problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Estimate the number of 2016-digit squares. There are at least
$$
10^{2016 / 2} - 10^{2015 / 2} - 1 > 10^{1000}.
$$
The number of different sets of 2016 digits is not more than the sets where each digit appears no more than 2016 times, i.e., $2017^{10}$. Therefore, there exists such a set of 2016 digits, permutations of which can produce at least $10^{1000} / 2017^{10} > 10^{1000} / 10^{100} = 10^{900}$ squares, and certainly 2016. | 2016 | 0.75 |
Let \( a, b, c, \) and \( d \) be positive integers such that \( a^5 = b^4 \) and \( c^3 = d^2 \) with \( c - a = 19 \). Find \( d - b \). | 757 | 0.875 |
\(A B C D\) and \(E F G H\) are squares of side length 1, and \(A B \parallel E F\). The overlapped region of the two squares has area \(\frac{1}{16}\). Find the minimum distance between the centers of the two squares. | \frac{\sqrt{14}}{4} | 0.25 |
Find the smallest number consisting only of zeros and ones that is divisible by 225. | 11111111100 | 0.5 |
If \( x_{1}=1, x_{2}=1-\mathrm{i}, x_{3}=1+\mathrm{i} \) (where \( \mathrm{i} \) is the imaginary unit) are the three solutions to the equation \( x^{3}+a x^{2}+b x+c=0 \), then find \( a+b-c \). | 3 | 0.875 |
Given the quadratic function \( f(x) \) such that \( f(-1) = 0 \) and \( x \leq f(x) \leq \frac{1}{2}\left(x^2 + 1\right) \) holds for all real numbers \( x \). Determine the explicit form of the function \( f(x) \). | \frac{1}{4}(x+1)^2 | 0.125 |
Let \( a \) and \( b \) be two distinct roots of the polynomial \( X^{3} + 3X^{2} + X + 1 \). Calculate \( a^{2}b + ab^{2} + 3ab \). | 1 | 0.75 |
If one side of square \(ABCD\) lies on the line \(y = 2x - 17\) and two other vertices lie on the parabola \(y = x^2\), then the minimum area of the square is \(\quad\). | 80 | 0.375 |
There are 2016 cards, each with a unique number from 1 to 2016. A certain number \( k \) of these cards are selected. What is the smallest \( k \) such that among these selected cards, there exist two cards with numbers \( a \) and \( b \) satisfying the condition \( |\sqrt[3]{a} - \sqrt[3]{b}| < 1 \)? | 13 | 0.875 |
Calculate the area of the figure bounded by the lines defined by the equations:
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\sqrt{2} \cdot \cos t \\
y=2 \sqrt{2} \cdot \sin t
\end{array}\right. \\
& y=2(y \geq 2)
\end{aligned}
$$ | \pi - 2 | 0.875 |
The nonzero numbers \( x, y \), and \( z \) satisfy the equations
\[
xy = 2(x + y), \quad yz = 4(y + z), \quad \text{and} \quad xz = 8(x + z).
\]
Solve for \( x \). | \frac{16}{3} | 0.875 |
Petya places "+" and "-" signs in all possible ways into the expression $1 * 2 * 3 * 4 * 5 * 6$ at the positions of the asterisks. For each arrangement of signs, he calculates the resulting value and writes it on the board. Some numbers may appear on the board multiple times. Petya then sums all the numbers on the board. What is the sum that Petya obtains? | 32 | 0.875 |
You have 5 identical buckets, each with a maximum capacity of some integer number of liters, and a 30-liter barrel containing an integer number of liters of water. All the water from the barrel was poured into the buckets, with the first bucket being half full, the second one-third full, the third one-quarter full, the fourth one-fifth full, and the fifth one-sixth full. How many liters of water were in the barrel? | 29 | 0.375 |
The sequence \( \{x_{n}\} \) is defined as follows:
\[
x_{1} = \frac{2}{3}, \quad x_{n+1} = \frac{x_{n}}{2(2n+1)x_{n} + 1} \quad (n \in \mathbb{Z}_{+}).
