Olympiad problems

1992 IMO Longlists, 66

A circle of radius $\rho$ is tangent to the sides $AB$ and $AC$ of the triangle $ABC$, and its center $K$ is at a distance $p$ from $BC$. [i](a)[/i] Prove that $a(p - \rho) = 2s(r - \rho)$, where $r$ is the inradius and $2s$ the perimeter of $ABC$. [i](b)[/i] Prove that if the circle intersect $BC$ at $D$ and $E$, then \[DE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}\] where $r_1$ is the exradius corresponding to the vertex $A.$

1978 IMO, 1

Tags: circumcircle , inradius , triangle , geometry , incenter

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

2018 Yasinsky Geometry Olympiad, 6

Tags: geometry , midpoint , circumcenter , inradius , incenter

Let $O$ and $I$ be the centers of the circumscribed and inscribed circle the acute-angled triangle $ABC$, respectively. It is known that line $OI$ is parallel to the side $BC$ of this triangle. Line $MI$, where $M$ is the midpoint of $BC$, intersects the altitude $AH$ at the point $T$. Find the length of the segment $IT$, if the radius of the circle inscribed in the triangle $ABC$ is equal to $r$. (Grigory Filippovsky)

2000 National Olympiad First Round, 1

Tags: circumcircle , inradius , geometry

If the incircle of a right triangle with area $a$ is the circumcircle of a right triangle with area $b$, what is the minimum value of $\frac{a}{b}$? $ \textbf{(A)}\ 3 + 2\sqrt2 \qquad\textbf{(B)}\ 1+\sqrt2 \qquad\textbf{(C)}\ 2\sqrt2 \qquad\textbf{(D)}\ 2+\sqrt3 \qquad\textbf{(E)}\ 2\sqrt3$

2012 Uzbekistan National Olympiad, 3

Tags: circumcircle , geometry , geometric transformation , inradius , reflection , perpendicular bisector

The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$.

2007 IberoAmerican, 2

Tags: circumcircle , incenter , trig identities , law of sines , geometry , inradius , trigonometry

Let $ ABC$ be a triangle with incenter $ I$ and let $ \Gamma$ be a circle centered at $ I$, whose radius is greater than the inradius and does not pass through any vertex. Let $ X_{1}$ be the intersection point of $ \Gamma$ and line $ AB$, closer to $ B$; $ X_{2}$, $ X_{3}$ the points of intersection of $ \Gamma$ and line $ BC$, with $ X_{2}$ closer to $ B$; and let $ X_{4}$ be the point of intersection of $ \Gamma$ with line $ CA$ closer to $ C$. Let $ K$ be the intersection point of lines $ X_{1}X_{2}$ and $ X_{3}X_{4}$. Prove that $ AK$ bisects segment $ X_{2}X_{3}$.

2001 Croatia National Olympiad, Problem 2

Tags: geometry , inradius

Let $S$ be the center of a square $ABCD$ and $P$ be the midpoint of $AB$. The lines $AC$ and $PD$ meet at $M$, and the lines $BD$ and $PC$ meet at $N$. Prove that the radius of the incircle of the quadrilateral $PMSN$ equals $MP-MS$.

2004 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: circumcircle , geometry , inradius , inequalities

In a convex hexagon $ABCDEF$, triangles $ACE$ and $BDF$ have the same circumradius $R$. If triangle $ACE$ has inradius $r$, prove that \[ \text{Area}(ABCDEF)\le\frac{R}{r}\cdot\text{Area}(ACE).\]

2008 Sharygin Geometry Olympiad, 3

Tags: geometry , inradius , trigonometry , inequalities

(R.Pirkuliev) Prove the inequality \[ \frac1{\sqrt {2\sin A}} \plus{} \frac1{\sqrt {2\sin B}} \plus{} \frac1{\sqrt {2\sin C}}\leq\sqrt {\frac {p}{r}}, \] where $ p$ and $ r$ are the semiperimeter and the inradius of triangle $ ABC$.

