Found problems: 23
2018-2019 SDML (High School), 6
For how many integers $n$, with $2 \leq n \leq 80$, is $\frac{(n-1)n(n+1)}{8}$ equal to an integer?
$ \mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 20 \qquad \mathrm {(C) \ } 39 \qquad \mathrm{(D) \ } 49 \qquad \mathrm{(E) \ } 59$
2018-2019 SDML (High School), 7
In a game of Shipbattle, Willis secretly places his aircraft carrier somewhere in a $9 \times 9$ grid, represented by five consecutive squares. Two example positions are shown below.
[asy]
size(5cm);
fill((2,7)--(7,7)--(7,8)--(2,8)--cycle);
fill((5,1)--(6,1)--(6,6)--(5,6)--cycle);
for (int i = 0; i <= 9; ++i)
{
draw((i,0)--(i,9));
draw((0,i)--(9,i));
}
[/asy]
Phyllis then takes shots at the grid, one square at a time, trying to hit Willis's aircraft carrier. What is the minimum number of shots that Phyllis must take to ensure that she hits the aircraft carrier at least once?
2018-2019 SDML (High School), 10
If $s$ and $d$ are positive integers such that $\frac{1}{s} + \frac{1}{2s} + \frac{1}{3s} = \frac{1}{d^2 - 2d},$ then the smallest possible value of $s + d$ is
$ \mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 8 \qquad \mathrm {(C) \ } 10 \qquad \mathrm{(D) \ } 50 \qquad \mathrm{(E) \ } 96$
2018-2019 SDML (High School), 3
In the diagram below, $\angle B = 43^\circ$ and $\angle D = 102^\circ$. Find $\angle A + \angle B + \angle C + \angle D + \angle E + \angle F$.
[NEEDS DIAGRAM]
2018-2019 SDML (High School), 7
Given $A = \left\{1,2,3,5,8,13,21,34,55\right\}$, how many of the numbers between $3$ and $89$ cannot be written as the sum of two elements of set $A$?
$ \mathrm{(A) \ } 34 \qquad \mathrm{(B) \ } 35 \qquad \mathrm {(C) \ } 43\qquad \mathrm{(D) \ } 51 \qquad \mathrm{(E) \ } 55$
2018-2019 SDML (High School), 15
Pentagon $ABCDE$ is such that all five diagonals $AC, BD, CE, DA,$ and $EB$ lie entirely within the pentagon. If the area of each of the triangles $ABC, BCD, CDE,$ and $DEA$ is equal to $1$ and the area of triangle $EAB$ is equal to $2$, the area of the pentagon $ABCDE$ is closest to
$ \mathrm{(A) \ } 4.42 \qquad \mathrm{(B) \ } 4.44 \qquad \mathrm {(C) \ } 4.46 \qquad \mathrm{(D) \ } 4.48 \qquad \mathrm{(E) \ } 4.5$
2018-2019 SDML (High School), 4
How many $3$-element subsets of $\left\{1, 2, 3, \dots, 11\right\}$ are there, such that the sum of the three elements is a multiple of $3$?
2018-2019 SDML (High School), 1
Find the remainder when $1! + 2! + 3! + \dots + 1000!$ is divided by $9$.
2018-2019 SDML (High School), 5
Let $f(x) = x^2 + ax + b$, where $a$ and $b$ are real numbers. If $f(f(1)) = f(f(2)) = 0$, then find $f(0)$.
2018-2019 SDML (High School), 11
For the system of equations $x^2 + x^2y^2 + x^2y^4 = 525$ and $x + xy + xy^2 = 35$, the sum of the real $y$ values that satisfy the equations is
$ \mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } \frac{5}{2} \qquad \mathrm {(C) \ } 5 \qquad \mathrm{(D) \ } 20 \qquad \mathrm{(E) \ } \frac{55}{2}$
2018-2019 SDML (High School), 12
How many ordered pairs $(s, d)$ of positive integers with $4 \leq s \leq d \leq 2019$ are there such that when $s$ silver balls and $d$ diamond balls are randomly arranged in a row, the probability that the balls on each end have the same color is $\frac{1}{2}$?
$ \mathrm{(A) \ } 58 \qquad \mathrm{(B) \ } 59 \qquad \mathrm {(C) \ } 60 \qquad \mathrm{(D) \ } 61 \qquad \mathrm{(E) \ } 62$
2018-2019 SDML (High School), 2
Given that $\frac{x}{\sqrt{x} + \sqrt{y}} = 18$ and $\frac{y}{\sqrt{x} + \sqrt{y}} = 2$, find $\sqrt{x} - \sqrt{y}$.
2018-2019 SDML (High School), 1
Seven children, each with the same birthday, were born in seven consecutive years. The sum of the ages of the youngest three children in $42$. What is the sum of the ages of the oldest three?
