id int64 0 10k | old_math_orz_id int64 1 56.9k | input stringlengths 20 3.72k | target stringlengths 1 18 | qwen2.5-3b-instruct-pass-at-10 int64 0 10 |
|---|---|---|---|---|
200 | 3,765 | Let M be a subst of {1,2,...,2006} with the following property: For any three elements x,y and z (x<y<z) of M, x+y does not divide z. Determine the largest possible size of M. Justify your claim. | 1004 | 0 |
201 | 33,841 | What is the smallest positive integer $n$ such that $5n$ is a perfect square and $3n$ is a perfect cube? | 1125 | 0 |
202 | 29,408 | Determine the number of roots of unity that are also roots of the polynomial equation $z^3 + az^2 + bz + c = 0$ for some integers $a$, $b$, and $c$. | 8 | 0 |
203 | 11,793 | Let $P_{1}: y=x^{2}+\frac{101}{100}$ and $P_{2}: x=y^{2}+\frac{45}{4}$ be two parabolas in the Cartesian plane. Let $\mathcal{L}$ be the common tangent line of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax+by=c$ for positive integers $a,b,c$ where $\gcd(a,b,c)=1$, find $a+b+... | 11 | 0 |
204 | 9,790 | Medians are drawn from point $A$ and point $B$ in this right triangle to divide segments $\overline{BC}$ and $\overline{AC}$ in half, respectively. The lengths of the medians are 6 and $2\sqrt{11}$ units, respectively. How many units are in the length of segment $\overline{AB}$?
[asy]
draw((0,0)--(7,0)--(0,4)--(0,0)--... | 8 | 0 |
205 | 4,588 | How many ordered triples $(a, b, c)$ of positive integers are there such that none of $a, b, c$ exceeds $2010$ and each of $a, b, c$ divides $a + b + c$ ? | 9045 | 0 |
206 | 42 | A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a s... | 945 | 0 |
207 | 54,538 | If real numbers \( x \) and \( y \) satisfy \( 4x^2 - 4xy + 2y^2 = 1 \), then the sum of the maximum and minimum values of \( 3x^2 + xy + y^2 \) is \(\qquad\) | 3 | 0 |
208 | 54,965 | Find the largest natural number \( n \) that is a divisor of \( a^{25} - a \) for every integer \( a \). | 2730 | 0 |
209 | 4,015 | For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$ . Find the minimum possible value for $|p(15)|$ . | 47 | 0 |
210 | 47,253 | Each chocolate costs 1 dollar, each licorice stick costs 50 cents, and each lolly costs 40 cents. How many different combinations of these three items cost a total of 10 dollars? | 36 | 0 |
211 | 3 | In [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$ with [right angle](https://artofproblemsolving.com/wiki/index.php/Right_angle) $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. [Points](https://artofproblemsolving.com/wiki/index.php/Point) $... | 737 | 0 |
212 | 48,430 | What is the average of five-digit palindromic numbers? (A whole number is a palindrome if it reads the same backward as forward.) | 55000 | 0 |
213 | 46,590 | Find the number of 5-digit numbers where the ten-thousands place is not 5, the units place is not 2, and all digits are distinct. | 21840 | 0 |
214 | 28,054 | Thirty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $\log_2 ... | 409 | 0 |
215 | 52,161 | Let \( t \) be a positive number greater than zero.
Quadrilateral \(ABCD\) has vertices \(A(0,3), B(0,k), C(t, 10)\), and \(D(t, 0)\), where \(k>3\) and \(t>0\). The area of quadrilateral \(ABCD\) is 50 square units. What is the value of \(k\)? | 13 | 0 |
216 | 55,041 | Let \( n \) be an integer and
$$
m = (n-1001)(n-2001)(n-2002)(n-3001)(n-3002)(n-3003).
