Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
13,700 | Maria ordered a certain number of televisions for the stock of a large store, paying R\$ 1994.00 per television. She noticed that in the total amount to be paid, the digits 0, 7, 8, and 9 do not appear. What is the smallest number of televisions she could have ordered? | 56 | 85.9375 |
13,701 | Let \( f(n) \) be the integer closest to \( \sqrt[4]{n} \). Then, \( \sum_{k=1}^{2018} \frac{1}{f(k)} = \) ______. | \frac{2823}{7} | 2.34375 |
13,702 |
Calculate the definite integral:
$$
\int_{\frac{\pi}{2}}^{2 \operatorname{arctg} 2} \frac{d x}{\sin x(1+\sin x)}
$$ | \ln 2 - \frac{1}{3} | 36.71875 |
13,703 | Some vertices (the vertices of the unit squares) of a \(6 \times 6\) grid are colored red. We need to ensure that for any sub-grid \(k \times k\) where \(1 \leq k \leq 6\), at least one red point exists on its boundary. Find the minimum number of red points needed to satisfy this condition. | 12 | 18.75 |
13,704 | ABCD is an isosceles trapezoid with \(AB = CD\). \(\angle A\) is acute, \(AB\) is the diameter of a circle, \(M\) is the center of the circle, and \(P\) is the point of tangency of the circle with the side \(CD\). Denote the radius of the circle as \(x\). Then \(AM = MB = MP = x\). Let \(N\) be the midpoint of the side... | 30 | 2.34375 |
13,705 | Given point \( A(2,0) \), point \( B \) lies on the curve \( y = \sqrt{1 - x^2} \), and the triangle \( \triangle ABC \) is an isosceles right triangle with \( A \) as the right angle vertex. Determine the maximum value of \( |OC| \). | 2\sqrt{2} + 1 | 0 |
13,706 | Find the probability that, when five different numbers are randomly chosen from the set $\{1, 2, \ldots, 20\}$, at least two of them are consecutive. | \frac{232}{323} | 42.96875 |
13,707 | Johan has a large number of identical cubes. He has made a structure by taking a single cube and then sticking another cube to each face. He wants to make an extended structure in the same way so that each face of the current structure will have a cube stuck to it. How many extra cubes will he need to complete his exte... | 18 | 17.96875 |
13,708 | A circle inscribed in triangle \( ABC \) touches side \( AB \) at point \( M \), and \( AM = 1 \), \( BM = 4 \). Find \( CM \) given that \( \angle BAC = 120^\circ \). | \sqrt{273} | 2.34375 |
13,709 |
Oleg drew an empty $50 \times 50$ table and wrote a non-zero number above each column and to the left of each row. It turned out that all 100 numbers written were different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrote the product of the numbers correspondin... | 1250 | 7.03125 |
13,710 | Let \( a, b, c \) be prime numbers such that \( a^5 \) divides \( b^2 - c \), and \( b + c \) is a perfect square. Find the minimum value of \( abc \). | 1958 | 42.1875 |
13,711 |
The pensioners on one of the planets of Alpha Centauri enjoy spending their free time solving numeric puzzles: they choose natural numbers from a given range $[A, B]$ such that the sum of any two chosen numbers is not divisible by a certain number $N$. Last week, the newspaper "Alpha Centaurian Panorama" offered its r... | 356 | 25 |
13,712 | The weight of grain in a sample of 256 grains is 18 grains, and the total weight of rice is 1536 dan. Calculate the amount of mixed grain in the total batch of rice. | 108 | 43.75 |
13,713 | Rearrange the digits of 124669 to form a different even number. | 240 | 0 |
13,714 | A pentagon is inscribed around a circle, with the lengths of its sides being whole numbers, and the lengths of the first and third sides equal to 1. Into what segments does the point of tangency divide the second side? | \frac{1}{2} | 6.25 |
13,715 | Three couples dine at the same restaurant every Saturday at the same table. The table is round and the couples agreed that:
(a) under no circumstances should husband and wife sit next to each other; and
(b) the seating arrangement of the six people at the table must be different each Saturday.
