Give a triangle $ABC$ with heights $h_a = 3$ cm, $h_b = 7$ cm and $h_c = d$ cm, where $d$ is an integer. Determine $d$.

In the cuboid $ABCDA'B'C'D'$ with $AB=a$, $AD=b$ and $AA'=c$ such that $a>b>c>0$, the points $E$ and $F$ are the orthogonal projections of $A$ on the lines $A'D$ and $A'B$, respectively, and the points $M$ and $N$ are the orthogonal projections of $C$ on the lines $C'D$ and $C'B$, respectively. Let $DF\cap BE=\{G\}$ and $DN\cap BM=\{P\}$. [list=a] [*] Show that $(A'AG)\parallel (C'CP)$ and determine the distance between these two planes; [*] Show that $GP\parallel (ABC)$ and determine the distance between the line $GP$ and the plane $(ABC)$. [/list] [i]Petre Simion, Nicolae Victor Ioan[/i]

The triangle $ABC$ is isosceles with $AB=AC$, and $\angle{BAC}<60^{\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\angle{ABD}\equiv\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\angle{BDC}$ and $\angle{ACB}$. Compute $\angle{COD}$.

Let $ABC$ be a right triangle with $\angle A= 90^{\circ}$. A circle $\omega$ centered on $BC$ is tangent to $AB$ at $D$ and $AC$ at $E$. Let $F$ and $G$ be the intersections of $\omega$ and $BC$ so that $F$ lies between $B$ and $G$. If lines $DG$ and $EF$ intersect at $X$, show that $AX=AD.$

Several fleas sit on the squares of a $10\times 10$ chessboard (at most one fea per square). Every minute, all fleas simultaneously jump to adjacent squares. Each fea begins jumping in one of four directions (up, down, left, right), and keeps jumping in this direction while it is possible; otherwise, it reverses direction on the opposite. It happened that during one hour, no two fleas ever occupied the same square. Find the maximal possible number of fleas on the board.

The altitudes $CC_1$ and $BB_1$ are drawn in the acute triangle $ABC$. The bisectors of angles $\angle BB_1C$ and $\angle CC_1B$ intersect the line $BC$ at points $D$ and $E$, respectively, and meet each other at point $X$. Prove that the intersection points of circumcircles of the triangles $BEX$ and $CDX$ lie on the line $AX$. [i](A. Voidelevich)[/i]

Let $ABC$ be a triangle with a circumcircle $(K)$. A circle touching the sides $AB,AC$ is internally tangent to $(K)$ at $K_a$; two other points $K_b,K_c$ are defined in the same manner. Prove that the area of triangle $K_aK_bK_c$ does not exceed that of triangle $ABC$. Nguyen Minh Ha, Hanoi University of Education, Xuan Thuy, Cau Giay, Hanoi.

Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.

In a triangle $ABC$ with $AB\not = AC$, the incircle touches the sides $BC, CA, AB$ at $D, E, F$ , respectively. Line $AD$ meets the incircle again at $P$ . The line $EF$ and the line through $P$ perpendicular to $AD$ meet at $Q$. Line $AQ$ intersects $DE$ at $X$ and $DF$ at $Y$ . Prove that $A$ is the midpoint of $XY$.

We color all points of a plane using $3$ colors. Prove that there are at least two points of the plane having same colours and with distance among them $1$.

Denote in the triangle $ABC$ by $T_A,T_B,T_C$ the touch points of the exscribed circles of $\vartriangle ABC$, tangent to sides $BC, AC$ and $AB$ respectively. Let $O$ be the center of the circumcircle of $\vartriangle ABC$, and $I$ is the center of it's inscribed circle. It is known that $OI\parallel AC$. Prove that $\angle T_A T_B T_C= 90^o - \frac12 \angle ABC$. (Anton Trygub)

Points $M$ and $N$ are the midpoints of sides $AC$ and $BC$ of a triangle $ABC$. It is known that $\angle MAN = 15^o$ and $\angle BAN = 45^o$. Find the value of angle $\angle ABM$. (A. Blinkov)

Let $ABC$ be a non-equilateral triangle and let $R$ be the radius of its circumcircle. The incircle of $ABC$ has $I$ as its centre and is tangent to side $CA$ in point $D$ and to side $CB$ in point $E$. Let $A_1$ be the point on line $EI$ such that $A_1I=R$, with $I$ being between $A_1$ and $E$. Let $B_1$ be the point on line $DI$ such that $B_1I=R$, with $I$ being between $B_1$ and $D$. Let $P$ be the intersection of lines $AA_1$ and $BB_1$. (a) Prove that $P$ belongs to the circumcircle of $ABC$. (b) Let us now also suppose that $AB=1$ and $P$ coincides with $C$. Determine the possible values of the perimeter of $ABC$.

