Found problems: 329
Given a triangle $ABC$, with right $\angle A$, we know: the point $T$ of tangency of the circumference inscribed in $ABC$ with the hypotenuse $BC$, the point $D$ of intersection of the angle bisector of $\angle B$ with side AC and the point E of intersection of the angle bisector of $\angle C$ with side $AB$ . Describe a construction with ruler and compass for points $A$, $B$, and $C$. Justify.
Given a triangle $ABC$ with $\angle C = 90^o$.
a) Construct the circle with center $C$, so that one of the tangents from $A$ to that circle is parallel to one of the tangents from $B$ to that circle.
b) A circle with center $C$ has two parallel tangents passing through A and go respectively. If $AC = b$ and $BC = a$, express the radius of the circle in terms of $a$ and $b$.
Given a triangle $ABC$, on the side $BC$ which marked the point $E$ such that $BE \ge CE$. Construct on the sides $AB$ and $AC$ the points $D$ and $F$, respectively, such that $\angle DEF = 90 {} ^ \circ$ and the segment $BF$ is bisected by the segment $DE $.
(Black Maxim)
Inside a given convex quadrilateral, find a point such that the segments connecting this point with the midpoints of the quadrilateral's sides divide the quadrilateral into four parts with equal areas.
Inside an acute angle is a circle. Investigate the possibility of constructing with only a compass and a ruler, a tangent to this circle that the point of contact will bisect the segment of the tangent that is cut off by the sides of the angle.
We want to construct an isosceles triangle using a compass and a straightedge. We are given two of the following four data: the length of the base of the triangle $(a),$ the length of the leg of the triangle $(b),$ the radius of the inscribed circle $(r),$ and the radius of the circumscribed circle $(R).$ In which of the six possible cases will we definitely be able to construct the triangle?
[i]Proposed by György Rubóczky, Budapest[/i]
There is a table with $n > 2$ cells in the first row, $n-1$ cells in the second row is a cell, $n-2$ in the third row, $\ldots$, $1$ cell in the $n$-th row. The cells are arranged as shown below.
[img]https://i.ibb.co/0Z1CR0c/UMO24-8-2.png[/img]
In each cell of the top row Petryk writes a number from $1$ to $n$, so that each number is written exactly once. For each other cell, if the cells directly above it contains numbers $a, b$, it contains number $|a-b|$. What is the largest number that can be written in a single cell of the bottom row?
[i]Proposed by Bogdan Rublov[/i]
Two circles meeting at points $A, B$ and a point $O$ lying outside them are given. Using a compass and a ruler construct a ray with origin $O$ meeting the first circle at point $C$ and the second one at point $D$ in such a way that the ratio $OC : OD$ be maximal.
Construct a quadrilateral with three sides $1$, $4$ and $3$ so that a circle could be circumscribed around it.
Construct a convex polyhedron of equal “bricks” shown in Figure.
[img]https://cdn.artofproblemsolving.com/attachments/6/6/75681a90478f978665b6874d0c0c9441ea3bd2.gif[/img]
Construct a square knowing the sum of the diagonal and the side.
Three segments $2$ cm, $5$ cm and $12$ cm long are constructed on the plane. Construct a trapezoid with bases of $2$ cm and $5$ cm, the sum of the sides of which is $12$ cm, and one of the angles is $60^o$.
(Bogdan Rublev)
A circle $k$ is drawn using a given disc (e.g. a coin). A point $A$ is chosen on $k$. Using just the given disc, determine the point $B$ on $k$ so that $AB$ is a diameter of $k$. (You are allowed to choose an arbitrary point in one of the drawn circles, and using the given disc it is possible to construct either of the two circles that passes through the points at a distance that is smaller than the radius of the circle.)
We are given two concentric circles, Construct a square whose two vertices lie on one circle and the other two on the other circle.
Given quadrilateral $ABCD$ and the directions of its sides. Inscribe a rectangle in $ABCD$.
In a triangle $ABC$ the bisectors $BB'$ and $CC'$ are drawn. After that, the whole picture except the points $A, B'$, and $C'$ is erased. Restore the triangle using a compass and a ruler.
(A.Karlyuchenko)
Construct a triangle, given the feet of its altitudes. Express the sides of a triangle $Y$ in terms of the sides of the triangle $X$ formed by the feet of the altitudes of $Y$.
A triangle $ABC$ $(a>b>c)$ is given. Its incenter $I$ and the touching points $K, N$ of the incircle with $BC$ and $AC$ respectively are marked. Construct a segment with length $a-c$ using only a ruler and drawing at most three lines.
Given a circle $k$ and two points $A$ and $B$ outside the circle. Specify how to can construct a circle with a compass and ruler, so that $A$ and $B$ lie on that circle and that circle is tangent to $k$.
Let $ABC$ be an acute-angled triangle. Construct points $A', B', C'$ on its sides $BC, CA, AB$ such that:
- $A'B' \parallel AB$,
- $C'C$ is the bisector of angle $A'C'B'$,
- $A'C' + B'C'= AB$.
Find all positive integers $n$ satisfying the following: there exists a way to fill in $1, \cdots, n^2$ into a $n \times n$ grid so that each block has exactly one number, each number appears exactly once, and:
1. For all positive integers $1 \leq i < n^2$, $i$ and $i + 1$ are neighbors (two numbers neighbor each other if and only if their blocks share a common edge.)
2. Any two numbers among $1^2, \cdots, n^2$ are not in the same row or the same column.
From the triangle $ABC$, are gicen only the incenter $I$, the touchpoint $K$ of the inscribed circle with the side $AB$, as well as the center $I_a$ of the exscribed circle, that touches the side $BC$ . Construct a triangle equal in size to triangle $ABC$.
(Gryhoriy Filippovskyi)
Circle $\omega$ is inscribed in the $\vartriangle ABC$, with center $I$. Using only a ruler, divide segment $AI$ in half.
(Grigory Filippovsky)
The cells of a $8 \times 8$ table are colored either black or white so that each row has a different number of black squares, and each column has a different number of black squares. What is the maximum number of pairs of adjacent cells of different colors?
At a position $O$ of an airport in a plateau there is a gun which can rotate arbitrarily. Two tanks moving along two given segments $AB$ and $CD$ attack the airport. Determine, by a ruler and a compass, the reach of the gun, knowing that the total length of the parts of the trajectories of the two tanks reachable by the gun is equal to a given length $\ell$.