All Questions ⁺⁰237293 Questions

-1

84

0

+1766

quadratic

Let $f(x) = x^2 + bx + 2 - 3x + 8$ for all real numbers $x$. For what real values of $b$ is $-2$ not in the range of the function $f(x)$? Express your answer in interval notation.

blackpanther Dec 11, 2023

-1

47

0

+1766

quadratic

Let $f(x) = x^2 + bx + 12 - 3x + 10$ for all real numbers $x$. Find the greatest integer value of $b$ such that $-4$ is not in the range of $f(x)$.

blackpanther Dec 11, 2023

0

78

0

+818

Algebra

Let $f(x)$ be a polynomial with integer coefficients. There exist distinct integers $p,$ $q,$ $r,$ $s,$ $t$ such that
\[f(p) = f(q) = f(r) = f(s) = 1\]
and $f(t) > 1.$ What is the smallest possible value of $f(t)?$

maximum Dec 11, 2023

0

66

1

+818

circles

The circles $x^2 + y^2 = 4$ and $(x - 5)^2 + (y - 8)^2 = 60$ intersect in two points $A$ and $B.$ Find the distance $AB$.

maximum Dec 11, 2023

0

29

1

+818

quadratic

The quadratic equation $x^2-5x+5=-4x+7$ has two roots. Find the absolute difference of these roots.

maximum Dec 11, 2023

0

58

1

+818

help wth coordinates

Find the number of lattice points that lie on the graph of $x^2-y^2=17-2xy$. (A lattice point is a point in the coordinate plane where both coordinates are integers.)

maximum Dec 11, 2023

0

27

1

+818

system

The real numbers x and y satisfy x (x+y) = 27 and (x+y) = 54. What is the value of (x+y)^2?

maximum Dec 11, 2023

0

36

1

+45

Geometry Problem

EFGH is a rectangle with area 48. If EGJI is a rectangle such that H is on JI, what is the area of EGJI?

LuckyLemonie Dec 11, 2023

Dec 10, 2023

-1

46

1

+1766

floor

Find the number of positive integers ${}n$ that satisfy $\lfloor \sqrt{n} \rfloor + \lfloor \sqrt{4n} \rfloor = 5.$

blackpanther Dec 10, 2023

-1

74

0

+1766

hard sum

Let
\[f(x) = \left\lfloor\frac{2 - 3x}{2x + 8}\right\rfloor.\]
Evaluate $f(1)+f(2) + f(3) + \dots + f(999)+f(1000).$ (This sum has $1000$ terms, one for the result when we input each integer from $1$ to $1000$ into ${}f$.)

blackpanther Dec 10, 2023

-1

81

0

+1766

help trig

In right triangle $BCD$ with $\angle D = 90^\circ$, we have $BC = 10$ and $BD = 8$. Find $\sin B$.

blackpanther Dec 10, 2023

0

34

2

+1678

Algebra

The quadratic equation $x^2-5x+t =3x$ has only positive integer roots. Find the average of all distinct possible values of $t$.

wiseowl Dec 10, 2023

0

58

1

+1678

quadratic

Let $a$ and $b$ be the roots of the quadratic equation $2x^2 - 7x + 2 = -x^2 + 4x + 9.$ Find $\frac{1}{a-1}+\frac{1}{b-1}.$

wiseowl Dec 10, 2023

0

82

0

+1678

quadratic

Find the non-zero value of $c$ for which there is exactly one positive value of $b$ for which there is one solution to the equation $x^2 + (b + 4c)x + c^2 = 0$.

wiseowl Dec 10, 2023

0

76

1

+1678

coordinates

There is exactly one value of $x$ for which the distance from $(5,6)$ to $(3x-1,ax+5)$ is $4$. If $a \neq 0,$ what is $a$?

wiseowl Dec 10, 2023

0

62

1

+1678

coordinates

The point $(x,y)$ in the coordinate has a distance of $6$ units from the $x$-axis, a distance of $15$ units from the point $(5,7)$, and a distance of $\sqrt{n}$ from the origin. If both $x$ and $y$ are negative, what is $n$?

wiseowl Dec 10, 2023

0

67

1

+4

You have three towels. The first towel is 3/4 meters long. It is 3/5 times length of the second towel.

You have three towels. The first towel is 3/4 meters long. It is 3/5 times length of the second towel. The length of the third towel is 5/12 of the sum of the length of the first and the length of the second. What part of the length of the third towel is read more ..

asedsc12 Dec 10, 2023

0

63

1

+4

PLS HELP ASAP

For an integer , the inequality
has no real solutions in . Find the number of different possible values of .

Ookbroo Dec 10, 2023

0

29

1

+1926

altitudes

Triangle $ABC$ has altitudes $\overline{AD},$ $\overline{BE},$ and $\overline{CF}.$ If $AD = 12,$ $BE = 20,$ and $CF$ is a positive integer, then find the largest possible value of $CF.$

tomtom Dec 10, 2023

0

80

0

+1926

help with triangles

In triangle $ABC,$ $\angle A > \angle B > \angle C.$ If $n$ is a positive integer, how many possible values of $n$ are there?

tomtom Dec 10, 2023

0

56

1

+1926

help with triangle inequality

There are four cities: Capital City, Little Village, Mytown, and Yourtown. The distance from Capital City to Little Village is $5$ miles. The distance from Capital City to Mytown is $2$ miles. The distance from Mytown to Yourtown is $3$ read more ..

tomtom Dec 10, 2023

0

77

0

+1926

triangle inequality

Two of the sides of an obtuse triangle are shown. The length of the third side is also a positive integer. How many different possible values are there for the third side length? (Assume that the triangle is non-degenerate.)

tomtom Dec 10, 2023

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