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Math

### Ruth told her mother that a circle is inscribed in triangle ABC so that the circle is tangent to side AB at P, side...

Denote the center of the circle as `O ` and consider a quadrilateral `O Q C R . ` It has two right angles, `Q ` and `R , ` because `C Q ` and `C R ` are tangent lines. The sum of all angles of a...

Math

### Mike asked his grandmother Becky, what is the remainder when x + x^9 + x^25 + x^49 + x^81 is divided by x^3 – x?

It is possible to find this remainder directly using polynomial division. First, take care of the greatest degree: `x ^ 81 = x ^ 78 ( x ^ 3 - x ) + x ^ 79 .` The first summand is divisible by `x ^...

Math

### Show that for every nonnegative number x, we have x + 3 ≥ 4∜x.

First, note that the right side, `f ( x ) = 4 root ( 4 ) ( x ) = 4 x^( 1 / 4 ) , ` is a function continuous on `[ 0 , + oo ) ` and infinitely differentiable on `( 0 , + oo ) . ` Find its first and...

Math

### Suzan told her best friend Jim, that there are 6 red and 8 green balls in a bag. A total of 5 balls are drawn...

Fortunately, the two quantities to add up are not independent. Denote the numbers of balls by GinG, RinG, GinR and RinR, then GinG + RinG = 9, GinR + RinR = 5 (if there are 5 balls in the red bag,...

Math

### Lin was told by her social studies teacher that a, b, and c form a triangle. How do we prove that for all n =...

For three positive numbers `a , ` `b , ` and `c ` to form a triangle (to be the length of the sides of some triangle), it is necessary and sufficient to satisfy the following triangle inequalities:...

Math

### Integral 3 ^ 1 f(x)dx= 6 and integral 3 ^ 1 g(x)dx= 3. Find integral 3 ^ 1 [g(x)-2f(x)+8]dx.

Probably you mean integral from 3 to 1, which means we need to compute the integral `int_3^1 [ g ( x ) - 2f ( x ) + 8 ] dx .` Usually definite integrals are defined on intervals `[ a, b ] ` , where...

Math

### Lora told her father that n, a natural number > 1, can be shown that 1/n + 1/n+1 + 1/n+2 + ... + 1/n^2 >1. Show...

Consider the sums for `n = 2 , 3 , 4 :` 1/2 + 1/3 + 1/4,1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9,1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16. Each sum has `n^2...

Math

### Gem told her father that the sum of the digits in the decimal representation of 4444^4444 is A. The sum of the digits...

Let's estimate how large A can be. It is not larger than the number of digits in `4444^4444 ` multiplied by `9 , ` i.e. `A lt= 9 * ( log_10 ( 4444^4444 ) + 1 ) = 9 * ( 4444 * log_10 ( 4444 ) + 1 )...

Math

### Prove the Following Trig identity: sin^ 2theta*tan^ 2theta*csc^ 2theta +cos^ 2theta*tan^ 2theta csc^ 2theta=sec^ 2theta

We are asked to prove the trigonometric identity `sin^2theta tan^2theta csc^2theta+cos^2theta tan^2theta csc^2theta=sec^2theta` A valid proof of a trigonometric identity involves using definitions...

Math

### What are all the positive integer pairs (m , n) that satisfy m^2 - n^2 - 2n = 2021?

We are asked to find all of the integer pairs (m, n) such that `m^2-n^2-2n=2021` `m^2-(n^2+2n)=2021` Complete the square on the expression in the parentheses: `m^2-(n^2+2n+1)=2021-1`...

Math

### Show that for every nonnegative number x, we have x + 3 `>=` 4`root(4)(x)`.

We are asked to show that `x+3>=4root[4](x)` for all nonnegative x. Divide both sides by 4: `(x+3)/4 >=root[4](x)` Since `x>=0` we can raise both sides to the fourth power without...

Math

### A square ABCD has side AB=1, and K is between B and C. E is on the extension of AK, with K between A and E. Lines EC...

