Found problems: 15
2016 ASMT, 6
Let $ABC$ be a triangle with $AB = 5$ and $AC = 4$. Let $D$ be the reflection of $C$ across $AB$, and let $E$ be the reflection of $B$ across $AC$. $D$ and $E$ have the special property that $D, A, E$ are collinear. Finally, suppose that lines $DB$ and $EC$ intersect at a point $F$. Compute the area of $\vartriangle BCF$.
2016 ASMT, 1
A circle is inscribed in a unit square, and a diagonal of the square is drawn. Find the total length of the segments of the diagonal not contained within the circle.
2016 ASMT, 8
In rectangle $ABCD$, point $E$ is chosen on $AB$ and $F$ is the foot of $E$ onto side $CD$ such that the circumcircle of $\vartriangle ABF$ intersects line segments $AD$ and $BC$ at points $G$ and $H$ respectively. Let $S$ be the intersection of $EF$ and $GH$, and $T$ the intersection of lines $EC$ and $DS$. If $\angle SF T = 15^o$ , compute the measure of $\angle CSD$.
2016 ASMT, General
[u]General Round[/u]
[b]p1.[/b] Alice can bake a pie in $5$ minutes. Bob can bake a pie in $6$ minutes. Compute how many more pies Alice can bake than Bob in $60$ minutes.
[b]p2.[/b] Ben likes long bike rides. On one ride, he goes biking for six hours. For the first hour, he bikes at a speed of $15$ miles per hour. For the next two hours, he bikes at a speed of $12$ miles per hour. He remembers biking $90$ miles over the six hours. Compute the average speed, in miles per hour, Ben biked during the last three hours of his trip.
[b]p3.[/b] Compute the perimeter of a square with area $36$.
[b]p4.[/b] Two ants are standing side-by-side. One ant, which is $4$ inches tall, casts a shadow that is $10$ inches long. The other ant is $6$ inches tall. Compute, in inches, the length of the shadow that the taller ant casts.
[b]p5.[/b] Compute the number of distinct line segments that can be drawn inside a square such that the endpoints of the segment are on the square and the segment divides the square into two congruent triangles.
[b]p6.[/b] Emily has a cylindrical water bottle that can hold $1000\pi$ cubic centimeters of water. Right now, the bottle is holding $100\pi$ cubic centimeters of water, and the height of the water is $1$ centimeter. Compute the radius of the water bottle.
[b]p7.[/b] Given that $x$ and $y$ are nonnegative integers, compute the number of pairs $(x, y)$ such that $5x + y = 20$.
[b]p8.[/b] A sequence an is recursively defined where $a_n = 3(a_{n-1}-1000)$ for $n > 0$. Compute the smallest integer $x$ such that when $a_0 = x$, $a_n > a_0$ for all $n > 0$.
[b]p9.[/b] Compute the probability that two random integers, independently chosen and both taking on an integer value between $1$ and $10$ with equal probability, have a prime product.
[b]p10.[/b] If $x$ and $y$ are nonnegative integers, both less than or equal to $2$, then we say that $(x, y)$ is a friendly point. Compute the number of unordered triples of friendly points that form triangles with positive area.
[b]p11.[/b] Cindy is thinking of a number which is $4$ less than the square of a positive integer. The number has the property that it has two $2$-digit prime factors. What is the smallest possible value of Cindy's number?
[b]p12.[/b] Winona can purchase a pencil and two pens for $250$ cents, or two pencils and three pens for $425$ cents. If the cost of a pencil and the cost of a pen does not change, compute the cost in cents of five pencils and six pens.
[b]p13.[/b] Colin has an app on his phone that generates a random integer betwen $1$ and $10$. He generates $10$ random numbers and computes the sum. Compute the number of distinct possible sums Colin can end up with.
[b]p14.[/b] A circle is inscribed in a unit square, and a diagonal of the square is drawn. Find the total length of the segments of the diagonal not contained within the circle.
[b]p15.[/b] A class of six students has to split into two indistinguishable teams of three people. Compute the number of distinct team arrangements that can result.
[b]p16.[/b] A unit square is subdivided into a grid composed of $9$ squares each with sidelength $\frac13$ . A circle is drawn through the centers of the $4$ squares in the outermost corners of the grid. Compute the area of this circle.
[b]p17.[/b] There exists exactly one positive value of $k$ such that the line $y = kx$ intersects the parabola $y = x^2 + x + 4$ at exactly one point. Compute the intersection point.
[b]p18.[/b] Given an integer $x$, let $f(x)$ be the sum of the digits of $x$. Compute the number of positive integers less than $1000$ where $f(x) = 2$.
[b]p19.[/b] Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let $BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$?
[b]p20.[/b] Compute all real solutions to $16^x + 4^{x+1} - 96 = 0$.
[b]p21.[/b] At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna is watching the conveyor belt, and has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportion of the M&Ms are red?
[b]p22.[/b] $ABCDEFGH$ is an equiangular octagon with side lengths $2$, $4\sqrt2$, $1$, $3\sqrt2$, $2$, $3\sqrt2$, $3$, and $2\sqrt2$,in that order. Compute the area of the octagon.
[b]p23.[/b] The cubic $f(x) = x^3 +bx^2 +cx+d$ satisfies $f(1) = 3$, $f(2) = 6$, and $f(4) = 12$. Compute $f(3)$.
[b]p24.[/b] Given a unit square, two points are chosen uniformly at random within the square. Compute the probability that the line segment connecting those two points touches both diagonals of the square.
[b]p25.[/b] Compute the remainder when: $$5\underbrace{666...6666}_{2016 \,\, sixes}5$$ is divided by $17$.
[u]General Tiebreaker [/u]
[b]Tie 1.[/b] Trapezoid $ABCD$ has $AB$ parallel to $CD$, with $\angle ADC = 90^o$. Given that $AD = 5$, $BC = 13$ and $DC = 18$, compute the area of the trapezoid.
[b]Tie 2.[/b] The cubic $f(x) = x^3- 7x - 6$ has three distinct roots, $a$, $b$, and $c$. Compute $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ .
[b]Tie 3.[/b] Ben flips a fair coin repeatedly. Given that Ben's first coin flip is heads, compute the probability Ben flips two heads in a row before Ben flips two tails in a row.
PS. You should use hide for answers.
2016 ASMT, Algebra
[u]Algebra Round[/u]
[b]p1.[/b] Given that $x$ and $y$ are nonnegative integers, compute the number of pairs $(x, y)$ such that $5x + y = 20$.
[b]p2.[/b] $f(x) = x^2 + bx + c$ is a function with the property that the $x$-coordinate of the vertex is equal to the positive difference of the two roots of $f(x)$. Given that $c = 48$, compute $b$.
[b]p3.[/b] Suppose we have a function $f(x)$ such that $f(x)^2 = f(x - 5)f(x + 5)$ for all integers $x$. Given that $f(1) = 1$ and $f(16) = 8$, what is $f(2016)$?
[b]p4.[/b] Suppose that we have the following set of equations
$$\log_2 x + \log_3 x + \log_4 x = 20$$
$$\log_4 y + \log_9 y + \log_{16} y = 16$$
Compute $\log_x y$.
[b]p5.[/b] Let $\{a_n\}$ be the arithmetic sequence defined as $a_n = 2(n - 1) + 6$ for all $n \ge 1$. Compute
$$\sum^{\infty}_{i=1} \frac{1}{a_ia_{i+2}}.$$
[b]p6.[/b] Let $a, b, c, d, e, f$ be non-negative real numbers. Suppose that $a + b + c + d + e + f = 1$ and $ad + be + cf \ge \frac{1}{18} $. Find the maximum value of $ab + bc + cd + de + ef + fa$.
[b]p7.[/b] Let f be a continuous real-valued function defined on the positive real numbers. Determine all $f$ such that for all positive real $x, y$ we have $f(xy) = xf(y) + yf(x)$ and $f(2016) = 1$.
[b]p8.[/b] Find the maximum of the following expression:
$$21 cos \theta + 18 sin \theta sin \phi + 14 sin \theta cos \phi $$
[b]p9.[/b] $a, b, c, d$ satisfy the following system of equations $$ab + c + d = 13$$
$$bc + d + a = 27$$
$$cd + a + b = 30$$
$$da + b + c = 17.$$ Compute the value of $a + b + c + d$.
[b]p10.[/b] Define a sequence of numbers $a_{n+1} = \frac{(2+\sqrt3)a_n+1}{(2+\sqrt3)-a_{n}}$ for $n > 0$, and suppose that $a_1 = 2$. What is $a_{2016}$?
[u]Algebra Tiebreakers[/u]
[b]Tie 1.[/b] Mark takes a two digit number $x$ and forms another two digit number by reversing the digits of $x$. He then sums the two values, obtaining a value which is divisible by $13$. Compute the smallest possible value of $x$.
[b]Tie 2.[/b] Let $p(x) = x^4 - 10x^3 + cx^2 - 10x + 1$, where $c$ is a real number. Given that $p(x)$ has at least one real root, what is the maximum value of $c$?
[b]Tie 3.[/b] $x$ satisfies the equation $(1 + i)x^3 + 8ix^2 + (-8 + 8i)x + 36 = 0$. Compute the largest possible value of $|x|$.
PS. You should use hide for answers.
2016 ASMT, 9
In quadrilateral $ABCD$, $AC = 5$, $CD = 7$, and $AD = 3$. The angle bisector of $\angle CAD$ intersects $CD$ at $E$. If $\angle CBD = 60^o$ and $\angle AED = \angle BEC$, compute the value of $AE + BE$.
2016 ASMT, 2
Points $D$ and $E$ are chosen on the exterior of $\vartriangle ABC$ such that $\angle ADC = \angle BEC = 90^o$. If $\angle ACB = 40^o$, $AD = 7$, $CD = 24$, $CE = 15$, and $BE = 20$, what is the measure of $\angle ABC $ in,degrees?
2016 ASMT, 4
Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let$ BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$?
2016 ASMT, 10
Circle $\omega_1$ has diameter $AB$, and circle $\omega_2$ has center $A$ and intersects $\omega_1$ at points $C$ and $D$. Let $E$ be the intersection of $AB$ and $CD$. Point $P$ is chosen on $\omega_2$ such that $P C = 8$, $P D = 14$, and $P E = 7$. Find the length of $P B$.
2016 ASMT, T2
Let $ABCD$ be a square, and let $E$ be a point external to $ABCD$ such that $AE = CE = 9$ and $BE = 8$. Compute the side length of $ABCD$.
2016 ASMT, 5
Plane $A$ passes through the points $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Plane $B$ is parallel to plane $A$, but passes through the point $(1, 0, 1)$. Find the distance between planes $A$ and $B$.
2016 ASMT, T1
Let $ABC$ be a triangle with $\angle BAC = 75^o$ and $\angle ABC = 45^o$. If $BC =\sqrt3 + 1$, what is the perimeter of $\vartriangle ABC$?
2016 ASMT, Discrete
[u]Discrete Math Round[/u]
[b]p1.[/b] A class of six students has to split into two indistinguishable teams of three people. Compute the number of distinct team arrangements that can result.
[b]p2.[/b] What is the probability that a randomly chosen factor of $2016$ is a perfect square?
[b]p3.[/b] Compute the remainder when $$5\underbrace{666...6666}_{2016 \,\, sixes}5$$ is divided by $17$.
[b]p4.[/b] At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna is watching the conveyor belt, and has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportion of the M&Ms are red?
[b]p5.[/b] Three cards are chosen from a standard deck of $52$ without replacing them. Given that the ace of spades was chosen, what is the expected number of aces chosen?
[b]p6.[/b] Moor decides that he needs a new email address, and forms the address by taking some permutation of the $12$ letters $MMMOOOOOORRR$. How many permutations of the letters will contain $MOOR$ in this exact order at least once?
[b]p7.[/b] Suppose that the $8$ corners of a cube can be colored either red, green, or blue. We call a coloring of the cube rotationally symmetric if the cube can be rotated along a single axis parallel to an edge of a cube either $90^o$, $180^o$, or $270^o$, and reach the original coloring. How many rotationally symmetric colorings exist using the $3$ colors? Assume that any colorings which are identical after rotation are equivalent.
[b]p8.[/b] Let $x = \frac{1}{9} + \frac{1}{99} + \frac{1}{999} + ...+ \frac{1}{999999999}$ . Compute the number of digits in the first $3000$ decimal places of the base $10$ representation of $x$ which are greater than or equal to $8$.
[b]p9.[/b] Two $20$-sided dice are rolled. Their outcomes are independent and take uniformly distributed integer values from $1$ to $20$, inclusive. For each roll, let $x$ be (the sum of the dice) $\times $ (the positive difference of the dice). What is the expected value of $x$?
[b]p10.[/b] Compute $$\sum^{1000}_{a=1} \sum^{1000}_{b=1} \sum^{1000}_{c=1} \left\lfloor \frac{1000}{lcm (a, b, c)} \right \rfloor \phi (a) \phi (b) \phi(c)$$ where $\phi (n) = | \{k : 1 \le k \le n, gcd (k, n) = 1\} |$ counts the integers coprime to $n$ that are less than or equal to $n$.
[u]Discrete Math Tiebreaker[/u]
[b]Tie 1.[/b] A certain elementary school has $48$ students in the third grade that must be organized into three classes of $16$ students each. There are three troublemakers in the grade. If the students are assigned independently and randomly to classes, what is the probability that all three trou blemakers are assigned to the same $16$ student class?
[b]Tie 2.[/b] A $4$-digit number $x$ has the property that the expected value of the integer obtained from switching any two digits in $x$ is $4625$. Given that the sum of the digits of $x$ is $20$, compute $x$.
[b]Tie 3.[/b] Let $S$ be the set of factors of $10^5$. The number of subsets of $S$ with a least common multiple of $10^5$ can be written as $2^n * m$, where $n$ and $m$ are positive integers and $m$ is not divisible by $2$. Compute $m + n$.
PS. You should use hide for answers.
2016 ASMT, 7
A circle intersects the $y$-axis at two points $(0, a)$ and $(0, b)$ and is tangent to the line $x+100y = 100$ at $(100, 0)$. Compute the sum of all possible values of $ab - a - b$.
2016 ASMT, T3
An ellipse lies in the $xy$-plane and is tangent to both the $x$-axis and $y$-axis. Given that one of the foci is at $(9, 12)$, compute the minimum possible distance between the two foci.