[b]p29.[/b] Consider pentagon $ABCDE$. How many paths are there from vertex $A$ to vertex $E$ where no edge is repeated and does not go through $E$. [b]p30.[/b] Let $a_1, a_2, ...$ be a sequence of positive real numbers such that $\sum^{\infty}_{n=1} a_n = 4$. Compute the maximum possible value of $\sum^{\infty}_{n=1}\frac{\sqrt{a_n}}{2^n}$ (assume this always converges). [b]p31.[/b] Define function $f(x) = x^4 + 4$. Let $$P =\prod^{2021}_{k=1} \frac{f(4k - 1)}{f(4k - 3)}.$$ Find the remainder when $P$ is divided by $1000$. [b]p32.[/b] Reduce the following expression to a simplified rational: $\cos^7 \frac{\pi}{9} + \cos^7 \frac{5\pi}{9}+ \cos^7 \frac{7\pi}{9}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given three points $H_1, H_2, H_3$ on a plane. The points are the reflections of the intersection point of the heights of the triangle $\vartriangle ABC$ through its sides. Construct $\vartriangle ABC$.

(a) Given a point $O$ inside an equilateral triangle $\vartriangle ABC$. Line $OG$ connects $O$ with the center of mass $G$ of the triangle and intersects the sides of the triangle, or their extensions, at points $A', B', C'$ . Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G} = 3.$$ (b) Point $G$ is the center of the sphere inscribed in a regular tetrahedron $ABCD$. Straight line $OG$ connecting $G$ with a point $O$ inside the tetrahedron intersects the faces at points $A', B', C', D'$. Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G}+ \frac{D'O}{D'G} = 4.$$

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

Let $k$ be a circle with diameter $AB$ and centre $O$. Let C be an arbitrary point on the circle different from $A$ and $B$. Let $D$ be the point for which $O$, $B$, $D$ and $C$ (in this order) are the four vertices of a parallelogram. Let $E$ be the intersection of the line $BD$ and the circle $k$, and let $F$ be the orthocenter of the triangle $OAC$. Prove that the points $O, D, E, C, F$ lie on a circle.

The diagonals of a cyclic quadrilateral $ABCD$ meet at point $P$. Points $K$ and $L$ lie on $AC$, $BD$ respectively in such a way that $CK=AP$ and $DL=BP$. Prove that the line joining the common points of circles $ALC$ and $BKD$ passes through the mass-center of $ABCD$. Proposed by:V.Konyshev

Given an isosceles triangle $ABC$ (with $AB=BC$). A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through the circumcentre of triangle $ABC$. [b]proposed by Sergej Berlov[/b]

Let a [i]triplet [/i] be some set of three distinct pairwise parallel lines. $20$ triplets are drawn on a plane. Find the maximum number of regions these $60$ lines can divide the plane into.

Let $ABC$ be a scalene triangle. Let $\ell$ be a line passing through point $B$ that lies outside of the triangle $ABC$ and creates different angles with sides $AB$ and $BC$ . The point $M$ is the midpoint of side $AC$ and the ponts $H_a$ and $H_c$ are the bases of the perpendicular lines on the line $\ell$ drawn from points $A$ and $C$ respectively. The circle circumscribing triangle $MBH_a$ intersects AB at the point $A_1$ and the circumscribed circle of triangle $MBH_c$ intersects $BC$ at point $C_1$. The point $A_2$ is symmetric to the point $A$ relative to the point $A_1$ and the point $C_2$ is symmetric to the point $C_1$ relative to the point $C_1$. Prove that the lines $\ell$, $AC_2$ and $CA_2$ intersect at one point. (Yana Kolodach)

(A.Zaslavsky, 8) A triangle can be dissected into three equal triangles. Prove that some its angle is equal to $ 60^{\circ}$.

$ ABCD$ is a square with side of unit length. Points $ E$ and $ F$ are taken respectively on sides $ AB$ and $ AD$ so that $ AE \equal{} AF$ and the quadrilateral $ CDFE$ has maximum area. In square units this maximum area is: $ \textbf{(A)}\ \frac12 \qquad \textbf{(B)}\ \frac {9}{16} \qquad \textbf{(C)}\ \frac{19}{32} \qquad \textbf{(D)}\ \frac {5}{8} \qquad \textbf{(E)}\ \frac23$

The number of regular triangles that three apexes are among eight vertex of a cube is $\text{(A)}4\qquad\text{(B)}8\qquad\text{(C)}12\qquad\text{(D)}24$