problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Given that there are three points on the coordinate plane: \( O(0,0) \), \( A(12,2) \), and \( B(0,8) \). A reflection of \( \triangle OAB \) along the straight line \( y = 6 \) creates \( \triangle PQR \). If the overlapped area of \( \triangle OAB \) and \( \triangle PQR \) is \( m \) square units, find the value of \( m \). | 8 | 0.375 |
Given that the quadratic equation $a x^{2}+b x+c=0$ has no real roots. Person A, due to an error in reading the coefficient of the $x^2$ term, incorrectly found the roots to be 2 and 4. Person B, due to an error in reading the sign of a certain coefficient, incorrectly found the roots to be -1 and 4. What is the value of $\frac{2 b+3 c}{a}$? | 6 | 0.75 |
A number is the product of four prime numbers. What is this number if the sum of the squares of the four prime numbers is 476? | 1989 | 0.375 |
Distribute 16 identical books among 4 classes, with each class receiving at least one book and each class receiving a different number of books. How many different distribution methods are there? (Answer in digits) | 216 | 0.625 |
Point \( O \) is the center of the circumscribed circle of the acute-angled triangle \( ABC \), and \( H \) is the orthocenter of the triangle. It turned out that line \( OH \) is parallel to side \( BC \). On the plane, a point \( K \) was marked such that \( ABHK \) is a parallelogram. The segments \( OK \) and \( AC \) intersect at point \( L \). In what ratio does the perpendicular dropped from point \( L \) to segment \( AH \) divide \( AH \)? | 1:1 | 0.625 |
Two cars simultaneously set off towards each other from cities $A$ and $B$, which are 250 km apart. The cars travel at speeds of 50 km/h and 90 km/h. At what distance from the halfway point between cities $A$ and $B$, denoted as point $C$, will the cars meet? Give the answer in kilometers, rounded to the nearest hundredth if necessary. | 35.71 | 0.625 |
Cover the following $2 \times 10$ grid with $1 \times 2$ tiles. How many different ways are there to cover it? | 89 | 0.5 |
Find all integer values that the expression
$$
\frac{p q + p^{p} + q^{q}}{p + q}
$$
where \(p\) and \(q\) are prime numbers. | 3 | 0.625 |
If \(n\) is a positive integer, the symbol \(n!\) (read "n factorial") represents the product of the integers from 1 to \(n\). For example, \(4! = (1)(2)(3)(4)\) or \(4! = 24\). Determine
$$
\frac{1}{\log_{2} 100!} + \frac{1}{\log_{3} 100!} + \cdots + \frac{1}{\log_{100} 100!}
$$ | 1 | 0.625 |
Solve the equation $\arcsin \frac{x \sqrt{5}}{3}+\arcsin \frac{x \sqrt{5}}{6}=\arcsin \frac{7 x \sqrt{5}}{18}$. | 0, \pm \frac{8}{7} | 0.125 |
Suppose \( x \) satisfies \( x^{3} + x^{2} + x + 1 = 0 \). What are all possible values of \( x^{4} + 2x^{3} + 2x^{2} + 2x + 1 \)? | 0 | 0.875 |
If the acute angle \(\alpha\) satisfies \(\frac{1}{\sqrt{\tan \frac{\alpha}{2}}}=\sqrt{2 \sqrt{3}} \sqrt{\tan 10^{\circ}}+\sqrt{\tan \frac{\alpha}{2}}\), then the measure of the angle \(\alpha\) in degrees is \(\qquad\) | 50^\circ | 0.875 |
Verify that 2 is a solution of the equation:
$$
(x+1)^{3}+(x+2)^{3}+(x+3)^{3}=(x+4)^{3}
$$
Does this equation have any other integer solutions? | 2 | 0.75 |
\(1.25 \times 67.875 + 125 \times 6.7875 + 1250 \times 0.053375\). | 1000 | 0.5 |
The three-tiered "pyramid" shown in the image is built from $1 \mathrm{~cm}^{3}$ cubes and has a surface area of $42 \mathrm{~cm}^{2}$. We made a larger "pyramid" based on this model, which has a surface area of $2352 \mathrm{~cm}^{2}$. How many tiers does it have? | 24 | 0.125 |
Let \( n \in \mathbf{N}^{*} \). Consider the set \( S = \{1, 2, \ldots, 2n\} \) and its \( k \) subsets \( A_{1}, A_{2}, \ldots, A_{k} \) that satisfy the following conditions:
1. For any \( i \neq j \) (with \( i, j \in \{1, 2, \ldots, k\} \)), the intersection \( A_{i} \cap A_{j} \) has exactly an odd number of elements.