Disregarding rotations o... | 16 | 4.6875 |
13,716 | A line parallel to the base of a triangle divides it into parts whose areas are in the ratio $2:1$, counting from the vertex. In what ratio does this line divide the sides of the triangle? | (\sqrt{6} + 2) : 1 | 0.78125 |
13,717 | Independent trials are conducted, in each of which event \( A \) can occur with a probability of 0.001. What is the probability that in 2000 trials, event \( A \) will occur at least two and at most four times? | 0.541 | 0 |
13,718 | Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property: for every $x,y\in A$ with $x\neq y$ , it holds that
\[\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}.\]
Determine the largest possible number of elements that the set $A$ can have. | 40 | 71.875 |
13,719 | The sides of triangle \(ABC\) are divided by points \(M, N\), and \(P\) such that \(AM : MB = BN : NC = CP : PA = 1 : 4\). Find the ratio of the area of the triangle bounded by lines \(AN, BP\), and \(CM\) to the area of triangle \(ABC\). | 3/7 | 3.90625 |
13,720 | Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$ , where $n$ is either $2012$ or $2013$ . | 338 | 6.25 |
13,721 | A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 5-primable positive integers are there that are less than 500? | 17 | 0 |
13,722 |
In the coordinate plane, a rectangle has vertices with coordinates $(34,0), (41,0), (34,9), (41,9)$. Find the smallest value of the parameter $a$ such that the line $y = ax$ divides this rectangle into two parts where the area of one part is twice the area of the other. If the answer is not an integer, write it as a d... | 0.08 | 7.8125 |
13,723 | On the Saturday of a weekend softball tournament, Team A plays Team D, Team B plays Team E, and Team C gets a bye (no match). The winner of Team A vs. Team D plays against Team C in the afternoon, while the winner of Team B vs. Team E has no further matches on Saturday. On Sunday, the winners of Saturday's afternoon ma... | 48 | 14.0625 |
13,724 | In triangle \( \triangle ABC \), \(\angle A\) is the smallest angle, \(\angle B\) is the largest angle, and \(2 \angle B = 5 \angle A\). If the maximum value of \(\angle B\) is \(m^{\circ}\) and the minimum value of \(\angle B\) is \(n^{\circ}\), then find \(m + n\). | 175 | 59.375 |
13,725 | In a circle, chords $A B$ and $C D$, which are not diameters, are drawn perpendicular to each other. Chord $C D$ divides chord $A B$ in the ratio $1:5$, and it divides the longer arc of $A B$ in the ratio $1:2$. In what ratio does chord $A B$ divide chord $C D$? | 1 : 3 | 0.78125 |
13,726 | A car and a truck start traveling towards each other simultaneously from points $A$ and $B$, respectively. It is known that the car's speed is twice the speed of the truck. The car arrives at point $C$ at 8:30, and the truck arrives at point $C$ at 15:00 on the same day. Both vehicles continue moving without stopping a... | 10:40 | 25 |
13,727 | We wrote the reciprocals of natural numbers from 2 to 2011 on a board. In one step, we erase two numbers, \( x \) and \( y \), and replace them with the number
$$
\frac{xy}{xy + (1 - x)(1 - y)}
$$
By repeating this process 2009 times, only one number remains. What could this number be? | \frac{1}{2010! + 1} | 0.78125 |
13,728 | Let \( M = \{1, 2, \cdots, 1995\} \). Suppose \( A \) is a subset of \( M \) that satisfies the condition: if \( x \in A \), then \( 15x \notin A \). What is the maximum number of elements in \( A \)? | 1870 | 57.03125 |
13,729 | There were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Determine the volume, in cubic light years, of the set of all possible locations for a base such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square lig... | \frac{27 \sqrt{6} \pi}{8} | 3.125 |
13,730 | The set \( S \) is given by \( S = \{1, 2, 3, 4, 5, 6\} \). A non-empty subset \( T \) of \( S \) has the property that it contains no pair of integers that share a common factor other than 1. How many distinct possibilities are there for \( T \)? | 27 | 92.1875 |
13,731 | a and b are real numbers for which the equation \(x^4 + ax^3 + bx^2 + ax + 1 = 0\) has at least one real solution. Find the least possible value of \(a^2 + b^2\). | 4/5 | 16.40625 |
13,732 | In some cases, it is not necessary to calculate the result to remove the absolute value. For example: $|6+7|=6+7$, $|6-7|=7-6$, $|7-6|=7-6$, $|-6-7|=6+7.\left(1\right)$ According to the above rule, express the following expressions in the form without absolute value symbols: <br/>①$|\frac{7}{17}-\frac{7}{18}|=$______；<... | \frac{505}{1011} | 45.3125 |
13,733 | Evaluate the sum $2345 + 3452 + 4523 + 5234$ and then subtract $1234$ from the result. | 14320 | 99.21875 |
13,734 | Let \( p(x) = 2x^3 - 3x^2 + 1 \). How many squares of integers are there among the numbers \( p(1), p(2), \ldots, p(2016) \)? | 32 | 69.53125 |
13,735 | It is known that the numbers \( x, y, z \) form an arithmetic progression in the given order with a common difference \( \alpha = \arccos \left(-\frac{1}{3}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{3}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression in the given order. Find \( \cos^2 y \). | \frac{4}{5} | 0 |
13,736 | Express the number $15.7$ billion in scientific notation. | 1.57\times 10^{9} | 0 |
13,737 | Given a trapezoid \( MNPQ \) with bases \( MQ \) and \( NP \). A line parallel to the bases intersects the lateral side \( MN \) at point \( A \), and the lateral side \( PQ \) at point \( B \). The ratio of the areas of the trapezoids \( ANPB \) and \( MABQ \) is \( \frac{2}{7} \). Find \( AB \) if \( NP = 4 \) and \(... | \frac{2\sqrt{46}}{3} | 6.25 |
13,738 | The numbers $a, b, c, d$ belong to the interval $[-8.5,8.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$. | 306 | 3.125 |
13,739 | In a certain country, there are 200 cities. The Ministry of Aviation requires that each pair of cities be connected by a bidirectional flight operated by exactly one airline, and that it should be possible to travel from any city to any other city using the flights of each airline (possibly with layovers). What is the ... | 100 | 39.84375 |
13,740 |
From May 1st to May 3rd, the provincial hospital plans to schedule 6 doctors to be on duty, with each person working 1 day and 2 people scheduled per day. Given that doctor A cannot work on the 2nd and doctor B cannot work on the 3rd, how many different scheduling arrangements are possible? | 42 | 55.46875 |
13,741 | Given \(3 \sin^{2} \alpha + 2 \sin^{2} \beta = 1\) and \(3 (\sin \alpha + \cos \alpha)^{2} - 2 (\sin \beta + \cos \beta)^{2} = 1\), find \(\cos 2 (\alpha + \beta) = \quad \) . | -\frac{1}{3} | 18.75 |
13,742 | Once, Carlson and Winnie the Pooh competed in the speed of eating honey and jam. Carlson, an expert in jam, eats a jar of jam in 2 minutes, while Winnie the Pooh takes a full 7 minutes to finish a jar of jam. Meanwhile, Winnie the Pooh can finish a pot of honey in 3 minutes, but Carlson requires 5 minutes to do the sam... | 48 | 21.09375 |
13,743 | An archipelago consists of \( N \geq 7 \) islands. Any two islands are connected by at most one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are always two that are connected by a bridge. What is the largest possible value of \( N \)? | 36 | 4.6875 |
13,744 | Let \( A \) and \( B \) be the endpoints of a semicircular arc of radius \( 3 \). The arc is divided into five congruent arcs by four equally spaced points \( C_1, C_2, C_3, C_4 \). All chords of the form \( \overline{AC_i} \) or \( \overline{BC_i} \) are drawn. Find the product of the lengths of these eight chords. | 32805 | 20.3125 |
13,745 | Determine the largest natural number \( n \) such that
\[ 4^{995} + 4^{1500} + 4^{n} \]
is a square number. | 2004 | 8.59375 |
13,746 | A target consists of five zones: the center circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the score for hitting each zone is inversely proportional to the probability of hitting that zone, and hitting the bullseye is worth 315 points. How many ... | 45 | 16.40625 |
13,747 | Find the smallest natural number $n$ with the following property: in any $n$-element subset of $\{1, 2, \cdots, 60\}$, there must be three numbers that are pairwise coprime. | 41 | 17.1875 |
13,748 | 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 \( f(n)=n \).
