geometry problems from Croatian Junior Mathematical Olympiads (HJMO) with aops links in the names
it started in 2017
collected inside aops here
2017-22
Above the sides $AC$ and $BC$ of triangle $ABC$, draw the squares $ACDE$ and $BCFG$. Let $P$ be the midpoint of the $AB$. Prove that the length of $DF$ is twice as long as the length of $CP$.
Let $AC$ be the diameter of the circle $k_1$ whose center is point $B$. The circle $k_2$ touches the line $AC$ at point $B$ and the circle $k_1$ at point $D$. The tangent from $A$ (different from $AC$) at circle $k_2$ touches that circle at point $E$ and intersects line $BD$ at point $F$. If $| AF | = 10$, calculate $| AB |$ .
Let $a, b$ and $c$ be the lengths of the sides of a right triangle and $a <b <c$, whose one angle is $75^o$ . Let $v$ be the length of the altitude to the side of length $c$. Calculate the ratios:
a) $\frac{a}{b}$ , b) $\frac{c}{v}$
Let $ABC$ be a right triangle with right angle at the vertex $C$. A circle $k$ with diameter $AC$ intersects the side $AB$ at the point $D$, and the tangent to the circle $k$ at the point $D$ intersects the side $BC$ at point $E$. The circle circumscribed around triangle $CDE$ intersects the side $AB$ at points $D$ and $F$. Determine the ratio of the areas of triangles $ABC$ and $BEF$.
Let $ABCD$ be a square and $k$ a circle with center at point $B$ passing through points $A$ and $C$, and let $T$ be a point on the circle $k$ within a given square. Tangent to a circle $k$ at a point $T$ intersects the segments $CD$ and $DA$ respectively at points $E$ and $F$. Let$ G$ and $H$ be the intersections of the segments $BE$ and $BF$ with segment $AC$ respectively . Prove that the lines $BT, EH$ and $FG$ pass through the same point.
The points $A,B,C$ and $D$ are given on the circle such that $| AB| = | BC | = | CD |$. Let the bisectors of the angles $\angle ACD$ and $\angle ABD$ intersect at the point $E$. If the lines $AE$ and $CD$ are parallel, determine $ \angle ABC$.
The circles $k_1$ and $k_2$ are externally tangent at the point $F$. Line $ t$ touches the circles $k_1$ and $k_2$ at points $A$ and $B$, respectively. The line $p$ parallel to the line $t$ touches the circle $k_2$ at the point $C$ and intersects $k_2$ at the points $D$ and $E$.
(a) Prove that the points $A,F$ and $C$ lie on the same line.
(b) Prove that the point $A$ is the center of the circumcircle of the triangle $BDE$.
Let $ABC$ be a triangle with an obtuse angle at the vertex $C$ and $k$ be a circle of diameter $AB$. Angle bisector of $\angle CAB$ intersects the circle $k$ at the point $D$ ($D\ne A$), and the angle bisector of $\angle ABC$ intersects that circle at the point $E$ ($E\ne B$). A circle inscribed in a triangle $ABC$ touches the sides $BC$ and $AC$ at points $F$ and $G$, respectively. Prove that the points $D,E,F$ and $G$ lie in the same line.
Let $ABC$ be an acute-angled triangle, $H$ its orthocenter, and $M$ the midpoint of the page $BC$. The point $N$ is the vertex of the perpendicular from $H$ to $AM$. Prove that the points $B,C,N$ and $H$ lie on the same circle.
Let $ABCD$ and $AEFG$ be rectangles such that the points $B, E, D, G$ are on the same line respectively). Let the point $T$ be the intersection of the lines $BC$ and $GF$ and let the point $H$ be the intersection of the lines $DC$ and $EF$. Prove that the points $A$, $H$ and $T$ are collinear.
source: https://natjecanja.math.hr/
No comments:
Post a Comment