\]
Find the sum \( x_{1} + x_{2} + \cdots + x_{2014} \). | \frac{4028}{4029} | 0.875 |
Determine the largest integer $x$ for which $4^{27} + 4^{1010} + 4^{x}$ is a perfect square. | 1992 | 0.625 |
On weekdays, the Absent-minded Scientist commutes to work on the ring light of the Moscow Metro from "Taganskaya" station to "Kievskaya" station, and back in the evening. Upon entering the station, he catches the first arriving train. It is known that in both directions trains run at approximately equal intervals. Via the northern route (through "Belorusskaya"), the train travels from "Kievskaya" to "Taganskaya" or back in 17 minutes, and via the southern route (through "Paveletskaya") in 11 minutes.
The Scientist has calculated through many years of observations that:
- A train going counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after a train going clockwise arrives. The same is true for "Taganskaya".
- The trip from home to work on average takes 1 minute less than the trip from work to home.
Find the expected interval between trains running in one direction. | 3 \text{ minutes} | 0.125 |
Find the relationship between the coefficients of the equation \(a x^{2}+b x+c=0\) if the ratio of the roots is 2. | 2b^2 = 9ac | 0.75 |
At 12 o'clock, the angle between the hour hand and the minute hand is 0 degrees. After that, at what time do the hour hand and the minute hand form a 90-degree angle for the 6th time? (12-hour format) | 3:00 | 0.375 |
A sphere is inscribed in a cone, and the surface area of the sphere is equal to the area of the base of the cone. Find the cosine of the angle at the vertex in the axial section of the cone. | \frac{7}{25} | 0.875 |
If \( p, q, \frac{2 p-1}{q}, \frac{2 q-1}{p} \) are all integers, and \( p > 1 \), \( q > 1 \), find the value of \( p+q \). | 8 | 0.75 |
The perimeter of the parallelogram \(ABCD\) is 20. The bisector of angle \(B\) intersects the lines \(AD\) and \(CD\) at points \(K\) and \(L\) respectively. Find \(CL\), given that \(DK = 4\). | 7 | 0.125 |
Compute the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{1+2+\ldots+n}{n-n^{2}+3}
\] | -\frac{1}{2} | 0.875 |
In the trapezoid \( A B C D \), sides \( A D \) and \( B C \) are parallel, and \( A B = B C = B D \). The height \( B K \) intersects the diagonal \( A C \) at point \( M \). Find \(\angle C D M\). | 90^\circ | 0.75 |
Pi Pi Lu wrote a 2020-digit number: \( 5368 \cdots \cdots \). If any four-digit number taken randomly from this multi-digit number is divisible by 11, what is the sum of the digits of this multi-digit number? | 11110 | 0.75 |
Given real numbers \(x_{1}, x_{2}, \cdots, x_{10}\) satisfying \(\sum_{i=1}^{10}\left|x_{i}-1\right| \leq 4\) and \(\sum_{i=1}^{10}\left|x_{i}-2\right| \leq 6\), find the average value \(\bar{x}\) of \(x_{1}, x_{2}, \cdots, x_{10}\). | 1.4 | 0.625 |
The circles $k_{1}$ and $k_{2}$, both with unit radius, touch each other at point $P$. One of their common tangents that does not pass through $P$ is the line $e$. For $i>2$, let $k_{i}$ be the circle different from $k_{i-2}$ that touches $k_{1}$, $k_{i-1}$, and $e$. Determine the radius of $k_{1999}$. | \frac{1}{1998^2} | 0.75 |
The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are 155 and 13 respectively, and its lateral sides are mutually perpendicular. Find the scalar product of the vectors \(\overrightarrow{AC}\) and \(\overrightarrow{BD}\). | -2015 | 0.625 |
16. Variance of the number of matches. A deck of playing cards is laid out on a table (for example, in a row). On top of each card, a card from another deck is placed. Some cards may match. Find:
a) the expected number of matches;
b) the variance of the number of matches. | 1 | 0.625 |
From the set \( M = \{1, 2, 3, \cdots, 2009\} \), remove all multiples of 3 and all multiples of 5. What is the number of remaining elements in \( M \)? | 1072 | 0.875 |
For any positive integer $n$, let $f_{1}(n)$ denote the square of the sum of the digits of $n$ plus $r+1$, where $r$ is the integer satisfying $n = 3q + r$ with $0 \leqslant r < 3$. For $k \geqslant 2$, let $f_{k}(n) = f_{1}(f_{k-1}(n))$. Find $f_{1990}(2345)$. | 3 | 0.75 |
Let $ABC$ be an equilateral triangle and $D$ and $E$ be two points on segment $[AB]$ such that $AD = DE = EB$. Let $F$ be a point on $BC$ such that $CF = AD$. Find the value of $\widehat{CDF} + \widehat{CEF}$. | 30^\circ | 0.25 |