(... | 3125 | 44.53125 |
13,749 | In the diagram, the circle has center \( O \) and square \( OPQR \) has vertex \( Q \) on the circle. If the area of the circle is \( 72 \pi \), the area of the square is: | 36 | 13.28125 |
13,750 | What is the maximum value that can be taken by the sum
$$
\left|x_{1}-1\right|+\left|x_{2}-2\right|+\ldots+\left|x_{63}-63\right|
$$
if $x_{1}, x_{2}, \ldots, x_{63}$ are some permutation of the numbers $1, 2, 3, \ldots, 63$? | 1984 | 46.875 |
13,751 | Given a function $f: \{1, 2, 3\} \rightarrow \{1, 2, 3\}$ that satisfies $f(f(x)) = f(x)$, determine the total number of such functions. | 10 | 95.3125 |
13,752 | Given the sequence ${a_n}$ satisfying $a_1=1$, $a_2=2$, $a_3=3$, $a_{n+3}=a_n$ ($n\in\mathbb{N}^*$). If $a_n=A\sin(\omega n+\varphi)+c$ $(ω>0,|\varphi|<\frac{\pi}{2})$, find the real number $A$. | -\frac{2\sqrt{3}}{3} | 0.78125 |
13,753 | The height of a cone and its slant height are 4 cm and 5 cm, respectively. Find the volume of a hemisphere inscribed in the cone, whose base lies on the base of the cone. | \frac{1152}{125} \pi | 0 |
13,754 | A string of 33 pearls has its middle pearl as the largest and most valuable. The values of the remaining pearls decrease by $3000 \mathrm{Ft}$ per pearl towards one end and by $4500 \mathrm{Ft}$ per pearl towards the other end. How much is the middle pearl worth if the total value of the string is 25 times the value of... | 90000 | 0 |
13,755 | Given that there are \( c \) prime numbers less than 100 such that their unit digits are not square numbers, find the values of \( c \). | 15 | 90.625 |
13,756 | The base of a prism is an equilateral triangle $ABC$. The lateral edges of the prism $AA_1$, $BB_1$, and $CC_1$ are perpendicular to the base. A sphere, whose radius is equal to the edge of the base of the prism, touches the plane $A_1B_1C_1$ and the extensions of the segments $AB_1$, $BC_1$, and $CA_1$ beyond the poin... | \sqrt{44} - 6 | 0 |
13,757 | A square board with three rows and three columns contains nine cells. In how many different ways can we write the three letters A, B, and C in three different cells, so that exactly one of these three letters is written in each row? | 162 | 9.375 |
13,758 | A package of seeds was passed around a table. The first person took 1 seed, the second person took 2 seeds, the third took 3 seeds, and so forth, with each subsequent person taking one more seed than the previous one. It is known that during the second round a total of 100 more seeds were taken than during the first ro... | 10 | 53.90625 |
13,759 |
In the prism \( A B C - A_{1} B_{1} C_{1} \), \( A B \) is perpendicular to the lateral face \( B B_{1} C_{1} C \). Point \( E \) lies on edge \( C C_{1} \) such that \( E \ne C \) and \( E \ne C_{1} \). Given that \( E A \perp E B_1 \), \( A B = \sqrt{2} \), \( B B_1 = 2 \), \( B C = 1 \), and \( \angle B C C_1 = \fr... | \frac{\sqrt{2}}{2} | 1.5625 |
13,760 | By permuting the digits of 20130518, how many different eight-digit positive odd numbers can be formed? | 3600 | 3.125 |
13,761 | Find all odd natural numbers greater than 500 but less than 1000, each of which has the property that the sum of the last digits of all its divisors (including 1 and the number itself) is equal to 33. | 729 | 96.09375 |
13,762 | Find the smallest natural decimal number \(n\) whose square starts with the digits 19 and ends with the digits 89. | 1383 | 100 |
13,763 | For the four-digit number \(\overline{abcd}\) where \(1 \leqslant a \leqslant 9\) and \(0 \leqslant b, c, d \leqslant 9\), if \(a > b, b < c, c > d\), then \(\overline{abcd}\) is called a \(P\)-type number. If \(a < b, b > c, c < d\), then \(\overline{abcd}\) is called a \(Q\)-type number. Let \(N(P)\) and \(N(Q)\) rep... | 285 | 100 |
13,764 | Three positive integers are each greater than $1$, have a product of $1728$, and are pairwise relatively prime. What is their sum? | 43 | 0 |
13,765 | Vasya wrote a note on a piece of paper, folded it into quarters, and labeled the top with "MAME". Then he unfolded the note, wrote something else on it, folded the note along the creases randomly (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the inscriptio... | 1/8 | 3.90625 |
13,766 | Given that two congruent 30°-60°-90° triangles with hypotenuses of 12 are overlapped such that their hypotenuses exactly coincide, calculate the area of the overlapping region. | 9 \sqrt{3} | 3.90625 |
13,767 | Find the cosine of the angle between the non-intersecting diagonals of two adjacent lateral faces of a regular triangular prism, where the lateral edge is equal to the side of the base. | \frac{1}{4} | 52.34375 |
13,768 | Given the hyperbola \( C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) with \( a > 0 \) and \( b > 0 \), the eccentricity is \( \frac{\sqrt{17}}{3} \). Let \( F \) be the right focus, and points \( A \) and \( B \) lie on the right branch of the hyperbola. Let \( D \) be the point symmetric to \( A \) with respect ... | \frac{1}{2} | 0.78125 |
13,769 | In a building with 10 mailboxes, a distributor places a flyer in 5 of the mailboxes. Later, another distributor also places a flyer in 5 of the mailboxes. What is the probability that at least 8 mailboxes receive a flyer? | 1/2 | 0.78125 |
13,770 | Let \( T \) be a right triangle with sides having lengths 3, 4, and 5. A point \( P \) is called awesome if \( P \) is the center of a parallelogram whose vertices all lie on the boundary of \( T \). What is the area of the set of awesome points? | 3/2 | 89.0625 |
13,771 | Given a connected simple graph \( G \) with \( e \) edges and pieces placed on each vertex of \( G \) (where each piece can only be placed on a single vertex of \( G \)), you are allowed to perform the following operation: if the number of pieces on a vertex \( v \) is at least the number of vertices adjacent to \( v \... | e | 5.46875 |
13,772 | Majka examined multi-digit numbers in which odd and even digits alternate regularly. Those that start with an odd digit, she called "funny," and those that start with an even digit, she called "cheerful" (for example, the number 32387 is funny, the number 4529 is cheerful).
Majka created one three-digit funny number a... | 635040 | 6.25 |
13,773 | Given \( 5 \sin 2 \alpha = \sin 2^\circ \), find the value of \( \frac{\tan (\alpha + 1^\circ)}{\tan (\alpha - 1^\circ)} \). | -\frac{3}{2} | 2.34375 |
13,774 | In tetrahedron \(ABCD\), \(AB = 1\), \(BC = 5\), \(CD = 7\), \(DA = 5\), \(AC = 5\), \(BD = 2\sqrt{6}\). Find the distance between skew lines \(AC\) and \(BD\). | \frac{3\sqrt{11}}{10} | 3.125 |
13,775 | Evaluate $(128)^{\frac{1}{3}}(729)^{\frac{1}{2}}$. | 108 \cdot 2^{\frac{1}{3}} | 19.53125 |
13,776 | A seven-digit phone number \(d_{1} d_{2} d_{3}-d_{4} d_{5} d_{6} d_{7}\) is called "memorable" if the initial three digits \(d_{1} d_{2} d_{3}\) match either the middle three digits \(d_{4} d_{5} d_{6}\) or the last three digits \(d_{5} d_{6} d_{7}\) (it is possible for all three groups to be the same). Each digit can ... | 19990 | 3.125 |
13,777 | Let \(\mathbb{N}\) be the set of all positive integers. A function \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfies \( f(m + n) = f(f(m) + n) \) for all \( m, n \in \mathbb{N} \), and \( f(6) = 2 \). Also, no two of the values \( f(6), f(9), f(12) \), and \( f(15) \) coincide. How many three-digit positive integers... | 225 | 38.28125 |
13,778 | During the first eleven days, 700 people responded to a survey question. Each respondent chose exactly one of the three offered options. The ratio of the frequencies of each response was \(4: 7: 14\). On the twelfth day, more people participated in the survey, which changed the ratio of the response frequencies to \(6:... | 75 | 0.78125 |
13,779 | Find a positive integer \( n \) with 1000 digits, none of which are 0, such that we can group the digits into 500 pairs so that the sum of the products of the numbers in each pair divides \( n \). | 111...111211221122112211221122112211221122112211221122112211221122112211221122112 | 0 |
13,780 | Two teams of 20 people each participated in a relay race from Moscow to Petushki. Each team divided the distance into 20 segments, not necessarily of equal length, and assigned them to participants such that each member runs exactly one segment (each participant maintains a constant speed, but the speeds of different p... | 38 | 5.46875 |
13,781 | A group of adventurers displays their loot. It is known that exactly 9 adventurers have rubies; exactly 8 have emeralds; exactly 2 have sapphires; exactly 11 have diamonds. Additionally, it is known that:
- If an adventurer has diamonds, they either have rubies or sapphires (but not both simultaneously);
- If an adven... | 17 | 8.59375 |
13,782 | Given the function \( f: \mathbf{R} \rightarrow \mathbf{R} \), for any real numbers \( x, y, z \), the inequality \(\frac{1}{3} f(x y) + \frac{1}{3} f(x z) - f(x) f(y z) \geq \frac{1}{9} \) always holds. Find the value of \(\sum_{i=1}^{100} [i f(i)]\), where \([x]\) represents the greatest integer less than or equal to... | 1650 | 62.5 |
13,783 | Given the quadratic function \( f(x) = a x^{2} + b x + c \) where \( a, b, c \in \mathbf{R}_{+} \), if the function has real roots, determine the maximum value of \( \min \left\{\frac{b+c}{a}, \frac{c+a}{b}, \frac{a+b}{c}\right\} \). | 5/4 | 10.15625 |
13,784 | The base of the pyramid is a parallelogram with adjacent sides of 9 cm and 10 cm, and one of the diagonals measuring 11 cm. The opposite lateral edges are equal, and each of the longer edges is 10.5 cm. Calculate the volume of the pyramid. | 200 | 5.46875 |
13,785 | Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 3,$ and $\|\mathbf{c}\| = 6,$ and
\[2\mathbf{a} + 3\mathbf{b} + 4\mathbf{c} = \mathbf{0}.\]Compute $2(\mathbf{a} \cdot \mathbf{b}) + 3(\mathbf{a} \cdot \mathbf{c}) + 4(\mathbf{b} \cdot \mathbf{c}).$ | -673/4 | 0 |
13,786 | Sasha wrote down numbers from one to one hundred, and Misha erased some of them. Among the remaining numbers, 20 contain the digit one, 19 contain the digit two, and 30 contain neither one nor two. How many numbers did Misha erase? | 33 | 6.25 |
13,787 | Two water particles fall freely in succession from a $300 \mathrm{~m}$ high cliff. The first one has already fallen $\frac{1}{1000} \mathrm{~mm}$ when the second one starts to fall.
