Given a random variable $X \sim B(n, 0.8)$, and $D(X) = 1.6$, calculate the value of $n$. Express your answer as a whole number.

Given a quintic polynomial $f(x) = 4x^5 - 3x^3 + 2x^2 + 5x + 1$, use the Horner's method to find the value of this polynomial when $x = 2$.

Let positive real numbers \( a, b, c \) satisfy $$ (a+c)\left(b^{2}+a c\right)=4a. $$ Find the maximum value of \( b+c \).

What is the maximum mass of ethanol that can be produced from 15.5 kg of glucose, given the balanced chemical equation C6H12O6 → 2C2H5OH + 2CO2 and the molar masses of glucose (180.2 g/mol) and ethanol (46.1 g/mol)? Express your answer in k

Given $$\frac {a+2i}{i}=b+i$$ (where $a$, $b$ are real numbers) and $i$ is the imaginary unit, find $a+b=$ \_\_\_\_\_.

Several schoolchildren went mushroom picking. The schoolchild who gathered the most mushrooms collected \( \frac{1}{5} \) of the total amount of mushrooms, while the one who gathered the least collected \( \frac{1}{7} \) of the total amount

At a university banquet, there are 2017 mathematicians, each with a menu that includes two different main courses, and no two mathematicians have the same pair of main courses. The price of each main course is equal to the number of mathema

Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 3.$  Find the maximum value of \[(xy + z)(xz + y).\]

A square \(ABCD\) is bisected by its diagonals. The vertices of the square are colored either red or yellow, and any triangle formed by vertices of the same color is called a homogeneous triangle. Determine the number of distinct ways to co

Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourist


Given that \( m \) and \( n \) are both positive integers and satisfy \( 24m = n^4 \), find the minimum value of \( m \).


The slope of the line $(2m^{2}-5m+2)x-(m^{2}-4)y+5m=0$ is $45^{\circ}$, then the value of $m$ is ____.


The foci of the ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$ are $F_{1}$ and $F_{2}$. If a point $P$ on the ellipse makes $P F_{1} \perp P F_{2}$, find the area of the triangle $\triangle P F_{1} F_{2}$.


At a dog show, each dog was assigned an ordinal number from 1 to 24 by lot. Due to health issues, one of the dogs could not participate in the competition. It turned out that among the remaining 23 dogs, one had a number equal to the arithm


If (10 ^ 4 * 3.456789) ^ 9 is written as a single term, how many digits would be to the right of the decimal place?


Jackson had 20 kilograms of meat. He used 1/4 of the meat to make meatballs, 15% of the remaining meat to make spring rolls, and 2 pounds of meat to prepare a stew. After that, he used 10% of the leftover meat to make kebabs. How many kilog


Apollo pulls the sun across the sky every night. Once a month, his fiery chariot’s wheels need to be replaced. He trades golden apples to three different blacksmiths to get them to make him new wheels.   Hephaestus charged three golden appl


Determine the smallest possible value of the expression  $$ \frac{a b+1}{a+b}+\frac{b c+1}{b+c}+\frac{c a+1}{c+a} $$  where \(a, b, c \in \mathbb{R}\) satisfy \(a+b+c=-1\) and \(a b c \leq -3\).

Peter has 18 oranges, 27 pears, and some bananas. He wants to make fruit baskets with the same number of each fruit in each basket. The greatest number of fruit baskets he can make is 3. How many bananas does Peter have?


a is the average (arithmetic mean) of the first 7 positive multiples of 8 and b is the median of the first 3 positive multiples of a certain positive integer. The value of a^2 - b^2 is zero. What is the value of that positive integer?


Danny collects bottle caps. He threw away 60 of the old ones at the park and found 58 new ones. He also traded 15 of his bottle caps with his friends, receiving 25 bottle caps in return. Now he has 67 bottle caps in his collection. How many


Given $$\frac {a+2i}{i}=b+i$$ (where $a$, $b$ are real numbers) and $i$ is the imaginary unit, find $a+b=$ \_\_\_\_\_.

