Let $ A$ and $ B$ be two Abelian groups, and define the sum of two homomorphisms $ \eta$ and $ \chi$ from $ A$ to $ B$ by \[ a( \eta\plus{}\chi)\equal{}a\eta\plus{}a\chi \;\textrm{for all}\ \;a \in A\ .\] With this addition, the set of homomorphisms from $ A$ to $ B$ forms an Abelian group $ H$. Suppose now that $ A$ is a $ p$-group ( $ p$ a prime number). Prove that in this case $ H$ becomes a topological group under the topology defined by taking the subgroups $ p^kH \;(k\equal{}1,2,...)$ as a neighborhood base of $ 0$. Prove that $ H$ is complete in this topology and that every connected component of $ H$ consists of a single element. When is $ H$ compact in this topology? [L. Fuchs]

The point $M$ is the center of a regular pentagon $ABCDE$. The point $P$ is an inner point of the line segment $[DM]$. The circumscribed circle of triangle $\vartriangle ABP$ intersects the side $[AE]$ at point $Q$ (different from $A$). The perpendicular from $P$ on $CD$ intersects the side $[AE] $ at point $S$. Prove that $PS$ is the bisector of $\angle APQ$.

The bisectors $AA_1$ and $CC_1$ of the right triangle $ABC$ ($\angle B = 90^o$) intersect at point $I$. The line passing through the point $C_1$ and perpendicular on the line $AA_1$ intersects the line that passes through $A_1$ and is perpendicular on $CC_1$, at the point $K$. Prove that the midpoint of the segment $KI$ lies on segment $AC$.

(a) Let $z_1,z_2$ be complex numbers such that $z_1z_2=1$ and $|z_1-z_2|=2$. Let $A,B,M_1,M_2$ denote the points in complex plane corresponding to $-1,1,z_1,z_2$, respectively. Show that $AM_1BM_2$ is a trapezoid and compute the lengths of its non-parallel sides. Specify the particular cases. (b) Let $\mathcal C_1$ and $\mathcal C_2$ be circles in the plane with centers $O_1$ and $O_2$, respectively, and with radius $d\sqrt2$, where $2d=O_1O_2$. Let $P$ and $Q$ be two variable points on $\mathcal C_1$ and $\mathcal C_2$ respectively, both on $O_1O_2$ on on different sides of $O_1O_2$, such that $PQ=2d$. Prove that the locus of midpoints $I$ of segments $PQ$ is the same as the locus of points $M$ with $MO_1\cdot MO_2=m$ for some $m$.

A beam of light strikes $\overline{BC}$ at point $C$ with angle of incidence $\alpha=19.94^\circ$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\overline{AB}$ and $\overline{BC}$ according to the rule: angle of incidence equals angle of reflection. Given that $\beta=\alpha/10=1.994^\circ$ and $AB=AC,$ determine the number of times the light beam will bounce off the two line segments. Include the first reflection at $C$ in your count. [asy] size(250);defaultpen(linewidth(0.7)); real alpha=24, beta=32; pair B=origin, C=(1,0), A=dir(beta), D=C+0.5*dir(alpha); pair EE=2*dir(180-alpha), E=intersectionpoint(C--EE, A--B); pair EEE=reflect(B,A)*EE, EEEE=reflect(C,B)*EEE, F=intersectionpoint(E--EEE, B--C), G=intersectionpoint(F--EEEE, A--B); draw((1.4,0)--B--1.4*dir(beta)); draw(D--C, linetype("4 4"),EndArrow(5)); draw(C--E, linetype("4 4"),EndArrow(5)); draw(E--F, linetype("4 4"),EndArrow(5)); draw(F--G, linetype("4 4"),EndArrow(5)); markscalefactor=0.01; draw(anglemark(C,B,A)); draw(anglemark((1.4,0), C,D)); label("$\beta$", 0.07*dir(beta/2), dir(beta/2), fontsize(10)); label("$\alpha$", C+0.07*dir(alpha/2), dir(alpha/2), fontsize(10)); label("$A$", A, dir(90)*dir(A)); label("$B$", B, dir(beta/2+180)); label("$C$", C, S);[/asy]

In triangle $ABC$ with incenter $I$, line $AI$ intersects the circumcircle of $ABC$ at $S\ne A$. Let the reflection of $I$ with respect to $BC$ be $J$, and suppose that line $SJ$ intersects the circumcircle of $ABC$ for the second time at point $P\ne S$. Show that $AI=PI.$ [i]József Mészáros[/i]

In $\triangle ABC, \angle BAC$ is a right angle. $BP$ and $CQ$ are bisectors of $\angle B$ and $\angle C$ respectively, which intersect $AC$ and $AB$ at $P$ and $Q$ respectively. Two perpendicular segments $PM$ and $QN$ are drawn on $BC$ from $P$ and $Q$ respectively. Find the value of $\angle MAN$ with proof.

Consider $ \triangle ABC$ and points $ M \in (AB)$, $ N \in (BC)$, $ P \in (CA)$, $ R \in (MN)$, $ S \in (NP)$, $ T \in (PM)$ such that $ \frac {AM}{MB} \equal{} \frac {BN}{NC} \equal{} \frac {CP}{PA} \equal{} k$ and $ \frac {MR}{RN} \equal{} \frac {NS}{SP} \equal{} \frac {PT}{TN} \equal{} 1 \minus{} k$ for some $ k \in (0, 1)$. Prove that $ \triangle STR \sim \triangle ABC$ and, furthermore, determine $ k$ for which the minimum of $ [STR]$ is attained.

