geometry problems from Peruvian Mathematical Olympiads (phase 4, all 3 levels)

with aops links in the names

In a quadrilateral $ABCD$, a circle is inscribed that is tangent to the sides $AB, BC, CD$ and $DA$ at points $M, N, P$ and $Q$, respectively. If $(AM) (CP) = (BN) (DQ)$, prove that $ABCD$ is an cyclic quadrilateral.

2010 Peru L3 p3

Consider $A, B$ and $C$ three collinear points of the plane such that $B$ is between $A$ and $C$. Let $S$ be the circle of diameter $AB$ and $L$ a line that passes through $C$, which does not intersect $S$ and is not perpendicular to line $AC$. The points $M$ and $N$ are, respectively, the feet of the altitudes drawn from $A$ and $B$ on the line $L$. From $C$ draw the two tangent lines to $S$, where $P$ is the closest tangency point to $L$. Prove that the quadrilateral $MPBC$ is cyclic if and only if the lines $MB$ and $AN$ are perpendicular.

2011 Peru L3 p3

Let $ABC$ be a right triangle, right in $B$. Inner bisectors are drawn $CM$ and $AN$ that intersect in $I$. Then, the $AMIP$ and $CNIQ$ parallelograms are constructed. Let $U$ and $V$ are the midpoints of the segments $AC$ and $PQ$, respectively. Prove that $UV$ is perpendicular to $AC$.

2012 Peru L3 p4

In a circle $S$, a chord $AB$ is drawn and let $M$ be the midpoint of the arc $AB$. Let $P$ be a point in segment $AB$ other than its midpoint. The extension of the segment $MP$ cuts $S$ in $Q$. Let $S_1$ be the circle that is tangent to the AP segments and $MP$, and also is tangent to $S$, and let $S_2$ be the circle that is tangent to the segments $BP$ and $MP$, and also tangent to $S$. The common outer tangent lines to the circles $S_1$ and $S_2$ are cut at $C$. Prove that $\angle MQC = 90^o$.

Let $ABCDEF$ be a convex hexagon. The diagonal $AC$ is cut by $BF$ and $BD$ at points $P$ and $Q$, respectively. The diagonal $CE$ is cut by $DB$ and $DF$ at points $R$ and $S$, respectively. The diagonal $EA$ is cut by $FD$ and $FB$ at points $T$ and $U$, respectively. It is known that each of the seven triangles $APB, PBQ, QBC, CRD, DRS, DSE$ and $AUF$ has area $1$. Find the area of the hexagon $ABCDEF$.

Let $A, B, C, D$ be points in a line $l$ in this order where $AB = BC$ and $AC = CD$. Let $w$ be a circle that passes in the points $B$ and $D$, a line that passes by $A$ intersects $w$ in the points $P$ and $Q$(the point $Q$ is in the segment $AP$). Let $M$ be the midpoint of $PD$ and $R$ is the symmetric of $Q$ by the line $l$, suppose that the segments $PR$ and $MB$ intersect in the point $N$. Prove that the quadrilateral $PMNC$ is cyclic.

Level 2

2017 Peru L2 p3

In a right triangle $ABC$, right in $B$, $M$ is the midpoint of the $AC$ side. Be $C_1$ the excircle of the triangle $ABM$ , opposite to the vertex $B$. Let $C_2$ be the excircle of the triangle $MBC$, opposite to vertex $B$. Prove that there is a perpendicular line a $AC$ that is tangent to $C_1$ and $C_2$.

2018 Peru L2 p1

Level 1

2015 Peru L1 p3

Ada drew a triangle, chose a point on each side and chose a point $P$ on the inside of the triangle. Then, drew segments that join $P$ with the other six points (the three vertices and the three points that are on the sides). In this way the initial triangle was divided into six isosceles triangles. Show, by example, how Ada could have achieved this.

2018 Peru L1 p2

a) Show that a paper triangle whose interior angles measure $100^o, 60^o$ and $20^o$ can be divide into two isosceles triangles by a straight cut.

b) Show that a paper triangle whose interior angles measure $100^o, 50^o$ and $30^o$ can be divided into three isosceles triangles by straight cuts

source: https://onemperu.wordpress.com

thanks to Jorge Tipe for his help

with aops links in the names

Olimpiada Nacional Escolar de Matemática (ONEM)

2004 -2018

Level 3

Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.

