### May Olympiad 1995 - 2019 levels 1-2 (Mayo) 54p

[a  Junior Competition]
with aops links in the names

1995 - 2019

level 2 (<=15yo)

Consider a pyramid whose base is an equilateral triangle $BCD$ and whose other faces are triangles isosceles, right at the common vertex $A$. An ant leaves the vertex $B$ arrives at a point $P$ of the $CD$ edge, from there goes to a point $Q$ of the edge $AC$ and returns to point $B$. If the path you made is minimal, how much is the angle $PQA$ ?

Let $ABCD$ be a rectangle. A line $r$ moves parallel to $AB$ and intersects diagonal $AC$ , forming two triangles opposite the vertex, inside the rectangle.  Prove that the sum of the areas of these triangles is minimal when $r$ passes through the midpoint of segment $AD$ .

Let $ABCD$ be a square and let point $F$ be any point on side $BC$. Let the line perpendicular to $DF$, that passes through $B$, intersect line $DC$ at $Q$. What is value of $\angle FQC$?

In a square $ABCD$ with side $k$, let $P$ and $Q$ in $BC$ and $DC$ respectively, where $PC = 3PB$ and $QD = 2QC$. Let $M$ be the point of intersection of the lines $AQ$ and $PD$, determine the area of $QMD$ in function of $k$
Let $ABC$ be an equilateral triangle. $N$ is a point on the  side $AC$ such that $\vec{AC} = 7\vec{AN}$, $M$ is a point on the side  $AB$  such that $MN$ is parallel to $BC$ and $P$ is a point on the side $BC$  such that $MP$ is parallel to $AC$. Find the ratio of areas $\frac{ (MNP)}{(ABC)}$

In a unit circle where $O$ is your circumcenter, let $A$ and $B$ points in the circle with $\angle BOA = 90$. In the arc $AB$(minor arc) we have the points $P$ and $Q$ such that $PQ$ is parallel to $AB$. Let $X$ and $Y$ be the points of intersections of the line $PQ$ with $OA$ and $OB$ respectively. Find the value of $PX^2 + PY^2$

Let $ABC$ be an equilateral triangle. $M$ is the midpoint of segment $AB$ and $N$ is the midpoint of segment $BC$. Let $P$ be the point outside $ABC$ such that the triangle $ACP$ is isosceles and right in $P$. $PM$ and $AN$ are cut in $I$. Prove that $CI$ is the bisector of the angle $MCA$ .

Given a parallelogram with area $1$ and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the $8$ lines formed we have  a octagon inside of the parallelogram. Determine the area of this octagon.

Let $S$ be a circle with radius $2$, let $S_1$ be a circle,with radius $1$ and tangent, internally to $S$ in $B$ and let $S_2$ be a circle, with radius $1$ and tangent to $S_1$ in $A$, but $S_2$ isn't tangent to $S$. If $K$ is the point of intersection of the line $AB$ and the circle $S$, prove that $K$ is in the circle $S_2$

On the trapezoid $ABCD$ , side $DA$  is perpendicular to the bases $AB$ and $CD$ . The base $AB$ measures $45$, the base $CD$ measures $20$ and the $BC$ side measures $65$. Let $P$ on the $BC$ side such that $BP$ measures $45$ and $M$ is the midpoint of $DA$. Calculate the measure of the $PM$ segment.

In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$  be the point on the extension of $BA$ such that the triangle $ADE$  is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$  and $Q$ the point of  intersection of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

Let $ABCD$ be a rectangle of sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from  $A$ cuts $BD$ at point $H$. We call $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the measure of the  segment $MN$.

An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.

We have a pool table $8$ meters long and $2$ meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling $29$ meters, it stops at a corner of the table. How many times did the ball hit the edges of the table?

Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.

In a triangle $ABC$ with $AB = AC$, let $M$ be the midpoint of $CB$ and let $D$ be a point in $BC$ such that $\angle BAD = \frac{\angle BAC}{6}$. The perpendicular line to $AD$ by $C$ intersects $AD$ in $N$ where $DN = DM$. Find the angles of the triangle $BAC$

Let $ABCD$ be a trapezoid of $AB$ and $CD$ bases. Let $O$ be the point of intersection of your diagonals $AC$ and $BD$. If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$, calculate the area of the triangle $BCO$.