We are given a square ABCD with AB=1. Point K is located between B and C, with point E on the extension of AK such that K is between A and E. Lines EC and AD intersect at F. The area of `Delta AEF`...

Math

### Given that log(sec x)-log(cos x)=1 and x is in the first quadrant, compute sin x.

We are given `"log"(secx)-"log"(cosx)=1` (where x is in the 1st quadrant), and we are asked to compute `sinx` . Use the property `log(A)-log(B)=log(A/B)` on the left side to get:...

Math

### In ABC, D is a point on side BA such that BD: DA = 1:2. E is a point on side CB so that CE: EB = 1:4. Segments DC and...

Denote `BD = x , ` `AD = 2x , ` `EC = y , ` `EB = 4y , ` `AC = b , ` `FC = z , ` `FD = az . ` Our aim is to find `a .` Recall that sinuses of supplementary angles are equal, in particular `sin CFA...

Math

### Renee said to her grandma, a college mathematic instructor, that 7`^^9999` ends in three digits. What are the last...

It is clear that the last digit of a natural number `n ` is its remainder of division by `10 , ` `n mod 10 . ` We can observe that the last digit of powers of `7 ` are repeated after some period....

Math

### Mary told her uncle George that a fair die is thrown repeatedly until the running total first exceeds 12. What is the...

The smallest natural number that exceeds 12 is 13. Consider what the running total may be before the last throw. It cannot be less than 7 because 6 + 6 = 12 < 13, and it could not be more than...

Math

### If a =log3(1/3 log3x) and b =1/3 log3(log3x), what is the value of a in terms of b?

I am assuming that 3 after the first log in both expressions is supposed to be the base of the logarithm. (The second 3 can be either a base of the log or a coefficient of x.) Then, the task is to...

Math

### John told his mother, who is a mathematics teacher that x is a positive real number and a = log3 (log27 x) b =...

Given `a=log_3(log_(27)x)` and `b=log_(27)(log_3 x)` with x a positive real number we are asked to find an equation for a in terms of b alone. Note that as x is a positive real number the...

Math

### A student believed that T is a root of x^4 - x^3 + x^2 - x + 1 = 0. Find the value of T^20 + T^10 + T^5 +1.

We are given that T is a root of `x^4-x^3+x^2-x+1=0` , and we are asked to find the value of `T^(20)+T^(10)+T^(5)+1` . To find the roots, we notice that the given expression is one of the factors...

Math

### What is the smallest positive value of x (x not equal to 0), such that sin 7x + sin 9x + sin 11x + sin 13x = 0?

To solve this problem, we can use the formula which transforms a sum of two sinuses into a product. Namely, the formula is `sin a + sin b = 2 sin ( ( a + b ) / 2 ) cos ( ( a - b ) / 2 ) ` and it...

Math

### A student told her mother that the number 100(101)(102)(103) + 1 is a perfect square. Without using a calculator,...

The square root is between `100 ^ 2 ` and `103 ^ 2 . ` Let's express factors as 100 `+-` something and 103 `+- ` something and group them: `100 * 101 * 102 * 103 + 1 = 100 * ( 100 + 1 ) * ( 103 - 1...

Math

### A student told her teacher that triangle ABC has angle A=75, angle C=45, and AC=20. A circle passing through point A...

It is known that if a circle with the center `O ` is tangent to a straight line and the point of tangency is `X , ` then the radius `O X ` is perpendicular to the tangent line. Because of this, the...

Math

### An American Airlines plane takes off and remains at an angle of 15 degrees while climbing. What distance has it flown...

We are told that a plane takes off at a `15^@` inclination which it maintains until it reaches its cruising altitude of 44km. We are asked to find the distance the plane has flown. We can impose a...

Math

### An investment pays $1,200 a quarter of the time, $1,000 half of the time, and $800 a quarter of the time. Calculate...