2005 Peru L3 p3

Let $A,B,C,D$, be four different points on a line $\ell$, so that $AB=BC=CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points of the plane be such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the angle $\angle MBN$.

Let $A,B,C,D$, be four different points on a line $\ell$, so that $AB=BC=CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points of the plane be such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the angle $\angle MBN$.

2007 Peru L3 p4

Let $ABCD$ be rhombus $ABCD$ where the triangles $ABD$ and $BCD$ are equilateral. Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, such that $\angle MAN = 30^o$. Let $X$ be the intersection point of the diagonals $AC$ and $BD$. Prove that $\angle XMN = \angle\ DAM$ and $\angle XNM = \angle BAN$.

Let $ABCD$ be rhombus $ABCD$ where the triangles $ABD$ and $BCD$ are equilateral. Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, such that $\angle MAN = 30^o$. Let $X$ be the intersection point of the diagonals $AC$ and $BD$. Prove that $\angle XMN = \angle\ DAM$ and $\angle XNM = \angle BAN$.

2008 Peru L3 p3

$ABC$ is an acute triangle with $\angle ACB = 45^o$. Let $D$ and $E$ be points on the sides $BC$ and $AC$, respectively, such that $AB = AD = BE$. Let $M,N$ and $X$ be the midpoints of $BD, AE$ and $AB$, respectively. Let lines $AM$ and $BN$ intersect at point $P$. Show that lines $XP$ and $DE$ are perpendicular.

2009 Peru L3 p2$ABC$ is an acute triangle with $\angle ACB = 45^o$. Let $D$ and $E$ be points on the sides $BC$ and $AC$, respectively, such that $AB = AD = BE$. Let $M,N$ and $X$ be the midpoints of $BD, AE$ and $AB$, respectively. Let lines $AM$ and $BN$ intersect at point $P$. Show that lines $XP$ and $DE$ are perpendicular.

In a quadrilateral $ABCD$, a circle is inscribed that is tangent to the sides $AB, BC, CD$ and $DA$ at points $M, N, P$ and $Q$, respectively. If $(AM) (CP) = (BN) (DQ)$, prove that $ABCD$ is an cyclic quadrilateral.

2010 Peru L3 p3

Consider $A, B$ and $C$ three collinear points of the plane such that $B$ is between $A$ and $C$. Let $S$ be the circle of diameter $AB$ and $L$ a line that passes through $C$, which does not intersect $S$ and is not perpendicular to line $AC$. The points $M$ and $N$ are, respectively, the feet of the altitudes drawn from $A$ and $B$ on the line $L$. From $C$ draw the two tangent lines to $S$, where $P$ is the closest tangency point to $L$. Prove that the quadrilateral $MPBC$ is cyclic if and only if the lines $MB$ and $AN$ are perpendicular.

2011 Peru L3 p3

Let $ABC$ be a right triangle, right in $B$. Inner bisectors are drawn $CM$ and $AN$ that intersect in $I$. Then, the $AMIP$ and $CNIQ$ parallelograms are constructed. Let $U$ and $V$ are the midpoints of the segments $AC$ and $PQ$, respectively. Prove that $UV$ is perpendicular to $AC$.

2012 Peru L3 p4

In a circle $S$, a chord $AB$ is drawn and let $M$ be the midpoint of the arc $AB$. Let $P$ be a point in segment $AB$ other than its midpoint. The extension of the segment $MP$ cuts $S$ in $Q$. Let $S_1$ be the circle that is tangent to the AP segments and $MP$, and also is tangent to $S$, and let $S_2$ be the circle that is tangent to the segments $BP$ and $MP$, and also tangent to $S$. The common outer tangent lines to the circles $S_1$ and $S_2$ are cut at $C$. Prove that $\angle MQC = 90^o$.

Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$. Find the measure of the angle $\angle PBC$ .

Let $ABC$ be an acute triangle with circumcenter $O$, on the sides $BC, CA$ and $AB$ they take the points $D, E$ and $F$, respectively, in such a way that $BDEF$ is a parallelogram. Supposing that $DF^2 = AE\cdot EC <\frac{AC^2}{4}$ show that the circles circumscribed to the triangles $FBD$ and $AOC$ are tangent.
Let $ABCD$ be a trapezoid of parallel bases $ BC$ and $AD$. If $\angle CAD = 2\angle CAB, BC = CD$ and $AC = AD$, determine all the possible values of the measure of the angle $\angle CAB$.

