geometry problems from Peruvian Mathematical Olympiads (phase 4, all 3 levels)
with aops links in the names
collected inside aops: here
2009 Peru L3 p2
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.
2009 Peru L3 p3
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
2019 Peru L2 p1
Let $ABCDEF$ be a regular hexagon, $C_1$ be the circle of diameter $AF$ and $C_2$ the circle of center $E$ and radius $EF$. Circles $C_1$ and $C_2$ intersect at points $F$ and $P$. $AP$ intersects $ED$ at $Q$. Determine the ratio of segments $DQ$ and $QE$.
with aops links in the names
Olimpiada Nacional Escolar de Matemática (ONEM)
collected inside aops: here
2020 was cancelled
2004 -2021
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 $\angle MBN$.
Find all values of $k$ by which it is possible to divide any triangular region in $k$ quadrilaterals of equal area.
Assuming that each point of a straight line is painted red or blue, arbitrarily, show that it is always possible to choose three points $A, B$ and $C$ in such a way straight, that are painted the same color and that: $$\frac{AB}{1}=\frac{BC}{2}=\frac{AC}{3}.$$
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.
$ABC$ 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.
All the points of the plane that have both integer coordinates are painted, using the colors red, green and yellow. If the points are painted so that there is at least one point of each color, proves that there are always three points $X,Y$, and $Z$, of different colors, such that $\angle XYZ = 45^o$.
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.
2009 Peru L3 p3
a) On a circumference $8$ points are marked. We say that Juliana does an “T-operration ” if she chooses three of these points and paint the sides of the triangle that they determine, so that each painted triangle has at most one vertex in common with a painted triangle previously. What is the greatest number of “T-operations” that Juliana can do?
b) If in part (a), instead of considering $8$ points, $7$ points are considered, what is the greatest number of “T operations” that Juliana can do?
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.
2010 Peru L3 p4
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.
2010 Peru L3 p4
A parallelepiped is said to be integer when at least one of its edges measures a integer number of units. We have a group of integer parallelepipeds with which a larger parallelepiped is assembled, which has no holes inside or on its edge. Prove that the assembled parallelepiped is also integer.
Example. The following figure shows an assembled parallelepiped with a certain group of integer parallelepipeds.
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 $ABC$ 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.If $C$ is a set of$ n$ points in the plane that has the following property: For each point $P$ of $C$, there are four points of $C$, each one distinct from $P$ , which are the vertices of a square. Find the smallest possible value of $n$.
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 $\angle ABC$.
2019 Peru L3 p3
In the trapezoid $ABCD$ , the base $AB$ is smaller than the $CD$ base. The point $K$ is chosen such that $AK$ is parallel to BC and $BK$ is parallel to $AD$. The points $P$ and $Q$ are chosen on the $AK$ and $BK$ rays respectively, such that $\angle ADP = \angle BCK$ and $\angle BCQ = \angle ADK$.
(a) Show that the lines $AD, BC$ and $PQ$ go through the same point.
(b) Assuming that the circumscribed circumferences of the $APD$ and $BCQ$ triangles intersect at two points, show that one of those points belongs to the line $PQ$.
In the trapezoid $ABCD$ , the base $AB$ is smaller than the $CD$ base. The point $K$ is chosen such that $AK$ is parallel to BC and $BK$ is parallel to $AD$. The points $P$ and $Q$ are chosen on the $AK$ and $BK$ rays respectively, such that $\angle ADP = \angle BCK$ and $\angle BCQ = \angle ADK$.
(a) Show that the lines $AD, BC$ and $PQ$ go through the same point.
(b) Assuming that the circumscribed circumferences of the $APD$ and $BCQ$ triangles intersect at two points, show that one of those points belongs to the line $PQ$.
Let $M,N,P$ be points in the sides $BC,AC,AB$ of $\triangle ABC$ respectively. The quadrilateral $MCNP$ has an incircle of radius $r$, if the incircles of $\triangle BPM$ and $\triangle ANP$ also have the radius $r$. Prove that
$$AP\cdot MP=BP\cdot NP$$
Let $n\geq 3$ be a positive integer and a circle $\omega$ is given. A regular polygon(with $n$ sides) $P$ is drawn and your vertices are in the circle $\omega$ and these vertices are red. One operation is choose three red points $A,B,C$, such that $AB=BC$ and delete the point $B$. Prove that one can do some operations, such that only two red points remain in the circle.
Level 2
Enrique drew $2n$ lines in the plane, where n is a positive integer, and noticed that there were not three concurrent lines (three lines with a common point). Then he colored each intersection point red and counted the number of red points on each intersection point of straights lines. If of these $2n$ sum , half of the sums are $2007$ and the other half are $2008$, how many lines did Enrique draw?
Iván marks some points of a straight line in such a way that the following is fulfilled property: “Whenever Iván chooses three marked points, there are two of them whose distance is less than $3$ and there are two of them whose distance is greater than $3$”. What is the largest number of numbers that Ivan can mark?
$N$ points are marked on a circumference ($N \ge 5$) so that the $N$ arcs formed have the same length. N tokens are placed on the $N$ marked points (one token for each point). Two players, Ricardo and Tomas, play by removing the placed chips, according to the following rules:
$\bullet$ Game turns are staggered.
$\bullet$ Ricardo begins.
$\bullet$ If on a player's turn there are three checkers such that the corresponding marked points form a non-obtuse triangle, the player must remove one of those checkers.
$\bullet$ The player who cannot remove any tile on his turn loses.
Does any player have a winning strategy? If so, what does such a strategy consist of?
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 $AF = BC$.
2013 Peru L2 p3
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 $AF = BC$.
a) Prove that for every even positive integer $n\ge 4$, there exists a convex polygon of $n$ sides in such a way that the number of isosceles triangles that can be formed with $3$ of its vertices is greater than or equal to$$\frac{(n - 2)(3n - 4)} {4}$$b) If $25$ points of a circumference are painted red, in such a way that any two segments that have their endpoints in red points are not perpendicular, at most, how many isosceles triangles have their vertices in three red points?
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
Let $ABCDEF$ be a regular hexagon, $C_1$ be the circle of diameter $AF$ and $C_2$ the circle of center $E$ and radius $EF$. Circles $C_1$ and $C_2$ intersect at points $F$ and $P$. $AP$ intersects $ED$ at $Q$. Determine the ratio of segments $DQ$ and $QE$.
Let $ABC$ be a triangle, the bisector of $\angle CAB$ intersects the circumcircle of the triangle $ABC$ at $D$. The line parallel to $AC$ through $D$ intersects side $AB$ at $L$ and side $BC$ at $K$. Let $M$ be the midpoint of $AC$ . If $MK = ML$, prove that $BL = KD$.
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
2019 Peru L1 p3
Let $ABC$ be a triangle. On the side $AB$ lies point $D$ is chosen and on the side $AC$ lies point $E$ such that $AE = ED = DB$ and $EC = CB =\frac{AB}{2}$. Find the measure of the angle $\angle BAC$.
Let $ABC$ be an isosceles triangle such that $AB = BC$. Let $D$ be a point on the side $AB$ such that $\angle ACD =\angle BCD$ and $E$ a point on side $AC$ such that $\angle AED = 90^o$. If $AE =\frac{BC}{2}$ , calculate $\angle ACD$.
thanks to Jorge Tipe for his help
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