In the triangle $ABC$ we have $\angle A = 2\angle C$ and $2\angle B = \angle A + \angle C$. The angle bisector of $\angle C$ intersects the segment $AB$ in $E$, let $F$ be the midpoint of $AE$, let $AD$ be the altitude of the triangle $ABC$. The perpendicular bisector of $DF$ intersects $AC$ in $M$. Prove that $AM = CM$

Let $ABCD$ be a rectangle  and  $P$ be a point on the side$AD$ such that $\angle BPC = 90^o$. The perpendicular from  $A$ on $BP$ cuts $BP$ at  $M$ and the perpendicular from $D$ on  $CP$ cuts $CP$ in $N$. Show that the center of the rectangle lies in the $MN$ segment.

Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.

Let $ABCD$ be a rectangle and the circle of center $D$ and radius $DA$, which cuts the extension of the side $AD$ at point $P$. Line $PC$ cuts the circle at point $Q$ and the extension of the side $AB$ at point $R$. Show that $QB = BR$.

In a right triangle rectangle $ABC$ such that $AB = AC$, $M$ is the midpoint of  $BC$. Let $P$ be a point on the perpendicular bisector of  $AC$, lying in the semi-plane determined by $BC$ that does not contain $A$. Lines $CP$ and $AM$ intersect at $Q$. Calculate the angles that form the lines $AP$ and $BQ$.

Given Triangle $ABC$, $\angle B= 2 \angle C$, and $\angle A>90^\circ$. Let $M$ be midpoint of $BC$. Perpendicular of $AC$ at $C$ intersects $AB$ at $D$. Show $\angle AMB = \angle DMC$

Construct the midpoint of a segment using an unmarked ruler and a trisector that marks in a segment the two points that divide the segment in three equal parts.

In a convex quadrilateral $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. If $MP$ and $NQ$ divide $ABCD$ in four quadrilaterals with the same area, prove that $ABCD$ is a parallelogram.

Let $ABCDEFGHI$ be a regular polygon of $9$ sides. The segments $AE$ and $DF$ intersect at $P$. Prove that $PG$ and $AF$ are perpendicular.

In a triangle $ABC$, let $D$ and $E$ be points of the sides $BC$ and $AC$ respectively. Segments $AD$ and $BE$ intersect at $O$. Suppose that the line connecting midpoints of the triangle and parallel to $AB$, bisects the segment $DE$. Prove that the triangle $ABO$ and the quadrilateral $ODCE$ have equal areas.

Let $ABCD$ be a quadrilateral such that $\angle ABC = \angle ADC = 90º$ and $\angle BCD$ > $90º$. Let $P$ be a point inside of the $ABCD$ such that $BCDP$ is parallelogram, the line $AP$ intersects $BC$ in $M$. If $BM = 2, MC = 5, CD = 3$. Find the length of $AM$.

In a parallelogram $ABCD$, let $M$ be the point on the $BC$ side such that $MC = 2BM$ and let $N$ be the point of side  $CD$ such that $NC = 2DN$. If the distance from point $B$ to the line $AM$ is $3$, calculate the distance from point $N$ to the line $AM$.

On the sides $AB, BC$ and $CA$ of a triangle $ABC$ are located the points $P, Q$ and $R$ respectively, such that $BQ = 2QC, CR = 2RA$ and $\angle PRQ = 90^o$. Show that $\angle APR =\angle RPQ$.

Level 1 (<=13yo)

We have four white equilateral triangles of 3 cm on each side and join them by their sides to obtain a triangular base pir'amide. At each edge of the pyramid we mark two red dots that divide it into three equal parts. Number the red dots, so that when you scroll them in the order they were numbered, result a path with the smallest possible perimeter. How much does that path measure?

A terrain ( $ABCD$ ) has a rectangular trapezoidal shape.  The angle in $A$ measures $90^o$. $AB$ measures $30$ m, $AD$ measures $20$ m and $DC$ measures 45 m. This land must be divided into two areas of the same area, drawing a parallel to the $AD$ side . At what distance from $D$ do we have to draw the parallel?
In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$, calculate the area of the rectangle $ABCD$.
$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles $DAF$ and $DCE$ have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle.
In a parallelogram $ABCD$ , $BD$ is the largest diagonal. By matching $B$ with $D$ by a bend, a regular pentagon is formed.  Calculate the measures of the angles formed by the diagonal $BD$ with each of the sides of the parallelogram.

Let $ABC$ be a right triangle in $A$ , whose leg measures $1$ cm. The bisector of the angle $BAC$ cuts the hypotenuse in $R$, the perpendicular to $AR$ on $R$ , cuts the side $AB$ at its midpoint. Find the measurement of the side  $AB$ .

