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All Questions
^{+0}
235663 Questions
0
2
1
+688
Algebra
Let a and b be integers such that the polynomial
x^4 + ax^3 + bx^2 + ax + 1 = 0
has four distinct positive real roots. Find the smallest possible value of a + b.
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booboo44
Oct 29, 2024
0
2
1
+688
Algebra
Let $d$ be a function taking the positive integers to the nonnegative integers such that $d(p) = 1$ for any prime $p,$ and
d(ab) = b \cdot d(a) + a \cdot d(b)
for all positive integers $a$ and $b.$ Find the number of positive integers
read more ..
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booboo44
Oct 29, 2024
0
1
1
+688
Algebra
Let r, s, and t be the real roots of
(x - \sqrt[3]{13})(x - \sqrt[3]{53})(x - \sqrt[3]{103}) = 0.
Compute r^3 + s^3 + t^3.
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booboo44
Oct 29, 2024
0
2
0
+293
Algebra
Points $A,$ $B,$ and $C$ are given in the coordinate plane. There exists a point $Q$ and a constant $k$ such that for any point $P$,
PA^2 + PB^2 + PC^2 = 3PQ^2 + k.
If $A = (7,-11),$ $B = (10,13),$ and $C = (18,-22)$, then find the constant
read more ..
MeIdHunter
Oct 29, 2024
0
2
0
+293
Algebra
Let $O$ be the origin. Points $P$ and $Q$ lie in the first quadrant. The slope of line segment $\overline{OP}$ is $4,$ and the slope of line segment $\overline{OQ}$ is $5.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$
Note:
read more ..
MeIdHunter
Oct 29, 2024
0
3
0
+293
Algebra
Let $a$ and $b$ be real numbers, where $a < b$, and let $A = (a,a^2)$ and $B = (b,b^2)$. The line $\overline{AB}$ (meaning the unique line that contains the point $A$ and the point $B$) has slope $2$. Find $a + b$.
MeIdHunter
Oct 29, 2024
0
2
0
+250
Number Theory
A four-digit hexadecimal integer is written on a napkin such that the units digit is illegible. The first three digits are 2, $F$, and 1. If the integer is a multiple of $19_{10}$, what is the units digit?
MEMEG0D
Oct 29, 2024
0
1
1
+250
Number Theory
The numbers $24^2 = 576$ and $56^2 = 3136$ are examples of perfect squares that have a units digits of $6.$
If the units digit of a perfect square is $5,$ then what are the possible values of the tens digit?
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MEMEG0D
Oct 29, 2024
0
3
0
+250
Number Theory
Which of the residues 0, 1, 2, ..., 11 satisfy the congruence 3x = 1 mod 12?
MEMEG0D
Oct 29, 2024
0
3
0
+250
Number Theory
Which of the residues 0, 1, 2, 3, 4 satisfy the congruence x^5 = 0 mod 5?
MEMEG0D
Oct 29, 2024
0
3
0
+830
Algebra
Let $x$ and $y$ be real numbers such that $x^3 + 3xy^2 = 679$ and $x^3 - 3xy^2 = 615.$ Find $x - y.$
magenta
Oct 29, 2024
0
2
0
+830
Algebra
When the same constant is added to the numbers 60, 120, and 160, a three-term geometric sequence arises. What is the common ratio of the resulting sequence?
magenta
Oct 29, 2024
0
3
0
+830
Algebra
Let a_1, a_2, a_3, \dots, a_{10}, a_{11}, a_{12} be an arithmetic sequence. If $a_1 + a_3 + a_5 + a_7 + a_9 + a_{11} = 0$ and $a_2 + a_4 + a_6 + a_8 + a_{10} + a_{12} = 0$, then find $a_1$.
magenta
Oct 29, 2024
0
2
0
+830
Algebra
Laverne starts counting out loud by 5's. She starts with 2. As Laverne counts, Shirley sums the numbers Laverne says. When the sum finally exceeds 20, Shirley runs screaming from the room. What number does Laverne say that sends
read more ..
magenta
Oct 29, 2024
0
1
2
+4
Any idea?
In trapezoid EFGH, \overline{EF} \parallel \overline{GH}, and P is the point on \overline{EH} such that EP:PH = 1:2. If the area of triangle PEF is 6, and the area of triangle PGH is 6, then find the area of trapezoid EFGH.
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Causeveras39
Oct 29, 2024
0
4
0
+688
Algebra
Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be an arithmetic sequence. Let $S_n$ denote the sum of the first $n$ terms. If $S_{20} = \frac{1}{5}$ and $S_{10} = 0,$ then find $S_{70}.$
booboo44
Oct 29, 2024
0
2
0
+688
Algebra
Let $f(x) = 3x - 8 + 4x^2$ and $g(x) = 15x + c - 3x^2$. Find $c$ if $(f \circ g)(x) = (g \circ f)(x)$ for all $x$.
booboo44
Oct 29, 2024
0
2
0
+688
Algebra
Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$
booboo44
Oct 29, 2024
0
4
0
+688
Algebra
Solve the system of equations
y = \log_2 (2x)
y = log_4 (16 + x)
booboo44
Oct 29, 2024
0
2
0
+806
Number Theory
In how many ways can the numbers 1, 2, 3, 4, 5, 6 be arranged in a row, so that the product of any two adjacent numbers is at least 5?
gnistory
Oct 29, 2024
0
2
0
+806
Number Theory
Find the number of $7$-digit numbers, where the sum of the digits is divisible by $11.$
gnistory
Oct 29, 2024
0
2
0
+806
Number Theory
Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.
gnistory
Oct 29, 2024
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