2007 Bulgaria National Olympiad, 1

Tags: reflection , circumcircle , geometry , rhombus , trigonometry , geometric transformation , inradius

The quadrilateral $ABCD$, where $\angle BAD+\angle ADC>\pi$, is inscribed a circle with centre $I$. A line through $I$ intersects $AB$ and $CD$ in points $X$ and $Y$ respectively such that $IX=IY$. Prove that $AX\cdot DY=BX\cdot CY$.

2005 Italy TST, 2

Tags: geometry , rearrangement inequality , inradius , inequalities

$(a)$ 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.

2003 Korea - Final Round, 1

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

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 $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

2002 USAMO, 2

Tags: geometry , inequalities , inradius

Let $ABC$ be a triangle such that \[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \] where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.

2007 Pre-Preparation Course Examination, 1

Tags: geometry , inradius , inequalities

$D$ is an arbitrary point inside triangle $ABC$, and $E$ is inside triangle $BDC$. Prove that \[\frac{S_{DBC}}{(P_{DBC})^{2}}\geq\frac{S_{EBC}}{(P_{EBC})^{2}}\]

Ukrainian TYM Qualifying - geometry, 2019.9

Tags: construction , isosceles , geometry , inradius

On the base $BC$ of the isosceles triangle $ABC$ chose a point $D$ and in each of the triangles $ABD$ and $ACD$ inscribe a circle. Then everything was wiped, leaving only two circles. It is known from which side of their line of centers the apex $A$ is located . Use a compass and ruler to restore the triangle $ABC$ , if we know that : a) $AD$ is angle bisector, b) $AD$ is median.

2014 Regional Olympiad of Mexico Center Zone, 3

Tags: inradius , excircle , geometry

Let $AB$ be a triangle and $\Gamma$ the excircle, relative to the vertex $A$, with center $D$. The circle $\Gamma$ is tangent to the lines $AB$ and $AC$ at $E$ and $F$, respectively. Let $P$ and $Q$ be the intersections of $EF$ with $BD$ and $CD$, respectively. If $O$ is the point of intersection of $BQ$ and $CP$, show that the distance from $O$ to the line $BC$ is equal to the radius of the inscribed circle in the triangle $ABC$.

2005 Germany Team Selection Test, 3

Tags: function , inequalities , geometry , inradius , triangle inequality , trigonometry

Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that [b](a)[/b] $\triangle ABC$ is acute. [b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.

1990 Mexico National Olympiad, 2

Tags: right triangle , inradius , geometry

$ABC$ is a triangle with $\angle B = 90^o$ and altitude $BH$. The inradii of $ABC, ABH, CBH$ are $r, r_1, r_2$. Find a relation between them.

2000 Tuymaada Olympiad, 2

Tags: geometry , rhombus , inradius

A tangent $l$ to the circle inscribed in a rhombus meets its sides $AB$ and $BC$ at points $E$ and $F$ respectively. Prove that the product $AE\cdot CF$ is independent of the choice of $l$.

Estonia Open Senior - geometry, 2007.2.5

Tags: geometry , circumcircle , inradius , trigonometry

Consider triangles whose each side length squared is a rational number. Is it true that (a) the square of the circumradius of every such triangle is rational; (b) the square of the inradius of every such triangle is rational?

Estonia Open Senior - geometry, 2002.1.2

Tags: geometry , right triangle , arithmetic progression , inradius

The sidelengths of a triangle and the diameter of its incircle, taken in some order, form an arithmetic progression. Prove that the triangle is right-angled.

VII Soros Olympiad 2000 - 01, 10.4

Tags: geometry , centroid , inradius

An acute-angled triangle is inscribed in a circle of radius $R$. The distance between the center of the circle and the point of intersection of the medians of the triangle is $d$. Find the radius of a circle inscribed in a triangle whose vertices are the feet of the altitudes of this triangle.