problem stringlengths 10 5.15k | answer stringlengths 0 1.22k |
|---|---|
A cuboid with dimensions corresponding to length twice the cube's edge is painted with stripes running from the center of one edge to the center of the opposite edge, on each of its six faces. Each face's stripe orientation (either horizontal-center or vertical-center) is chosen at random. What is the probability that ... | \frac{1}{16} |
What is the maximum area of a triangle if none of its side lengths exceed 2? | \sqrt{3} |
Cinderella and her fairy godmother released a collection of seven new models of crystal slippers. The storybook heroines held a presentation of the collection for some guests: the audience had to state which slippers they liked. The guests wrote in a survey which models they considered the best. It is known that no two... | 128 |
A fruit store sells a type of fruit. The purchase price is $30$ yuan per kilogram. According to a survey, when the selling price is $40$ yuan per kilogram, the store can sell an average of $500$ kilograms per day. For every $1$ yuan increase in the selling price, the store sells $10$ kilograms less. Let the selling pri... | 60 |
In the sequence $\{a_n\}$, $a_{n+1} = 2(a_n - n + 3)$, $a_1 = -1$. If the sequence $\{a_n - pn + q\}$ is a geometric sequence, where $p$, $q$ are constants, then $a_{p+q} = \_\_\_\_\_\_\_\_\_\_\_\_\_.$ | 40 |
For natural numbers _m_ greater than or equal to 2, the _n_-th power of _m_ can be decomposed as follows:
2<sup>2</sup> = 1 + 3, 3<sup>2</sup> = 1 + 3 + 5, 4<sup>2</sup> = 1 + 3 + 5 + 7…
2<sup>3</sup> = 3 + 5, 3<sup>3</sup> = 7 + 9 + 11…
2<sup>4</sup> = 7 + 9…
According to this pattern, the third number in the ... | 125 |
Grandpa is twice as strong as Grandma, Grandma is three times as strong as Granddaughter, Granddaughter is four times as strong as Dog, Dog is five times as strong as Cat, Cat is six times as strong as Mouse. Grandpa, Grandma, Granddaughter, Dog, and Cat together with Mouse can pull up the Turnip, but without the Mous... | 1237 |
For each positive integer $n$ , let $g(n)$ be the sum of the digits when $n$ is written in binary. For how many positive integers $n$ , where $1\leq n\leq 2007$ , is $g(n)\geq 3$ ? | 1941 |
Martin is playing a game. His goal is to place tokens on an 8 by 8 chessboard in such a way that there is at most one token per square, and each column and each row contains at most 4 tokens.
a) How many tokens can Martin place, at most?
b) If, in addition to the previous constraints, each of the two main diagonals c... | 32 |
A secret agent is trying to decipher a passcode. So far, he has obtained the following information:
- It is a four-digit number.
- It is not divisible by seven.
- The digit in the tens place is the sum of the digit in the units place and the digit in the hundreds place.
- The number formed by the first two digits of t... | 4583 |
The circle, which has its center on the hypotenuse $AB$ of the right triangle $ABC$, touches the two legs $AC$ and $BC$ at points $E$ and $D$ respectively.
Find the angle $ABC$, given that $AE = 1$ and $BD = 3$. | 30 |
Let $A$ be a positive integer which is a multiple of 3, but isn't a multiple of 9. If adding the product of each digit of $A$ to $A$ gives a multiple of 9, then find the possible minimum value of $A$ . | 138 |
Determine the minimum possible value of the sum \[\frac{a}{3b} + \frac{b}{5c} + \frac{c}{6a},\] where \( a, b, \) and \( c \) are positive real numbers. | \frac{3}{\sqrt[3]{90}} |
Roll a die twice in succession, observing the number of points facing up each time, and calculate:
(1) The probability that the sum of the two numbers is 5;
(2) The probability that at least one of the two numbers is odd;
(3) The probability that the point (x, y), with x being the number of points facing up on th... | \frac{2}{9} |
Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $29x + y = 2035$. Find the number of such distinct triangles whose area is a positive integer. | 1225 |
There exists \( x_{0} < 0 \) such that \( x^{2} + |x - a| - 2 < 0 \) (where \( a \in \mathbb{Z} \)) is always true. Find the sum of all values of \( a \) that satisfy this condition. | -2 |
Consider a parabola with vertex V and a focus F. There exists a point B on the parabola such that BF = 25 and BV = 24. Determine the sum of all possible values of the length FV. | \frac{50}{3} |
The price (in euros) of a diamond corresponds to its mass (in grams) squared and then multiplied by 100. The price (in euros) of a crystal corresponds to three times its mass (in grams).
