Consider two spheres $\Sigma_1$ and $\Sigma_2$ and a line $\Delta$ not meeting them. Let $C_i$ and $r_i$ be the center and radius of $\Sigma_i$, and let $H_i$ and $d_i$ be the orthogonal projection of $C_i$ onto $\Delta$ and the distance of $C_i$ from $\Delta~(i=1,2)$. For a point $M$ on $\Delta$, let $\delta_i(M)$ be the length of a tangent $MT_i$ to $\Sigma_i$, where $T_i\in\Sigma_i~(i=1,2)$. Find $M$ on $\Delta$ for which $\delta_1(M)+\delta_2(M)$ is minimal.

suppose that $0\le p \le 1$ and we have a wooden square with side length $1$. in the first step we cut this square into $4$ smaller squares with side length $\frac{1}{2}$ and leave each square with probability $p$ or take it with probability $1-p$. in the next step we cut every remaining square from the previous step to $4$ smaller squares (as above) and take them with probability $1-p$. it's obvios that at the end what remains is a subset of the first square. [b]a)[/b] show that there exists a number $0<p_0<1$ such that for $p>p_0$ the probability that the remainig set is not empty is positive and for $p<p_0$ this probability is zero. [b]b)[/b] show that for every $p\neq 1$ with probability $1$, the remainig set has size zero. [b]c)[/b] for this statement that the right side of the square is connected to the left side of the square with a path, write anything that you can.

Claudia and Adela are betting to see which one of them will ask the boy they like for his telephone number. To decide they roll dice. If none of the dice are a multiple of 3, Claudia will do it. If exactly one die is a multiple of 3, Adela will do it. If 2 or more of the dice are a multiple of 3 neither one of them will do it. How many dice should be rolled so that the risk is the same for both Claudia and Adela?

Connie multiplies a number by 2 and gets 60 as her answer. However, she should have divided the number by 2 to get the correct answer. What is the correct answer? $\textbf{(A)}\ 7.5 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 30 \qquad \textbf{(D)}\ 120 \qquad \textbf{(E)}\ 240$

There are $2n+1$ segments on the line. Any segment intersects at with at least $n$ others. Prove that there is a segment that intersects all the others.

$ O $ is the center of a parallelogram $ ABCD. $ Let $ G $ on the segment $ OB $ (excluding its endpoints), $ N $ on the line $ DC $ and $ M $ on the segment $ AD $ (excluding its endpoints) such that $ CN>ND, AM=6MD $ and so that there exists a natural number $ n\ge 3 $ such that $ OB=nGO. $ Show that $ G,M,N $ are collinear if and only if $$ \left( \frac{CN}{ND} -6 \right) (n+1)=2. $$

Find all real solutions of the equation $$x^2+y^2+z^2+t^2=xy+yz+zt+t-\frac25.$$

Find the sum of all positive divisors of $40081$.

Let $l,\ m$ be the tangent lines passing through the point $A(a,\ a-1)$ on the line $y=x-1$ and touch the parabola $y=x^2$. Note that the slope of $l$ is greater than that of $m$. (1) Exress the slope of $l$ in terms of $a$. (2) Denote $P,\ Q$ be the points of tangency of the lines $l,\ m$ and the parabola $y=x^2$. Find the minimum area of the part bounded by the line segment $PQ$ and the parabola $y=x^2$. (3) Find the minimum distance between the parabola $y=x^2$ and the line $y=x-1$.

Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the from $\pi(a-b\sqrt{c})$, where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$. [asy] real r=2-sqrt(3); draw(Circle(origin, 1)); int i; for(i=0; i<12; i=i+1) { draw(Circle(dir(30*i), r)); dot(dir(30*i)); } draw(origin--(1,0)--dir(30)--cycle); label("1", (0.5,0), S);[/asy]

Let $ABC$ be a right angled triangle with hypotenuse $AC$, and let $H$ be the foot of the altitude from $B$ to $AC$. Knowing that there is a right-angled triangle with side-lengths $AB, BC, BH$, determine all the possible values ​​of $\frac{AH}{CH}$

Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.

In the grid to the right, the shortest path through unit squares between between the pair of 2's has length 2. Fill in some of the unit squares in the grid so that (i) exactly half of the squares in each row and column contain a number, (ii) each of the number 1 through 12 appears exactly twice, and (iii) for $n=1,2,\cdot\cdot\cdot,12$, the shortest path between the pair of $n$'s has length exactly $n$.