Expected value (EV) or mean is the expected outcome of a given investment and is calculated as the weighted average of all the possible values of an outcome based on their probabilities of...

Math

### Jill told her teacher that a square ABCD is cut out of cardboard with AB=1 and isosceles right triangles are cut off...

The main task is to determine the length of a leg of each triangle cut. When we'll know this length, denote it `x , ` we'll easily find that the area of the octagon as the area of the square minus...

Math

### What is 33 x 100?

Multiplying or dividing with powers of 10 is easy. If we are multiplying with positive powers, the decimal needs to move to the right by the same number of digits as the power. For example, if we...

Math

### A student stated that each side of an equilateral triangle has a semi-circle constructed exterior to the triangle...

We are given an equilateral triangle with semicircles erected on the sides. We are told that the area of the figure is equal to its perimeter. The length of the side of the equilateral triangle...

Math

### For k∈N, find the gcd in the following two cases: A. (9k+4,2k+1), and B. (9k+4,2k-1).

In the first case, the answer is always `1 . ` Indeed, if we have two numbers, say `a ` and `b , ` their gcd is the same as of `a ` and `a - b , ` or of `a - b ` and `b . ` The Euclidean algorithm...

Math

### There will be positive `ZZ` m,n, and there will be dm>0 devider of m and dn>0 devider of n. Prove or disprove:...

So, it is given that `m , n \in NN , ` `d_m ` divides `m ` and `d_n ` divides `n . ` It is also given that `d_m * d_n gt 1 / 2 mn . ` Prove that this implies `d_m = m ` and `d_n = n .` First, prove...

Math

### Ho: m1 £ m2 H1: m1 > m2. 1. Is this a one-tailed or a tow-tailed test? 2. State the decision rule. 3. Compute the...

If the test is based on two normal populations with known standard deviations, then let `\bar{X_{1}}` be the mean of the first random sample of size `n_{1}` from a normal population with a mean of...

Math

### The distance around a neighborhood is 4 miles. When measuring it, Andy counted 3.8 miles. What is Andy's percent error?

Knowing how to calculate percent error is a valuable skill, especially in fields like statistics, physics, and chemistry. Your percent error tells you how close your results were to the true value...

Math

### A circular grass field has a radius of 20 yards. A hungry goat is chained to a point on the edge of the field by a...

To solve, draw a circle that has a radius of 20 yards. This circle represents the circular grass field. At its edge, draw a point representing where the goat is tied (see Figure 1). The goat is...

Math

### Prove that to every m`epsilon` `NN` that exists (m!+1,(m+1)!+1)=1.

This statement may be proved with the use of the Euclidean algorithm. It is based on the fact that the greatest common divisor of two numbers remains the same if we subtract one of them from the...

Math

### Find all the n `in` `ZZ` positive that exist for n+1|n^2+1.

Let's express `n ^ 2 + 1 ` in terms of `n + 1 ` (we can tell this polynomial division): `n ^ 2 + 1 = n ^ 2 - 1 + 2 = ( n - 1 ) ( n + 1 ) + 2 .` Now we see that the greatest common divisor of `n ^ 2...

Math

### If the US dollar–British pound exchange rate is $1.50 per pound, the US dollar–euro rate is $0.90 per euro. What is...

The current question relates to currency exchange rates, and the concerned currencies are US Dollars (USD), the British pound, and the euro. The USD to pound exchange rate is $1.5 per pound. This...

Math

### If the original image is 2 inches and the scale drawing is 3.5 inches, what is the scale factor?

A scale factor expresses the relationship between two lengths. It can be expressed as a ratio or a positive number. Let's look at some examples. If the original image is 4 inches, and the scale...

Math

### Gina has a road trip and is reading a map that has a scale of 1 in to 30 miles. Her trip measures 3.5 inches on the...

The best way to solve a problem like this one is to set up a proportion with the missing information designated by x and then solve for x. Let's look at a couple of examples. If a map has a scale...

