Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .

Find all solutions for (x,y) , both integers such that: $xy=3(\sqrt{x^2+y^2}-1)$

Find all triples $(a,b,c)$ satisfying the following conditions: (i) $a,b,c$ are prime numbers, where $a<b<c<100$. (ii) $a+1,b+1,c+1$ form a geometric sequence.

Let $ G(V,E)$ be a connected graph, and let $ d_G(x,y)$ denote the length of the shortest path joining $ x$ and $ y$ in $ G$. Let $ r_G(x)\equal{} \max \{ d_G(x,y) : \; y \in V \ \}$ for $ x \in V$, and let $ r(G)\equal{} \min \{ r_G(x) : \;x \in V\ \}$. Show that if $ r(G) \geq 2$, then $ G$ contains a path of length $ 2r(G)\minus{}2$ as an induced subgraph. [i]V. T. Sos[/i]

The base of the pyramid is a triangle $ABC$, in which $\angle ACB= 30^o$, and the length of the median from the vertex $B$ is twice less than the side $AC$ and is equal to $\alpha$ . All side edges of the pyramid are inclined to the plane of the base at an angle $a$. Determine the cross-sectional area of ​​the pyramid with a plane passing through the vertex $B$ parallel to the edge $AD$ and inclined to the plane of the base at an angle of $\beta$,

[b]Q13.[/b] Determine the greatest value of the sum $M=11xy+3x+2012yz$, where $x,y,z$ are non negative integers satisfying condition $x+y+z=1000.$

In the city of Urbextorto, the sales tax is $25\%$. A certain clothing store in the city is currently giving an $n\%$ discount on all items, and $n$ is special in that, after both the sales tax and discount are applied, a $\$20$ shirt ends up costing $\$20$. Find the value of $n$. $\textbf{(A) }5\qquad\textbf{(B) }10\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }50$

$201$ positive integers are written on a line, such that both the first one and the last one are equal to $19999$. Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the second-to-last term on the line.

Let $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $A+B=AB+BA$. Prove that: a) if $n$ is odd, then $\det(AB-BA)=0$; b) if $\text{tr}(A)\neq \text{tr}(B)$, then $\det(AB-BA)=0$.

Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)

In a circle $C$ with centre $O$ and radius $r$, let $C_1$, $C_2$ be two circles with centres $O_1$, $O_2$ and radii $r_1$, $r_2$ respectively, so that each circle $C_i$ is internally tangent to $C$ at $A_i$ and so that $C_1$, $C_2$ are externally tangent to each other at $A$. Prove that the three lines $OA$, $O_1 A_2$, and $O_2 A_1$ are concurrent.

A square is subdivided into $K^2$ equal smaller squares. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself). Find the minimum number of links in this broken line. (A Andjans, Riga)

How many reorderings of $2,3,4,5,6$ have the property that every pair of adjacent numbers are relatively prime? [i]2021 CCA Math Bonanza Individual Round #3[/i]

In the coordinate plane a rectangle with vertices $ (0, 0),$ $ (m, 0),$ $ (0, n),$ $ (m, n)$ is given where both $ m$ and $ n$ are odd integers. The rectangle is partitioned into triangles in such a way that [i](i)[/i] each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form $ x \equal{} j$ or $ y \equal{} k,$ where $ j$ and $ k$ are integers, and the altitude on this side has length 1; [i](ii)[/i] each “bad” side (i.e., a side of any triangle in the partition that is not a “good” one) is a common side of two triangles in the partition. Prove that there exist at least two triangles in the partition each of which has two good sides.

Determine the minimum possible amount of distinct prime divisors of $19^{4n}+4$, for a positive integer $n$.