Let's take a $ABCD$ rectangle of paper; the side $AB$  measures $5$ cm and the side $BC$  measures $9$ cm. We do three folds:
1.We take the $AB$ side on the $BC$ side and call $P$ the point on the $BC$ side that coincides with $A$. A right trapezoid $BCDQ$ is then formed.
2. We fold so that $B$ and $Q$ coincide. A $5$-sided polygon $RPCDQ$ is formed.
3. We fold again by matching $D$ with $C$ and $Q$ with $P$. A new right trapezoid $RPCS$.
After making these folds, we make a cut perpendicular to $SC$ by its midpoint $T$, obtaining the right trapezoid $RUTS$. Calculate the area of the figure that appears as we unfold the last trapezoid $RUTS$.

A rectangular sheet of paper (white on one side and gray on the other) was folded three times, as shown in the figure:

Rectangle $1$, which was white after the first fold, has $20$ cm more perimeter than rectangle $2$, which was white after the second fold, and this in turn has $16$ cm more perimeter than rectangle $3$, which was white after the third fold. Determine the area of the sheet.

The triangle $ABC$ is right in $A$ and $R$ is the midpoint of the hypotenuse $BC$ . On the major leg $AB$ the point $P$ is marked such that $CP = BP$ and on the segment $BP$ the point $Q$ is marked such that the triangle $PQR$ is equilateral.  If the area of triangle $ABC$ is $27$, calculate the area of triangle $PQR$ .

In a square $ABCD$ of diagonals $AC$ and $BD$, we call $O$ at the center of the square. A square $PQRS$ is constructed with sides parallel to those of $ABCD$ with $P$ in segment $AO, Q$ in segment $BO, R$ in segment $CO, S$ in segment $DO$.  If area of $ABCD$ equals two times the  area of $PQRS$,  and $M$ is the midpoint of the $AB$ side, calculate the measure of the angle $\angle AMP$.

There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.

A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate your area.

You have a paper pentagon, $ABCDE$, such that $AB = BC = 3$ cm, $CD = DE= 5$ cm, $EA = 4$ cm, $\angle ABC = 100^o , \angle CDE = 80^o$. You have to divide the pentagon into four triangles, by three straight cuts, so that with the four triangles assemble a rectangle, without gaps or overlays. (The triangles can be rotated  and / or turned around.)

Let $ABF$ be a right-angled triangle with $\angle AFB = 90$, a square $ABCD$ is externally to the triangle. If $FA = 6$, $FB = 8$ and $E$ is the circumcenter of the square $ABCD$, determine the value of $EF$

Three circumferences are tangent to each other, as shown in the figure. The region of the outer circle that is not covered by the two inner circles has an area equal to $2$ p. Determine the length of the $PQ$ segment
A closed container in the shape of a rectangular parallelepiped contains $1$ liter of water. If the container rests horizontally on three different sides, the water level is $2$ cm, $4$ cm and $5$ cm. Calculate the volume of the parallelepiped.

In the rectangle $ABCD, BC = 5, EC = 1/3 CD$ and $F$ is the point where $AE$ and $BD$ are cut. The triangle $DFE$ has area $12$ and the triangle $ABF$ has area $27$. Find the area of the quadrilateral $BCEF$ .
From a paper quadrilateral like the one in the figure, you have to cut out a new quadrilateral whose area is equal to half the area of the original quadrilateral.You can only bend one or more times and cut by some of the lines of the folds. Describe the folds and cuts and justify that the area is half.
Let $ABCD$ be a square of side paper $10$ and $P$ a point on side $BC$. By folding the paper along the $AP$ line, point $B$ determines the point $Q$, as seen in the figure. The line $PQ$ cuts the side $CD$  at $R$. Calculate the perimeter of the$PCR$ triangle.
Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint  of $AC$. Let $P$ be such that $MNP$ is an equilateral triangle with $P$ inside the quadrilateral $MBCN$.  Calculate the measure of $\angle CAP$

In the quadrilateral $ABCD$, we have $\angle C$ is triple of $\angle A$, let $P$ be a point in the side $AB$ such that  $\angle DPA = 90º$ and let $Q$ be a point in the segment $DA$ where $\angle BQA = 90º$ the segments $DP$ and $CQ$ intersects in $O$ such that $BO = CO = DO$, find $\angle A$ and $\angle C$.

In a triangle $ABC$, let $D$ and $E$ point in the sides $BC$ and $AC$ respectively. The segments $AD$ and $BE$ intersects in $O$, let $r$ be line (parallel to $AB$) such that $r$  intersects $DE$ in your midpoint, show that the triangle $ABO$ and the quadrilateral $ODCE$ have the same area.

Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$. On the side $AB$ construct the rhombus  $BAFC$ outside $ABCD$ and such that the side $AF$  is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$.
Let $ABCDEFGHIJ$ be a regular $10$-sided polygon that has all its vertices in one circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the measure of the  segment $PQ$.