Found problems: 405
2015 China Team Selection Test, 4
Prove that : For each integer $n \ge 3$, there exists the positive integers $a_1<a_2< \cdots <a_n$ , such that for $ i=1,2,\cdots,n-2 $ , With $a_{i},a_{i+1},a_{i+2}$ may be formed as a triangle side length , and the area of the triangle is a positive integer.
1977 Yugoslav Team Selection Test, Problem 3
Assume that the equality $2BC=AB+AC$ holds in $\triangle ABC$. Prove that:
(a) The vertex $A$, the midpoints $M$ and $N$ of $AB$ and $AC$ respectively, the incenter $I$, and the circumcenter $O$ belong to a circle $k$.
(b) The line $GI$, where $G$ is the centroid of $\triangle ABC$ is a tangent to $k$.
2018 Pan-African Shortlist, G5
Let $ABC$ be a triangle with $AB \neq AC$. The incircle of $ABC$ touches the sides $BC$, $CA$, $AB$ at $X$, $Y$, $Z$ respectively. The line through $Z$ and $Y$ intersects $BC$ extended in $X^\prime$. The lines through $B$ that are parallel to $AX$ and $AC$ intersect $AX^\prime$ in $K$ and $L$ respectively. Prove that $AK = KL$.
1976 Euclid, 1
Source: 1976 Euclid Part B Problem 1
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Triangle $ABC$ has $\angle{B}=30^{\circ}$, $AB=150$, and $AC=50\sqrt{3}$. Determine the length of $BC$.
2002 India IMO Training Camp, 11
Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.
2021 EGMO, 4
Let $ABC$ be a triangle with incenter $I$ and let $D$ be an arbitrary point on the side $BC$. Let the line through $D$ perpendicular to $BI$ intersect $CI$ at $E$. Let the line through $D$ perpendicular to $CI$ intersect $BI$ at $F$. Prove that the reflection of $A$ across the line $EF$ lies on the line $BC$.
1979 IMO Longlists, 47
Inside an equilateral triangle $ABC$ one constructs points $P, Q$ and $R$ such that
\[\angle QAB = \angle PBA = 15^\circ,\\ \angle RBC = \angle QCB = 20^\circ,\\ \angle PCA = \angle RAC = 25^\circ.\]
Determine the angles of triangle $PQR.$
2000 Croatia National Olympiad, Problem 2
Two squares $ACXE$ and $CBDY$ are constructed in the exterior of an acute-angled triangle $ABC$. Prove that the intersection of the lines $AD$ and $BE$ lies on the altitude of the triangle from $C$.
2001 Moldova National Olympiad, Problem 4
In a triangle $ABC$, $BC=a$, $AC=b$, $\angle B=\beta$ and $\angle C=\gamma$. Prove that the bisector of the angle at $A$ is equal to the altitude from $B$ if and only if $b=a\cos\frac{\beta-\gamma}2$.
2004 IMO Shortlist, 7
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$.
2009 Serbia National Math Olympiad, 1
In a scalene triangle $ABC$, $\alpha$ and $\beta$ respectively denote the interior angles at vertixes $A$ and $B$. The bisectors of these two angles meet the opposite sides of the triangle at points $D$ and $E$, respectively. Prove that the acute angle between the lines $DE$ and $AB$ does not exceed $ \frac{ | \alpha - \beta |}{3}$ .
[i]Proposed by Dusan Djukic[/i]
1978 IMO, 1
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 $ABC.$
2008 Grigore Moisil Intercounty, 3
Let $ A_1,B_1,C_1 $ be points on the sides (excluding their endpoints) $ BC,CA,AB, $ respectively, of a triangle $ ABC, $ such that $ \angle A_1AB =\angle B_1BC=\angle C_1CA. $ Let $ A^* $ be the intersection of $ BB_1 $ with $ CC_1,B^* $ be the intersection of $ CC_1 $ with $ AA_1, $ and $ C^* $ be the intersection of $ AA_1 $ with $ BB_1. $ Denote with $ r_A,r_B,r_C $ the inradii of $ A^*BC,AB^*C,ABC^*, $ respectively. Prove that
$$ \frac{r_A}{BC}=\frac{r_B}{CA}=\frac{r_C}{AB} $$
if and only if $ ABC $ is equilateral.
[i]Daniel Văcărețu[/i]
1985 IMO, 5
A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.
2003 IMO Shortlist, 3
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]
1975 Putnam, A6
Given three points in space forming an acute-angled triangle, show that we can find two further points such that no three of the five points are collinear and the line through any two is normal to the plane through the other three.
1983 Czech and Slovak Olympiad III A, 6
Consider a circle $k$ with center $S$ and radius $r$. Denote $\mathsf M$ the set of all triangles with incircle $k$ such that the largest inner angle is twice bigger than the smallest one. For a triangle $\mathcal T\in\mathsf M$ denote its vertices $A,B,C$ in way that $SA\ge SB\ge SC$. Find the locus of points $\{B\mid\mathcal T\in\mathsf M\}$.
1969 IMO Shortlist, 57
Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively.
If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .
2003 German National Olympiad, 2
There are four circles $k_1 , k_2 , k_3$ and $k_4$ of equal radius inside the triangle $ABC$. The circle $k_1$ touches the sides $AB, CA$ and the circle $k_4 $, $k_2$ touches the sides $AB,BC$ and $k_4$, and $k_3$ touches the sides $AC, BC$ and $k_4.$ Prove that the center of $k_4$ lies on the line connecting the incenter and circumcenter of $ABC.$
1975 Czech and Slovak Olympiad III A, 5
Let a square $\mathbf P=P_1P_2P_3P_4$ be given in the plane. Determine the locus of all vertices $A$ of isosceles triangles $ABC,AB=BC$ such that the vertices $B,C$ are points of the square $\mathbf P.$
2007 IMO Shortlist, 8
Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear.
[i]Author: Waldemar Pompe, Poland[/i]
1980 Bundeswettbewerb Mathematik, 2
In a triangle $ABC$, the bisectors of angles $A$ and $B$ meet the opposite sides of the triangle at points $D$ and $E$, respectively. A point $P$ is arbitrarily chosen on the line $DE$. Prove that the distance of $P$ from line $AB$ equals the sum or the difference of the distances of $P$ from lines $AC$ and $BC$.
1986 IMO Shortlist, 17
Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.
2019 Czech and Slovak Olympiad III A, 4
Let be $ABC$ an acute-angled triangle. Consider point $P$ lying on the opposite ray to the ray $BC$ such that $|AB|=|BP|$. Similarly, consider point $Q$ on the opposite ray to the ray $CB$ such that $|AC|=|CQ|$. Denote $J$ the excenter of $ABC$ with respect to $A$ and $D,E$ tangent points of this excircle with the lines $AB$ and $AC$, respectively. Suppose that the opposite rays to $DP$ and $EQ$ intersect in $F\neq J$. Prove that $AF\perp FJ$.
2001 IMO Shortlist, 3
Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.