2. For any \( i \) (where \( i = 1, 2, \ldots, k \)), \( i \notin A_{i} \).
3. If \( i \in A_{j} \), then \( j \in A_{i} \).
Determine the maximum value of \( k \). | 2n - 1 | 0.625 |
Find the number of different monic quadratic trinomials (i.e. with the leading coefficient equal to 1) with integer coefficients that have two different roots, which are powers of the number 3 with natural exponents, and their coefficients in absolute value do not exceed $27^{45}$. | 4489 | 0.125 |
For the positive integer \( n \), if the expansion of \( (xy - 5x + 3y - 15)^n \) is combined and simplified, and \( x^i y^j \) (where \( i, j = 0, 1, \ldots, n \)) has at least 2021 terms, what is the minimum value of \( n \)? | 44 | 0.875 |
Circles \( C_1, C_2, C_3 \) have radius 1 and centers \( O, P, Q \) respectively. \( C_1 \) and \( C_2 \) intersect at \( A \), \( C_2 \) and \( C_3 \) intersect at \( B \), \( C_3 \) and \( C_1 \) intersect at \( C \), in such a way that \( \angle A P B = 60^\circ \), \( \angle B Q C = 36^\circ \), and \( \angle C O A = 72^\circ \). Find angle \( A B C \) (degrees). | 90^\circ | 0.375 |
Given a sequence \( A = (a_1, a_2, \cdots, a_{10}) \) that satisfies the following four conditions:
1. \( a_1, a_2, \cdots, a_{10} \) is a permutation of \{1, 2, \cdots, 10\};
2. \( a_1 < a_2, a_3 < a_4, a_5 < a_6, a_7 < a_8, a_9 < a_{10} \);
3. \( a_2 > a_3, a_4 > a_5, a_6 > a_7, a_8 > a_9 \);
4. There does not exist \( 1 \leq i < j < k \leq 10 \) such that \( a_i < a_k < a_j \).
Find the number of such sequences \( A \). | 42 | 0.375 |
For a positive integer \( n \), if there exist positive integers \( a \) and \( b \) such that \( n = a + b + a \times b \), then \( n \) is called a "good number". For example, \( 3 = 1 + 1 + 1 \times 1 \), so 3 is a "good number". Among the 100 positive integers from 1 to 100, there are \(\qquad\) "good numbers". | 74 | 0.625 |
Find the valuation...
(i) 3-adic valuation of \( A = 2^{27} + 1 \).
(ii) 7-adic valuation of \( B = 161^{14} - 112^{14} \).
(iii) 2-adic valuation of \( C = 7^{20} + 1 \).
(iv) 2-adic valuation of \( D = 17^{48} - 5^{48} \). | 6 | 0.125 |
Alice, Bob, and Charlie each pick a 2-digit number at random. What is the probability that all of their numbers' tens digits are different from each other's tens digits and all of their numbers' ones digits are different from each other's ones digits? | \frac{112}{225} | 0.375 |
An archipelago consisting of an infinite number of islands stretches along the southern shore of a boundless sea. The islands are connected by an infinite chain of bridges, with each island connected to the shore by a bridge. In the event of a severe earthquake, each bridge independently has a probability $p=0.5$ of being destroyed. What is the probability that after the severe earthquake, it will be possible to travel from the first island to the shore using the remaining intact bridges? | \frac{2}{3} | 0.5 |
Let \(\mathcal{P}\) be a regular 17-gon. We draw in the \(\binom{17}{2}\) diagonals and sides of \(\mathcal{P}\) and paint each side or diagonal one of eight different colors. Suppose that there is no triangle (with vertices among vertices of \(\mathcal{P}\)) whose three edges all have the same color. What is the maximum possible number of triangles, all of whose edges have different colors? | 544 | 0.375 |
Find all pairs of positive numbers \(a\) and \(b\) for which the numbers \(\sqrt{ab}, \frac{a+b}{2},\) and \(\sqrt{\frac{a^{2}+b^{2}}{2}}\) can form an arithmetic progression. | a = b | 0.75 |
The side \( AB \) of triangle \( ABC \) is equal to 3, \( BC = 2AC \), \( E \) is the point of intersection of the extension of the angle bisector \( CD \) of this triangle with the circumscribed circle around it, \( DE = 1 \). Find \( AC \). | \sqrt{3} | 0.625 |
Given that $a \in A$, and $a-1 \notin A$ and $a+1 \notin A$, $a$ is called an isolated element of set $A$. How many four-element subsets of the set $M=\{1,2, \cdots, 9\}$ have no isolated elements? | 21 | 0.375 |
Let \( p \) and \( q \) be two distinct odd prime numbers. Determine the (multiplicative) order of \( 1 + pq \) modulo \( p^2 q^3 \). | pq^2 | 0.5 |
Divide each natural number by the sum of the squares of its digits (for single-digit numbers, divide by the square of the number). Is there a smallest quotient among the obtained quotients, and if so, which one is it? | \frac{1}{9} | 0.75 |
Petya came up with the reduced quadratic equation \(x^{2} + px + q\), whose roots are \(x_{1}\) and \(x_{2}\). He informed Vasya of three out of the four numbers \(p, q, x_{1}, x_{2}\) without specifying which is which. These numbers turned out to be \(1, 2, -6\). What was the fourth number? | -3 | 0.625 |
The Pythagorean school believed that numbers are the origin of all things, and they called numbers like $1, 3, 6, 10, \cdots$ triangular numbers. What is the sum of the first 100 triangular numbers? | 171700 | 0.875 |
Solve the equation \((9-3x) \cdot 3^x - (x-2)\left(x^2 - 5x + 6\right) = 0\) in the set of real numbers. | x = 3 | 0.875 |
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 $\{x\}=x-[x]$, and $[x]$ denotes the greatest integer not exceeding $x$. Find the minimum perimeter of such a triangle.
| 3003 | 0.375 |
Let's conduct \( DL \perp AC \), \(LK \parallel CC_1\) (where \( K \in AC_1 \)), and \( PK \parallel DL \). By marking segment \( BQ = PD_1 \) on the lateral edge \( BB_1 \), we form a parallelogram \( PAQC_1 \), which will be the intersection of minimum area; here, \( AC_1 \) is the larger diagonal, and \( PQ \) is the smaller diagonal, both given in the problem statement.
Given:
\[ AC_1 = 3, \quad PQ = \sqrt{3}, \quad \varphi = 30^\circ \]
\[ \sin \varphi = \frac{1}{2}, \quad \cos \varphi = \frac{\sqrt{3}}{2} \]
Calculations:
\[ DL = PK = \frac{1}{2} \cdot PQ \cdot \sin \varphi = \frac{1}{2} \cdot \sqrt{3} \cdot \frac{1}{2} = \frac{\sqrt{3}}{4} \]
\[ OK = \frac{1}{2} \cdot PQ \cdot \cos \varphi = \frac{1}{2} \cdot \sqrt{3} \cdot \frac{\sqrt{3}}{2} = \frac{3}{4} \]
\[ \frac{CL}{AL} = \frac{C_1 K}{AK} = \frac{C_1 O - OK}{C_1 O + OK} = \frac{3/2 - 3/4}{3/2 + 3/4} = \frac{1}{3} \]
Let \( CL = x \), then \( AL = 3x \), \( AC = 4x \).
\[ DL^2 = CL \cdot AL \]
\[ \frac{3}{16} = 3x^2 \]
\[ x = \frac{1}{4}, \quad AC = 1 \]
\[ CC_1 = \sqrt{AC_1^2 - AC^2} = \sqrt{9 - 1} = 2\sqrt{2} \]
Volume \( V = AC \cdot DL \cdot CC_1 = 1 \cdot \frac{\sqrt{3}}{4} \cdot 2\sqrt{2} = \sqrt{\frac{3}{2}} = \frac{\sqrt{6}}{2} \). | \frac{\sqrt{6}}{2} | 0.875 |
Dad, Masha, and Yasha are walking to school. While Dad takes 3 steps, Masha takes 5 steps. While Masha takes 3 steps, Yasha takes 5 steps. Masha and Yasha counted that together they have taken 400 steps. How many steps did Dad take? | 90 | 0.875 |
Five individuals are assigned to five specific seats, but they do not know their assigned numbers. When these five individuals sit randomly in these five seats, find:
(1) The probability that exactly 3 out of the 5 individuals sit in their assigned seats.
(2) If the probability that all 5 individuals sit in their assigned seats is not less than $\frac{1}{6}$, what is the maximum number of people who can sit in their assigned seats? | 2 | 0.875 |
In a math competition, there are 8 fill-in-the-blank questions worth 4 points each if answered correctly, and 6 short-answer questions worth 7 points each if answered correctly. If 400 people participated in the competition, what is the minimum number of people who have the same total score? | 8 | 0.375 |
Find the smallest real number \( A \) such that for every quadratic polynomial \( f(x) \) satisfying the condition
\[
|f(x)| \leq 1 \quad (0 \leq x \leq 1),
\]
the inequality \( f^{\prime}(0) \leq A \) holds. | 8 | 0.625 |
Find all real numbers \( p \) such that the cubic equation \( 5x^{3} - 5(p+1)x^{2} + (71p - 1)x + 1 = 66p \) has three roots, all of which are positive integers. | 76 | 0.375 |
On the segment \( A C \), point \( B \) is marked. Semicircles are constructed on the segments \( A B \), \( B C \), and \( A C \) as diameters in the same half-plane. A circle with center at point \( O \) is tangent to all these semicircles (see figure). Find the radius of this circle, given that \( A B = 4 \) and \( B C = 2 \). If necessary, round your answer to the nearest hundredth. | 0.86 | 0.625 |
Given an isosceles triangle \(ABC\) with \(AB = AC\) and \(\angle ABC = 53^\circ\), find the measure of \(\angle BAM\). 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 \(AC\);
- \(KM = AB\);
- \(\angle MAK\) is the maximum possible.
