Beaker \(A\) contains \(500\) mL of \(20\%\) salt solution, and
beaker \(B\) contains \(800\) mL of \(50\%\) salt solution. A
lab tech pours some of each of these solutions into beakers
\(C\) and \(D\) so that beaker \(C\) contains \(100\) mL of
\(30\%\) salt solution, and beaker \(D\) contains \(200\) mL of
\(40\%\) salt solution. How many milliliters remain in beaker
\(B\) after this is done?
The widths and lengths of two distinct rectangles form a
sequence of four consecutive odd integers. The perimeter of the
first rectangle is \(44\) less than twice the perimeter of the
second rectangle, and the sum of their areas is less than
\(150\). Find the sum of their areas.
The equation \(a^2 + b^2 + c^4 = 2020\) has exactly one solution
in the positive integers for which \(a > b\). Find
\(a + b + c + d\) for this solution.
Consider a balance scale where weights may be placed on either
side. We can use this scale to weigh a \(3\) pound object by
placing it on one side and placing a \(3\) pound weight on the
opposite side. Another way would be to place a \(4\) pound
weight on the same side as the object and a \(2\) pound and a
\(5\) pound weight on the opposite side. Suppose you need to be
able to weigh objects with any whole number weight from \(1\)
to \(40\) pounds. What is the least number of weights that are
needed?
The region inside a circle of radius \(1\) centered at the
origin is painted blue. Then the regions inside two circles of
radius \(1\) centered at \((-1,1)\) and \((-1,-1)\) are painted
red. The regions that are painted twice will now be purple.
What is the area of the remaining blue region?
Let \(K\) be an integer that is greater than \(1\), a perfect
square, and equal to \(\sum_{i=1}^D i\) for some integer \(D\).
Find \([ 1 + 2 + \cdots + ( \sqrt{K} -1 ) ] - [ ( \sqrt{K}+1 ) +
( \sqrt{K} +2 ) + \cdots + D ]\).
In a track race between Achilles, a tortoise, and a hare, the
hare gives the tortoise a head start of \(1000\) meters and
gives Achilles a head start of \(100\) meters. If Achilles, the
tortoise, and the hare move at \(1000\), \(10\), and \(1050\)
meters per minute, respectively, for how many minutes will
Achilles hold the lead?
A collection of \(62\) coins consists of \(D\) dimes, \(N\)
nickels, and \(Q\) quarters. The total value is \(\$8.30\). Find
the sum of all possible values of \(N\).
Kara looks at a wall clock (with constant velocity hands)
sometimes between 3 and 4 o'clock and observes that the angle
between the hour-hand and the minute-hand is 30 degrees. Ten
minutes later, she observes that the angle between the hour-hand
and the minute-hand is 85 degrees. Find the time when she first
looked at the clock to the nearest second. Write your answer in
hr:min:sec format first, then give hr + min + sec as your final
answer.
Let \(\#\) be the binary operation on all real \(2 \times 2\)
matrices defined by \(A \# B = AB + BA\).
(i) Is \(\#\) commutative for all real \(2\times 2\) matrices?
(ii) Is \(\#\) associative for all real \(2\times 2\) matrices?
Now, let \(x = 1\) if (i) is true, otherwise let \(x = 0\). And
let \(y = 1\) if (ii) is true, otherwise let \(y = 0\). What is
\(2^x + 3^y\)?
Three people \((X,Y,Z)\) are in a room with you. One is a knight
(knights always tell the truth), one is a knave (knaves always
lie), and the other is a spy (spies may either lie or tell the
truth). \(X\) says, "If you asked me who the spy is, I would say
that \(Z\) is the spy." \(Y\) says, "\(Z\) is the spy." \(Z\)
says, "I am the spy." Now, assign each person \(X,Y,Z\) a
numeric value: 1 if knight, 2 if knave and 3 if spy. What is
\(100 X + 10 Y + Z\)?
Refer to the figure as shown. Find the measure of \(\angle
AFB\) in degrees if \(AB = BC = CD = DE = EF\) and \(\triangle
ABF\) is isosceles.
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How many 5-digit palindromic numbers (of the form \(abcba\) with
\(a \neq 0\)) are divisible by \(37\)? Note \(a, b, c\) need not
be distinct.