There is a 6 × 6 grid of lights. There is a switch at the top of each column and on the left of each row. A light will only turn on if the switches corresponding to both its column and its row are in the "on" position. Compute the number of different configurations of lights. | 3970 | 0.75 |
Given \( n > 2 \) natural numbers such that there are no three equal numbers among them, and the sum of any two of these numbers is a prime number. What is the largest possible value of \( n \)? | 3 | 0.625 |
Given a natural number \( n \geq 3 \) and real numbers \( x_{1}, x_{2}, \cdots, x_{n} \) satisfying \( x_{1} + x_{2} + \cdots + x_{n} = n \) and \( x_{1}^{2} + x_{2}^{2} + \cdots + x_{n}^{2} = n^{2} \), find the minimum value of \( \sum_{i=1}^{n} x_{i}^{3} \) and the corresponding values of \( \left(x_{1}, x_{2}, \cdots, x_{n}\right) \) that achieve this minimum. | -n^3 + 6n^2 - 4n | 0.75 |
In a convex quadrilateral inscribed around a circle, the products of opposite sides are equal. The angle between a side and one of the diagonals is $20^{\circ}$. Find the angle between this side and the other diagonal. | 70^\circ | 0.5 |
Of all the three-digit numbers formed by using the four digit cards $0$, $1$, $2$, and $2$, arrange them from largest to smallest. The 2nd number is $\qquad$. The difference between the 4th number and the 8th number is $\qquad$. | 90 | 0.625 |
Let the four vertices of a regular tetrahedron be \( A, B, C, D \) with each edge length being 1 meter. A bug starts at point \( A \) and moves according to the following rule: at each vertex, it randomly chooses one of the three edges passing through that vertex and crawls to the end of that edge. What is the probability that, after crawling a total of 7 meters, the bug will be at vertex \( A \)? | \frac{182}{729} | 0.375 |
Alloy $A$ of two metals has a mass of 6 kg, with the first metal being twice as abundant as the second metal. When placed in a container of water, it exerts a force of $30\ \mathrm{N}$ on the bottom. Alloy $B$ of the same metals has a mass of 3 kg, with the first metal being five times less abundant than the second metal. When placed in a container of water, it exerts a force of $10\ \mathrm{N}$ on the bottom. What force (in newtons) will the third alloy, obtained by combining the original alloys, exert on the bottom? | 40 \text{ N} | 0.75 |
Given an isosceles triangle \(ABC\) with \(AB = AC\) and \(\angle ABC = 53^\circ\). Point \(K\) is such that \(C\) is the midpoint of segment \(AK\). Point \(M\) is chosen such that:
- \(B\) and \(M\) are on the same side of line \(AC\);
- \(KM = AB\);
- The angle \(\angle MAK\) is the largest possible.
How many degrees is the angle \(\angle BAM\)? | 44^\circ | 0.75 |
There are at least 150 boys in a school, and the number of girls is 15% more than the number of boys. When the boys went on a trip, 6 buses were required, with each bus carrying the same number of students. How many students are there in the school, given that the total number of students is not more than 400? | 387 | 0.875 |
How many roots does the equation \(\sqrt{14-x^{2}}(\sin x-\cos 2x)=0\) have? | 6 | 0.5 |
Rectangle \(ABCD\) has diagonal \(BD\) with endpoints \(B(4,2)\) and \(D(12,8)\). Diagonal \(AC\) lies on the line with equation \(x + 2y - 18 = 0\). Determine the area of \(ABCD\). | 20 \sqrt{5} | 0.75 |
A circle \( C \) touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that the radius of \( C \) must be larger than the radius of the middle of the three circles. | R > r_2 | 0.125 |
In the quadrilateral \(ABCD\), if \(\overrightarrow{AB} \cdot \overrightarrow{CD} = -3\) and \(\overrightarrow{AD} \cdot \overrightarrow{BC} = 5\), then \(\overrightarrow{AC} \cdot \overrightarrow{BD} =\) _______. | 2 | 0.75 |
In a rectangle \(ABCD\) with an area of 1 (including its boundary), there are 5 points such that no three points are collinear. Determine the minimum number of triangles, formed by these 5 points as vertices, that have an area of at most \(\frac{1}{4}\). | 2 | 0.25 |
Little Abel received a $2 \times n$ board and $n$ $2 \times 1$ tiles as a gift. For example, the figure below shows the case where $n=10$, that is, when Abel has a $2 \times 10$ board and 10 $2 \times 1$ tiles.