Several schoolchildren went mushroom picking. The schoolchild who gathered the most mushrooms collected \( \frac{1}{5} \) of the total amount of mushrooms, while the one who gathered the least collected \( \frac{1}{7} \) of the total amount

At a university banquet, there are 2017 mathematicians, each with a menu that includes two different main courses, and no two mathematicians have the same pair of main courses. The price of each main course is equal to the number of mathema


Given that \( x \) and \( y \) are real numbers such that \( |x| + x + y = 10 \) and \( |y| + x - y = 10 \), if \( P = x + y \), find the value of \( P \).


Given the function f(x) = x(2017 + ln(x))e, where e is the base of the natural logarithm, find the value of x_0 such that f'(x_0) = 2018. Express your answer as a single numerical value.

Calculate the sum of the squares of the numbers 17, 19, 23, and 29. Express your answer as a single integer.


Samantha bought 8 packages of cat food, 5 packages of dog food, and 3 packages of bird food. Each package of cat food contained 12 cans, each package of dog food contained 7 cans, and each package of bird food contained 4 cans. If she gave


Two circles with radii 13 and 5 intersect, and the distance between their centers is 12. Find the length of their common chord. Express your answer as a single number.


The foci of the ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$ are $F_{1}$ and $F_{2}$. If a point $P$ on the ellipse makes $P F_{1} \perp P F_{2}$, find the area of the triangle $\triangle P F_{1} F_{2}$.

At a dog show, each dog was assigned an ordinal number from 1 to 24 by lot. Due to health issues, one of the dogs could not participate in the competition. It turned out that among the remaining 23 dogs, one had a number equal to the arithm


Consider all prisms whose base is a convex 2015-gon.  What is the maximum number of edges of such a prism that can be intersected by a plane not passing through its vertices?


Grandma baked 21 batches of pies with $N$ pies in each batch, where $N > 70$. Then she distributed all the pies onto several trays, with 70 pies on each tray. What is the smallest possible value of $N$?

Given $f(x) = x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1$, calculate $v_2$ using the Horner's method when $x = 2$.


We successively write all the numbers from 1 to one million. Then, we replace each number with the sum of its digits. We repeat this process until only single-digit numbers remain. Which single-digit number appears most frequently?


Find the smallest positive angle $x$ that satisfies $\sin 2x \sin 3x = \cos 2x \cos 3x,$ in degrees.


A tailor trims 4 feet from opposite edges of a square piece of cloth, and 3 feet from the other two edges. If 120 square feet of cloth remain, what was the length of a side of the original piece of cloth?


There are 12 students playing a card game where each game involves 4 players, and any two students play together at most once. What is the maximum number of games they can play?


A number, when divided by 44, gives 432 as quotient and a certain remainder. When dividing the same number by 31, the remainder is 5. What is the remainder when the number is divided by 44?


Walter fell from the eighth platform of some scaffolding and fell past David after falling 4 meters. He fell for an additional multiple of that depth before hitting the ground, and the platforms are evenly spaced out in height. David was on


How many monomials of degree 7 contain the letters a, b, and c, and have a coefficient of 1? Express your answer as a whole number.


Given the function f(x) = x(2017 + ln(x))e, where e is the base of the natural logarithm, find the value of x_0 such that f'(x_0) = 2018. Express your answer as a single numerical value.


Three cars are traveling in the same direction along the same route. The red car travels at a constant speed of 40 miles per hour, the black car is traveling at a varying speed of Y miles per hour, and the green car is traveling at a consta


The highest common factors of all the pairs chosen from the positive integers Q, R, and S are three different primes. What is the smallest possible value of Q + R + S? Express your answer as a single integer.


There are 10 "Triple-A Student" quota spots to be divided among three classes A, B, and C, with each class getting at least one spot. Calculate the number of possible distributions of these spots among the three classes.


The foci of the ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$ are $F_{1}$ and $F_{2}$. If a point $P$ on the ellipse makes $P F_{1} \perp P F_{2}$, find the area of the triangle $\triangle P F_{1} F_{2}$.


Given real numbers \( x \) and \( y \) satisfy \( x^2 + y^2 = 20 \). Find the maximum value of \( xy + 8x + y \).