How many degrees is \(\angle BAM\)? | 44^\circ | 0.625 |
Find the smallest 10-digit number such that the sum of its digits is greater than that of any smaller number. | 1999999999 | 0.25 |
If \( a_{1} = 1 \), \( a_{2} = 0 \), and \( a_{n+1} = a_{n} + \frac{a_{n+2}}{2} \) for all \( n \geq 1 \), compute \( a_{2004} \). | -2^{1002} | 0.875 |
Find the smallest positive integer n such that n has exactly 144 positive divisors including 10 consecutive integers. | 110880 | 0.125 |
Determine the number $ABCC$ (written in decimal system) given that
$$
ABCC = (DD - E) \cdot 100 + DD \cdot E
$$
where $A, B, C, D,$ and $E$ are distinct digits. | 1966 | 0.125 |
Let \( n \) be a positive integer greater than 1. If \( 2n \) is divided by 3, the remainder is 2. If \( 3n \) is divided by 4, the remainder is 3. If \( 4n \) is divided by 5, the remainder is 4. If \( 5n \) is divided by 6, the remainder is 5. What is the least possible value of \( n \)? | 61 | 0.875 |
At a math competition, three problems were presented: \(A\), \(B\), and \(C\). There were 25 students, each of whom solved at least one problem. Among the students who did not solve problem \(A\), twice as many solved problem \(B\) as those who solved problem \(C\). Exactly one more student solved only problem \(A\) than the number of other students who also solved \(A\). Half of the students who solved only one problem did not solve problem \(A\). How many students solved only problem \(B\)? | 6 | 0.625 |
In the equation \(\overline{ABC} \times \overline{AB} + C \times D = 2017\), the same letters represent the same digits, and different letters represent different digits. If the equation holds true, what two-digit number does \(\overline{\mathrm{AB}}\) represent? | 14 | 0.5 |
A $7 \times 7$ grid is colored black and white. If the number of columns with fewer black squares than white squares is $m$, and the number of rows with more black squares than white squares is $n$, find the maximum value of $m + n$. | 12 | 0.5 |
The function \( g \) defined on the set of integers satisfies the following conditions:
1) \( g(1) - 1 > 0 \)
2) \( g(x) g(y) + x + y + xy = g(x+y) + x g(y) + y g(x) \) for any \( x, y \in \mathbb{Z} \)
3) \( 3 g(x+1) = g(x) + 2x + 3 \) for any \( x \in \mathbb{Z} \).
Find \( g(-6) \). | 723 | 0.875 |
\[ S_{ABCD} = \frac{1}{2} \cdot 6 \sqrt{3} \cdot (8 + 20) = 84 \sqrt{3}, \text{ then } S_{MNKP} = 42 \sqrt{3}. \] | 42 \sqrt{3} | 0.625 |
Given the plane point set \( A = \{ (x, y) \mid x = 2 \sin \alpha + 2 \sin \beta, y = 2 \cos \alpha + 2 \cos \beta \} \) and \( B = \{ (x, y) \mid \sin (x + y) \cos (x + y) \geq 0 \} \), find the area of the region represented by \( A \cap B \). | 8\pi | 0.5 |
Point \( K \) lies on side \( BC \) of parallelogram \( ABCD \), and point \( M \) lies on its side \( AD \). Segments \( CM \) and \( DK \) intersect at point \( L \), and segments \( AK \) and \( BM \) intersect at point \( N \). Find the maximum value of the ratio of the areas of quadrilateral \( KLMN \) to \( ABCD \). | \frac{1}{4} | 0.625 |
Find the base of an isosceles triangle if its side is equal to \( a \), and the height dropped onto the base is equal to the segment connecting the midpoint of the base with the midpoint of the side. | x = a \sqrt{3} | 0.625 |
Find the smallest real number \( r \) such that there exists a sequence of positive real numbers \(\{x_n\}\) satisfying \(\sum_{i=1}^{n+1} x_i \leq r \cdot x_n\) for any \( n \in \mathbb{N}^+ \). | 4 | 0.75 |
Calculate the limit of the numerical sequence:
\[ \lim _{n \rightarrow \infty} \sqrt{n^{3}+8}\left(\sqrt{n^{3}+2}-\sqrt{n^{3}-1}\right) \] | \frac{3}{2} | 0.875 |
29 boys and 15 girls attended a ball. Some boys danced with some girls (at most once per pair). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers that the children could have mentioned? | 29 | 0.125 |
There are 17 people standing in a circle: each of them is either truthful (always tells the truth) or a liar (always lies). All of them said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle? | 11 | 0.5 |
Using the digits from 1 to 9 exactly once, form the smallest nine-digit number that is divisible by 11. | 123475869 | 0.125 |
