of these numbers with the sum of the other two. After performing this operation 9 times, what is the maximum possible value of the largest number among the resulting three natural numbers?
|
233
| 0.5 |
Bing Dwen Dwen cut 25 square pieces of paper, having a total of 100 corners. Xue Rong Rong cut a triangle from each square piece of paper. In the end, 50 pieces of paper had a total of 170 corners. How many more triangular pieces of paper are there than pentagonal pieces of paper among these 50 pieces?
|
30
| 0.625 |
Andrei was asked to name the apartment number his family received in a new building. He replied that this number is expressed by a number that is 17 times the digit in the units place of the number. What is this apartment number?
|
85
| 0.75 |
Find the smallest number, written using only ones and zeros, that would be divisible by 225.
|
11111111100
| 0.625 |
A person picks \( n \) different prime numbers each less than 150 and finds that they form an arithmetic sequence. What is the greatest possible value of \( n \)?
|
5
| 0.75 |
Using the seven digits $1, 2, 3, 4, 5, 6, 7$ to appropriately arrange them into a 7-digit number so that it is a multiple of 11, how many such numbers can be formed?
|
576
| 0.125 |
Consider a table with \( m \) rows and \( n \) columns. In how many ways can this table be filled with all zeros and ones so that there is an even number of ones in every row and every column?
|
2^{(m-1)(n-1)}
| 0.375 |
Let the sequence \(\{x_n\}\) be defined as follows: \(x_{1}=\frac{1}{2}\), and
\[ x_{k+1}=x_{k}+x_{k}^{2} \quad \text{for} \quad k=1,2, \ldots \]
Find the integer part of \(\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{100}+1}\).
|
1
| 0.75 |
In the Cartesian coordinate system, circle \( C_1 \) and circle \( C_2 \) intersect at points \( P \) and \( Q \), where the coordinates of point \( P \) are \( (3, 2) \). The product of the radii of the two circles is \( \frac{13}{2} \). If the line \( y = kx \) (where \( k > 0 \)) is tangent to both circles \( C_1 \) and \( C_2 \) and also tangent to the x-axis, find the value of \( k \).
|
2 \sqrt{2}
| 0.125 |
In a triangle, the lengths of the three sides are integers \( l, m, n \), with \( l > m > n \). It is known that \( \left\{ \frac{3^{l}}{10^{4}} \right\} = \left\{ \frac{3^{m}}{10^{4}} \right\} = \left\{ \frac{3^{n}}{10^{4}} \right\} \), where \( \{x\} \) denotes the fractional part of \( x \) and \( [x] \) denotes the greatest integer less than or equal to \( x \). Determine the smallest possible value of the perimeter of such a triangle.
|
3003
| 0.5 |
Let \( f(x) = 1 + x + x^2 + x^3 + \ldots \). Compute the formal power series \( g(x) \) such that \( f(x) g(x) = 1 \).
|
g(x) = 1 - x
| 0.875 |
Euler's inequality: Let $ \triangle ABC $ have a circumradius $ R $ and an inradius $ r $. Then $ R \geq 2r $.
|
R \geq 2r
| 0.5 |
Ludvík noticed in a certain division problem that when he doubles the dividend and increases the divisor by 12, he gets his favorite number as the result. He would get the same number if he reduced the original dividend by 42 and halved the original divisor. Determine Ludvík’s favorite number.
|
7
| 0.875 |
In triangle \( ABC \), it is known that \(\angle BAC = 75^\circ\), \( AB = 1 \), and \( AC = \sqrt{6} \). On the side \( BC \), a point \( M \) is chosen such that \(\angle BAM = 30^\circ\). The line \( AM \) intersects the circumcircle of triangle \( ABC \) at a point \( N \) different from \( A \). Find \( AN \).
|
2
| 0.75 |
Compose the equation of the plane that passes through the line of intersection of the planes \(x + 3y + 5z - 4 = 0\) and \(x - y - 2z + 7 = 0\) and is parallel to the y-axis.
|
4x - z + 17 = 0
| 0.875 |
Find all pairs of integers \((x, y)\) for which \(x^2 + xy = y^2\).
|
(0,0)
| 0.75 |
Let \( f: \mathbf{R} \rightarrow \mathbf{R} \) be a smooth function such that \( f^{\prime}(x)^{2} = f(x) f^{\prime \prime}(x) \) for all \( x \). Suppose \( f(0) = 1 \) and \( f^{(4)}(0) = 9 \). Find all possible values of \( f^{\prime}(0) \).
|
\pm \sqrt{3}
| 0.875 |
To obtain the summary of a number with up to 9 digits, you must write how many digits it has, then how many of those digits are odd, and finally how many are even. For example, the number 9103405 has 7 digits, 4 of which are odd and 3 are even, so its summary is 743.
a) Find a number whose summary is 523.
b) Find a number that is equal to its own summary.
c) For any number with up to 9 digits, we can calculate the summary of the summary of its summary. Show that this procedure always leads to the same result regardless of the initial number.
|
321
| 0.5 |
Let \( n \) be a positive integer, and let \( s \) be the sum of the digits of the base-four representation of \( 2^n - 1 \). If \( s = 2023 \) (in base ten), compute \( n \) (in base ten).
|
1349
| 0.875 |
Martin is playing a game. His goal is to place tokens on an 8x8 chessboard such that there is at most one token per square, and each row and column contains at most 4 tokens.
a) How many tokens can Martin place at most?
b) If, in addition to the previous constraints, each of the two main diagonals can contain at most 4 tokens, how many tokens can Martin place at most?
The main diagonals of a chessboard are the two diagonals going from one corner of the board to the opposite corner.
|
32
| 0.5 |
Find all functions \( f: \mathbb{Q} \rightarrow \{-1, 1\} \) such that for all distinct \( x, y \in \mathbb{Q} \) satisfying \( xy = 1 \) or \( x + y \in \{0, 1\} \), we have \( f(x) f(y) = -1 \).
Intermediate question: Let \( f \) be a function having the above property and such that \( f(0) = 1 \). What is \( f\left(\frac{42}{17}\right) \) ?
|
-1
| 0.5 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.