What is the largest integer less than or equal to $$\frac{3^{31}+2^{31}}{3^{29}+2^{29}} \,\,\, ?$$

Construct a convex polyhedron of equal “bricks” shown in Figure. [img]https://cdn.artofproblemsolving.com/attachments/6/6/75681a90478f978665b6874d0c0c9441ea3bd2.gif[/img]

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

Find the number of different pairs of positive integers $(a,b)$ for which $a+b\le100$ and \[\dfrac{a+\frac{1}{b}}{\frac{1}{a}+b}=10\]

Let $\sigma$ be a permutation of the set $S := \{1, 2, \ldots , 100\},$ such that $\sigma(a+b) \equiv \sigma(a)+\sigma(b) \pmod{100}$ whenever $a, b, a + b \in S.$ Denote by $f(s)$ the sum of the distinct values $\sigma(s)$ can take over all possible $\sigma$s satisfying the given condition. What is the nonnegative difference between the maximum and minimum value $f$ takes on when ranging over all $s \in S$?

$ABCD$ is a cyclic quadrilateral with $AC \perp BD$; $AC$ meets $BD$ at $E$. Prove that \[ EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2 \] where $R$ is the radius of the circumscribing circle.

At least one of the coefficients of a polynomial $P(x)$ is negative. Can all of the coefficients of all of its powers $(P(x))^n$, $n > 1$, be positive? (0 Kryzhanovskij)

Let $\Omega$ be a circle with center $O$. Let $\omega_1$ and $\omega_2$ be circles with centers $O_1$ and $O_2$, respectively, internally tangent to $\Omega$ at points $A$ and $B$, respectively, such that $O_1$ is on $\overline{OA}$, and $O_2$ is on $\overline{OB}$ and $\omega_1$. There exists a point $P$ on line $AB$ such that $P$ is on both $\omega_1$ and $\omega_2$. Let the external tangent of $\omega_1$ and $\omega_2$ on the same side of line $AB$ as $O$ hit $\omega_1$ at $X$ and $\omega_2$ at $Y$, and let lines $AX$ and $BY$ intersect at $N$. Given that $O_1X = 81$ and $O_2Y = 18$, the value of $NX \cdot NA$ can be written as $a\sqrt{b} + c$, where $a$, $b$, and $c$ are positive integers, and $b$ is not divisible by the square of a prime. Find $a+b+c$. [i]Proposed by Kevin Zhao[/i]

In trapezoid $ABCD$, $BC \parallel AD$, $AB = 13$, $BC = 15$, $CD = 14$, and $DA = 30$. Find the area of $ABCD$.

Let $AH_1$ and $BH_2$ be the altitudes of triangle $ABC$. Let the tangent to the circumcircle of $ABC$ at $A$ meet $BC$ at point $S_1$, and the tangent at $B$ meet $AC$ at point $S_2$. Let $T_1$ and $T_2$ be the midpoints of $AS_1$ and $BS_2$ respectively. Prove that $T_1T_2$, $AB$ and $H_1H_2$ concur.

Source: 2018 Canadian Open Math Challenge Part C Problem 1 ----- At Math-$e^e$-Mart, cans of cat food are arranged in an pentagonal pyramid of 15 layers high, with 1 can in the top layer, 5 cans in the second layer, 12 cans in the third layer, 22 cans in the fourth layer etc, so that the $k^{\text{th}}$ layer is a pentagon with $k$ cans on each side. [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNC9lLzA0NTc0MmM2OGUzMWIyYmE1OGJmZWQzMGNjMGY1NTVmNDExZjU2LnBuZw==&rn=YzFhLlBORw==[/img][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9hLzA1YWJlYmE1ODBjMzYwZDFkYWQyOWQ1YTFhOTkzN2IyNzJlN2NmLnBuZw==&rn=YzFiLlBORw==[/img][/center] $\text{(a)}$ How many cans are on the bottom, $15^{\text{th}}$, [color=transparent](A.)[/color]layer of this pyramid? $\text{(b)}$ The pentagonal pyramid is rearranged into a prism consisting of 15 identical layers. [color=transparent](B.)[/color]How many cans are on the bottom layer of the prism? $\text{(c)}$ A triangular prism consist of indentical layers, each of which has a shape of a triangle. [color=transparent](C.)[/color](the number of cans in a triangular layer is one of the triangular numbers: 1,3,6,10,...) [color=transparent](C.)[/color]For example, a prism could be composed of the following layers: [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi85L2NlZmE2M2Y3ODhiN2UzMTRkYzIxY2MzNjFmMDJkYmE0ZTJhMTcwLnBuZw==&rn=YzFjLlBORw==[/img][/center] Prove that a pentagonal pyramid of cans with any number of layers $l\ge 2$ can be rearranged (without a deficit or leftover) into a triangluar prism of cans with the same number of layers $l$.