Indonesia MO Province 2002-20 (OSP SMA) 24p

geometry problems from Province (=Regional) Level of Indonesian National Science Olympiads (OSP)

with aops links in the names

Olimpiade Sains Provinsi, OSP SMA

collected inside aops here

                                            2002 - 2020 

2002 Indonesia MO Province P2 q4

Given an equilateral triangle $ABC$ and a point $P$ so that the distances $P$  to $A$ and to $C$ are not farther than the distances $P$ to $B$. Prove that $PB = PA + PC$ if and only if $P$ lies on the circumcircle  of $\vartriangle ABC$.

2003 Indonesia MO Province P2 q3

The points $P$ and $Q$ are the midpoints of the edges $AE$ and $CG$ on the cube $ABCD.EFGH$ respectively. If the length of the cube edges is $1$ unit, determine the area of the quasrilateral $DPFQ$ .

2004 Indonesia MO Province P2 q2

Triangle $ABC$ is given. The points $D, E$, and $F$ are located on the sides $BC, CA$ and $AB$ respectively so that the lines $AD, BE$ and $CF$ intersect at point $O$. Prove that  $\frac{AO}{AD} +   \frac{BO}{BE} +   \frac{CO}{ CF}=2$

2004 Indonesia MO Province P2 q5

The lattice point on the plane is a point that has coordinates in the form of a pair of integers. Let $P_1, P_2, P_3, P_4, P_5$ be five different lattice points on the plane.Prove that there is a pair of points $(P_i, P_j), i \ne j$, so that the line segment $P_iP_j$ contains a lattice point other than $P_i$ and $P_j$.

2005 Indonesia MO Province P2 q1

The length of the largest side of the cyclic quadrilateral $ABCD$ is $a$, while the radius of the circumcircle of $\vartriangle ACD$ is $1$. Find the smallest possible value for $a$. Which cyclic quadrilateral $ABCD$ gives the value $a$ equal to the smallest value?

2005 Indonesia MO Province P2 q4

The lengths of the three sides $a, b, c$ with $a \le b \le c$, of a right triangle is an integer. Find all the sequences $(a, b, c)$ so that the values of perimeter and area of the triangle are the same.

2006 Indonesia MO Province P2 q1

Suppose triangle $ABC$ is right-angled at $B$. The altitude from $B$ intersects the side $AC$ at point $D$. If points $E$ and $F$ are the midpoints of $BD$ and $CD$, prove that $AE \perp BF$.

2007 Indonesia MO Province P2 q1

Let $ABCD$ be a quadrilateral with $AB = BC = CD = DA$.

(a) Prove that point A must be outside of triangle $BCD$.

(b) Prove that each pair of opposite sides on $ABCD$ is always parallel.

2007 Indonesia MO Province P2 q4

In acute triangles $ABC$,  $AD, BE ,CF$ are altitudes, with $D, E, F$ on the sides $BC, CA, AB$, respectively. Prove that $DE + DF  \le BC$

2008 Indonesia MO Province P2 q3

The incircle of triangle $ABC$, is tangent to the sides $BC, CA$, and $AB$ at $D, E$, and $F$, respectively. Through $D$, the perpendicular line  on $EF$ intersects $EF$ at $G$. Prove that $\frac{FG}{EG} = \frac{BF}{CE}$.

2009 Indonesia MO Province P2 q3

Given triangle $ABC$ and point $D$ on the $AC$ side. Let $r_1, r_2$ and $r$ denote the radii of the incircle of the triangles $ABD, BCD$, and $ABC$, respectively. Prove that $r_1 + r_2> r$.

2010 Indonesia MO Province P2 q1

Given triangle $ABC$. Suppose $P$ and $P_1$ are points on $BC, Q$  lies on $CA, R$ lies  on $AB$, such that $$\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP_1}{P_1B}$$ Let $G$ be the centroid of triangle $ABC$ and $K = AP_1  \cap RQ$.  Prove that points $P,G$, and $K$ are collinear.

2011 Indonesia MO Province P2 q4

Given a rectangle $ABCD$ with $AB = a$ and $BC = b$. Point $O$ is the intersection of the two diagonals. Extend the side of the $BA$ so that $AE = AO$, also extend the diagonal of $BD$ so that $BZ = BO.$  Assume that triangle $EZC$ is equilateral. Prove that

(i) $b = a\sqrt3$

(ii) $EO$ is perpendicular to $ZD$

2012 Indonesia MO Province P2 q4

Given an acute triangle $ABC$. Point $H$ denotes the foot of the altitude drawn from $A$. Prove that $$AB + AC \ge  BC cos \angle BAC + 2AH sin  \angle BAC$$