He plays by filling the board using the $n$ tiles. For example, for $n=10$ Abel could fill it in the ways illustrated below:
Note, however, that there are many other ways for Abel to fill his board.
a) Calculate the total number of ways in which Abel can fill his board for the cases where $n=1, n=2,$ and $n=3$, that is, when the boards have dimensions $2 \times 1, 2 \times 2,$ and $2 \times 3$.
b) Let $a_{n}$ be the number of ways in which Abel can fill a $2 \times n$ board using $n$ $2 \times 1$ tiles. Show that $a_{10}=a_{9}+a_{8}$.
c) Calculate the total number of ways in which Abel can fill his board when $n=10$. | 89 | 0.75 |
As shown in Figure 2.10.5, let $\triangle ABC$ have a perimeter of $2p$. A tangent line to the incircle of the triangle, parallel to side $AC$, is drawn as $DE$. Find the maximum length of the segment of this tangent line intercepted by the other two sides of the triangle. | \frac{p}{4} | 0.125 |
Given that \( a \), \( b \), and \( c \) are three distinct prime numbers, and \( a + b \times c = 37 \), what is the maximum value of \( a + b - c \)? | 32 | 0.25 |
Let \(ABC\) be a triangle and \(\Gamma\) be its circumcircle. The tangent to \(\Gamma\) at \(A\) intersects \((BC)\) at \(D\). The bisector of \(\widehat{CDA}\) intersects \((AB)\) at \(E\) and \((AC)\) at \(F\). Show that \(AE = AF\). | AE = AF | 0.75 |
The circle \(\gamma_{1}\) centered at \(O_{1}\) intersects the circle \(\gamma_{2}\) centered at \(O_{2}\) at two points \(P\) and \(Q\). The tangent to \(\gamma_{2}\) at \(P\) intersects \(\gamma_{1}\) at the point \(A\) and the tangent to \(\gamma_{1}\) at \(P\) intersects \(\gamma_{2}\) at the point \(B\), where \(A\) and \(B\) are distinct from \(P\). Suppose \(PQ \cdot O_{1}O_{2} = PO_{1} \cdot PO_{2}\) and \(\angle APB\) is acute. Determine the size of \(\angle APB\) in degrees. | 30 | 0.125 |
How many four-digit numbers greater than 5000 can be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if only the digit 4 may be repeated? | 2645 | 0.5 |
Find the number of eight-digit numbers where the product of the digits is 64827. Provide the answer as an integer. | 1120 | 0.125 |
Given that \( \mathrm{f}(x) \) is a polynomial of degree 2012, and that \( \mathrm{f}(k) = \frac{2}{k} \) for \( k = 1,2,3, \cdots, 2013 \), find the value of \( 2014 \times \mathrm{f}(2014) \). | 4 | 0.875 |
Xiao Ming departs from his home to his grandmother's house on a bicycle at a speed of 12 km/h. After 2.5 hours, his father realizes that Xiao Ming forgot to bring his homework and starts chasing him on a motorcycle at a speed of 36 km/h. Half an hour after Xiao Ming reaches his grandmother's house, his father arrives. What is the distance between Xiao Ming's home and his grandmother's house in kilometers? | 36 \ \text{km} | 0.625 |
Find all natural numbers $p$ such that both $p$ and $5p + 1$ are prime. | 2 | 0.875 |
Let \( f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z} \) be a function with the following properties:
(i) \( f(1)=0 \),
(ii) \( f(p)=1 \) for all prime numbers \( p \),
(iii) \( f(xy)=y f(x)+x f(y) \) for all \( x, y \in \mathbb{Z}_{>0} \).
Determine the smallest integer \( n \geq 2015 \) that satisfies \( f(n)=n \).
(Gerhard J. Woeginger) | 3125 | 0.375 |
A segment can slide along a line; a circular arc can slide along a circle; do other plane curves exist that have this property? | \text{No} | 0.125 |
On the sides \( AB \) and \( BC \) of an equilateral triangle \( ABC \), points \( P \) and \( Q \) are chosen such that \( AP:PB = BQ:QC = 2:1 \). Find \( \angle AKB \), where \( K \) is the intersection point of segments \( AQ \) and \( CP \). | 90^\circ | 0.75 |
Is it possible to find at least one cube of an integer among the numbers $2^{2^{n}}+1$ for $n=0,1,2, \ldots$? | \text{No} | 0.75 |
If \( a, b, c, d \) are distinct positive integers such that \( a \times b \times c \times d = 357 \), then \( a + b + c + d = \) $\qquad$. | 28 | 0.875 |
Petya and Masha take turns taking candy from a box. Masha took one candy, then Petya took 2 candies, Masha took 3 candies, Petya took 4 candies, and so on. When the number of candies in the box became insufficient for the next turn, all the remaining candies went to the person whose turn it was to take candies. How many candies did Petya receive if Masha got 101 candies? | 110 | 0.5 |
In a checkers tournament, students from 10th and 11th grades participated. Each player played against every other player exactly once. A win earned a player 2 points, a draw earned 1 point, and a loss earned 0 points. The number of 11th graders was 10 times the number of 10th graders, and together they scored 4.5 times more points than all the 10th graders combined. How many points did the most successful 10th grader score? | 20 | 0.625 |
On his birthday, the last guest to arrive was Yana, who gave Andrey a ball, and the second last was Eduard, who gave him a calculator. While using the calculator, Andrey noticed that the product of the total number of gifts he received and the number of gifts he had before Eduard arrived is exactly 16 more than the product of his age and the number of gifts he had before Yana arrived. How many gifts does Andrey have? | 18 | 0.875 |
A true-false test has ten questions. If you answer five questions "true" and five "false," your score is guaranteed to be at least four. How many answer keys are there for which this is true? | 22 | 0.75 |
Find all strictly positive integers $x, y, z$ such that $5^{x} - 3^{y} = z^{2}$. | (2, 2, 4) | 0.75 |
A regular triangular pyramid \(SABC\) is given, with its base edge equal to 1. Medians of the lateral faces are drawn from the vertices \(A\) and \(B\) of the base \(ABC\), and these medians do not intersect. It is known that the lines containing these medians also contain edges of a certain cube. Find the length of the lateral edge of the pyramid. | \frac{\sqrt{6}}{2} | 0.25 |
In the first grade of the school, between 200 and 300 children enrolled. It was decided to form classes with 25 students each, but it turned out that there would be no place for 10 children. Then they formed classes with 30 students each, but one of the classes ended up having 15 fewer students. How many children enrolled in the first grade? | 285 | 0.875 |
There is an urn with 10 balls, among which there are 2 red, 5 blue, and 3 white balls. Find the probability that a randomly drawn ball will be a colored one (event $A$). | 0.7 | 0.375 |
In triangle \(ABC\), \(AC = 1\), \(AB = 2\), and \(O\) is the point where the angle bisectors intersect. A segment passing through point \(O\) parallel to side \(BC\) intersects sides \(AC\) and \(AB\) at points \(K\) and \(M\) respectively. Find the perimeter of triangle \(AKM\). | 3 | 0.625 |
For any subset \( S \subseteq \{1, 2, \ldots, 15\} \), a number \( n \) is called an "anchor" for \( S \) if \( n \) and \( n+|S| \) are both members of \( S \), where \( |S| \) denotes the number of members of \( S \). Find the average number of anchors over all possible subsets \( S \subseteq \{1, 2, \ldots, 15\} \). | 13/8 | 0.75 |
Given the expression \(1.24 \frac{\sqrt{\frac{a b c + 4}{a} + 4 \sqrt{\frac{b c}{a}}}}{\sqrt{a b c} + 2}\) where \( a = 0.04 \). | 6.2 | 0.625 |
Let \( S = \{1, 2, \ldots, 2016\} \), and let \( f \) be a randomly chosen bijection from \( S \) to itself. Let \( n \) be the smallest positive integer such that \( f^{(n)}(1) = 1 \), where \( f^{(i)}(x) = f\left(f^{(i-1)}(x)\right) \). What is the expected value of \( n \)? | \frac{2017}{2} | 0.125 |
Amelia wrote down a sequence of consecutive positive integers, erased one integer, and scrambled the rest, leaving the sequence below. What integer did she erase?