Let $ABC$ be an acute triangle such that $BA = BC$. On the sides $BA$ and $BC$ points $D$ and $E$ are chosen respectively, such that $DE$ and $AC$ are parallel. Let $H$ be the orthocenter of the triangle $DBE$ and $M$ be the midpoint of $AE$. If $\angle HMC = 90^o$, determine the measure of angle $\angle ABC$.

2012 Peru L2 p1

In a right triangle $ABC$ (right in $B$) its inscribed circle has been drawn, which is tangent to the side $AB$ at $D$, to the side $BC$ at $E$ and to the side $AC$ at $F$. If $\angle FDC = 2 \angle DCB$, show that $AF = BC$.

Let $P$ be an interior point of a triangle ABC such that $\angle PAB = \angle PCA = \angle PBC - 60^o$ and $PC = BC =\frac{AB}{\sqrt2}$ . Find the measure of the angle $\angle PAB$.In a right triangle $ABC$ (right in $B$) its inscribed circle has been drawn, which is tangent to the side $AB$ at $D$, to the side $BC$ at $E$ and to the side $AC$ at $F$. If $\angle FDC = 2 \angle DCB$, show that $AF = BC$.

Given a quadrilateral $ABCD$ such that $AB = AD, \angle CBD + \angle ABC = \angle ADB + \angle ADC = 180^o$ and $\angle BAD> 60^o$. Let $M$ be any point of segment $AB$ ($M \ne A$ and $M \ne B$).

a) Prove that there is an $N$ point in the $CD$ segment such that $BM = DN$ and a point $X$ in segment $BC$ such that $MX = XN$.

b) Prove that the measure of the angle $\angle XAN$ is always the same no matter what the point $M$.

2015 Peru L2 p3

Let $ABCD$ be a trapezoid with parallel sides $AD$ and $BC$, circumscribed around a circle of center $O$, which is tangent to $BC$ at point $E$. Prove that if $AD = 2BC$, then $O$ is the orthocenter of the triangle $AED$.

a) Prove that there is an $N$ point in the $CD$ segment such that $BM = DN$ and a point $X$ in segment $BC$ such that $MX = XN$.

b) Prove that the measure of the angle $\angle XAN$ is always the same no matter what the point $M$.

2015 Peru L2 p3

Let $ABCD$ be a trapezoid with parallel sides $AD$ and $BC$, circumscribed around a circle of center $O$, which is tangent to $BC$ at point $E$. Prove that if $AD = 2BC$, then $O$ is the orthocenter of the triangle $AED$.

2017 Peru L2 p3

In a right triangle $ABC$, right in $B$, $M$ is the midpoint of the $AC$ side. Be $C_1$ the excircle of the triangle $ABM$ , opposite to the vertex $B$. Let $C_2$ be the excircle of the triangle $MBC$, opposite to vertex $B$. Prove that there is a perpendicular line a $AC$ that is tangent to $C_1$ and $C_2$.

Let $ABC$ be a triangle and let $D, E$ and $F$ be points on the sides $BC, CA$ and $AB$, respectively, such that $DE$ is perpendicular to $AC$ and $\angle BAC = 2\angle BFD$. If $AE = EC + BD$ and $CD = DB + AF$, prove that triangle $ABC$ is equilateral.

Level 1

2015 Peru L1 p3

Let $ABCD$ be a parallelogram, $E$ a point of the segment $BD$, and $F$ be a point of the segment $AD$, such that $BC = CE = ED = EF$. If it is true that $AB = AF + 2BE$, calculate the measure of angle $\angle BAD$.

2016 Peru L1 p2Ada drew a triangle, chose a point on each side and chose a point $P$ on the inside of the triangle. Then, drew segments that join $P$ with the other six points (the three vertices and the three points that are on the sides). In this way the initial triangle was divided into six isosceles triangles. Show, by example, how Ada could have achieved this.

Let $D$ and $E$ be points on the sides $AC$ and $BC$ of a triangle $ABC$, respectively, such that $AB = BD = DE = EC$. If the triangle $ABD$ and $DEC$ have the same area, find the measure of angle $\angle DBC$.

a) Show that a paper triangle whose interior angles measure $100^o, 60^o$ and $20^o$ can be divide into two isosceles triangles by a straight cut.

b) Show that a paper triangle whose interior angles measure $100^o, 50^o$ and $30^o$ can be divided into three isosceles triangles by straight cuts

thanks to Jorge Tipe for his help

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