At a market in Egypt, a tourist is negotiating with a seller for a souvenir worth 10,000 Egyptian pounds. The tourist first reduces the price by \( x \) percent \((0 < x < 100)\), then the seller increases the price by \( x \) percent, and so on. The value of \( x \) does not change during the bargaining process, and the seller increases the price at least once. The negotiation continues until one of the participants gets a non-integer value for the price of the souvenir. Find the largest possible number of price changes during such a negotiation (including the last non-integer price change). | 5 | 0.625 |
Is it possible to place 8 different numbers from the range 1 to 220 at the vertices of a cube so that the numbers at adjacent vertices have a common divisor greater than 1, while the numbers at non-adjacent vertices do not have one? | \text{No} | 0.75 |
Suppose that the pair of positive integers \((x, y)\) satisfies \(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}} = \frac{1}{\sqrt{20}}\). How many different possible values are there for \(xy\)? | 2 | 0.875 |
Given two sets \( A = \{1, 2, 3, \ldots, 15\} \) and \( B = \{0, 1\} \), find the number of mappings \( f: A \rightarrow B \) with 1 being the image of at least two elements of \( A \). | 32752 | 0.875 |
Given \( f_{1}(x)=|1-2 x| \) for \( x \in [0,1] \) and \( f_{n}(x)=f_{1}(f_{n-1}(x)) \), determine the number of solutions to the equation \( f_{2005}(x)=\frac{1}{2} x \). | 2^{2005} | 0.625 |
Calculate the volumes of the solids formed by rotating the regions bounded by the graphs of the functions around the $y$-axis.
$$
y = x^3, \quad y = x^2
$$ | \frac{\pi}{10} | 0.75 |
There is a basket of apples. The first time, half of the apples plus 2 were taken out. The second time, half of the remaining apples minus 3 were taken out. There are 24 apples left in the basket. How many apples were originally in the basket? | 88 | 0.875 |
Find the largest root of the equation
$$
3 \sqrt{x-2} + 2 \sqrt{2x+3} + \sqrt{x+1} = 11
$$ | 3 | 0.875 |
Find the sum of all even factors of 1152. | 3302 | 0.5 |
In the nodes of a grid, numbers \( 1, 2, 3, \ldots \) are placed in a spiral order. Then, at the center of each cell, the sum of the numbers in its nodes is written (see Fig. 4). Is it true that the numbers divisible by 52 will appear infinitely many times at the centers of the cells? | Yes | 0.75 |
A cell can divide into 42 or 44 smaller cells. How many divisions are needed to obtain exactly 1993 cells from a single cell? | 48 | 0.625 |
Compute the number of sequences of numbers \( a_{1}, a_{2}, \ldots, a_{10} \) such that:
I. \( a_{i}=0 \) or 1 for all \( i \),
II. \( a_{i} \cdot a_{i+1}=0 \) for \( i=1,2, \ldots, 9 \),
III. \( a_{i} \cdot a_{i+2}=0 \) for \( i=1,2, \ldots, 8 \). | 60 | 0.375 |
There is a caravan with 100 camels, consisting of both one-humped and two-humped camels, with at least one of each kind. If you take any 62 camels, they will have at least half of the total number of humps in the caravan. Let \( N \) be the number of two-humped camels. How many possible values can \( N \) take within the range from 1 to 99?
| 72 | 0.625 |
Find all natural numbers whose own divisors can be paired such that the numbers in each pair differ by 545. An own divisor of a natural number is a natural divisor different from one and the number itself. | 1094 | 0.5 |
Find the largest natural number that cannot be represented as the sum of two composite numbers. (Recall that a natural number is called composite if it is divisible by some natural number other than itself and one.) | 11 | 0.625 |
A technique often used to calculate summations is the Telescoping Sum. It involves "decomposing" the terms of a sum into parts that cancel out. For example,
$$
\begin{aligned}
& \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}= \\
& \left(\frac{1}{1}-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)= \\
& \frac{1}{1}-\frac{1}{5}= \\
& \frac{4}{5}
\end{aligned}
$$
Using this technique, we can find a way to sum consecutive odd numbers. Let's see:
a) By listing the odd numbers one by one and starting from 1, verify that the number in position \( m \) is equal to \( m^{2}-(m-1)^{2} \).
