Olympiad problems

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]

2008 SDMO (Middle School), 3

Tags: geometry , NEEDS DIAGRAM , SDMO Middle School , 2008 SDMO

In the diagram, $AD:DB=1:1$, $BE:EC=1:2$, and $CF:FA=1:3$. If the area of triangle $ABC$ is $120$, then find the area of triangle $DEF$. (will insert image here later)

2011-2012 SDML (High School), 4

Tags: geometry , NEEDS DIAGRAM , 2011-12 SDML Round 2 , SDML High School

In triangle $ABC$, $AB=3$, $AC=5$, and $BC=4$. Let $P$ be a point inside triangle $ABC$, and let $D$, $E$, and $F$ be the projections of $P$ onto sides $BC$, $AC$, and $AB$, respectively. If $PD:PE:PF=1:1:2$, then find the area of triangle $DEF$. (Express your answer as a reduced fraction.) (will insert image here later)

2012-2013 SDML (Middle School), 11

Tags: geometry , 3D geometry , NEEDS DIAGRAM , SDML Middle School , 2012-13 SDML Round 2

Six different-sized cubes are glued together, one on top of the other. The bottom cube has edge length $8$. Each of the other cubes has four vertices at the midpoints of the edges of the cube below it as shown. The entire solid is then dipped in red paint. What is the total area of the red-painted surface on the solid? (will insert image here later) $\text{(A) }630\qquad\text{(B) }632\qquad\text{(C) }648\qquad\text{(D) }694\qquad\text{(E) }756$

2016-2017 SDML (Middle School), 1

Tags: SDML Middle School , 2016-17 SDML Round 2 , NEEDS DIAGRAM

A "domino" is made up of two small squares: [asy] unitsize(10); draw((0,0) -- (2,0) -- (2,1) -- (0,1) -- cycle); fill((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle); [/asy] Which of the "checkerboards" illustrated below CANNOT be covered exactly and completely by a whole number of non-overlapping dominoes? [diagram requires in-line asy]

2018-2019 SDML (High School), 5

Tags: NEEDS DIAGRAM , 2018-2019 Round 2 , SDML High School

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), 4

Tags: 2018-2019 Round 2 , SDML High School , NEEDS DIAGRAM

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), 8

Tags: NEEDS DIAGRAM , SDML High School , 2018-2019 Round 2 , geometry , rhombus

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

Tags: NEEDS DIAGRAM , SDML High School , 2018-2019 Round 2 , geometry , 3D geometry

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$