Olympiad problems

2017 Puerto Rico Team Selection Test, 6

Miguel has a square piece of paper $ABCD$ that he folded along a line $EF$, $E$ on $AB$, and $F$ on $CD$. This fold sent $A$ to point $A'$ on $BC$, distinct from $B$ and $C$. Also, it brought $D$ to point $D'$. $G$ is the intersection of $A'D'$ and $DC$. Prove that the inradius of $GCA'$ is equal to the sum of the inradius of $D'GF$ and $A'BE$.

2019 Jozsef Wildt International Math Competition, W. 69

Tags: angle bisector , triangle , geometric inequality , circumradius , inradius , inequalities , geometry

Denote $\overline{w_a}, \overline{w_b}, \overline{w_c}$ the external angle-bisectors in triangle $ABC$, prove that $$\sum \limits_{cyc} \frac{1}{w_a}\leq \sqrt{\frac{(s^2 - r^2 - 4Rr)(8R^2 - s^2 - r^2 - 2Rr)}{8s^2R^2r}}$$

1980 Canada National Olympiad, 3

Tags: geometry , inradius , perimeter

Among all triangles having (i) a fixed angle $A$ and (ii) an inscribed circle of fixed radius $r$, determine which triangle has the least minimum perimeter.

1987 Tournament Of Towns, (160) 4

Tags: perpendicular , altitude , inradius , geometry

From point $M$ in triangle $ABC$ perpendiculars are dropped to each altitude. It can be shown that each of the line segments of altitudes, measured between the vertex and the foot of the perpendicular drawn to it, are of equal length. Prove that these lengths are each equal to the diameter of the circle inscribed in the triangle.

2001 AIME Problems, 7

Tags: incenter , ratio , inradius , number theory , geometry

Triangle $ABC$ has $AB=21$, $AC=22$, and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2009 CHKMO, 3

Tags: incenter , trigonometry , parallelogram , circumcircle , inradius , ratio , geometry

$ \Delta ABC$ is a triangle such that $ AB \neq AC$. The incircle of $ \Delta ABC$ touches $ BC, CA, AB$ at $ D, E, F$ respectively. $ H$ is a point on the segment $ EF$ such that $ DH \bot EF$. Suppose $ AH \bot BC$, prove that $ H$ is the orthocentre of $ \Delta ABC$. Remark: the original question has missed the condition $ AB \neq AC$

1987 ITAMO, 3

Tags: construction , circumradius , inradius , geometry

Show how to construct (by a ruler and a compass) a right-angled triangle, given its inradius and circumradius.

2016 Czech And Slovak Olympiad III A, 2

Tags: right triangle , exradius , inradius , geometry

Let us denote successively $r$ and $r_a$ the radii of the inscribed circle and the exscribed circle wrt to side BC of triangle $ABC$. Prove that if it is true that $r+r_a=|BC|$ , then the triangle $ABC$ is a right one

2004 France Team Selection Test, 2

Tags: incenter , inequalities , cyclic quadrilateral , inradius , perimeter , trigonometry , geometry

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

2004 Brazil National Olympiad, 1

Tags: incenter , rhombus , hyperbola , symmetry , conic , inradius , geometry

Let $ABCD$ be a convex quadrilateral. Prove that the incircles of the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a point in common if, and only if, $ABCD$ is a rhombus.

2011 Albania Team Selection Test, 2

Tags: geometry , perimeter , trigonometry , integration , algorithm , inradius , calculus

The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.

2013 Romania National Olympiad, 3

Tags: geometry , perimeter , analytic geometry , inradius , circumcircle

Given $P$ a point m inside a triangle acute-angled $ABC$ and $DEF$ intersections of lines with that $AP$, $BP$, $CP$ with$\left[ BC \right],\left[ CA \right],$respective $\left[ AB \right]$ a) Show that the area of the triangle $DEF$ is at most a quarter of the area of the triangle $ABC$ b) Show that the radius of the circle inscribed in the triangle $DEF$ is at most a quarter of the radius of the circle circumscribed on triangle $4ABC.$

2005 Italy TST, 2

Tags: inequalities , geometry , rearrangement inequality , inradius

$(a)$ Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality. $(b)$ Determine the points in a triangle that minimize the sum of the distances to the sides.

1998 Belarus Team Selection Test, 2

Tags: perimeter , incircle , geometric inequality , inradius , geometry

The incircle of the triangle $ABC$ touches its sides $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. If $r$ is the inradius of $\vartriangle ABC, P,P_1$ are the perimeters of $\vartriangle ABC, \vartriangle A_1B_1C_1$ respectively, prove that $P+P_1 \ge 9 \sqrt3 r$. I. Voronovich

2006 Tournament of Towns, 4

Tags: tangent , hexagon , inradius , geometry

A circle of radius $R$ is inscribed into an acute triangle. Three tangents to the circle split the triangle into three right angle triangles and a hexagon that has perimeter $Q$. Find the sum of diameters of circles inscribed into the three right triangles. (6)

2014 India PRMO, 12

Tags: sum , right angle , inradius , geometry

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle B DC = 90^o$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$ ?

2011 Morocco TST, 3

Tags: incenter , triangle , geometry , inradius

For a given triangle $ ABC$, let $ X$ be a variable point on the line $ BC$ such that $ C$ lies between $ B$ and $ X$ and the incircles of the triangles $ ABX$ and $ ACX$ intersect at two distinct points $ P$ and $ Q.$ Prove that the line $ PQ$ passes through a point independent of $ X$.

2016 India IMO Training Camp, 1

Tags: geometric inequality , circumcircle , inradius , geometry

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.

1982 IMO Longlists, 35

Tags: geometric transformation , circumcircle , homothety , inradius , trigonometry , geometry , inequalities

If the inradius of a triangle is half of its circumradius, prove that the triangle is equilateral.

2019 Yasinsky Geometry Olympiad, p2

Tags: rectangle , inradius , circumcircle , geometry

Given the equilateral triangle $ABC$. It is known that the radius of the inscribed circle is in this triangle is equal to $1$. The rectangle $ABDE$ is such that point $C$ belongs to its side $DE$. Find the radius of the circle circumscribed around the rectangle $ABDE$.

Ukraine Correspondence MO - geometry, 2009.3

Tags: geometry , compass , right triangle , inradius

A right triangle is drawn on the plane. How to use only a compass to mark two points, such that the distance between them is equal to the diameter of the circle inscribed in this triangle?

1998 Austrian-Polish Competition, 9

Tags: geometry , circumcircle , midpoint , arc midpoint , inradius

Given a triangle $ABC$, points $K,L,M$ are the midpoints of the sides $BC,CA,AB$, and points $X,Y,Z$ are the midpoints of the arcs $BC,CA,AB$ of the circumcircle not containing $A,B,C$ respectively. If $R$ denotes the circumradius and $r$ the inradius of the triangle, show that $r+KX+LY+MZ=2R$.

2005 MOP Homework, 4

Tags: geometry , inradius , trigonometry , inequalities

Let $ABC$ be an obtuse triangle with $\angle A>90^{\circ}$, and let $r$ and $R$ denote its inradius and circumradius. Prove that \[\frac{r}{R} \le \frac{a\sin A}{a+b+c}.\]

1989 IMO, 2

Tags: geometry , geometric inequality , inradius , circumcircle , area

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

2009 India IMO Training Camp, 10

Tags: geometry , inradius

For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle.