geometry problems from Iceland Math Olympiads (Mathematics competition for upper secondary school students) with aops links
collected inside aops here
2010-22 (-2019)
In triangle $ABC$, $\angle ABC = 2\angle ACB$. Prove that:
(a) $AC^2 = AB^2 + AB\cdot BC$
(b) $AB + BC <2AC$
In triangle $ABC$, point $P$ divides side $AB$ in the ratio $AP: P B =1:4$. The perpendicular bisector of the segment $PB$ ntersects the side $BC$ at point $Q$. Find $BC$ if it is known that $AC = 7$ and $F(PQC) =\frac{4}{25} F(ABC)$, where $F(XYZ)$ dentoes the area of the triangle $XYZ$.
The rectangle $PQRS$ represents the A4 sheet, that is $PQ: PS =\sqrt2:1$. The rectangle (the paper) is folded such that after the folding the point $Q$ coincides with the point $X$ on the side $SR$ and the broken line $PY$ runs along the point $P$. The length of the side $PS$ is $1$. Find the sidelengths of the triangle $RXY$.
In the quadrilateral $ABCD$ , the angles ∠ADC and ∠BCD are greater than $90^o$ . Let $F$ be the intersection point of line $AC$ with the line through $B$ parallel to $AD$ . Let $E$ be the intersection poin of line $BD$ with the line through $A$ parallel to $BC$. Prove that $EF$ is parallel $CD$.
In the trapezoid $ABCD$, $AB\parallel CD$ , $| AB | = a$, $| CD | = b$ and the diagonals $AC , BD$ intersect at point $E$. The line $FG$ passes through $E$ parallel to $AB$ and has its endpoints on the sides $BC$ and $AD$ respectively. Express $|FG|$ in terms of $a$ and $b$.
In the acute triangle $ABC$, $AD, BE$ and $CF$ are altitudes and $H$ their intersection. Prove that
$$\frac{| AH |}{| AD |}+\frac{| BH |}{| BE |}+\frac{| CH |}{| CF |}=2$$
Let $P, Q, R$ and $S$ be four points on a line (in that order) so that $PQ = RS$. Above of the line are drawn semicircles with the diameters $PQ$, $RS$ and $PS$. Below the line is drawn semicircle with diameters $QR$. These four semicircles delimit an area called the salinon. Let the symmetrical axis of the salinon intersect its edges at points $M$ and $N$. Show that the area of the salinon is equal to the area of the circle with the diameter $MN$.
The quadrilateral $ABCD$ has its vertices in a circle. The chords $AC$ and $BD$ intersect at point $Q$. The ray from $D$ through $A$ and the ray from $C$ through $B$ intersect at $P$. Given $CD = CP = DQ$. Find the angle $\angle CAD$.
Equilateral triangle $RST$ is inscribed in equilateral triangle $ABC$ so that $RS$ is perpendicular at $AB$. Find the area of $RST$ with respect to the area of $ABC$.
Let $C$ be a circle with the center of $M$ and $P \ne M$ be a point in the plane. Each line that passes through $P$ and intersects the circle $C$, determines a chord with the circle. Show that the centers of all these chords lie on one and the same circle.
The triangle $ABC$ is inscribed in a circle with a radius $R$. Show that $| AB | \cdot | AC | = 2R \cdot h_{BC}$, where $h_{BC}$ is the altitude of the triangle respective to $BC$.
The triangle $ABC$ has side lengths $AC = 31$ and $AB = 22$. The medians $CC'$ and $BB'$ are perpendicular to each other. Find the length $BC$.
2018 missing
Triangle $ABC$ has sidelengths $7, 8$ and $9$. A line, which contains the center of the inscribed circle and is parallel to the shortest side of the triangle, intersect the other two sides of the triangle at points $D$ and $E$. What is the length of the segment $DE$?
Triangle $ABC$ is inscribed in a circle. A circle with the center $O$ is inscribed in the triangle $ABC$. The line $AO$ extended intersects the larger circle at point $D$ . Prove that $CD = OD = BD$.
In the acute triangle $ABC$, $D$ is the foot of the altitude from $A$ and $E$ the foot of the altitude from $B$. Show that the triangles $ABC$ and $DEC$ are similar.
Given a circle $\Omega$ and a point $A$ on $\Omega$. Another point $B$ lies on the circle $\Omega$ and the circle $\Gamma_1$ with center $B$ and radius $|AB|$ intersects $\Omega$ again at $C$. The circle $\Gamma_2$ with center $A$ and radius $|AC|$ intersects $\Gamma_1$ again at $C$. Show that the line $AD$ is tangent to $\Omega$.
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