$$
Given that \( m \) is positive, find the minimum number of digits of \( m \). | 11 | 0 |
217 | 3,166 | Three concentric circles have radii $3$ , $4$ , and $5$ . An equilateral triangle with one vertex on each circle has side length $s$ . The largest possible area of the triangle can be written as $a+\frac{b}{c}\sqrt{d}$ , where $a,b,c$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and ... | 41 | 0 |
218 | 48,564 | From the set of three-digit numbers that do not contain the digits 0, 6, 7, 8, or 9, several numbers were written on paper in such a way that no two numbers can be obtained from each other by swapping two adjacent digits. What is the maximum number of such numbers that could have been written? | 75 | 0 |
219 | 44,809 | Compute the sum of all integers \(1 \leq a \leq 10\) with the following property: there exist integers \(p\) and \(q\) such that \(p, q, p^{2}+a\) and \(q^{2}+a\) are all distinct prime numbers. | 20 | 0 |
220 | 3,631 | Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$ , if $p = kq + r, 0 \leq r < q$ , there does not exist an integer $q > 1$ such that $a^{2} \mid r$ . | 13 | 0 |
221 | 18,643 | Given the following data, determine how much cheaper, in cents, is the cheapest store's price for Camera $Y$ compared to the most expensive one?
\begin{tabular}{|l|l|}
\hline
\textbf{Store} & \textbf{Sale Price for Camera $Y$} \\ \hline
Mega Deals & $\$12$~off the list price~$\$52.50$ \\ \hline
Budget Buys & $30\%$~of... | 375 | 0 |
222 | 1,160 | Fuzzy draws a segment of positive length in a plane. How many locations can Fuzzy place another point in the same plane to form a non-degenerate isosceles right triangle with vertices consisting of his new point and the endpoints of the segment?
*Proposed by Timothy Qian* | 6 | 0 |
223 | 799 | Let $S$ be a set of $2020$ distinct points in the plane. Let
\[M=\{P:P\text{ is the midpoint of }XY\text{ for some distinct points }X,Y\text{ in }S\}.\]
Find the least possible value of the number of points in $M$ . | 4037 | 0 |
224 | 37,482 | Find the $50^{\text{th}}$ term in the sequence of all positive integers which are either powers of 3 or sums of distinct powers of 3. | 327 | 0 |
225 | 48,724 | There is a peculiar four-digit number (with the first digit not being 0). It is a perfect square, and the sum of its digits is also a perfect square. Dividing this four-digit number by the sum of its digits results in yet another perfect square. Additionally, the number of divisors of this number equals the sum of its ... | 2601 | 0 |
226 | 54,951 | A natural number \( n \) is such that the number \( 36 n^{2} \) has exactly 51 distinct natural divisors. How many natural divisors does the number \( 5n \) have? | 16 | 0 |
227 | 41,804 | Consider a solar system where the planets have the following numbers of moons:
| Planet | Number of Moons |
|----------|-----------------|
| Mercury | 0 |
| Venus | 0 |
| Earth | 3 |
| Mars | 4 |
| Jupiter | 18 |
| Saturn | 24 ... | 7.78 | 0 |
228 | 8,285 | Let $N$ be the number of ways to write $2010$ in the form $2010 = a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0$, where the $a_i$'s are integers, and $0 \le a_i \le 99$. An example of such a representation is $1\cdot 10^3 + 3\cdot 10^2 + 67\cdot 10^1 + 40\cdot 10^0$. Find $N$.
| 202 | 0 |
229 | 45,831 | What is the minimum number of cells that need to be marked in a $7 \times 7$ grid so that in each vertical or horizontal $1 \times 4$ strip there is at least one marked cell? | 12 | 0 |
230 | 22,235 | If $m$ is an even integer and $n$ is an odd integer, how many terms in the expansion of $(m+n)^8$ are even? | 8 | 0 |
231 | 55,595 | The figure \( F \) is bounded by the line \( y=x \) and the parabola \( y=2^{-100} x^{2} \). How many points of the form \( (2^{m}, 2^{n}) \) (where \( m, n \in \mathbf{N}_{+} \)) are inside the figure \( F \) (excluding the boundary)? | 2401 | 0 |
232 | 53,262 | In the number $2 * 0 * 1 * 6 * 0 * 2 *$, each of the 6 asterisks needs to be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may repeat), so that the resulting 12-digit number is divisible by 75. How many ways can this be done? | 2592 | 0 |
233 | 49,197 | In a certain game, the "magician" asks a person to randomly think of a three-digit number ($abc$), where $a, b, c$ are the digits of this number. They then ask this person to arrange the digits into 5 different numbers: $(acb)$, $(bac)$, $(bca)$, $(cab)$, and $(cba)$. The magician then asks for the sum of these 5 numbe... | 358 | 0 |
234 | 45,918 | In triangle $ABC$, the angles $\angle B = 30^{\circ}$ and $\angle A = 90^{\circ}$ are known. Point $K$ is marked on side $AC$, and points $L$ and $M$ are marked on side $BC$ such that $KL = KM$ (point $L$ lies on segment $BM$).