The acute triangle $ABC$ satisfies $\overline {AB}<\overline {BC}<\overline {CA}$. Denote the foot of perpendicular from $A,B,C$ to opposing sides as $D,E,F$. Let $P$ a foot of perpendicular from $F$ to $DE$, and $Q(\neq F)$ a intersection point of line $FP$ and circumcircle of $BDF$. Prove that $\angle PBQ=\angle PAD$.

Let $C_1$ and $C_2$ be given circles. Let $A_1$ on $C_1$ and $A_2$ on $C_2$ be fixed points. If chord $A_1P_1$ of $C_1$ is parallel to chord $A_2P_2$ of $C_2$, find the locus of the midpoint of $P_1P_2$.

[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].

Lines $l$ and $m$ are perpendicular. Line $l$ partitions a convex polygon into two parts of equal area, and partitions the projection of the polygon onto $m$ into two line segments of length $a$ and $b$ respectively. Determine the maximum value of $\left\lfloor \frac{1000a}{b} \right\rfloor$. (The floor notation $\lfloor x \rfloor$ denotes largest integer not exceeding $x$)

Let $AL$ be the bisector of triangle $ABC$. Circle $\omega_1$ is circumscribed around triangle $ABL$. Tangent to $\omega_1$ at point $B$ intersects the extension of $AL$ at point $K$. The circle $\omega_2$ circumscribed around the triangle $CKL$ intersects $\omega_1$ a second time at point $Q$, with $Q$ lying on the side $AC$. Find the value of the angle $ABC$. (Vladislav Radomsky)

A semicircle is inscribed in a semicircle of radius $2$ as shown. Find the radius of the smaller semicircle. [img]https://cdn.artofproblemsolving.com/attachments/c/1/c60cd40eaecfe417aca46ce4fd386fe22af85b.png[/img]

In a regular hexagonal pyramid $SABCDEF$, a plane is drawn through vertex $A$ and the midpoints of edges $SC$ and $CE$. Find the ratio in which this plane divides the volume of the pyramid.

We call a bar of width ${w}$ on the surface of the unit sphere ${\Bbb{S}^2}$, a spherical segment, centered at the origin, which has width ${w}$ and is symmetric with respect to the origin. Prove that there exists a constant ${c>0}$, such that for any positive integer ${n}$ the surface ${\Bbb{S}^2}$ can be covered with ${n}$ bars of the same width so that any point is contained in no more than ${c\sqrt{n}}$ bars.

Let $ABC$ be a triangle with angles $\angle A = 80^\circ, \angle B = 70^\circ, \angle C = 30^\circ$. Let $P$ be a point on the bisector of $\angle BAC$ satisfying $\angle BPC =130^\circ$. Let $PX, PY, PZ$ be the perpendiculars drawn from $P$ to the sides $BC, AC, AB$, respectively. Prove that the following equation with segment lengths is satisfied $$AY^3+BZ^3+CX^3=AZ^3+BX^3+CY^3.$$

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. Prove that there exists a point $J$ such that for any point $X$ inside $ABC$ if $AX,BX,CX$ intersect $\omega$ in $A_1,B_1,C_1$ and $A_2,B_2,C_2$ be reflections of $A_1,B_1,C_1$ in midpoints of $BC,AC,AB$ respectively then $A_2,B_2,C_2,J$ lie on a circle.

Let $ABC$ be an acute triangle with $AB<AC$ and let $O$ be its circumcenter. Let $D$ be a point on the segment $AC$ such that $AB=AD$. Let $E$ be the intersection of the line $AB$ with the perpendicular line to $AO$ through $D$. Let $F$ be the intersection of the perpendicular line to $OC$ through $C$ with the line parallel to $AC$ and passing through $E$. Finally, let the lines $CE$ and $DF$ intersect in $G$. Show that $AG$ and $BF$ are parallel.

Let $ ABCD$ be a convex quadrilateral with opposite side not parallel. The line through $ A$ parallel to $ BD$ intersect line $ CD$ in $ F$, but parallel through $ D$ to $ AC$ intersect line $ AB$ at $ E$. Denote by $ M,N,P,Q$ midpoints of the segments $ AC,BD,AF,DE$. Prove that lines $ MN,PQ$ and $ AD$ are concurrent.

Triangle $ABC$ is the right angled triangle with the vertex $C$ at the right angle. Let $P$ be the point of reflection of $C$ about $AB$. It is known that $P$ and two midpoints of two sides of $ABC$ lie on a line. Find the angles of the triangle.

The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$. [i]Proposed by Evan Chen[/i]

Two circles $\Gamma_1$ and $\Gamma_2$ have common external tangents $\ell_1$ and $\ell_2$ meeting at $T$. Suppose $\ell_1$ touches $\Gamma_1$ at $A$ and $\ell_2$ touches $\Gamma_2$ at $B$. A circle $\Omega$ through $A$ and $B$ intersects $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$, such that quadrilateral $ABCD$ is convex. Suppose lines $AC$ and $BD$ meet at point $X$, while lines $AD$ and $BC$ meet at point $Y$. Show that $T$, $X$, $Y$ are collinear. [i]Merlijn Staps[/i]

A storage depot is a pyramid with height $30$ and a square base with side length $40$. Determine how many cubical $3\times 3\times 3$ boxes can be stored in this depot if the boxes are always packed so that each of their edges is parallel to either an edge of the base or the altitude of the pyramid.