$$
6,12,1,3,11,10,8,15,13,9,7,4,14,5,2
$$ | 16 | 0.5 |
Let \( x[n] \) denote \( x \) raised to the power of \( x \), repeated \( n \) times. What is the minimum value of \( n \) such that \( 9[9] < 3[n] \)?
(For example, \( 3[2] = 3^3 = 27 \); \( 2[3] = 2^{2^2} = 16 \).) | 10 | 0.125 |
Given a triangle $ABC$ with $\angle BAC = 60^\circ$, let $H$, $I$, and $O$ denote its orthocenter, its incenter, and its circumcenter respectively. Show that $IO = IH$. | IO = IH | 0.875 |
Several young men and women are seated around a round table. It is known that to the left of exactly 7 women, there are women, and to the left of 12 women, there are men. It is also known that for 75% of the young men, there are women to their right. How many people are seated at the table? | 35 | 0.625 |
Find the largest six-digit number in which all digits are distinct, and each digit, except for the extreme ones, is equal either to the sum or the difference of its neighboring digits. | 972538 | 0.125 |
Let \( a, b, c \in \left[\frac{1}{2}, 1\right] \). Define \( s = \frac{a+b}{1+c} + \frac{b+c}{1+a} + \frac{c+a}{1+b} \). What is the range of possible values for \( s \)? | [2, 3] | 0.625 |
Let \( p_{1}, p_{2}, \ldots, p_{97} \) be prime numbers (not necessarily distinct). What is the maximum integer value that the expression
$$
\sum_{i=1}^{97} \frac{p_{i}}{p_{i}^{2}+1}=\frac{p_{1}}{p_{1}^{2}+1}+\frac{p_{2}}{p_{2}^{2}+1}+\ldots+\frac{p_{97}}{p_{97}^{2}+1}
$$
can take? | 38 | 0.875 |
Determine the prime numbers $p$ for which $p^{2}+8$ is also prime. | 3 | 0.875 |
Given a triangle \( ABC \), a circle passing through \( B \) and \( C \) intersects \( AB \) and \( AC \) at \( B' \) and \( C' \) respectively. Let \( O \) be the center of the circumcircle of \( AB'C' \). Show that \( AO \) is perpendicular to \( BC \). | AO \perp BC | 0.875 |
The bases of a trapezoid are 8 and 2. The angles adjacent to the larger base are each $45^{\circ}$. Find the volume of the solid formed by rotating the trapezoid about its larger base. | 36\pi | 0.625 |
Let \( x_{1} \) and \( x_{2} \) be two real numbers that satisfy \( x_{1} x_{2} = 2013 \). What is the minimum value of \( (x_{1} + x_{2})^{2} \)? | 8052 | 0.5 |
174. \(n^{2}\) integers from 1 to \(n^{2}\) are written in a square table of size \(n \times n\): number 1 is in any position in the table; number 2 belongs to the row whose number is equal to the column number containing 1; number 3 belongs to the row whose number matches the column number containing 2, and so on. By how much does the sum of the numbers in the row containing number 1 differ from the sum of the numbers in the column containing the number \(n^{2}\)? | n^2 - n | 0.125 |
Let $D$ be an interior point of the acute triangle $\triangle ABC$ such that $\angle ADB = \angle ACB + 90^{\circ}$ and $AC \cdot BD = AD \cdot BC$. Find the value of $\frac{AB \cdot CD}{AC \cdot BD}$. | \sqrt{2} | 0.5 |
Determine all functions \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) such that for all \( m, n \in \mathbb{N}^{*} \), the number \( f(m) + f(n) - mn \) is nonzero and divides \( m f(m) + n f(n) \). | f(n) = n^2 | 0.75 |
The sequence \(a_{1}, a_{2}, \ldots\) is defined by the equations
\[ a_{1}=100, \quad a_{n+1}=a_{n}+\frac{1}{a_{n}}, \quad n \in \mathbb{N} \]
Find the integer closest to \(a_{2013}\). | 118 | 0.875 |