Consider the rational function \(f(x) = \frac{n(x)}{m(x)}\)
where \(n(x)\) and \(m(x)\) are both polynomials of degree 3 or
lower with real coefficients and a leading coefficient of 1. If
\(g(x)\) has a removable discontinuity at \(x = 2\), a vertical
asymptote of \(x = 7\), is continuous everywhere, has exactly
two real zeroes of \(x = 4\) and \(x = -3\), no nonreal complex
zeros, and has a slant asymptote. Let \(N\) and \(M\) be the
absolute values of the constant terms of \(n(x)\) and \(m(x)\).
Find \(M + N\).
What is the hundreds digit in the product of \(5^{94}\) and
\(98,777,782,163\)?
Let \(M\) be the unique whole number less than \(200\) that has
exactly \(18\) whole number factors. Find the sum of all \(18\)
factors of \(M\).
Let \(r, s, t\) be nonnegative integers. How many triples
\((r, s, t)\) satisfy the system \(\begin{cases} rs + t = 14 \\
r + st = 13 \end{cases}\) ?
Find the radius of the circle inscribed in an isosceles triangle
with two sides of length \(20\) and a base of length \(24\).
Consider \(f(x)\cos(\frac{f(x)x}{2}) + 2\sin(x) = f(x)\) where
\(f(x) = \begin{cases} 2\sin(x) & |x| \le 2 \\ 2 & |x| > 2
\end{cases}\). How many solutions does this equation have in the
interval \((-2\pi, 2\pi)\)?
Suppose a biased six-sided dice is rolled until either two 1s
are obtained on successive rolls or until a 1 and then a 2 are
obtained on successive rolls. The die will show a 1 with
probability 50%, a 2 with probability 20%, and something else
with probability 30%. What is the expected number of times the
die is rolled?
A store sells rope by the whole foot. If a landscaper needs a
rope at least 16,800 mm long, what is the least number in feet
she must purchase? Use 1 inch = 2.54 cm.
Let \(A = \{1, 2, 3, 4\}\). Let \(M =\) the number of distinct
proper subsets of \(A\). Let \(N =\) the number of distinct
differences that can be found by subtracting two distinct
elements of \(A\) (for example, \(1\) would be one such
difference since \(3 - 2 = 1\)). Find \(M + N\).
How many different ways can a cashier break (return an
equivalent dollar amount in smaller denominations) a \(\$50\)
bill if there are an unlimited number of \(\$20, \$10, \$5\),
and \(\$1\) bills available to the cashier? Assume bills of the
same denomination are indistinguishable.
An isosceles triangle has two sides of length 40 and a base of
length 48. A circle circumscribes the triangle. What is the
radius of the circle?
Let \(M\) be the number of digits \(\{0,1\}\) required to
express the largest prime factor of 2019 in base 2. Let \(N\) be
the number of hex digits \(\{0,1,2,\ldots,E,F\}\) required to
express 2019 in base 16. Find \(M-N\).
Triangle \(ABC\) has vertices at \((8,8)\), \((6,4)\), and
\((10,7)\). Find the sum of the lengths of the three altitudes
of this triangle, rounded to the nearest tenth.
The polynomial \(2x^3 + x^2 + cx + d\) is divisible by \(x+1\).
If \(d\) and \(c\) are integers with \(d+c = 29\), find the sum
of the two non-real roots of this polynomial.
Let \(N\) be the smallest integer greater than 2 such that
\(N^{N-1}\) is not the square of an integer. Find the product of
all rational numbers that could be roots of
\(5x^4 + bx^3 + cx^2 + dx + N\), where \(b, c\), and \(d\) can
be any integers. Round your answer to the nearest hundredth.
Three people \(\{X, Y, Z\}\) are in a room with you. One is a
knight (knights always tell the truth), one is a knave (knaves
always lie), and the other is a spy (spies may either lie or
tell the truth). \(X\) says "I am a spy." \(Y\) says "\(X\) is
telling the truth." \(Z\) says "I am not a spy." If we assign
each of \(X,Y,Z\) a value, where \(1\) is a knight, \(2\) is a
knave, and \(3\) is a spy, then what is \(100 X+ 10Y + Z\)?
