maximum

260 Questions
128 Answers

0

142

0

+834

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 11.12.2023

0

95

1

+834

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 11.12.2023

0

69

1

+834

quadratic

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

maximum 11.12.2023

0

108

1

+834

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 11.12.2023

0

60

1

+834

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 11.12.2023

0

131

0

+834

plz help quick

In how many ways can three pairs of siblings from different families be seated in two rows of three chairs, if siblings may sit next to each other in the same row, but no child may sit directly in front of their sibling?

maximum 28.11.2023

0

133

0

+834

counting

Find the number of positive integers that satisfy both the following conditions:
Each digit is a 1 or a 2 or a 3
The sum of the digits is 5

maximum 28.11.2023

0

104

1

+834

counting problem

In the array below, in how many different ways can we start with the letter A and move from letter to letter (horizontally, vertically, or diagonally), to spell the word "ARCH"?
A
RRR
&mehr ..

maximum 28.11.2023

0

147

0

+834

need help

Fill in the blanks with numbers to make a true equation
3x^2 + 12x + 4 - x^3 + 8x - 5x + 7x^2 + 11 = __ (x + __)^3 + __(x + __)^2 + __x + ___

maximum 21.11.2023

0

120

1

+834

graphing question

The parabola $y = ax^2 + bx + c$ is graphed below. Find $a + b + c$. (The grid lines are one unit apart.)

maximum 21.11.2023

+834

-1

Do not post AoPS homework.

maximum20.01.2025