$ \mathrm{(A) \ } 51 \qquad \mathrm{(B) \ } 54 \qquad \mathrm {(C) \ } 57 \qquad \mathrm{(D) \ } 60 \qquad \mathrm{(E) \ } 63$
2018-2019 SDML (High School), 5
The graph of the equation $y = ax^2 + bx + c$ is shown in the diagram. Which of the following must be positive?
[DIAGRAM NEEDED]
$ \mathrm{(A) \ } a \qquad \mathrm{(B) \ } ab^2 \qquad \mathrm {(C) \ } b - c \qquad \mathrm{(D) \ } bc \qquad \mathrm{(E) \ } c - a$
2018-2019 SDML (High School), 3
How many three-digit positive integers $x$ are there with the property that $x$ and $2x$ have only even digits? (One such number is $x = 220$, since $2x = 440$ and each of $x$ and $2x$ has only even digits.)
$ \mathrm{(A) \ } 16 \qquad \mathrm{(B) \ } 18 \qquad \mathrm {(C) \ } 64 \qquad \mathrm{(D) \ } 100 \qquad \mathrm{(E) \ } 125$
2018-2019 SDML (High School), 14
A square array of dots with $7$ rows and $7$ columns is given. Each dot is colored either blue or red. Whenever two dots of the same color are adjacent in the same row or column, they are joined by a line segment of the same color as the dots. If they are adjacent but of difference colors, they are then joined by a purple line segment. There are $20$ red line segments and $19$ blue line segments. Find the positive difference between the maximum and minimum number of red dots.
[asy]
size(4cm);
for (int i = 0; i <= 7; ++i) {
for (int j = 0; j <= 7; ++j) {
dot((i,j));
}
}
[/asy]
$ \mathrm{(A) \ } 4 \qquad \mathrm{(B) \ } 5 \qquad \mathrm {(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ } 8$
2018-2019 SDML (High School), 4
A beam of light shines from point $L$, reflects off a reflector at point $S$, and reaches point $D$ so that $\overline{SD}$ is perpendicular to $\overline{ML}$. Then $x$ is
[DIAGRAM NEEDED]
$ \mathrm{(A) \ } 13^\circ \qquad \mathrm{(B) \ } 26^\circ \qquad \mathrm {(C) \ } 32^\circ \qquad \mathrm{(D) \ } 58^\circ \qquad \mathrm{(E) \ } 64^\circ$
2018-2019 SDML (High School), 2
When a positive integer $N$ is divided by $60$, the remainder is $49$. When $N$ is divided by $15$, the remainder is
$ \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 3 \qquad \mathrm {(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ } 8$
2018-2019 SDML (High School), 8
Five consecutive positive integers have the property that the sum of the second, third, and fourth is a perfect square, while the sum of all five is a perfect cube. If $m$ is the first of these five integers, then the minimum possible value of $m$ satisfies
$ \mathrm{(A) \ } m \leq 200 \qquad \mathrm{(B) \ } 200 < m \leq 400 \qquad \mathrm {(C) \ } 400 < m \leq 600 \qquad \mathrm{(D) \ } 600 < m \leq 800 \qquad \mathrm{(E) \ } 800 < m$
2018-2019 SDML (High School), 9
Triangle $ABC$ is isosceles with $AB + AC$ and $BC = 65$ cm. $P$ is a point on $\overline{BC}$ such that the perpendicular distances from $P$ to $\overline{AB}$ and $\overline{AC}$ are $24$ cm and $36$ cm, respectively. The area of $\triangle ABC$, in cm$^2$, is
$ \mathrm{(A) \ } 1254 \qquad \mathrm{(B) \ } 1640 \qquad \mathrm {(C) \ } 1950 \qquad \mathrm{(D) \ } 2535 \qquad \mathrm{(E) \ } 2942$
2018-2019 SDML (High School), 8
The figure below consists of five isosceles triangles and ten rhombi. The bases of the isosceles triangles are $12$, $13$, $14$, $15$, as shown below. The top rhombus, which is shaded, is actually a square. Find the area of this square.
[DIAGRAM NEEDED]
2018-2019 SDML (High School), 13
A steel cube has edges of length $3$ cm, and a cone has a diameter of $8$ cm and a height of $24$ cm. The cube is placed in the cone so that one of its interior diagonals coincides with the axis of the cone. What is the distance, in cm, between the vertex of the cone and the closest vertex of the cube?
[NEEDS DIAGRAM]
$ \mathrm{(A) \ } \frac{12\sqrt6-3\sqrt3}{4} \qquad \mathrm{(B) \ } \frac{9\sqrt6-3\sqrt3}{2} \qquad \mathrm {(C) \ } 5\sqrt3 \qquad \mathrm{(D) \ } 6\sqrt6 - \sqrt3 \qquad \mathrm{(E) \ } 6\sqrt6$
2018-2019 SDML (High School), 6
Find the largest prime $p$ less than $210$ such that the number $210 - p$ is composite.