Four cones are given with a common vertex and the same generatrix, but with, generally speaking, different radii of the bases. Each of them is tangent to two others. Prove that the four tangent points of the circles of the bases of the cones lie on the same circle.

Let $ABC$ be such a triangle that $\sphericalangle BAC = 45^{\circ}$ and $ \sphericalangle ACB > 90^{\circ}.$ Show that $BC + (\sqrt{2} - 1)\cdot CA < AB.$

Through each vertex of a tetrahedron with a given volume $ V $, a plane is drawn parallel to the opposite face of the tetrahedron. Calculate the volume of the tetrahedron formed by these planes.

Matías has a rectangular sheet of paper $ABCD$, with $AB<AD$.Initially, he folds the sheet along a straight line $AE$, where $E$ is a point on the side $DC$ , so that vertex $D$ is located on side $BC$, as shown in the figure. Then folds the sheet again along a straight line $AF$, where $F$ is a point on side $BC$, so that vertex $B$ lies on the line $AE$; and finally folds the sheet along the line $EF$. Matías observed that the vertices $B$ and $C$ were located on the same point of segment $AE$ after making the folds. Calculate the measure of the angle $\angle DAE$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/b9ab717e1806c6503a9310ee923f20109da31a.png[/img]

A circle passes through the vertices of a triangle with side-lengths of $7\tfrac{1}{2},10,12\tfrac{1}{2}$. The radius of the circle is: $\textbf{(A)}\ \dfrac{15}{4} \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ \dfrac{25}{4} \qquad \textbf{(D)}\ \dfrac{35}{4} \qquad \textbf{(E)}\ \dfrac{15\sqrt2}{2} $

$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.

Let a regular polygon with $p$ vertices be given, where $p$ is an odd prime number. At every vertex there is one monkey. An owner of monkeys takes $p$ peanuts, goes along the perimeter of polygon clockwise and delivers to the monkeys by the following rule: Gives the first peanut for the leader, skips the two next vertices and gives the second peanut to the monkey at the next vertex; skip four next vertices gives the second peanut for the monkey at the next vertex ... after giving the $k$-th peanut, he skips the $2 \cdot k$ next vertices and gives $k+1$-th for the monkey at the next vertex. He does so until all $p$ peanuts are delivered. [b]I.[/b] How many monkeys are there which does not receive peanuts? [b]II.[/b] How many edges of polygon are there which satisfying condition: both two monkey at its vertex received peanut(s)?

Let $I$ be the incircle and $O$ a circumcenter of triangle $ABC$ such that $\angle ACB=30^{\circ}$. On sides $AC$ and $BC$ there are points $E$ and $D$, respectively, such that $EA=AB=BD$. Prove that $DE=IO$ and $DE \perp IO$

A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k,\thinspace 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules: $\bullet$ Any cube may be the bottom cube in the tower. $\bullet$ The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$ Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by 1000?

Prove that the numbers $A,B$ and $C$ are equal, where: - $A=$ number of ways that we can cover a $2 \times n$ rectangle with $2 \times 1$ retangles. - $B=$ number of sequences of ones and twos that add up to $n$ - $C= \sum^m_{k=0} \binom{m + k}{2 \cdot k}$ if $n = 2 \cdot m,$ and - $C= \sum^m_{k=0} \binom{m + k + 1}{2 \cdot k + 1}$ if $n = 2 \cdot m + 1.$

Points $A, S$ are given in plane such that $AS = a > 0$ as well as positive numbers $b, c$ satisfying $b < a < c$. Construct an equilateral triangle $ABC$ with the property $BS = b$, $CS = c$. Discuss conditions of solvability.

Let there be a quadrilateral $ABCD$ such that $\angle CAD=45, \angle ACD=30, \angle BAC=\angle BCA=15$. Find $\angle DBC$.

The incircle of triangle $ABC$ is tangent to $BC,CA,AB$ at $D,E,F$ respectively. Consider the triangle formed by the line joining the midpoints of $AE,AF$, the line joining the midpoints of $BF,BD$, and the line joining the midpoints of $CD,CE$. Prove that the circumcenter of this triangle coincides with the circumcenter of triangle $ABC$.

Ten chairs are evenly spaced around a round table and numbered clockwise from $ 1$ through $ 10$. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or directly across from his or her spouse. How many seating arrangements are possible? $ \textbf{(A)}\ 240\qquad \textbf{(B)}\ 360\qquad \textbf{(C)}\ 480\qquad \textbf{(D)}\ 540\qquad \textbf{(E)}\ 720$

Given a quadrilateral $ABCD$, around which you can circumscribe a circle. The perpendicular bisectors of sides $AD$ and $CD$ intersect at point $Q$ and intersect sides $BC$ and $AB$ at points $P$ and $K$ resepctively. It turned out that the points $K, B, P, Q$ lie on the same circle. Prove that the points $A, Q, C$ lie on one line. (Olena Artemchuk)

Two points $K$ and $L$ are given on a circle. Construct a triangle $ABC$ so that its vertex $C$ and the intersection points of its medians $AK$ and $BL$ both lie on the circle, $K$ and $L$ being the midpoints of its sides $BC$ and $AC$.

$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?

Let $ABC$ be an equilateral triangle and $\Gamma$ its incircle. If $D$ and $E$ are points on the segments $AB$ and $AC$ such that $DE$ is tangent to $\Gamma$, show that $\frac{AD}{DB}+\frac{AE}{EC}=1$.

Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$. Let $I$ be the center of $\omega$, and let $IA=12,$ $IB=16,$ $IC=14,$ and $ID=11$. Let $M$ be the midpoint of segment $AC$. Compute the ratio $\frac{IM}{IN}$, where $N$ is the midpoint of segment $BD$.

Consider a cyclic quadrilateral $ABCD$, and let $S$ be the intersection of $AC$ and $BD$. Let $E$ and $F$ the orthogonal projections of $S$ on $AB$ and $CD$ respectively. Prove that the perpendicular bisector of segment $EF$ meets the segments $AD$ and $BC$ at their midpoints.

Let $ABC$ be a triangle in which $AB=AC$. A point $I$ lies inside the triangle such that $\angle ABI=\angle CBI$ and $\angle BAI=\angle CAI$. Prove that \[\angle BIA=90^o+\dfrac{\angle C}{2}\]

[b]p1.[/b] In the picture below, the two parallel cuts divide the square into three pieces of equal area. The distance between the two parallel cuts is $d$. The square has length $s$. Find and prove a formula that expresses $s$ as a function of $d$. [img]https://cdn.artofproblemsolving.com/attachments/c/b/666074d28de50cdbf338a2c667f88feba6b20c.png[/img] [b]p2.[/b] Let $S$ be a subset of $\{1, 2, 3, . . . 10, 11\}$. We say that $S$ is lucky if no two elements of $S$ differ by $4$ or $7$. (a) Give an example of a lucky set with five elements. (b) Is it possible to find a lucky set with six elements? Explain why or why not.[/quote] [b]p3.[/b] Find polynomials $p(x)$ and $q(x)$ with real coefficients such that (a) $p(x) - q(x) = x^3 + x^2 - x - 1$ for all real $x$, (b) $p(x) > 0$ for all real $x$, (c) $q(x) > 0$ for all real $x$. [b]p4.[/b] A permutation on $\{1, 2, 3, …, n\}$ is a rearrangement of the symbols. For example $32154$ is a permutation on $\{1, 2, 3, 4, 5\}$. Given a permutation $a_1a_2a_3…a_n$, an inversion is a pair of $a_i$ and $a_j$ such that $a_i > a_j$ but $i < j$. For example, $32154$ has $4$ inversions. Suppose you are only allowed to exchange adjacent symbols. For any permutation, show that the minimum number of exchanges required to put all the symbols in their natural positions (that is, $123 …n$) is the number of inversions. [b]p5.[/b] We say a number $N$ is a nontrivial sum of consecutive positive integers if it can be written as the sum of $2$ or more consecutive positive integers. What is the set of numbers from $1000$ to $2000$ that are NOT nontrivial sums of consecutive positive integers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the diagonals $AC$ and $CE$ of a regular hexagon $ABCDEF$ with side $1$ we mark points$ M$ and $N$ such that $AM = CN = a$. Find $a$ if the points $B, M, N$ lie on the same line.

Triangle $ABC$ is inscribed in circle $(O)$. $A$ varies on $(O)$ such that $AB>BC$. $M$ is the midpoint of $AC$. The circle with diameter $BM$ intersects $(O)$ at $R$. $RM$ intersects $(O)$ at $Q$ and intersects $BC$ at $P$. The circle with diameter $BP$ intersects $AB, BO$ at $K,S$ in this order. a. Prove that $SR$ passes through the midpoint of $KP$. b. Let $N$ be the midpoint of $BC$. The radical axis of circles with diameters $AN, BM$ intersects $SR$ at $E$. Prove that $ME$ always passes through a fixed point.

On the side $AC$ of the triangle $ABC$, choose any point $D$ different from the vertices $A$ and C. Let $O_1$ and $O_2$ be circumcenters the triangles $ABD$ and $CBD$, respectively. Prove that the triangles $O_1DO_2$ and $ABC$ are similar.