b) Calculate the sum of all the odd numbers between 1000 and 2014. | 764049 | 0.625 |
In a regular polygon with 1994 sides, a positive number is written at each vertex. Each number is either the arithmetic mean or the geometric mean of its two adjacent numbers. If one of these numbers is 32, what is the number adjacent to it? | 32 | 0.875 |
Let \( F_{1} \) and \( F_{2} \) be the foci of the ellipse \(\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1\). \( P \) is a point on the ellipse such that \(\left|P F_{1}\right| : \left|P F_{2}\right| = 2 : 1\). Find the area of triangle \(\triangle P F_{1} F_{2}\). | 4 | 0.75 |
How many six-digit numbers exist such that the sum of their digits is 51? | 56 | 0.125 |
Find the smallest natural number \( N \) such that \( N+2 \) is divisible by 2, \( N+3 \) is divisible by 3, \ldots, \( N+10 \) is divisible by 10. | 2520 | 0.625 |
In triangle \( A B C \), the angle \( \angle B \) is equal to \( \frac{\pi}{3} \). A circle with radius 3 cm is drawn through points \( A \) and \( B \), touching the line \( A C \) at point \( A \). Another circle with radius 4 cm is drawn through points \( B \) and \( C \), touching the line \( A C \) at point \( C \). Find the length of side \( A C \). | 6 | 0.25 |
A four-digit number satisfies the following conditions:
(1) If you simultaneously swap its unit digit with the hundred digit and the ten digit with the thousand digit, the value increases by 5940;
(2) When divided by 9, the remainder is 8.
Find the smallest odd four-digit number that satisfies these conditions.
(Shandong Province Mathematics Competition, 1979) | 1979 | 0.375 |
Find the number of pairs of integers $(a, b)$ such that \(1 \leq a \leq 80\) and \(1 \leq b \leq 30\), and the area \( S \) of the figure defined by the system of inequalities
\[
\begin{cases}
\frac{x}{a} + \frac{y}{b} \geq 1 \\
x \leq a \\
y \leq b
\end{cases}
\]
is such that the number \( 2S \) is a multiple of 5. | 864 | 0.75 |
A chord of a circle is at a distance \( h \) from the center. In each of the segments subtended by the chord, a square is inscribed such that two adjacent vertices of the square lie on the arc, and the other two lie on the chord or its extension. What is the difference in the side lengths of these squares? | \frac{8h}{5} | 0.5 |
On a rotating round table, there are 8 white teacups and 7 black teacups. Fifteen dwarves wearing hats (8 white hats and 7 black hats) are sitting around the table. Each dwarf picks a teacup of the same color as their hat and places it in front of them. After this, the table is rotated randomly. What is the maximum number of teacups that can be guaranteed to match the color of the dwarf's hat after the table is rotated? (The dwarves are allowed to choose their seating, but they do not know how the table will be rotated.) | 7 | 0.5 |
What will the inflation be over two years:
$$
\left((1+0,015)^{\wedge} 2-1\right)^{*} 100 \%=3,0225 \%
$$
What will be the real yield of a bank deposit with an extension for the second year:
$$
(1,07 * 1,07 /(1+0,030225)-1) * 100 \%=11,13 \%
$$ | 11.13\% | 0.5 |
Given the sequence \(\left\{x_{n}\right\}\) satisfies:
\[ x_{1}=a, \, x_{2}=b, \, x_{n+2}=3 x_{n+1}+2 x_{n}, \]
if there exists an integer \(k \geq 3\) such that \(x_{k}=2019\), then determine the number of such positive integer ordered pairs \((a, b)\). | 370 | 0.125 |