Find the length of segment $LM$ if it is known that $AK = 4$, $BL = 31$, and $MC = 3$. | 14 | 0 |
235 | 1,446 | For positive integer $n,$ let $s(n)$ be the sum of the digits of n when n is expressed in base ten. For example, $s(2022) = 2 + 0 + 2 + 2 = 6.$ Find the sum of the two solutions to the equation $n - 3s(n) = 2022.$ | 4107 | 0 |
236 | 24,880 | Calculate the sum of the series $2 + 4 + 8 + 16 + 32 + \cdots + 512 + 1000$. | 2022 | 0 |
237 | 34,644 | For an upcoming French exam, I need to cover 750 words. The exam grading system is different: I get full points for each word I recall correctly, and for every incorrectly guessed word, I receive half the points if they are semantically or contextually related, assuming 30% of my wrong guesses are relevant. What is the... | 574 | 0 |
238 | 51,838 | A seven-digit number \(\overline{m0A0B9C}\) is a multiple of 33. We denote the number of such seven-digit numbers as \(a_{m}\). For example, \(a_{5}\) represents the count of seven-digit numbers of the form \(\overline{50A0B9C}\) that are multiples of 33. Find the value of \(a_{2}-a_{3}\). | 8 | 0 |
239 | 23,699 | Find the 150th term of the sequence that consists of all those positive integers which are either powers of 3 or sums of distinct powers of 3. | 2280 | 0 |
240 | 52,141 | Let \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) be a non-decreasing sequence of positive integers. For \( m \geqslant 1 \), define \( b_{m}=\min \left\{n ; a_{n} \geqslant m\right\} \), i.e., \( b_{m} \) is the smallest value of \( n \) such that \( a_{n} \geqslant m \). Given that \( a_{19}=85 \), find the maximum value... | 1700 | 0 |
241 | 38,697 | If \( f(x) \) is a monic quartic polynomial such that \( f(-2)=-4 \), \( f(1)=-1 \), \( f(-4)=-16 \), and \( f(5)=-25 \), find \( f(0) \). | 40 | 0 |
242 | 45,316 | Split the numbers \(1, 2, \ldots, 10\) into two groups and let \(P_{1}\) be the product of the first group and \(P_{2}\) the product of the second group. If \(P_{1}\) is a multiple of \(P_{2}\), find the minimum value of \(\frac{P_{1}}{\P_{2}}\). | 7 | 0 |
243 | 34,241 | Let $\triangle PQR$ be a right triangle with $Q$ as a right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If $PS = 3$ and $QS = 9$, find the length of $RS$. | 27 | 0 |
244 | 4,476 | Four men are each given a unique number from $1$ to $4$ , and four women are each given a unique number from $1$ to $4$ . How many ways are there to arrange the men and women in a circle such that no two men are next to each other, no two women are next to each other, and no two people with the same number are ne... | 12 | 0 |
245 | 1,087 | We delete the four corners of a $8 \times 8$ chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?
*Proposed by Evan Chen* | 21600 | 0 |
246 | 14,986 | How many whole numbers between 200 and 500 contain the digit 3? | 138 | 0 |
247 | 28,857 | In a modified game of Deal or No Deal, participants choose a box at random from a set of $30$ boxes, each containing one of the following values: \begin{tabular}{|c|c|}\hline\$.01&\$2,000\\\hline\$1&\$10,000\\\hline\$10&\$50,000\\\hline\$100&\$100,000\\\hline\$500&\$200,000\\\hline\$1,000&\$300,000\\\hline\$5,000&\$400... | 18 | 0 |
248 | 39,390 | Let $\alpha$, $\beta$, $\gamma$ be the roots of the cubic polynomial $x^3 - 3x - 2 = 0.$ Find
\[
\alpha(\beta - \gamma)^2 + \beta(\gamma - \alpha)^2 + \gamma(\alpha - \beta)^2.