How far apart will the two particles be at the moment when the first particle reaches the base of the cliff? (The result should be calcul... | 34.6 | 3.125 |
13,788 | Given that $\sum_{k=1}^{36}\sin 4k=\tan \frac{p}{q},$ where angles are measured in degrees, and $p$ and $q$ are relatively prime positive integers that satisfy $\frac{p}{q}<90,$ find $p+q.$ | 73 | 1.5625 |
13,789 | In a city, there are 10,000 bicycles with all possible numbers from 1 to 10,000. What is the probability that the number of the first bicycle encountered does not contain the digit 8? | 0.6561 | 16.40625 |
13,790 | Fill in the appropriate numbers in the parentheses:
(1) 7÷9= $$\frac {( )}{( )}$$
(2) $$\frac {12}{7}$$=\_\_\_\_÷\_\_\_\_\_
(3) 3 $$\frac {5}{8}$$= $$\frac {( )}{( )}$$
(4) 6= $$\frac {()}{11}$$ | \frac {66}{11} | 85.9375 |
13,791 | Determine the coefficient of \(x^{29}\) in the expansion of \(\left(1 + x^{5} + x^{7} + x^{9}\right)^{16}\). | 65520 | 62.5 |
13,792 | Given that $\overline{2 a 1 b 9}$ represents a five-digit number, how many ordered digit pairs $(a, b)$ are there such that
$$
\overline{2 a 1 b 9}^{2019} \equiv 1 \pmod{13}?
$$ | 23 | 43.75 |
13,793 | Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$ . What is the expected value of the number of games they will play? | 14 | 50 |
13,794 | In a regular triangular prism \(ABC-A_1B_1C_1\), all 9 edges are equal in length. Point \(P\) is the midpoint of \(CC_1\). The dihedral angle \(B-A_1P-B_1 = \alpha\). Find \(\sin \alpha\). | \frac{\sqrt{10}}{4} | 36.71875 |
13,795 | Define a function \( f \), whose domain is positive integers, such that:
$$
f(n)=\begin{cases}
n-3 & \text{if } n \geq 1000 \\
f(f(n+7)) & \text{if } n < 1000
\end{cases}
$$
Find \( f(90) \). | 999 | 84.375 |
13,796 | Find the number of eight-digit integers comprising the eight digits from 1 to 8 such that \( (i+1) \) does not immediately follow \( i \) for all \( i \) that runs from 1 to 7. | 16687 | 1.5625 |
13,797 | Which integers from 1 to 60,000 (inclusive) are more numerous and by how much: those containing only even digits in their representation, or those containing only odd digits in their representation? | 780 | 57.03125 |
13,798 | In a board game played with dice, our piece is four spaces away from the finish line. If we roll at least a four, we reach the finish line. If we roll a three, we are guaranteed to finish in the next roll.
What is the probability that we will reach the finish line in more than two rolls? | 1/12 | 0 |
13,799 | A pedestrian traffic light allows pedestrians to cross the street for one minute and prohibits crossing for two minutes. Find the average waiting time for a pedestrian who approaches the intersection. | 40 | 0.78125 |
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