Given a complex number \( z \) such that \( |z| = 1 \), and \( u = z^4 - z^3 - 3z^2 i - z + 1 \). Find the maximum value of \( |u| \) and determine the complex number \( z \) when this maximum value is achieved. | 5 | 0.75 |
Three rods (with negligible thickness) are fixed perpendicularly to each other at a common point at one end. The lengths of the rods are 1, 2, and 3. This construction is placed on a table in such a way that the free ends of the rods lie on the plane of the tabletop. Determine the exact height of the fixed point above the table. | \frac{6}{7} | 0.625 |
The function \( f(x) = a^{2x} + 3a^x - 2 \) (where \( a > 0 \) and \( a \neq 1 \)) has a maximum value of 8 on the interval \( x \in [-1,1] \). Determine its minimum value on this interval. | -\frac{1}{4} | 0.75 |
Consider the set \( S = \{1, 2, \cdots, 15\} \). Let \( A = \{a_1, a_2, a_3\} \) be a subset of \( S \), with \( (a_1, a_2, a_3) \) satisfying:
\[
1 \leq a_1 < a_2 < a_3 \leq 15, \quad a_3 - a_2 \leq 6.
\]
Determine the number of subsets \( A \) that satisfy these conditions. | 371 | 0.625 |
Find the sum of the digits in the number $\underbrace{44 \ldots 4}_{2012 \text{ times}} \cdot \underbrace{99 \ldots 9}_{2012 \text{ times}}$. | 18108 | 0.75 |
All natural numbers from 1 to 1000 inclusive are divided into two groups: even and odd. In which group is the sum of all the digits used to write the numbers greater and by how much? | 499 | 0.625 |
$M$ is the midpoint of the height $B D$ in an isosceles triangle $A B C$. Point $M$ serves as the center of a circle with radius $M D$. Find the angular measure of the arc of the circle enclosed between the sides $B A$ and $B C$, given that $\angle B A C = 65^{\circ}$. | 100^\circ | 0.125 |
The perimeter of a trapezoid inscribed in a circle is 40. Find the midsegment of the trapezoid. | 10 | 0.75 |
The sequence \(\left\{a_n\right\}\) satisfies \(a_1 = 1\) and \(a_{n+1} = \frac{(n^2 + n) a_n}{3 a_n + n^2 + n}\) for \(n \in \mathbb{N}^*\). Find the general term \(a_n\). | \frac{n}{4n-3} | 0.875 |
The perimeter of triangle \(ABC\) is \(2p\). What is the maximum length that a segment parallel to \(BC\) and tangent to the inscribed circle of \(ABC\) can have inside the triangle? For what triangle(s) is this value achieved? | \frac{p}{4} | 0.375 |
Let \( x \) and \( y \) be real numbers with \( x > y \) such that \( x^{2} y^{2} + x^{2} + y^{2} + 2xy = 40 \) and \( xy + x + y = 8 \). Find the value of \( x \). | 3 + \sqrt{7} | 0.875 |
Calculate the limit of the function:
$$\lim _{x \rightarrow \pi} \frac{\sin ^{2} x-\operatorname{tg}^{2} x}{(x-\pi)^{4}}$$ | -1 | 0.875 |
Two straight roads intersect perpendicularly. On one road, a car starts from a distance of $l_{1}=80 \mathrm{~km}$ from the intersection and heads towards the intersection at a constant speed of $v_{1}=80 \mathrm{~km/h}$. On the other road, another car starts at the same time from a distance of $l_{2}=60 \mathrm{~km}$ from the intersection, traveling towards the intersection at a speed of $v_{2}=60 \mathrm{~km/h}$. How fast is the distance between the cars changing? What is the situation if the initial distances are $l_{1}=l_{2}^{\prime}=80 \mathrm{~km}$? How much time after departure will the two cars be closest to each other, and what is this distance in both cases? | 16 \text{ km} | 0.5 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.