\] | -18 | 0 |
249 | 44,998 | Given a polygon drawn on graph paper with a perimeter of 2014 units, and whose sides follow the grid lines, what is the maximum possible area of this polygon? | 253512 | 0 |
250 | 2,286 | Compute the maximum integer value of $k$ such that $2^k$ divides $3^{2n+3}+40n-27$ for any positive integer $n$ . | 6 | 0 |
251 | 21,748 | The Hoopers, coached by Coach Loud, have 15 players. George and Alex are the two players who refuse to play together in the same lineup. Additionally, if George plays, another player named Sam refuses to play. How many starting lineups of 6 players can Coach Loud create, provided the lineup does not include both George... | 3795 | 0 |
252 | 8,112 | Find the value of $x$ between 0 and 180 such that
\[\tan (120^\circ - x^\circ) = \frac{\sin 120^\circ - \sin x^\circ}{\cos 120^\circ - \cos x^\circ}.\] | 100 | 0 |
253 | 31,607 | Let $d$ be a complex number. Suppose there exist distinct complex numbers $x$, $y$, and $z$ such that for every complex number $w$, we have
\[
(w - x)(w - y)(w - z) = (w - dx)(w - dy)(w - dz).
\]
Compute the number of distinct possible values of $d$. | 4 | 0 |
254 | 53,445 |
An investor has an open brokerage account with an investment company. In 2021, the investor received the following income from securities:
- Dividends from shares of the company PAO “Winning” amounted to 50,000 rubles.
- Coupon income from government bonds OFZ amounted to 40,000 rubles.
- Coupon income from corporate... | 11050 | 0 |
255 | 49,894 | If there is a positive integer \( m \) such that the factorial of \( m \) has exactly \( n \) trailing zeros, then the positive integer \( n \) is called a "factorial trailing number." How many positive integers less than 1992 are non-"factorial trailing numbers"? | 396 | 0 |
256 | 44 | What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be
\[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\] | 337 | 0 |
257 | 9,991 | Point $B$ is on $\overline{AC}$ with $AB = 9$ and $BC = 21.$ Point $D$ is not on $\overline{AC}$ so that $AD = CD,$ and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\triangle ACD$. Find $s.$
| 380 | 0 |
258 | 726 | The smallest three positive proper divisors of an integer n are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 = 57$ . Find the sum of the possible values of $d_2$ . | 42 | 0 |
259 | 41,664 | A root of unity is a complex number that satisfies $z^n = 1$ for some positive integer $n$. Determine the number of roots of unity that also satisfy $z^3 + az^2 + bz + c = 0$ for integers $a$, $b$, and $c$. | 8 | 0 |
260 | 3,556 | Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions:
1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$
2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$ 3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$ | 9 | 0 |
261 | 43,814 | A positive integer cannot be divisible by 2 or 3, and there do not exist non-negative integers \(a\) and \(b\) such that \(|2^a - 3^b| = n\). Find the smallest value of \(n\). | 35 | 0 |
262 | 44,090 | If a positive integer cannot be written as the difference of two square numbers, then the integer is called a "cute" integer. For example, 1, 2, and 4 are the first three "cute" integers. Find the \(2010^{\text{th}}\) "cute" integer. | 8030 | 0 |
263 | 30,058 | How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | 39 | 0 |
264 | 45,324 | The following sequence of numbers is written: \(\sqrt{7.301}, \sqrt{7.302}, \sqrt{7.303}, \ldots, \sqrt{16.002}, \sqrt{16.003}\) (under the square root are consecutive terms of an arithmetic progression with a difference of 0.001). Find the number of rational numbers among the written numbers. | 13 | 0 |
265 | 4,434 | Angela, Bill, and Charles each independently and randomly choose a subset of $\{ 1,2,3,4,5,6,7,8 \}$ that consists of consecutive integers (two people can select the same subset). The expected number of elements in the intersection of the three chosen sets is $\frac{m}{n}$ , where $m$ and $n$ are relatively prim... | 421 | 0 |
266 | 30,200 | A solid rectangular block is composed of $N$ congruent 1-cm cubes glued together face to face. If this block is viewed so that three of its faces are visible, exactly $143$ of the 1-cm cubes are not visible. Determine the smallest possible value of $N$. | 336 | 0 |
267 | 16,385 | Seven test scores have a mean of $85$, a median of $88$, and a mode of $90$. Calculate the sum of the three lowest test scores. | 237 | 0 |
268 | 45,261 | The pages in a book are numbered as follows: the first sheet contains two pages (numbered 1 and 2), the second sheet contains the next two pages (numbered 3 and 4), and so on. A mischievous boy named Petya tore out several consecutive sheets: the first torn-out page has the number 185, and the number of the last torn-o... | 167 | 0 |
269 | 1,581 | Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$ ), for how many integers $1\leq n\leq2017$ does $5^n$ start with $1$ ?
