By straightedge and compass, reconstruct a right triangle $ABC$ ($\angle C = 90^o$), given the vertices $A, C$ and a point on the bisector of angle $B$.

The radii of the circumcircle and the incircle of a right triangle are given. Cconstruct that triangle with compass and ruler, describe the construction and justify why it is correct.

A point $P$ lies on side $AB$ of a convex quadrilateral $ABCD$. Let $\omega$ be the inscribed circumference of triangle $CPD$ and $I$ the centre of $\omega$. It is known that $\omega$ is tangent to the inscribed circumferences of triangles $APD$ and $BPC$ at points $K$ and $L$ respectively. Let $E$ be the point where the lines $AC$ and $BD$ intersect, and $F$ the point where the lines $AK$ and $BL$ intersect. Prove that the points $E, I, F$ are collinear.

$ABC$ is a triangle. Take $a = BC$ etc as usual. Take points $T_1, T_2$ on the side $AB$ so that $AT_1 = T_1T_2 = T_2B$. Similarly, take points $T_3, T_4$ on the side BC so that $BT_3 = T_3T_4 = T_4C$, and points $T_5, T_6$ on the side $CA$ so that $CT_5 = T_5T_6 = T_6A$. Show that if $a' = BT_5, b' = CT_1, c'=AT_3$, then there is a triangle $A'B'C'$ with sides $a', b', c'$ ($a' = B'C$' etc). In the same way we take points $T_i'$ on the sides of $A'B'C' $ and put $a'' = B'T_6', b'' = C'T_2', c'' = A'T_4'$. Show that there is a triangle $A'' B'' C'' $ with sides $a'' b'' , c''$ and that it is similar to $ABC$. Find $a'' /a$.

$\vartriangle A_0B_0C_0$ has side lengths $A_0B_0 = 13$, $B_0C_0 = 14$, and $C_0A_0 = 15$. $\vartriangle A_1B_1C_1$ is inscribed in the incircle of $\vartriangle A_0B_0C_0$ such that it is similar to the first triangle. Beginning with $\vartriangle A_1B_1C_1$, the same steps are repeated to construct $\vartriangle A_2B_2C_2$, and so on infinitely many times. What is the value of $\sum_{i=0}^{\infty} A_iB_i$?

Triangle $ABC$ is inscribed in a circle with center $O'$. A circle with center $O$ is inscribed in triangle $ABC$. $AO$ is drawn, and extended to intersect the larger circle in $D$. Then, we must have: $\text{(A)}\ CD=BD=O'D \qquad \text{(B)}\ AO=CO=OD \qquad \text{(C)}\ CD=CO=BD \qquad\\ \text{(D)}\ CD=OD=BD \qquad \text{(E)}\ O'B=O'C=OD $ [asy] size(200); defaultpen(linewidth(0.8)+fontsize(12pt)); pair A=origin,B=(15,0),C=(5,9),O=incenter(A,B,C),Op=circumcenter(A,B,C); path incirc = incircle(A,B,C),circumcirc = circumcircle(A,B,C),line=A--3*O; pair D[]=intersectionpoints(circumcirc,line); draw(A--B--C--A--D[0]^^incirc^^circumcirc); dot(O^^Op,linewidth(4)); label("$A$",A,dir(185)); label("$B$",B,dir(355)); label("$C$",C,dir(95)); label("$D$",D[0],dir(O--D[0])); label("$O$",O,NW); label("$O'$",Op,E);[/asy]

The diagonals $AC$ and $BD$ of a convex cyclic quadrilateral $ABCD$ intersect at point $E$. Given that $AB = 39, AE = 45, AD = 60$ and $BC = 56$, determine the length of $CD.$

Let $ABC$ be a triangle and the points $M$ and $N$ on the sides $AB$ respectively $BC$, such that $2 \cdot \frac{CN}{BC} = \frac{AM}{AB}$. Let $P$ be a point on the line $AC$. Prove that the lines $MN$ and $NP$ are perpendicular if and only if $PN$ is the interior angle bisector of $\angle MPC$.

Prove that for every positive integer $m$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $m$ points in $S$ at unit distance from $A$.

Let $ABCD$ be a square of side length $1$, and let $P_1, P_2$ and $P_3$ be points on the perimeter such that $\angle P_1P_2P_3 = 90^\circ$ and $P_1, P_2, P_3$ lie on different sides of the square. As these points vary, the locus of the circumcenter of $\triangle P_1P_2P_3$ is a region $\mathcal{R}$; what is the area of $\mathcal{R}$?

Let a right cone of the base radius $r=3$ and height greater than $6$ be inscribed in a sphere of radius $R=6$. The volume of the cone can be expressed as $\pi(a\sqrt{b}+c)$, where $b$ is square free. Find $a+b+c$.

Let $ABC$ be a right triangle with hypotenuse $AC$. Let $G$ be the centroid of this triangle and suppose that we have $AG^2 + BG^2 + CG^2 = 156$. Find $AC^2$.

Let $ABC$ be an acute triangle with $AB>AC$. Prove that there exists a point $D$ with the following property: whenever two distinct points $X$ and $Y$ lie in the interior of $ABC$ such that the points $B$, $C$, $X$, and $Y$ lie on a circle and $$\angle AXB-\angle ACB=\angle CYA-\angle CBA$$ holds, the line $XY$ passes through $D$.

$ABC$ is a non-isosceles triangle. $O, I, H$ are respectively the center of its circumscribed circle, the inscribed circle and its orthocenter. prove that $\widehat{OIH}$ is obtuse.

A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$.