Given the hyperbola \( C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (where \( a > 0 \) and \( b > 0 \)), its eccentricity is \( \frac{\sqrt{5}}{2} \). The points \( F_{1} \) and \( F_{2} \) are the left and right foci of \( C \), respectively. A line \( l \) passing through \( F_{2} \) intersects \( C \) at points \( A \) and \( B \) (with point \( A \) in the first quadrant), and \(\overrightarrow{A F_{2}} = 3 \overrightarrow{F_{2} B} \). Given that the area of triangle \( \triangle F_{1} A B \) is \( \frac{32}{3} \), find the radius of the incircle of triangle \( \triangle F_{1} A B \). | 1 | 0.5 |
In triangle \( \triangle ABC \), \( |\overrightarrow{AB}| = 2 \), \( |\overrightarrow{BC}| = 3 \), \( |\overrightarrow{CA}| = 4 \). Find the value of \( \overrightarrow{AB} \cdot \overrightarrow{BC} + \overrightarrow{BC} \cdot \overrightarrow{CA} + \overrightarrow{CA} \cdot \overrightarrow{AB} \). | -\frac{29}{2} | 0.875 |
Let \( f(x) \) be an even function and \( g(x) \) be an odd function, and suppose \( f(x) = -g(x+c) \) where \( c > 0 \). Determine the smallest positive period of the function \( f(x) \). | 4c | 0.375 |
Vasya has selected 8 squares on a chessboard such that no two squares are in the same row or column. On his turn, Petya places 8 rooks on the board in such a way that none of them attack each other, and then Vasya points out all the rooks that are standing on the selected squares. If the number of rooks pointed out by Vasya on this turn is even (i.e., 0, 2, 4, 6, or 8), Petya wins; otherwise, all pieces are removed from the board and Petya takes the next turn. What is the minimum number of turns in which Petya can guarantee a win?
(I. Bogdanov) | 2 | 0.375 |
The increasing sequence $1, 3, 4, 9, 10, 12, 13, \cdots$ consists of positive integers that are either powers of 3 or the sum of some distinct powers of 3. Find the 100th term of this sequence. | 981 | 0.375 |
There are 22 people standing in a circle, and each of them is either a knight (who always tells the truth) or a liar (who always lies). Each person says: "The next 10 people clockwise from me are liars." How many liars are there among these 22 people? | 20 | 0.625 |
Let \( a \in \mathbf{R} \). The equation \( ||x-a|-a|=2 \) has exactly three distinct solutions. Find the value of \( a \). | 2 | 0.75 |
Find the largest even three-digit number \( x \) that leaves a remainder of 2 when divided by 5 and satisfies the condition \(\text{GCD}(30, \text{GCD}(x, 15)) = 3\). | 972 | 0.75 |
On an $8 \times 8$ board, several dominoes (rectangles of $2 \times 1$) can be placed without overlapping. Let $N$ be the number of ways to place 32 dominoes in this manner, and $T$ be the number of ways to place 24 dominoes in this manner. Which is greater: $-N$ or $M$? Configurations that can be derived from one another by rotating or reflecting the board are considered distinct. | T | 0.75 |
Let \([x]\) denote the greatest integer less than or equal to \(x\). Find the value of \(\sum_{k=0}^{\infty}\left[\frac{n+2^{k}}{2^{k+1}}\right]\), where \(n\) is any natural number. | n | 0.75 |
Find the sum of all irreducible fractions with a denominator of 3 that are between the positive integers \( m \) and \( n \) (where \( m < n \)). | n^2 - m^2 | 0.375 |
Twenty-five coins are being separated into piles in the following manner. First, they are randomly divided into two groups. Then, any one of the existing groups is again divided into two groups, and so on until each group consists of one coin. During each division of a group into two, the product of the number of coins in the two resulting groups is recorded. What could be the sum of all the recorded numbers? | 300 | 0.875 |
Which is greater: \(7^{92}\) or \(8^{91}\)? | 8^{91} | 0.625 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.