geometry problems from Izumrud Olympiad Emerald by UrFU - Ural Federal University (Russia) with aops links
collected inside aops here
2016- 2022
Given are six pencils in the form of identical right circular cylinders. Place them in space so that each pencil has a common endpoint with any other pencil.
Prove that if $a$ and $b$ are legs, $c$ is the hypotenuse of a right triangle, then the radius of a circle inscribed in this triangle can be found by the formula $r = \frac12 (a + b - c)$.
Let $A$ be the intersection point of two circles. From this point along each circle, clockwise, with constant speeds, points $X_1$ and $X_2$ begin to move. After one rotation, both points again appear in $A$. Prove that there is always such a fixed point $B$ that the equality $X_1B = X_2B$ holds throughout the motion.
The diagonals of a quadrilateral are equal, and the lengths of its midlines are $p$ and $q$. Find the area of the quadrilateral.
In an isosceles triangle with a lateral side equal to $b$, the bisectors of the angles at the base are drawn. The segment of a straight line between the points of intersection of the bisectors with the lateral sides is equal to $m$. Determine the base of the triangle.
Three circles are inscribed in the angle, a small, a medium and a large. The large circle goes through the center of the medium one, and the medium one goes through the center of the small one. Define the radii of the medium and large circles, if the radius of the smaller one is $r$ and the distance from its center to the vertice of the angle is equal to $a$.
On each median of the triangle, a point is taken dividing the median in a ratio of $1: 3$, counting from the vertex. How many times is the area of the triangle with vertices at these points less than the area of the original triangle?
To two externally tangent circles of radii $R$ and $r$, a secant is constructed so that the circles cut off three equal segments on it. Find the lengths of these lines.
On the segment $AC$ of $12$ cm long, point $B$ is marked such that $AB = 4$ cm. Using the segments $AB$ and $AC$ as diameters in the same half-plane with the segment $AC$ , semicircles are constructed. Calculate the radius of the circle tangent to the constructed circles and the $AC$.
In a triangle with a base equal to $a$, a square is inscribed, one of the sides of which lies on the base of the triangle. The area of the square is $\frac16$ of the area of the triangle. Determine the altitude of the triangle and the side of the square.
A straight line parallel to the bases of this right trapezoid cuts it into two trapezoids, into each of which you can inscribe a circle. Find the bases of the original trapezoid if its lateral sides are equal to $c$ and $d$, and $c <d$.
In triangles $ABC$ and $A_1B_1C_1$, the segments $CD$ and $C_1D_1$ are the bisectors of the angles $C$ and $C_1$, respectively. It is known that $AB = A_1B_1$, $CD = C_1D_1$ and $\angle ADC = \angle A_1D_1C_1$. Are triangles $ABC$ and $A_1B_1C_1$ congruent?
An equilateral triangle $ABC$ is inscribed in a circle. On an arc $AB$ not containing a point $C$, a point $M$ is chosen that is different from fixed points $A$ and $B$. Let lines $AC$ and $BM$ intersect at point $K$, let lines $BC$ and $AM$ intersect at point $N$. Prove that the product of the lengths of the segments $AK$ and $BN$ does not depend from the choice of point $M$.
An ant sits in a rectangular matchbox measuring $1 \times 2 \times 3$ cm, and a net of sugar crumbs. If we introduce a coordinate system with axes parallel to the edges of the box so that one vertex of the box is at the origin, and the second at point with coordinates (1$0$ mm, $20$ mm, $30$ mm), then the ant will sit at point with coordinates ($1$ mm, $2$ mm, $0$ mm), and the crumb will be coordinates at point with coordinates ($9$ mm, $3$ mm, $30$ mm). What is the shortest distance that the ant will have to crawl around the sugar crumbs if it can move only over the surface of the box?
The three faces of the tetrahedron are right-angled triangles, and the fourth face is not an obtuse triangle. Prove that a necessary and sufficient condition for the fourth face to be a right-angled triangle is the proposition that exactly two of the plane angles at one vertex of the tetrahedron are right.
The point of intersection of the medians of the triangle was connected by segments with each of its vertices, dividing it into three triangles. One of the smaller triangles turned out to be similar to the original one. Find the largest side of the original triangle if its the smallest side is $10\sqrt3$.
On the sides $AB$ and $AC$ of the triangle $ABC$ , respectively, mark points $M$ and $N$ so that $AM=MC=CB$ and $MB=BN=NA$ . Find the angles of the triangle $ABC$.
On the plane, there is a triangle $ABC$ and a point $M$. Feet of the perpendiculars drawn from the point $M$ on the lines $AB$ and $AC$ lie outside the triangle. The foot of the perpendicular drawn from the point $M$ on line $BC$ lies on side $BC$, and the points $A$ and $M$ lie on opposite sides of $BC$. It is known that the distance from point $M$ to side $BC$ is equal to the length of the side $BC$, and the distances to the lines containing the sides $AB$ and $AC$ are equal respectively to the lengths of the sides $AB$ and $AC$. Find all the values that can take tangent of anlge $A$.
Three circles $\omega_1, \omega_2,\omega_3$ of increasing radii are inscribed in one angle. Let $A_1,A_2,A_3$ be the touch points on one side of the angle , and $B_1,B_2,B_3$ lie on the other side. Let the circles $\omega_1$ and $\omega_2$ meet at the points $M$ and $N$, and the circles $\omega_2$ and $\omega_3$ at points $T$ and $S$. It turned out that the points $A_1,M,S,B_3$ lie on one straight line. Find the aspect ratio of the segments $A_1M:MS:MB_3$ .
There is a regular octagonal prism, all edges of which are equal to $2$ m. A spider sits in the center of one of the side faces. It can move on the surface prisms until its $3$ m long cobweb ends. The spider wondered if there were on the bases of the prism, points to which it can reach with at least two different shortest paths, and at the same time spending all the cobweb. Help the spider count the number of such points.
In the triangle $ABC$, intersecting at one point are the altitude $AH$, median $BM$ and angle bisector $CL$. Point $K$ is the foot of the altitude drawn from the vertice $B$. Prove that $KH=BL$.
Point $M$ has been marked inside triangle $ABC$. It is known that $AM=a$, $BM=b$,$CM=c$. Could the areas of triangles $AMB$ ,$BMC$, $CMA$ be equal to $a^2$, $b^2$ ,$c^2$?
In a triangle, the median, angle bisector and altitude , drawn from different vertices intersect at one point, and the lengths of the altitude and the angle bisector are equal. Is the triangle nessessarily equilateral?
You are given a square pyramid $SABCD$. All the edges of the pyramid have the same length equal to $10$. The cockroach, moving only along the surface of the pyramid, moved from the midpoint of the $SA$ to the midpoint of the $SC$ , while having time to visit the base of $ABCD$. What is the minimum possible length of the path traveled by a cockroach?
The inscribed quadrilateral $ABCD$ is such that $AB\cdot CD = BC \cdot AD$. Rays $AB$ and $DC$ intersect at point $E$, and rays $BC$ and $AD$ intersect at point $F$. It turned out that $BE=CD$, $DF=BC$ . Prove that a circle can be inscribed in a quadrilateral $ABCD$ .
In an acute-angled triangle $ABC$, the altitude $AK$ was drawn. Let $H$ be the intersection point of the altitudes of the triangle $ABC$. It is known that the areas of triangles $AKC, AHB$ and $BHK$ turned out to be equal. Find $\angle ABC$.
Let $M$ be the intersection point of the medians of the triangle $ABC$ . It turned out that $\angle ABM= \angle BCM$, $\angle BAM= \angle ACM$. Is it true that triangle $ABC$ is equilateral?
In the triangle $ABC$, the medians drawn from the vertices $A $and $B$ are perpendicular and intersect at point $M$. Points $P$ and $Q$ are marked on side $AB$ so that $AP=PQ=QB$. Prove that the perimeter of the triangle $CPQ$ is less than twice the perimeter of the triangle $ABM$.
In the cube $ABCDA_1B_1C_1D_1$, all edges of which are equal to one, the point $M$ is the midpoint of the edge $CC_1$, point $O$ is the center of face $ABB_1A1$. A set of points lying on the face $CBB_1C_1$ is such that for any point $X$ of this set, the plane $XOM$ intersects the edge $AD$. Find the area of this set.
In an acute-angled triangle$ ABC$, the altitudes $AA_1$ and $BB_1$ meet at the point $H$. Points $M$ and $N$ are the midpoints of altitudes $AA_1$ and $BB_1$. It turned out that the center $I$ of the circle inscribed in the triangle $HMN$ lies on the bisector of the angle $MCN$. Prove that the triangle $ABC$ isosceles.
The figure shows three equal squares. Prove that $AB>BC$.
Two congruent triangles $ABC$ and $A'B'C' $ are depicted on the plane. On the extensions of the sides of the triangle $ABC$, we took the points $A_1,B_1,C_1$ , and on the extensions of the sides of the triangle $A'B'C'$, we took the points $A_2,B_2,C_2$, and then marked with strokes all the same segments (see figure). Prove that the areas of triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal.
In an isosceles triangle $ABC$ , the angle bisector $AD$ intersects the circumcircle of the triangle at point $P$ . Point $I$ is the center of the inscribed circle of triangle $ABC$ . It turned out that $ID=DP$. Find the ratio $AI:ID$.
In an isosceles triangle $ABC$ , the point $K$ is the midpoint of the side $AB$ , $M$ is the intersection point of the medians, $I$ is the center of the inscribed circle. It is known that $\angle KIB=90^o$ . Prove that $MI \perp BC$.
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