Suppose that  $x$  is measured in radians. Find the maximum value of \[\frac{\sin2x+\sin4x+\sin6x}{\cos2x+\cos4x+\cos6x}\] for  $0\le x\le \frac{\pi}{16}$


How many airplanes? (From the 11th grade, 8 points)  The Absent-minded Scientist's work involves long business trips, and therefore he often flies with the same airline. This airline has many identical airplanes, and they all have names. Si


Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$  Find the minimum value of \[\frac{1}{a} + \frac{2}{b}.\]


What is the largest integer \( n \) such that the quantity \[ \frac{50!}{(5!)^{n}} \] is an integer? Note: Here \( k! = 1 \times 2 \times 3 \times \cdots \times k \) is the product of all integers from 1 to \( k \). For example, \( 4! = 1 \


A tailor trims 4 feet from opposite edges of a square piece of cloth, and 3 feet from the other two edges. If 120 square feet of cloth remain, what was the length of a side of the original piece of cloth?

\(\frac{\sin 6 \alpha}{\sin 2 \alpha} + \frac{\cos (6 \alpha - \pi)}{\cos 2 \alpha}\).


Given that \( x \) is a positive real number and \( x \cdot 3^{x} = 3^{18} \). If \( k \) is a positive integer and \( k < x < k + 1 \), find the value of \( k \).


On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{2}$, meaning that if the length of the object was originally $x,$ then it is now $x + \frac{1}{2} x.$  On the second day he increased the ob


Calculate the value of \\(\\tan 15^{\\circ} + \\cot 15^{\\circ}\\). Express your answer as a single number.


If three piles of toys added together make 240 toys in total, with the larger pile being twice the size of the smaller pile and the third pile being three times the size of the smaller pile, how many toys are in the largest pile?


Let $f(x)=x^{3}+x+1$. Suppose $g$ is a cubic polynomial such that $g(0)=-1$, and the roots of $g$ are the squares of the roots of $f$. Find $g(9)$.


The numbers from 1 to 8 are arranged at the vertices of a cube so that the sum of the numbers at any three vertices lying on one face is at least 10. What is the minimal possible sum of the numbers on one face of the cube?

Given vectors $\overrightarrow{v} = \left(a_{n+1} - \frac{a_n}{2}, \frac{a_{n+1}^2}{2a_n}\right)$ and $\overrightarrow{\mu} = (3, 3)$, and $\overrightarrow{v}$ is parallel to $\overrightarrow{\mu}$, if $a_1 = 5$, find the sum of the first 1


Given that function f(x) is an odd function defined on R, g(x) = f(x) + ax^3 + 2. If g(2) = 6, then g(-2) = _____.


Let \(d\) and \(f\) be positive integers and \(a_{1} = 0.9\). If \(a_{i+1} = a_{i}^{2}\) and \(\prod_{i=1}^{4} a_{i} = \frac{3^{d}}{f}\), determine the smallest possible value of \(d\).


At a monthly meeting, 3/5 of the attendees were males and 7/8 of the male attendees arrived on time. Some fraction of the female attendees arrived on time, and 0.115 fraction of the attendees did not arrive on time. What fraction of the fem


Given $f(x) = x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1$, calculate $v_2$ using the Horner's method when $x = 2$.


During the holiday, a school organizes a trip for 360 teachers and students. A bus rental company offers two types of buses for hire: Type A buses have 40 seats each and a rental fee of 400 yuan; Type B buses have 50 seats each and a rental


Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$  Find the minimum value of \[\frac{1}{a} + \frac{2}{b}.\]

A line passing through $(2,1)$ intersects the coordinate axes at points $A$ and $B$. The area of $\triangle BO$ (where $O$ is the origin) is exactly $4$. Find the number of lines $l$ that satisfy the given conditions.


What is the value of y in the expression ( ( 2 ^ 5 ) * ( y ) ) / ( ( 8 ^ 2 ) * ( 3 ^ 5 ) ) if the result is 0.16666666666666666?


A rectangular prism has a volume of $12 \mathrm{~cm}^{3}$. A new prism is formed by doubling the length, doubling the width, and tripling the height of the original prism. What is the volume of this new prism?


In a class, there are a total of 8 rows of desks and each desk can seat one student. There are 10 desks in the first row. In each subsequent row, the number of desks is determined by the formula: d_n = d_(n-1) + n, where d_n represents the


The greatest number that divides 178340 and 253785 leaving remainders 20 and 35 respectively, and also divides 375690 leaving a remainder of 50 is: