Consider a box of dimensions $10$ cm $\times 16$ cm $\times 1$ cm. Determine the maximum number of balls of diameter $ 1$ cm that the box can contain.

Seven spheres are situated in space such that no three centers are collinear, no four centers are coplanar, and every pair of spheres intersect each other at more than one point. For every pair of spheres, the plane on which the intersection of the two spheres lies in is drawn. What is the least possible number of sets of four planes that intersect in at least one point?

There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.

Three rays are going out from point $O$ in space, forming pairwise angles $\alpha, \beta$ and $\gamma$ with $0^o<\alpha \le \beta \le \gamma <180^o$. Prove that $\sin \frac{\alpha}{2}+ \sin \frac{\beta}{2} > \sin \frac{\gamma}{2}$.

Problem 4 Let $ABCD$ be a rectangle with diagonals $AC$ and $BD$. Let $E$ be the intersection of the bisector of $\angle CAD$ with segment $CD$, $F$ on $CD$ such that $E$ is midpoint of $DF$, and $G$ on $BC$ such that $BG = AC$ (with $C$ between $B$ and $G$). Prove that the circumference through $D$, $F$ and $G$ is tangent to $BG$.

in a quadrilateral $ABCD$ digonals are perpendicular to each other. let $S$ be the intersection of digonals. $K$,$L$,$M$ and $N$ are reflections of $S$ to $AB$,$BC$,$CD$ and $DA$. $BN$ cuts the circumcircle of $SKN$ in $E$ and $BM$ cuts the circumcircle of $SLM$ in $F$. prove that $EFLK$ is concyclic.(20 points)

Consider a parabola. The [i]parabolic length[/i] of a segment is the length of the projection of this segment on a straight line perpendicular to the axis of symmetry of the parabola. In the parabola, two chords $AB$ and $CD$ are drawn, intersecting at the point $N{}$. Prove that the product of the parabolic lengths of the segments $AN$ and $BN$ is equal to the product of the parabolic lengths of the segments $CN$ and $DN$. [i]Proposed by M. Panov[/i]

Two rays start from a common point and have an angle of $60$ degrees. Circle $C$ is drawn with radius $42$ such that it is tangent to the two rays. Find the radius of the circle that has radius smaller than circle $C$ and is also tangent to $C$ and the two rays.

Let $ A $ be a finite set of points in space having the property that for any of its points $ P, Q $ there is an isometry of space that transforms the set $ A $ into the set $ A $ and the point $ P $ into the point $ Q $. Prove that there is a sphere passing through all points of the set $ A $.

Consider a right-angled triangle $ABC$ with $\angle C = 90^o$. Suppose that the hypotenuse $AB$ is divided into four equal parts by the points $D,E,F$, such that $AD = DE = EF = FB$. If $CD^2 +CE^2 +CF^2 = 350$, find the length of $AB$.

Given a regular polygon with apothem $ A $ and circumradius $ R $. Find for a regular polygon of equal perimeter and with double number of sides, the apothem $ a $ and the circumcircle $ r $ in terms of $A,R$

The angles of an acute triangle $ABC$ at $\alpha , \beta, \gamma$. Let $AD$ be a height, $CF$ a median, and $BE$ the bisector of $\angle B$. Show that $AD,CF$ and $BE$ are concurrent if and only if $\cos \gamma \tan\beta = \sin \alpha$ .

Let $M$ be a point in the plane, distinct from the vertices of $\triangle ABC$. Consider $N,P,Q$ the reflections of $M$ with respect to lines $AB, BC$ and $CA$, in this order. a) Prove that $N, P ,Q$ are collinear if and only if $M$ lies on the circumcircle of $\triangle ABC$. b) If $M$ does not lie on the circumcircle of $\triangle ABC$ and the centroids of triangles $\triangle ABC$ and $\triangle NPQ$ coincide, prove that $\triangle ABC$ is equilateral.