2013 Indonesia MO Province P2 q5

Given an acute triangle $ABC$. The longest line of altitude is the one from vertex $A$ perpendicular to $BC$, and it's length is equal to the length of the median of vertex $B$. Prove that $\angle ABC \le 60^o$

2014 Indonesia MO Province P2 q2

Given an acute triangle $ABC$ with $AB <AC$. The ex-circles of triangle $ABC$ opposite $B$ and $C$ are centered on $B_1$ and $C_1$, respectively. Let $D$ be the midpoint of $B_1C_1$. Suppose that $E$ is the point of intersection of $AB$ and $CD$, and $F$ is the point of intersection of $AC$ and $BD$. If $EF$ intersects $BC$ at point $G$, prove that $AG$ is the bisector of $\angle BAC$.

2014 Indonesia MO Province P2 q4

Let $\Gamma$ be the circumcircle of triangle $ABC$. One circle $\omega$is tangent to $\Gamma$ at $A$ and tangent to $BC$ at $N$. Suppose that the extension of $AN$ crosses  $\Gamma$ again at $E$. Let $AD$ and $AF$ be respectively the line of altitude $ABC$ and diameter of  $\Gamma$, show that $AN \times AE = AD \times AF = AB \times AC$

2015 Indonesia MO Province P2 q3

Given the isosceles triangle $ABC$, where $AB = AC$. Let $D$ be a point in the segment $BC$ so that $BD = 2DC$. Suppose also that point $P$ lies on the segment $AD$ such that: $\angle BAC = \angle BP D$. Prove that $\angle BAC = 2\angle DP C$.

2016 Indonesia MO Province P2 q4

Let $PA$ and $PB$ be the tangent of a circle $\omega$ from a point $P$ outside the circle. Let $M$ be any point on $AP$ and $N$ is the midpoint of segment $AB$. $MN$ cuts $\omega$ at $C$ such that $N$ is between $M$ and $C$. Suppose $PC$ cuts $\omega$ at $D$ and $ND$ cuts $PB$ at $Q$. Prove $MQ$ is parallel to $AB$.

2017 Indonesia MO Province P2 q3

Given triangle $ABC$, the three  altitudes intersect at point $H$. Determine all points $X$ on the side $BC$ so that the symmetric of $H$ wrt  point $X$ lies on the circumcircle of triangle $ABC$.

2018 Indonesia MO Province P2 q3

Let $\Gamma_1$ and  $\Gamma_2$ be two different circles with the radius of same length and centers at points $O_1$ and $O_2$, respectively. Circles $\Gamma_1$ and  $\Gamma_2$  are tangent at point $P$. The line $\ell$  passing through $O_1$ is tangent to  $\Gamma_2$ at point $A$. The line $\ell$  intersects $\Gamma_1$ at point $X$ with $X$ between $A$ and $O_1$. Let $M$ be the midpoint  of $AX$ and $Y$ the intersection of $PM$ and $\Gamma_2$ with $Y\ne P$. Prove that $XY$ is parallel to $O_1O_2$.

2019 Indonesia MO Province P2 q1

Given cube $ABCD.EFGH$ with $AB = 4$ and $P$ midpoint of the side $EFGH$. If $M$ is the midpoint of $PH$, find the length of segment $AM$.

2019 Indonesia MO Province P2 q5

Given triangle $ABC$, with $AC> BC$, and the it's circumcircle centered at $O$. Let $M$ be the point on the circumcircle of triangle $ABC$ so that $CM$ is the bisector of $\angle ACB$. Let $\Gamma$ be a circle with diameter $CM$. The bisector of $BOC$ and bisector of $AOC$ intersect $\Gamma$ at $P$ and $Q$, respectively. If $K$ is the midpoint of $CM$, prove that $P, Q, O, K$ lie at one point of the circle.

2020 Indonesia MO Province P2 q1

In the figure, point $P, Q,R,S$ lies on the side of the rectangle $ABCD$. 

If it is known that the area of the small square is $1$ unit, determine the area of the rectangle $ABCD$.

2020 Indonesia MO Province P2 q5

It is known that triangle $ABC$ is not isosceles with altitudes  of $AA_1, BB_1$, and $CC_1$. Suppose $B_A$ and $C_A$ respectively points on $BB_1$ and $CC_1$ so that $A_1B_A$ is perpendicular on  $BB_1$ and $A_1C_A$ is perpendicular on $CC_1$. Lines $B_AC_A$ and $BC$ intersect at the point $T_A$ . Define in the same way the points $T_B$ and $T_C$ . Prove that points $T_A, T_B$, and $T_C$ are collinear.

sources: 

https://www.tomatalikuang.com/2016/12/download-soal-dan-pembahasan-osn-sma-matematika.html