Math

### The expression √(6+3(√3))-√(6-3(√3)) can be simplified to √(m). Find m.

We are asked to write the expression `sqrt(6+3sqrt(3))-sqrt(6-3sqrt(3))` in the form `sqrt(m)` . Let x be the given expression. Then, `x=sqrt(6+3sqrt(3))-sqrt(6-3sqrt(3))`. Squaring both sides...

Math

### What is the minimum value of x^2 +y^2, if 20x+21y=1247?

To find the minimum value of the given expression, let's first rewrite is as a function of x, given the constraint 20x + 21y = 1247. From here, we can find y: `y = (1247 - 20x)/21` and substitute...

Math

### What is the sum of all the even three-digit numbers which are multiples of 3?

All even multiples of 3 are multiples of 6. So the question is really what is the sum of all of the multiples of 6 between 100 and 999, inclusive. That makes this an arithmetic sequence question....

Math

### Find the length of the third median of a scalene triangle with an area of `3sqrt(15)` and medians of lengths 3 and 6....

We are given a scalene triangle with medians of length 3 and 6 and area `3sqrt(15)` We are asked to find the length of the third median. We use the fact that the area of a triangle can be given...

Math

### Each bag of cereal should contain 350 grams. A box was selected at random and weighed 342 grams. What is the...

The percentage error of a measurement is calculated using the following steps: 1) Subtract the experimental (observed) value from the accepted (true or theoretical) value 2) Determine the absolute...

Math

### Find T if 19x^3+92x^2+23x+T=0 has three roots in geometric progression.

If the polynomial has three roots `a , b , c , ` then it has the form `p ( x ) = 19 ( x - a ) ( x - b ) ( x - c ) . ` It, in turn, is equal to `19 x^3 - 19 ( a + b + c ) x^2 + 19 ( ab + bc + ac ) x...

Math

### Calculate the monthly payment for a 30-year mortgage, where the amount borrowed is $100,000 and the annual interest...

To determine the periodic payment of a mortgage, apply the formula `PMT = (PV(r/n))/([1-(1+r/n)^(-nt)])` where PMT is the periodic payment, PV is the principal value, r is the annual rate in...

Math

### Calculate the price of a zero-coupon bond that has an interest rate of 6.65% (.0665), a face value of $100.00, and...

A zero-coupon bond is a bond that doesn't pay any interest and trades at a discount to its face value. It is also referred to as a pure discount bond or simply discount bond because it does not pay...

Math

### On a sidewalk, a woman is walking at 1.80 m/s to the east. A man passes her on the right running at 6.40 m/s to the...

The relationship between velocities in different reference frames is `vecv_(mo) = vecv_(mw)+vecv_(wo)` Here, the subscript m stands for "man," w for "woman," and o for stationary observer. This...

Math

### If a bond has a face value of $1000 and a coupon rate of 4.25%, calculate the annual coupon payments.

A coupon payment is the amount of interest which a bond issuer pays to a bondholder from the issue date until it matures . The annual coupon payment 'C', is given by the mathematical expression:...

Math

### The original image is 3 cm and the scale drawing is 5 cm. What is the scale factor?

Scale factor is the number that we have to multiply to the dimension of a figure to increase or decrease its size. It has a form: `y=c*y_o` where c is the scale factor, y_o is the original...

Math

### In the quadrilateral ABCD, AB=BC=CD, and AC=BD=AD. What is AD/AB?

We are given quadrilateral ABCD, with AB=BC=CD and AC=BD=AD. We are asked to find the ratio of side AD to side AB. (See attached diagram.) Note that ABCD is an isosceles trapezoid with the legs...

Math

### A map has a scale of 1/2inch = 20 miles. Quinn wants to travel an actual distance of 70 miles. What will the length...

This kind of problem is best solved by using a ratio. Let's look at a couple examples to give you a pattern to follow. If a map has a scale of ½ inch = 40 miles, and someone wants to travel 90...

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