[b]7.1 / 6.3[/b] All integers from 1 to 1966 are written on the board. Allowed is to erase any two numbers by writing their difference instead. Prove that repeating such an operation many times cannot ensure that There are only zeros left on the board. [b]7.2 [/b] Prove that the radius of a circle is equal to the difference between the lengths of two chords, one of which subtends an arc of $1/10$ of a circle, and the other subtends an arc in $3/10$ of a circle. [b]7.3[/b] Prove that for any natural number $n$ the number $ n(2n+1)(3n+1)...(1966n + 1) $ is divisible by every prime number less than $1966$. [b]7.4[/b] What number needs to be put in place * so that the next the problem had a unique solution: [i]“There are n straight lines on the plane, intersecting at * points. Find n.” ?[/i] [b]7.5 / 6.4[/b] Black paint was sprayed onto a white surface. Prove that there are three points of the same color lying on the same line, and so, that one of the points lies in the middle between the other two. [b]7.6 [/b] There are $n$ points on the plane so that any triangle with vertices at these points has an area less than $1$. Prove that all these points can be enclosed in a triangle of area $4$. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].

A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.

Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom? [img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]

$ABC$ is triangle. Point $L$ is inside $ABC$ and lies on bisector of $\angle B$. $K$ is on $BL$. $\angle KAB=\angle LCB= \alpha$. Point $P$ inside triangle is such, that $AP=PC$ and $\angle APC=2\angle AKL$. Prove that $\angle KPL=2\alpha$

In a triangle $ABC$, with $\widehat{A} > 90^\circ$, let $O$ and $H$ denote its circumcenter and orthocenter, respectively. Let $K$ be the reflection of $H$ with respect to $A$. Prove that $K, O$ and $C$ are collinear if and only if $\widehat{A} - \widehat{B} = 90^\circ$.

Let $X_1$, $Z_2$, $Y_1$, $X_2$, $Z_1$, $Y_2$ be six points lying on the periphery of a circle (in this order). Let the chords $Y_1Y_2$ and $Z_1Z_2$ meet at a point $A$; let the chords $Z_1Z_2$ and $X_1X_2$ meet at a point $B$; let the chords $X_1X_2$ and $Y_1Y_2$ meet at a point $C$. Prove that $\left( BX_2-CX_1\right) \cdot BC+\left( CY_2-AY_1\right) \cdot CA+\left( AZ_2-BZ_1\right) \cdot AB=0$. [i]Comment on the source.[/i] The problem is inspired by Stergiu's proof in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=326112#p326112]http://www.mathlinks.ro/Forum/viewtopic.php?t=50262 post #5[/url]. Darij

In the triangle $ABC$ we have $|AB|<|AC|$. The bisectors of the angles at $B$ and $C$ meet $AC$ and $AB$ at $D$ and $E$ respectively. $BD$ and $CE$ intersect at the incenter $I$ of $\triangle ABC$. Prove that $\angle BAC=60^\circ$ if and only if $|IE|=|ID|$

Line $PQ$ is tangent to the incircle of triangle $ABC$ in such a way that the points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively. On the sides $AB$ and $AC$ are selected points $M$ and $N$, respectively, so that $AM = BP$ and $AN = CQ$. Prove that all lines constructed in this manner $MN$ pass through one point

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

$ ABC$ is an acute-angled triangle. $ M$ is the midpoint of $ BC$ and $ P$ is the point on $ AM$ such that $ MB \equal{} MP$. $ H$ is the foot of the perpendicular from $ P$ to $ BC$. The lines through $ H$ perpendicular to $ PB$, $ PC$ meet $ AB, AC$ respectively at $ Q, R$. Show that $ BC$ is tangent to the circle through $ Q, H, R$ at $ H$. [i]Original Formulation: [/i] For an acute triangle $ ABC, M$ is the midpoint of the segment $ BC, P$ is a point on the segment $ AM$ such that $ PM \equal{} BM, H$ is the foot of the perpendicular line from $ P$ to $ BC, Q$ is the point of intersection of segment $ AB$ and the line passing through $ H$ that is perpendicular to $ PB,$ and finally, $ R$ is the point of intersection of the segment $ AC$ and the line passing through $ H$ that is perpendicular to $ PC.$ Show that the circumcircle of $ QHR$ is tangent to the side $ BC$ at point $ H.$

Find all positive integers $n \ge 3$ such that there exists an $n$-gon with vertices on lattice points of the coordinate plane and all sides of equal length.

Two circles $w_{1}$ and $w_{2}$ of different radii touch externally at $L$. A line touches $w_{1}$ at $A$ and $w_{2}$ at $B$ (the points $A$ and $B$ are different from $L$). A point $X$ is chosen in the plane. $Y$ and $Z$ are the second points of intersection of the lines $XA$ and $XB$ with $w_{1}$ and $w_{2}$ respectively. Prove that all $X$ such that $AB||Y Z$ belong to one circle.