Martin and Théodore unearth a treasure consisting of precious stones that are either diamonds or crystals and whose total value is €... | 2000000 |
Consider a string of $n$ $7$s, $7777\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression. How many values of $n$ are possible if the inserted $+$ signs create a sum of $7350$ using groups of $7$s, $77$s, $777$s, and possibly $7777$s? | 117 |
Determine the number of ways to arrange the letters of the word MOREMOM. | 420 |
Initially, the fairy tale island was divided into three counties: in the first county lived only elves, in the second - only dwarves, and in the third - only centaurs.
- During the first year, each county where there were no elves was divided into three counties.
- During the second year, each county where there were ... | 54 |
Three tenths plus four thousandths is equal to | 0.304 |
Determine the minimum of the following function defined in the interval $45^{\circ}<x<90^{\circ}$:
$$
y=\tan x+\frac{\tan x}{\sin \left(2 x-90^{\circ}\right)}
$$ | 3\sqrt{3} |
Given a polynomial $f(x) = 2x^7 + x^6 + x^4 + x^2 + 1$, calculate the value of $V_2$ using the Horner's method when $x=2$. | 10 |
Five students, $A$, $B$, $C$, $D$, and $E$, entered the final of a school skills competition and the rankings from first to fifth were determined (with no ties). It is known that students $A$ and $B$ are neither first nor last. Calculate the number of different arrangements of the final rankings for these 5 students. | 36 |
In one month, three Wednesdays fell on even dates. On which day will the second Sunday fall in this month? | 13 |
Given vectors $\overrightarrow{m}=( \sqrt {3}\sin x-\cos x,1)$ and $\overrightarrow{n}=(\cos x, \frac {1}{2})$, and the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$,
(1) Find the interval(s) where the function $f(x)$ is monotonically increasing;
(2) If $a$, $b$, $c$ are the sides opposite to angles $A$, ... | 2 \sqrt {3} |
In a certain middle school, 500 eighth-grade students took the biology and geography exam. There were a total of 180 students who scored between 80 and 100 points. What is the frequency of this score range? | 0.36 |
Given $a+b+c=0$ and $a^2+b^2+c^2=1$, find the values of $ab+bc+ca$ and $a^4+b^4+c^4$. | \frac{1}{2} |
Two particles move along the sides of a right $\triangle ABC$ with $\angle B = 90^\circ$ in the direction \[A\Rightarrow B\Rightarrow C\Rightarrow A,\] starting simultaneously. One starts at $A$ moving at speed $v$, the other starts at $C$ moving at speed $2v$. The midpoint of the line segment joining the two particles... | \frac{1}{4} |
There is a ten-digit number. From left to right:
- Its first digit indicates the number of zeros in the ten-digit number.
- Its second digit indicates the number of ones in the ten-digit number.
- Its third digit indicates the number of twos in the ten-digit number.
- ...
