Boseo

Problem 1:

The function simplifies to f(x) = 2(x + 5)/(x + 3).  You can then see that the function is not defined for x = -3.

We can analyze the system of quadratic equations to determine the number of real solutions for different values of c. Here's how:

Analyzing the Discriminant:

The discriminant of a quadratic equation determines the nature of its roots (solutions). It is denoted by the symbol b2−4ac. In this case, considering the first equation (y = 6x^2 - 9x + c):

a = 6

b = -9

c (variable)

The discriminant (d) for the first equation is:

d = (-9)^2 - 4 * 6 * c

The number of real solutions depends on the value of the discriminant:

d > 0: Two real and distinct solutions (roots)

d = 0: One repeated real solution (root)

d < 0: No real solutions (complex roots)

Relating Discriminant to c:

We want to find the values of c that correspond to each case.

(a) Exactly one real solution:

For exactly one real solution (repeated root), the discriminant needs to be zero.

Therefore, we need to solve:

0 = (-9)^2 - 4 * 6 * c

This simplifies to:

c = \frac{81}{24} = \dfrac{7}{2}

(b) More than one real solution:

For more than one real solution (distinct roots), the discriminant needs to be positive.

Therefore, we need to solve:

0 < (-9)^2 - 4 * 6 * c

This simplifies to:

c < \dfrac{81}{24} = \dfrac{7}{2}

(c) No real solutions:

For no real solutions (complex roots), the discriminant needs to be negative.

Therefore, we need to solve:

0 > (-9)^2 - 4 * 6 * c

This simplifies to:

c > \dfrac{81}{24} = \dfrac{7}{2}

Summary:

(a) Exactly one real solution: c = dfrac{7}{2}

(b) More than one real solution: c < dfrac{7}{2}

(c) No real solutions: c > dfrac{7}{2}

Unfortunately, with the given information (side lengths PQ=8, QR=5, and PR=1), it is impossible to determine the length of XY. Here's why:

Triangle Inequality Violation: The side lengths provided violate the triangle inequality. The triangle inequality states that the sum of any two side lengths in a triangle must be greater than the third side length. In this case, PQ+PR=8+1=9, which is not greater than QR=5. Therefore, a triangle with these side lengths cannot exist.

Ambiguous Angle Bisector: Even if the side lengths were valid, the information provided wouldn't be sufficient to determine the length of XY. The angle bisector of angle P divides angle P into two ratios, which depend on the original measure of angle P. Without knowing the angle measure or additional information about the triangle (like area), it's impossible to determine the specific location of point X on side QR​. Consequently, the distance XY cannot be determined.

Since P, Q, and R are the midpoints of segments in triangle MNO, then triangle PQR is similar to triangle MNO with a scale factor of 21​. Therefore, the area of triangle PQR is (21​)2 times the area of triangle MNO, or 41​ the area of triangle MNO. We are given that the area of triangle PQR is 10, so the area of triangle MNO is 10⋅4=40.

Similarly, triangle JQR is similar to triangle MNO with a scale factor of 21​, so the area of triangle JQR is (21​)2 times the area of triangle MNO, or 41​ the area of triangle MNO. We know the area of MNO is 40, so the area of triangle JQR is 41​⋅40=10​.

Since ∠TSU=60∘ and ∠STU=45∘, then ∠SUT=75∘. Triangle STU is a 45-45-90 triangle, so SU=UT=2​⋅ST. Also, since M and N are midpoints, then SM=MT=NU=UT=2​⋅ST and TN=NS=US=2​⋅ST.

Triangle UMN is isosceles with UM=MN because they are both half of sides of an isosceles triangle. Similarly, triangle PNM is isosceles with PN=MN. Since ∠SUT=75∘, then ∠SUM=∠SUN=(180∘−75∘)/2=52.5∘. Likewise, ∠PNM=∠PMN=52.5∘.

Looking at triangle ZNM, we see that ∠ZNM+∠ZMN+∠MZN=180∘. Since ∠MZN is a right angle, then ∠ZNM+∠ZMN=90∘. Similarly, for triangle PNM, we have ∠PNM+∠PMN=90∘.

Adding all the angle measures together, we get the answer: $ \angle ZNM + \angle ZMN + \angle PNM + \angle PMN + \angle NZM + \angle NPM = 90^\circ + 90^\circ + 52.5^\circ + 52.5^\circ + \angle NZM + \angle NPM. $

Since ∠NZM and ∠NPM are supplementary angles (they form a straight line), then ∠NZM+∠NPM=180∘. Therefore, the final answer is $ 90^\circ + 90^\circ + 52.5^\circ + 52.5^\circ + 180^\circ = \boxed{465^\circ}$.

We can solve for a and b by utilizing the given information about A and its effect on x and y, along with the property of matrix multiplication.

Here's how we proceed:

Analyze the equation (A^5)x = ax + by:

This equation states that applying matrix A to vector x five times consecutively (A raised to the power of 5) results in a linear combination of x and y, with coefficients a and b.

Utilize the information about A:

We are given that Ax = y and Ay = x + 2y. These equations define how A transforms x and y.

Express (A^5)x in terms of x and y:

We can't directly expand (A^5) as it's a high power. However, we can use the given information about A iteratively.

Start with the first equation: (A^2)x = A(Ax) = A(y).

Substitute from the first given equation: (A^2)x = Ay.

Use the second given equation: (A^2)x = x + 2y.

Similarly, we can continue:

(A^3)x = A((A^2)x) = A(x + 2y) = Ax + 2Ay (using the definition of matrix multiplication).

Substitute from the first given equation: (A^3)x = y + 2(x + 2y) = 3x + 4y.

We can repeat this process further, but the pattern should be clear.

Find an expression for (A^4)x and (A^5)x:

Following the established pattern, we can see that:

(A^4)x = 3(A^3)x = 3(3x + 4y) = 9x + 12y.

(A^5)x = 3(A^4)x = 3(9x + 12y) = 27x + 36y.

Substitute (A^5)x in the original equation:

The original equation is: (A^5)x = ax + by.

Substitute the expression we found for (A^5)x: 27x + 36y = ax + by.

Solve for a and b:

We want to isolate a and b. Since x and y are not multiples of each other, we can treat them as independent variables.

If we set y = 0, the equation becomes 27x = ax, which implies a = 27.

If we set x = 0, the equation becomes 36y = by, which implies b = 36.

Therefore, in the equation (A^5)x = ax + by, the values of a and b are:

a = 27

b = 36

Since O is the origin (0,0), the coordinates of P are of the form (a,4a) for some positive value a. Similarly, the coordinates of Q are of the form (b,5b) for some positive value b. We are given that OP=OQ, so a2+(4a)2​=b2+(5b)2​, which simplifies to a=b.

Therefore, the coordinates of Q are (a,5a), which means the slope of PQ​ is [\frac{(5a) - (4a)}{a - (0)} = \boxed{1}.]

We can use the given information about A and B acting on x and y to find the expressions for (AB)x and (BA)x.

Finding (AB)x:

We know (AB)x = B(Ax) since matrix multiplication is associative.

From the givens, Ax = y. Substitute: (AB)x = B(y).

We are also given By = 2y. Substitute: (AB)x = B(2y) = 2(By) = 2(2y) = 4y. Therefore, we can write (AB)x = 4y. This translates to a = 0 and b = 4.

Finding (BA)x:

Similar to (AB)x, we know (BA)x = A(Bx).

From the givens, Bx = x + y. Substitute: (BA)x = A(x + y).

We are also given Ay = x + 2y. Substitute: (BA)x = A(x + y) = Ay - 2y = (x + 2y) - 2y = x. Therefore, we can write (BA)x = x. This translates to c = 1

and d = 0.

In conclusion, the scalars that satisfy the equations are:

a = 0

b = 4

c = 1

d = 0

This means:

(AB)x = 4y

(BA)x = x

Analyzing the function f(x) = floor((2 - 3x) / (x + 3)):

The denominator (x + 3) is 0 when x = -3. This means f(x) is undefined at x = -3.

We will need to consider different cases based on the sign of the denominator (x + 3) and the relative values of 2 - 3x compared to 0.

Cases for f(x):

x < -3: In this case, both denominator (x + 3) and numerator (2 - 3x) are negative. Dividing two negative numbers results in a positive value.

Since we take the floor (greatest integer less than or equal to), f(x) will be -1.

-3 < x < 2/3: Here, the denominator (x + 3) is positive, but the numerator (2 - 3x) is negative.

Dividing a positive by a negative results in a negative number.

Taking the floor of a negative number keeps it negative, so f(x) will be -2.

x = 2/3: At this specific point, the numerator becomes 0, and the result of the division is 0. The floor of 0 is 0, so f(x) = 0.

x > 2/3: In this case, both the numerator (2 - 3x) and denominator (x + 3) are positive. Dividing two positive numbers results in a positive value.

Taking the floor doesn't change the positive sign, so f(x) will be 1.

Evaluating the sum:

The key to evaluating the sum efficiently is to recognize that for a large range of x values (between -3 and 2/3), f(x) will be -2.

We can exploit this by calculating the number of terms that fall into this range and summing the contributions from the remaining terms separately.

Number of terms where f(x) = -2:

We know x = -3 falls outside this range (f(x) is undefined).

The range ends when x = 2/3, which is between terms 1000 and 1001 (1000th term is x = 999 and 1001st term is x = 1000).

Therefore, there are 1000 - (-3) + 1 = 1004 terms where f(x) = -2.

Contribution from terms where f(x) = -2:

Each term contributes -2 to the sum.

Total contribution = -2 * (number of terms) = -2 * 1004 = -2008

Remaining terms:

We need to consider terms for x < -3 (f(x) = -1), x = 2/3 (f(x) = 0), and x > 2/3 (f(x) = 1).

There are very few terms less than -3 (all negative x values), and they can be ignored for a large sum like this (their contribution will be negligible).

There's only one term for x = 2/3, contributing f(2/3) = 0.

The remaining terms from x slightly greater than 2/3 to x = 1000 will all have f(x) = 1. The exact number of these terms depends on the specific values, but there will be significantly fewer compared to the 1004 terms with f(x) = -2.

Overall Sum:

Sum from terms with f(x) = -2: -2008

Contribution from f(2/3) (x = 2/3): 0

Contribution from remaining terms with f(x) = 1 (positive but less than those with -2): + (positive value)

Since the number of terms with f(x) = 1 is significantly less than those with -2, and there's a negligible contribution from terms less than -3, the positive value from the remaining terms will be much smaller than 2008.

Therefore, the overall sum f(1) + f(2) + ... + f(999) + f(1000) is equal to -2008.