[b]p1.[/b] Consider this statement: An equilateral polygon circumscribed about a circle is also equiangular. Decide whether this statement is a true or false proposition in euclidean geometry. If it is true, prove it; if false, produce a counterexample. [b]p2.[/b] Show that the fraction $\frac{x^2-3x+1}{x-3}$ has no value between $1$ and $5$, for any real value of $x$. [b]p3.[/b] A man walked a total of $5$ hours, first along a level road and then up a hill, after which he turned around and walked back to his starting point along the same route. He walks $4$ miles per hour on the level, three miles per hour uphill, and $r$ miles per hour downhill. For what values of $r$ will this information uniquely determine his total walking distance? [b]p4.[/b] A point $P$ is so located in the interior of a rectangle that the distance of $P$ from one comer is $5$ yards, from the opposite comer is $14$ yards, and from a third comer is $10$ yards. What is the distance from $P$ to the fourth comer? [b]p5.[/b] Each small square in the $5$ by $5$ checkerboard shown has in it an integer according to the following rules: $\begin{tabular}{|l|l|l|l|l|} \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline \end{tabular}$ i. Each row consists of the integers $1, 2, 3, 4, 5$ in some order. ii. Тhе order of the integers down the first column has the same as the order of the integers, from left to right, across the first row and similarly fог any other column and the corresponding row. Prove that the diagonal squares running from the upper left to the lower right contain the numbers $1, 2, 3, 4, 5$ in some order. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A convex polygon of sides $\ell_1, \ell_2, ..., \ell_n$ is called [i]ordered [/i] if for all reordering $( \sigma (1), \sigma (2), ..., \sigma (n))$ of the set $(1, 2,..., n)$ there exists a point $P$ inside the polygon such that $d_{\sigma (1)} < _{\sigma (2)} <...< d_{\sigma (n)}$ , where $d_i$ represents the distance between $P$ and side $\ell_i$. Find all the convex ordered polygons.

Let $ ABC$ be an acute triangle, and $ M_a$, $ M_b$, $ M_c$ be the midpoints of the sides $ a$, $ b$, $ c$. The perpendicular bisectors of $ a$, $ b$, $ c$ (passing through $ M_a$, $ M_b$, $ M_c$) intersect the boundary of the triangle again in points $ T_a$, $ T_b$, $ T_c$. Show that if the set of points $ \left\{A,B,C\right\}$ can be mapped to the set $ \left\{T_a, T_b, T_c\right\}$ via a similitude transformation, then two feet of the altitudes of triangle $ ABC$ divide the respective triangle sides in the same ratio. (Here, "ratio" means the length of the shorter (or equal) part divided by the length of the longer (or equal) part.) Does the converse statement hold?

Consider the triangle $ ABC$ with $ m(\angle BAC \equal{} 90^\circ)$ and $ AC \equal{} 2AB$. Let $ P$ and $ Q$ be the midpoints of $ AB$ and $ AC$,respectively. Let $ M$ and $ N$ be two points found on the side $ BC$ such that $ CM \equal{} BN \equal{} x$. It is also known that $ 2S[MNPQ] \equal{} S[ABC]$. Determine $ x$ in function of $ AB$.

Given triangle $ ABC$ with $ C$ a right angle, show that the diameter of the incenter is $ a\plus{}b\minus{}c$, where $ a\equal{}BC$, $ b\equal{}CA$, and $ c\equal{}AB$.

Suppose that the sides $AB$ and $DC$ of a convex quadrilateral $ABCD$ are not parallel. On the sides $BC$ and $AD$, pairs of points $(M,N)$ and $(K,L)$ are chosen such that $BM=MN=NC$ and $AK=KL=LD$. Prove that the areas of triangles $OKM$ and $OLN$ are different, where $O$ is the intersection point of $AB$ and $CD$.

Let $ABC$ be an acute, non-isosceles triangle with the circumcircle $(O)$. Denote $D, E$ as the midpoints of $AB,AC$ respectively. Two circles $(ABE)$ and $(ACD)$ intersect at $K$ differs from $A$. Suppose that the ray $AK$ intersects $(O)$ at $L$. The line $LB$ meets $(ABE)$ at the second point $M$ and the line $LC$ meets $(ACD)$ at the second point $N$. a) Prove that $M, K, N$ collinear and $MN$ perpendicular to $OL$. b) Prove that $K$ is the midpoint of $MN$

Let $ABCD$ be an inscribed trapezoid such that the sides $[AB]$ and $[CD]$ are parallel. If $m(\widehat{AOD})=60^\circ$ and the altitude of the trapezoid is $10$, what is the area of the trapezoid?

The winners of first AGMC in 2019 gifts the person in charge of the organiser, which is a polyhedron formed by $60$ congruent triangles. From the photo, we can see that this polyhedron formed by $60$ quadrilateral spaces. (Note: You can find the photo in 3.4 of [url]https://files.alicdn.com/tpsservice/18c5c7b31a7074edc58abb48175ae4c3.pdf?spm=a1zmmc.index.0.0.51c0719dNAbw3C&file=18c5c7b31a7074edc58abb48175ae4c3.pdf[/url]) A quadrilateral space is the plane figures that we fold the figures following the diagonal on a $n$ sides on a plane (i.e. form an appropriate dihedral angle in where the chosen diagonal is). "Two figure spaces are congruent" means they can coincide completely by isometric transform in $\mathbb{R}^3$. A polyhedron is the bounded space region, whose boundary is formed by the common edge of finite polygon. (a) We know that $2021=43\times 47$. Does there exist a polyhedron, whose surface can be formed by $43$ congruent $47$-gon? (b) Prove your answer in (a) with logical explanation.

The circumradius $R$ of a triangle with side lengths $a, b, c$ satisfies $R =\frac{a\sqrt{bc}}{b+c}$. Find the angles of the triangle.