- Its tenth digit indicates the number of nines... | 6210001000 |
The sequence $\left\{a_{n}\right\}$ consists of 9 terms, where $a_{1} = a_{9} = 1$, and for each $i \in \{1,2, \cdots, 8\}$, we have $\frac{a_{i+1}}{a_{i}} \in \left\{2,1,-\frac{1}{2}\right\}$. Find the number of such sequences. | 491 |
Given that the positive real numbers \(a_{1}, a_{2}, a_{3}, a_{4}\) satisfy the conditions \(a_{1} \geqslant a_{2} a_{3}^{2}, a_{2} \geqslant a_{3} a_{4}^{2}, a_{3} \geqslant a_{4} a_{1}^{2}, a_{4} \geqslant a_{1} a_{2}^{2}\), find the maximum value of \(a_{1} a_{2} a_{3} a_{4}\left(a_{1}-a_{2} a_{3}^{2}\right)\left(a_... | 1/256 |
A convoy of cars is moving on a highway at a speed of 80 km/h with a distance of 10 meters between the cars. Upon passing a speed limit sign, all cars reduce their speed to 60 km/h. What will be the distance between the cars in the convoy after passing the sign? | 7.5 |
What is $\sqrt{123454321}$? | 11111 |
The equation $x^3 - 6x^2 - x + 3 = 0$ has three real roots $a$, $b$, $c$. Find $\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$. | \frac{37}{9} |
There is a magical tree with 58 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if on any given day the number of fruits on the tree is less than the number of fruits that should fall on that ... | 12 |
In the side face $A A^{\prime} B^{\prime} B$ of a unit cube $A B C D - A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, there is a point $M$ such that its distances to the two lines $A B$ and $B^{\prime} C^{\prime}$ are equal. What is the minimum distance from a point on the trajectory of $M$ to $C^{\prime}$? | \frac{\sqrt{5}}{2} |
Calculate the lengths of the arcs of the curves given by the equations in polar coordinates.
$$
\rho = 2 \varphi, \; 0 \leq \varphi \leq \frac{4}{3}
$$ | \frac{20}{9} + \ln 3 |
Find the value of $$\frac{\tan 7.5^\circ \cdot \tan 15^\circ}{\tan 15^\circ - \tan 7.5^\circ}$$ + $$\sqrt{3}(\sin^2 7.5^\circ - \cos^2 7.5^\circ)$$. | -\sqrt{2} |
\(1.25 \times 67.875 + 125 \times 6.7875 + 1250 \times 0.053375\). | 1000 |
Circles $A$ and $B$ each have a radius of 1 and are tangent to each other. Circle $C$ has a radius of 2 and is tangent to the midpoint of $\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$
A) $1.16$
B) $3 \pi - 2.456$
C) $4 \pi - 4.912$
D) $2 \pi$
E) $\pi + 4.912$ | 4 \pi - 4.912 |
If \( k \) is the smallest positive integer such that \(\left(2^{k}\right)\left(5^{300}\right)\) has 303 digits when expanded, then the sum of the digits of the expanded number is | 11 |
Given that Ben spent some amount of money and David spent $0.5 less for each dollar Ben spent, and Ben paid $16.00 more than David, determine the total amount they spent together in the bagel store. | 48.00 |
In $\triangle ABC$, $a=1$, $B=45^{\circ}$, $S_{\triangle ABC}=2$, find the diameter of the circumcircle of $\triangle ABC$. | 5 \sqrt {2} |
Simplify: $-{-\left[-|-1|^2\right]^3}^4$. | -1 |
In triangle $ABC$, angle $B$ equals $120^\circ$, and $AB = 2 BC$. The perpendicular bisector of side $AB$ intersects $AC$ at point $D$. Find the ratio $CD: DA$. | 3:2 |
The sequence $(x_n)$ is defined by $x_1 = 150$ and $x_k = x_{k - 1}^2 - x_{k - 1}$ for all $k \ge 2.$ Compute
\[\frac{1}{x_1 + 1} + \frac{1}{x_2 + 1} + \frac{1}{x_3 + 1} + \dots.\] | \frac{1}{150} |
Given vectors $a=(\cos α, \sin α)$ and $b=(\cos β, \sin β)$, with $|a-b|= \frac{2 \sqrt{5}}{5}$, find the value of $\cos (α-β)$.
(2) Suppose $α∈(0,\frac{π}{2})$, $β∈(-\frac{π}{2},0)$, and $\cos (\frac{5π}{2}-β) = -\frac{5}{13}$, find the value of $\sin α$. | \frac{33}{65} |
Given the sequence ${a_n}$, where $a_1=1$, and $P(a_n,a_{n+1})(n∈N^{+})$ is on the line $x-y+1=0$. If the function $f(n)= \frac {1}{n+a_{1}}+ \frac {1}{n+a_{2}}+ \frac {1}{n+a_{3}}+…+ \frac {1}{n+a_{n}}(n∈N^{\*})$, and $n\geqslant 2$, find the minimum value of the function $f(n)$. | \frac {7}{12} |
Given the function $f(x)= \sqrt {x^{2}-4x+4}-|x-1|$:
1. Solve the inequality $f(x) > \frac {1}{2}$;
2. If positive numbers $a$, $b$, $c$ satisfy $a+2b+4c=f(\frac {1}{2})+2$, find the minimum value of $\sqrt { \frac {1}{a}+ \frac {2}{b}+ \frac {4}{c}}$. | \frac {7}{3} \sqrt {3} |
(Full score for this problem is 12 points) Given $f(x) = e^x - ax - 1$.
(1) Find the intervals where $f(x)$ is monotonically increasing.
(2) If $f(x)$ is monotonically increasing on the domain $\mathbb{R}$, find the range of possible values for $a$.
(3) Does there exist a value of $a$ such that $f(x)$ is monotonically... | a = 1 |
Find $\frac{a^{8}-6561}{81 a^{4}} \cdot \frac{3 a}{a^{2}+9}$, given that $\frac{a}{3}-\frac{3}{a}=4$. | 72 |
A granite pedestal. When constructing a square foundation and a cubic pedestal for a monument, granite cubic blocks of size \(1 \times 1\) meter were used. The pedestal used exactly as many blocks as the square foundation upon which it stood. All the blocks were used whole and uncut.
Look at the picture and try to det... | 128 |
Given \( a > b \), the quadratic inequality \( ax^{2}+2x+b \geqslant 0 \) holds for all real numbers \( x \), and there exists \( x_{0} \in \mathbb{R} \) such that \( ax_{0}^{2}+2x_{0}+b=0 \) is satisfied. Find the minimum value of \( 2a^{2}+b^{2} \). | 2\sqrt{2} |
Determine how much money the Romanov family will save by using a multi-tariff meter over three years.
The cost of the meter is 3500 rubles. The installation cost is 1100 rubles. On average, the family's electricity consumption is 300 kWh per month, with 230 kWh used from 23:00 to 07:00.
Electricity rates with a mult... | 3824 |
Given $A=3x^{2}-x+2y-4xy$ and $B=2x^{2}-3x-y+xy$.
$(1)$ Simplify $2A-3B$.
$(2)$ When $x+y=\frac{6}{7}$ and $xy=-1$, find the value of $2A-3B$.
$(3)$ If the value of $2A-3B$ is independent of the value of $y$, find the value of $2A-3B$. | \frac{49}{11} |
Compute: $\frac{\cos 10^{\circ} - 2\sin 20^{\circ}}{\sin 10^{\circ}} = \_\_\_\_\_\_ \text{.}$ | \sqrt{3} |
Alice starts to make a list, in increasing order, of the positive integers that have a first digit of 2. She writes $2, 20, 21, 22, \ldots$ but by the 1000th digit she (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last three digits she wrote (th... | 216 |
How many distinct terms are in the expansion of \[(a+b+c+d)(e+f+g+h+i)\] assuming that terms involving the product of $a$ and $e$, and $b$ and $f$ are identical and combine into a single term? | 19 |
Given the sequence $\{a_n\}$ satisfying $(\log_3{a_n}+1=\log_3{a_{n+1}}\ (n\in \mathbb{N}^*)$, and $(a_2+a_4+a_6=9$, find the value of $(\log_{\frac{1}{3}}(a_5+a_7+a_9))$. | -5 |
(1) Given $\cos (α+ \frac {π}{6})- \sin α= \frac {3 \sqrt {3}}{5}$, find the value of $\sin (α+ \frac {5π}{6})$;
(2) Given $\sin α+ \sin β= \frac {1}{2}, \cos α+ \cos β= \frac { \sqrt {2}}{2}$, find the value of $\cos (α-β)$. | -\frac {5}{8} |
What is the smallest positive odd number that has the same number of divisors as 360? | 3465 |
Fill in the table with the numbers $0, 1, 2, \cdots, 14, 15$ so that for each row and each column, the remainders when divided by 4 are exactly $0, 1, 2, 3$ each, and the quotients when divided by 4 are also exactly $0, 1, 2, 3$ each, and determine the product of the four numbers in the bottom row of the table. | 32760 |
The coefficient of $x^{3}$ in the expansion of $(2x^{2}+x-1)^{5}$ is _______. | -30 |
If you set the clock back by 10 minutes, the number of radians the minute hand has turned is \_\_\_\_\_\_. | \frac{\pi}{3} |
We build a $4 \times 4 \times 4$ cube out of sugar cubes. How many different rectangular parallelepipeds can the sugar cubes determine, if the rectangular parallelepipeds differ in at least one sugar cube? | 1000 |
Given that the polar coordinate equation of curve $C\_1$ is $ρ=2\sin θ$, and the polar coordinate equation of curve $C\_2$ is $θ =\dfrac{π }{3}(ρ \in R)$, curves $C\_1$ and $C\_2$ intersect at points $M$ and $N$. The length of chord $MN$ is _______. | \sqrt {3} |
Find all real numbers \( k \) such that the inequality
$$
a^{3}+b^{3}+c^{3}+d^{3}+1 \geqslant k(a+b+c+d)
$$
holds for any \( a, b, c, d \in [-1, +\infty) \). | \frac{3}{4} |
Find the principal (smallest positive) period of the function
$$
y=(\arcsin (\sin (\arccos (\cos 3 x))))^{-5}
$$ | \frac{\pi}{3} |
In the Cartesian coordinate system $(xOy)$, the parametric equations of the curve $C$ are given by $\begin{cases} x=3\cos \alpha \\ y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of the li... | \dfrac{18\sqrt{2}}{5} |
Given the function f(x) = 2x^3 - ax^2 + 1, where a ∈ R.
(I) When a = 6, the line y = -6x + m is tangent to f(x). Find the value of m.
(II) If the function f(x) has exactly one zero in the interval (0, +∞), find the monotonic intervals of the function.
(III) When a > 0, if the sum of the maximum and minimum values of th... | \frac{1}{2} |
In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 16. What is the greatest possible perimeter of the triangle? | 41 |
How many three-digit numbers remain if we exclude all three-digit numbers in which all digits are the same or the middle digit is different from the two identical end digits? | 810 |
Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$. | 14 |
What is the base 4 representation of the base 2 number $101010101_2$? | 11111_4 |
Find all real numbers \( x \) that satisfy the equation
\[
\frac{x-2020}{1}+\frac{x-2019}{2}+\cdots+\frac{x-2000}{21}=\frac{x-1}{2020}+\frac{x-2}{2019}+\cdots+\frac{x-21}{2000},
\]
and simplify your answer(s) as much as possible. Justify your solution. | 2021 |
In the diagram, if the area of $\triangle ABC$ is 36 where $A(3, 15)$, $B(15, 0)$, and $C(0, q)$ lie on a Cartesian plane. Determine the value of $q$.
[asy]
size(5cm);defaultpen(fontsize(9));
pair a = (3, 15); pair b = (15, 0); pair c = (0, 12);pair d= (3, 0);
draw(a--b--c--cycle);
label("$A(3, 15)$", a, N);
label("$B... | 12.75 |
If the graph of the power function $y=mx^{\alpha}$ (where m and $\alpha \in \mathbb{R}$) passes through the point $(8, \frac{1}{4})$, then $\alpha$ equals \_\_\_\_\_\_. | -\frac{2}{3} |
In a right triangle $ABC$ (right angle at $C$), the bisector $BK$ is drawn. Point $L$ is on side $BC$ such that $\angle C K L = \angle A B C / 2$. Find $KB$ if $AB = 18$ and $BL = 8$. | 12 |
In the quadrilateral pyramid \( P-ABCD \), \( BC \parallel AD \), \( AD \perp AB \), \( AB=2\sqrt{3} \), \( AD=6 \), \( BC=4 \), \( PA = PB = PD = 4\sqrt{3} \). Find the surface area of the circumscribed sphere of the triangular pyramid \( P-BCD \). | 80\pi |
\(\cos \frac{\pi}{15} - \cos \frac{2\pi}{15} - \cos \frac{4\pi}{15} + \cos \frac{7\pi}{15} =\) | -\frac{1}{2} |
Find the value of $\sin \frac{\pi}{7} \sin \frac{2\pi}{7} \sin \frac{3\pi}{7}$. | \frac{\sqrt{7}}{8} |
The divisors of a natural number \( n \) (including \( n \) and 1) which has more than three divisors, are written in ascending order: \( 1 = d_{1} < d_{2} < \ldots < d_{k} = n \). The differences \( u_{1} = d_{2} - d_{1}, u_{2} = d_{3} - d_{2}, \ldots, u_{k-1} = d_{k} - d_{k-1} \) are such that \( u_{2} - u_{1} = u_{3... | 10 |