*2018 CCA Math Bonanza Tiebreaker Round #3* | 607 | 0 |
270 | 10,566 | In triangle $ABC$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O$, and that $\frac{AO}{OA'}+\frac{BO}{OB'}+\frac{CO}{OC'}=92$, find $\frac{AO}{OA'}\cdot \frac{BO}{OB'}\cdot \frac{CO}{OC'}$.
| 94 | 0 |
271 | 27,854 | The equations $x^3 + Cx^2 + 20 = 0$ and $x^3 + Dx + 100 = 0$ have two roots in common. Then the product of these common roots can be expressed in the form $a \sqrt[b]{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified. Find $a + b + c.$ | 15 | 0 |
272 | 39,413 | Eight students participate in a fruit eating contest, where they choose between apples and bananas. The graph shows the number of apples eaten by four students (Sam, Zoe, Mark, Anne) and the number of bananas eaten by another four students (Lily, John, Beth, Chris). In the apple contest, Sam ate the most apples and Mar... | 5 | 0 |
273 | 2,834 | Let $P\left(X\right)=X^5+3X^4-4X^3-X^2-3X+4$ . Determine the number of monic polynomials $Q\left(x\right)$ with integer coefficients such that $\frac{P\left(X\right)}{Q\left(X\right)}$ is a polynomial with integer coefficients. Note: a monic polynomial is one with leading coefficient $1$ (so $x^3-4x+5$ is one ... | 12 | 0 |
274 | 49,224 |
On the side \( AC \) of an equilateral triangle \( ABC \), a point \( D \) is marked. On the segments \( AD \) and \( DC \), equilateral triangles \( ADE \) and \( DCF \) are constructed outward from the original triangle. It is known that the perimeter of triangle \( DEF \) is 19, and the perimeter of pentagon \( ABC... | 7 | 0 |
275 | 16,670 | A digital watch in a 24-hour format displays hours and minutes. What is the largest possible sum of the digits in this display? | 24 | 0 |
276 | 26,945 | Complex numbers $a$, $b$, $c$ form an equilateral triangle with side length 24 in the complex plane. If $|a + b + c| = 48$, find $|ab + ac + bc|$. | 768 | 0 |
277 | 54,800 | There are 62 boxes in total, with small boxes being 5 units per ton and large boxes being 3 units per ton. A truck needs to transport these boxes. If the large boxes are loaded first, after loading all the large boxes, exactly 15 small boxes can still be loaded. If the small boxes are loaded first, after loading all th... | 27 | 0 |
278 | 31,215 | Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the diagram. If $AB = 35$ units and $CD = 65$ units, what is the area of square $BCFE$? | 2275 | 0 |
279 | 25,624 | Alice conducted a survey about common misconceptions regarding foxes among a group of people. She found that $75.6\%$ of the participants believed that foxes are typically domestic animals. Among those who held this belief, $30\%$ thought that foxes could be trained like domestic dogs. Since foxes are not typically tra... | 119 | 0 |
280 | 289 | Let $ a$ , $ b$ , $ c$ , $ x$ , $ y$ , and $ z$ be real numbers that satisfy the three equations
\begin{align*}
13x + by + cz &= 0
ax + 23y + cz &= 0
ax + by + 42z &= 0.
\end{align*}Suppose that $ a \ne 13$ and $ x \ne 0$ . What is the value of
\[ \frac{13}{a - 13} + \frac{23}{b - 23} + \... | -2 | 0 |
281 | 22,421 | A set of seven positive integers has a median of 5 and a mean of 14. What is the maximum possible value of the list's largest element? | 80 | 0 |
282 | 29,750 | Triangle $DEF$ has side lengths $DE = 15$, $EF = 39$, and $FD = 36$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \theta$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \... | 229 | 0 |
283 | 39,542 | Let $x$ and $y$ be positive numbers such that $x^2 y$ is constant. Suppose $y = 8$ when $x = 3$. Find the value of $x$ when $y$ is $72$ and $x^2$ has increased by a factor of $z = 4$. | 1 | 0 |
284 | 39,753 | The letters of the English alphabet are assigned numeric values based on the following conditions:
- Only the numeric values of $-2,$ $-1,$ $0,$ $1,$ and $2$ are used.
- Starting with A, a numeric value is assigned to each letter based on the pattern:
$$ 1, 2, 1, 0, -1, -2 $$
This cycle repeats throughout the alphabet... | 0 | 0 |
285 | 16,850 | Given $T' = \{(0,0),(5,0),(0,4)\}$ be a triangle in the coordinate plane, determine the number of sequences of three transformations—comprising rotations by $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, and reflections across the lines $y = x$ and $y = -x$—that will map $T'$ onto itse... | 12 | 0 |
286 | 48,329 | At the namesake festival, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady attended. At the beginning of the festival, all of them lined up so that no two people with the same name stood next to each other. What is the minimum number of Gennadys that could have attended the festival? | 49 | 0 |
287 | 18,710 | Jacob has run fifteen half-marathons in his life. Each half-marathon is $13$ miles and $193$ yards. One mile equals $1760$ yards. If the total distance Jacob covered in these half-marathons is $m$ miles and $y$ yards, where $0 \le y < 1760$, what is the value of $y$? | 1135 | 0 |
288 | 45,624 | 200 people stand in a circle. Each of them is either a liar or a conformist. Liars always lie. A conformist standing next to two conformists always tells the truth. A conformist standing next to at least one liar can either tell the truth or lie. 100 of the standing people said: "I am a liar," the other 100 said: "I am... | 150 | 0 |
289 | 24,616 | Suppose we have a right triangle $DEF$ with the right angle at $E$ such that $DF = \sqrt{85}$ and $DE = 7.$ A circle is drawn with its center on $DE$ such that the circle is tangent to $DF$ and $EF.$ If $Q$ is the point where the circle and side $DF$ meet, what is $FQ$? | 6 | 0 |
290 | 46,329 | Find the largest three-digit number that is equal to the sum of its digits and the square of twice the sum of its digits. | 915 | 0 |
291 | 49,660 | In a class of 45 students, all students participate in a tug-of-war. Among the remaining three events, each student participates in at least one event. It is known that 39 students participate in the shuttlecock kicking event and 28 students participate in the basketball shooting event. How many students participate in... | 22 | 0 |
292 | 55,130 | Find the largest possible number in decimal notation where all the digits are different, and the sum of its digits is 37. | 976543210 | 0 |
293 | 54,315 | The distance between two cells on an infinite chessboard is defined as the minimum number of moves a king needs to travel between these cells. Three cells on the board are marked, and the pairwise distances between them are all equal to 100. How many cells exist such that the distance from these cells to each of the th... | 1 | 0 |
294 | 6,956 | Find the coefficient of $x^{70}$ in the expansion of
\[(x - 1)(x^2 - 2)(x^3 - 3) \dotsm (x^{11} - 11)(x^{12} - 12).\] | 4 | 0 |
295 | 45,741 | 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... | 0 | 0 |
296 | 5,754 | The $52$ cards in a deck are numbered $1, 2, \cdots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let $... | 263 | 0 |
297 | 22,219 | What is the sum of all integer values of $n$ such that $\frac{36}{2n - 1}$ is an integer? | 3 | 0 |
298 | 16,662 | Given that the erroneous product of two positive integers $a$ and $b$, where the digits of the two-digit number $a$ are reversed, is $189$, calculate the correct value of the product of $a$ and $b$. | 108 | 0 |
299 | 52,611 | Several island inhabitants gather in a hut, with some belonging to the Ah tribe and the rest to the Uh tribe. Ah tribe members always tell the truth, while Uh tribe members always lie. One inhabitant said, "There are no more than 16 of us in the hut," and then added, "All of us are from the Uh tribe." Another said, "Th... | 15 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.