\(\def\dot{\raise 0.2 ex {\,\rule{3pt}{3pt}\,}}
\def\dash{\raise 0.2 ex {\,\rule{9pt}{3pt}\,}}\)
Morse code involves transmitting dots "\(\dot\)" and dashes
"\(\dash\)". An agent attempted to send a five-character code
five different times, but only one of the five transmissions was
correct. However, it was known that each erroneous transmission
had a different number of errors than the others, and no
transmission had five errors. The first transmission was (1)
\(\dot \dash \dash \dot \dash\), which was not correct. The
other four transmissions are (2) \(\dot\dash\dash\dash\dot\) (3)
\(\dash\dash\dot\dot\dash\) (4) \(\dot\dash\dot\dash\dot\) (5)
\(\dot\dot\dot\dot\dot\). Which one is correct, or is it
impossible to tell? Enter the number choice, or 0 if impossible
to tell.
A checker is placed on the top left of a \(5\times 5\)
checkerboard as pictured. The checker may be moved one square at
a time but only to the left or down. Also, the checker may not
move to any of the three black squares. In how many different
ways can the checker be moved to the lower left corner of the
board?
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The function \(P(t) = \cos(8t)\) can be written as sums and
differences of powers of \(\cos t\) only. When \(P(t)\) is
written this way, what is the coefficient of \((\cos t)^4\)?
Find the length of the shortest line segment with one endpoint
on the line passing through \((7,8)\) that is parallel to the
vector \(\langle 3, -4\rangle\) and the other endpoint on the
circle with equation \((x+3)^2 + (y-2)^2 = 3\). Round your
answer to the nearest tenth.
Let \(a\) and \(b\) be positive integers with \(a^2 + b^2 =
2019^2\). Find \(a+b\).
Which describes the graph (in \(\mathbb R^3)\) of all solutions
of this system \(\begin{cases}
2x - 6y - 8z = 15 \\
-8x - 8y + 6z = -65 \\
x - 19y - 17z = 5 \end{cases}\):
(1) A point, (2) a line, (3) two lines, (4) a plane, (5) two
planes. Answer your number choice.
The graph of \(f(x) ax^2 + bx + c\) is symmetric about the
\(y\)-axis and its \(x\)- and \(y\)-intercepts from an
equilateral triangle. If the maximum value of \(f(x)\) is \(4\),
find \(a+b+c\).
How many of the following are both a continuous function on
\(\mathbb R\) and also one-to-one? \(g(x) = \ln(e^x)\),
\(h(x)= x|x|\), \(k(x) = x^2\), \(m(x) = \frac{1}{x+1}\),
\(n(x) = \frac{x}{x^2+1}\), \(p(x)=\sin(x)\),
\(q(x) = \arctan(x)\), \(r(x) = \frac{x}{|x|+1}\).
Let \((a_n)\) be an arithmetic sequence with initial value \(m\)
and common difference \(d\). Let \((g_n)\) be a geometric
sequence with initial value \(k\) and common ratio \(2\). The
sum of the first \(100\) terms of \((a_n)\) and the sum of the
first \(10\) terms of \((g_n)\) are equal. If \(m, d\), and
\(k\) are all positive integers, which of the following numbers
must divide \(m\)?
Some hikers set out on a hike on noon. At some point, they turn
around and follow the same path back to where they began, and
arrive there at 8:00 p.m. Their speed is 4 mi/hr on level
ground, 3 mi/hr uphill and 6 mi/hr downhill. How many miles did
they hike?
In the game of craps, a player (known as the shooter) rolls two
fair six-sided dice. The shooter immediately loses if the sum of
the dice is 2, 3, or 12, and immediately wins if the sum of the
dice is 7 or 11 on the first roll. If the sum is anything else
(4, 5, 6, 8, 9, or 10), that number becomes the point and the
shooter rolls again. The shooter now wins by rolling that same
point again and loses by rolling a 7. If any other number is
rolled, the shooter rolls again and keeps rolling until the
shooter wins by rolling the point or loses by rolling a 7.
Find the probability that the shooter wins.