[b]8.1[/b] $x$ and $y$ are the roots of the equation $t^2-ct-c=0$. Prove that holds the inequality $x^3 + y^3 + (xy)^3 \ge 0.$ [b]8.2.[/b] Two circles touch internally at point $A$ . Through a point $B$ of the inner circle, different from $A$, a tangent to this circle intersecting the outer circle at points C and $D$. Prove that $AB$ is a bisector of angle $CAD$. [img]https://cdn.artofproblemsolving.com/attachments/2/8/3bab4b5c57639f24a6fd737f2386a5e05e6bc7.png[/img] [b]8.3[/b] Prove that $2^{3^{100}} + 1$ is divisible by $3^{101}$. [b]8.4 / 7.5[/b] An entire arc of circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest. [img]https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png [/img] [b]8.5[/b] In a certain group of people, everyone has one enemy and one Friend. Prove that these people can be divided into two companies so that in every company there will be neither enemies nor friends. [b]8.6[/b] Numbers $a_1, a_2, . . . , a_{100}$ are such that $$a_1 - 2a_2 + a_3 \le 0$$ $$a_2-2a_3 + a_ 4 \le 0$$ $$...$$ $$a_{98}-2a_{99 }+ a_{100} \le 0$$ and at the same time $a_1 = a_{100}\ge 0$. Prove that all these numbers are non-negative. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth? $ \textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 41 $

Points $A,B,Q,D,$ and $C$ lie on the circle shown and the measures of arcs $\widehat{BQ}$ and $\widehat{QD}$ are $42^\circ$ and $38^\circ$ respectively. The sum of the measures of angles $P$ and $Q$ is $\textbf{(A) }80^\circ\qquad\textbf{(B) }62^\circ\qquad\textbf{(C) }40^\circ\qquad\textbf{(D) }46^\circ\qquad \textbf{(E) }\text{None of these}$ [asy] size(3inch); draw(Circle((1,0),1)); pair A, B, C, D, P, Q; P = (-2,0); B=(sqrt(2)/2+1,sqrt(2)/2); D=(sqrt(2)/2+1,-sqrt(2)/2); Q = (2,0); A = intersectionpoints(Circle((1,0),1),B--P)[1]; C = intersectionpoints(Circle((1,0),1),D--P)[0]; draw(B--P--D); draw(A--Q--C); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SW); label("$D$",D,SE); label("$P$",P,W); label("$Q$",Q,E); //Credit to chezbgone2 for the diagram[/asy]

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \ge 2017$, the integer $P(n)$ is positive and $S(P(n)) = P(S(n))$.

3. Consider all 100-digit positive integers such that each decimal digit of these equals $2,3,4,5,6$, or 7 . How many of these integers are divisible by $2^{100}$ ? Pavel Kozhevnikov

Determine the smallest positive integer $\ N $ such that there exists 6 distinct integers $\ a_1, a_2, a_3, a_4, a_5, a_6 > 0 $ satisfying: (i) $\ N = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 $ (ii) $\ N - a_i$ is a perfect square for $\ i = 1,2,3,4,5,6 $.

There are two urns each containing an arbitrary number of balls. We are allowed two types of operations: (a) remove an equal number of balls simultaneously from both the urns, (b) double the number of balls in any of them Show that after performing these operations finitely many times, both the urns can be made empty.

Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$, $$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$ [i]Proposed by Morteza Saghafian[/i]

Each square of $1\times 1$, of a $n\times n$ grid is colored using red or blue, in such way that between all the $2\times 2$ subgrids, there are all the possible colorations of a $2\times 2$ grid using red or blue, (colorations that can be obtained by using rotation or symmetry, are said to be different, so there are 16 possibilities). Find: a) The minimum value of $n$. b) For that value, find the least possible number of red squares.

Let $k$ be a positive integer. For each nonnegative integer $n$, let $f(n)$ be the number of solutions $(x_1,\ldots,x_k)\in\mathbb{Z}^k$ of the inequality $|x_1|+...+|x_k|\leq n$. Prove that for every $n\ge1$, we have $f(n-1)f(n+1)\leq f(n)^2$. (Proposed by Esteban Arreaga, Renan Finder and José Madrid, IMPA, Rio de Janeiro)