The product of two positive integers plus their sum is 119. The integers are relatively prime and each is less than 30. What is the sum of the two integers? | 20 |
Consider a cube PQRSTUVW with a side length s. Let M and N be the midpoints of edges PU and RW, and let K be the midpoint of QT. Find the ratio of the area of triangle MNK to the area of one of the faces of the cube. | \frac{1}{4} |
It is known that the numbers \(x, y, z\) form an arithmetic progression in the given order with a common difference \(\alpha = \arccos \frac{5}{9}\), and the numbers \(1 + \cos x, 1 + \cos y, 1 + \cos z\) form a non-constant geometric progression in the given order. Find \(\cos y\). | -\frac{7}{9} |
Seven thousand twenty-two can be written as | 7022 |
Given two integers \( m \) and \( n \) which are coprime, calculate the GCD of \( 5^m + 7^m \) and \( 5^n + 7^n \). | 12 |
There are two rows of seats, with 6 seats in the front row and 7 seats in the back row. Arrange seating for 2 people in such a way that these 2 people cannot sit next to each other. Determine the number of different seating arrangements. | 134 |
Cut a 12cm long thin iron wire into three segments with lengths a, b, and c,
(1) Find the maximum volume of the rectangular solid with lengths a, b, and c as its dimensions;
(2) If these three segments each form an equilateral triangle, find the minimum sum of the areas of these three equilateral triangles. | \frac {4 \sqrt {3}}{3} |
What is the probability, expressed as a decimal, of drawing one marble which is either green or white from a bag containing 4 green, 3 white, and 8 black marbles? | 0.4667 |
Given the function $$f(x)=\sin^{2}x+ \sqrt {3}\sin x\cos x+2\cos^{2}x，x∈R$$.
(I) Find the smallest positive period and the interval where the function is monotonically increasing;
(II) Find the maximum value of the function on the interval $$[- \frac {π}{3}， \frac {π}{12}]$$. | \frac { \sqrt {3}+3}{2} |
Choose one of the following three conditions:①$a_{2}=60$, ②the sum of binomial coefficients is $64$, ③the maximum term of the binomial coefficients is the $4$th term. Fill in the blank below. Given ${(1-2x)}^{n}={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+…+{a}_{n}{x}^{n}(n∈{N}_{+})$,_____, find:<br/>$(1)$ the value of $n$;<br/>$(... | 63 |
Find the smallest natural number ending in the digit 4 that becomes 4 times larger when its last digit is moved to the beginning of the number. | 102564 |
Two parallel chords of a circle have lengths 24 and 32 respectively, and the distance between them is 14. What is the length of another parallel chord midway between the two chords? | 2\sqrt{249} |
There are three kinds of saltwater solutions: A, B, and C, with concentrations of 5%, 8%, and 9% respectively, and their weights are 60 grams, 60 grams, and 47 grams. Now, we want to prepare 100 grams of 7% saltwater solution. What is the maximum and minimum amount of solution A that can be used? | 35 |
A total of $960$ people are randomly numbered from $1$ to $960$. Using systematic sampling, $32$ people are selected for a survey. Find the number of people to be selected from those with numbers falling within $[450,750]$. | 10 |
Among the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, draw one at random. The probability of drawing a prime number is ____, and the probability of drawing a composite number is ____. | \frac{4}{9} |
Given that $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$ are distinct positive integers whose sum equals 159, find the maximum value of the smallest number $a_1$. | 19 |
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