The Fahrenheit temperature ($F$) is related to the Celsius temperature ($C$) by $F = \tfrac{9}{5} \cdot C + 32$. What is the temperature in Fahrenheit degrees that is one-fifth as large if measured in Celsius degrees?

Let be the sequence of sets $ \left(\left\{ A\in\mathcal{M}_2\left(\mathbb{R} \right) | A^{n+1} =2007^nA\right\}\right)_{n\ge 1} . $ [b]a)[/b] Prove that each term of the above sequence hasn't a finite cardinal. [b]b)[/b] Determine the intersection of the fourth element of the above sequence with the $ 2007\text{th} $ element. [i]Gheorghe Iurea[/i] [hide=Note]Similar with [url]https://artofproblemsolving.com/community/c7h1928039p13233629[/url].[/hide]

In this infinite tree, degree of each vertex is equal to 3. A real number $ \lambda$ is given. We want to assign a real number to each node in such a way that for each node sum of numbers assigned to its neighbors is equal to $ \lambda$ times of the number assigned to this node. Find all $ \lambda$ for which this is possible.

What is the smallest positive integer $m$ such that $15! \, m$ can be expressed in more than one way as a product of $16$ distinct positive integers, up to order?

Consider the diagonals $A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_4$ and $A_6A_2$ of a convex hexagon $A_1A_2A_3A_4A_5A_6$. The hexagon whose vertices are the points of intersection of the diagonals is regular. Can we conclude that the hexagon $A_1A_2A_3A_4A_5A_6$ is also regular?

Alice and Bob play a game, in which they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment shares at least one point - except for its starting point - with one of the previously drawn segments, one has lost. a) Show that both Alice and Bob could force the game to end, if they don’t care who wins. b) Is there a winning strategy for one of them?

Let $ABCD$ be a convex quadrilateral with $AD = BC$, $\angle BAC+\angle DCA = 180^{\circ}$, and $\angle BAC \neq 90^{\circ}.$ Let $O_1$ and $O_2$ be the circumcenters of triangles $ABC$ and $CAD$, respectively. Prove that one intersection point of the circumcircles of triangles $O_1BC$ and $O_2AD$ lies on $AC$.

Four congruent semicircular half-disks are arranged inside a circle with radius $4$ so that each semicircle is internally tangent to the circle, and the diameters of the semicircles form a $2\times 2$ square centered at the center of the circle as shown. The radius of each semicircular half-disk is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/f/e/8c0b9fdd69f6b54d39708da94ef2b2d039cb1e.png[/img]

Let $a$ and $b$ be distinct real numbers. Prove that there exist integers $m$ and $n$ such that $am+bn<0$, $bm+an>0$.

MTRPia in $2044$ is highly advanced and a lot of the work is done by disc-shaped robots, each of radius $1$ unit. In order to not collide with each other, there robots have a smaller $360$-degree camera mounted on top, as shown in the figure (robot $r_1$ 'sees' robot $r_2$). Each of there cameras themselves are smaller discs of radius $c$. Suppose there are three robots $r_1, r_2, r_3$ placed 'consecutively' such that $r_2$ is roughly in the middle. The angle between the lines joining the centres of $r_1, r_2$ and $r_2, r_3$ is given to be $\theta$. The distance between the centres of $r_1,r_2 = $ distance between centres of $r_2,r_3 = d$. Show (with the aid of clear diagrams) that $r_1$ and $r_3$ can see each other iff $\sin{\theta} > \frac{1-c}{d}$. As a bonus, try to show that in a longer 'chain' of such robots (same $d$, $\theta$), if $\sin{\theta} > \frac{1-c}{d}$ then all robots can see each other.

Let $n$ be a positive integer, $k\in \mathbb{C}$ and $A\in \mathcal{M}_n(\mathbb{C})$ such that $\text{Tr } A\neq 0$ and $$\text{rank } A +\text{rank } ((\text{Tr } A) \cdot I_n - kA) =n.$$ Find $\text{rank } A$.

Let $x,y$, and $z$ be positive real numbers such that $xy+yz+zx=3xyz$. Prove that $$x^2y+y^2z+z^2x \geq 2(x+y+z)-3.$$ In which cases do we have equality?

There are $100$ points of general position marked on the plane (i.e. no three lie on the same straight line). Prove that it is possible to select three marked points $A, B, C$ so that for any point $D$ of the remaining $97$ marked points, the lines $AD$ and $CD$ would not contain points lying inside the triangle $ABC$.

Evaluate the product \[\left(1 + \frac{2}{3}\right)\left(1 + \frac{2}{4}\right)\left(1 + \frac{2}{5}\right) \cdots \left(1 + \frac{2}{98}\right).\]

Farmer John owns 2013 cows. Some cows are enemies of each other, and Farmer John wishes to divide them into as few groups as possible such that each cow has at most 3 enemies in her group. Each cow has at most 61 enemies. Compute the smallest integer $G$ such that, no matter which enemies they have, the cows can always be divided into at most $G$ such groups?

Find the sum of all positive integers n with the property that the digits of n add up to 2015−n.

Which positive integers cannot be represented as a sum of (two or more) consecutive integers?

The $n^\text{th}$ term of a sequence is $a_n=(-1)^n(4n+3)$. Compute the sum \[a_1+a_2+a_3+\cdots+a_{2008}.\]

In triangle $ABC$, medians $AA'$, $BB'$, $CC'$ are extended until they intersect with the circumcircle at points $A_0$, $B_0$, $C_0$, respectively. It is known that the intersection point M of the medians of triangle $ABC$ divides the segment $AA_0$ in half. Prove that the triangle $A_0B_0C_0$ is isosceles.

Ada and Charles play the following game:at the beginning, an integer n>1 is written on the blackboard.In turn, Ada and Charles remove the number k that they find on the blackboard.In turn Ad and Charles remove the number k that they find on the blackboard and they replace it : 1 -either with a positive divisor k different from 1 and k 2- or with k+1 At the beginning each players have a thousand points each.When a player choses move 1, he/she gains one point;when a player choses move 2, he/she loses one point.The game ends when one of the tho players is left with zero points and this player loses the game.Ada moves first.For what values Chares has a winning strategy?

Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. If Bob and Chandra both read the whole book, Bob will spend how many more seconds reading than Chandra? $ \textbf{(A)}\ 7,600 \qquad \textbf{(B)}\ 11,400 \qquad \textbf{(C)}\ 12,500 \qquad \textbf{(D)}\ 15,200 \qquad \textbf{(E)}\ 22,800$

Determine if there exist pairwise distinct positive integers $a_1,a_2,\ldots,a_{101}$, $b_1$, $b_2$, \ldots, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1,2,\ldots,101\}$ the sum $\sum\limits_{i\in S}a_i$ divides $\left( 100!+\sum\limits_{i\in S}b_i \right)$.

Find all quadruples $(x,y,z,w)$ of integers satisfying the sys­tem of equations $$x + y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1$$

Mazo performs the following operation on triplets of non-negative integers: If at least one of them is positive, it chooses one positive number, decreases it by one, and replaces the digits in the units place with the other two numbers. It starts with the triple $x$, $y$, $z$. Find a triple of positive integers $x$, $y$, $z$ such that $xy + yz + zx = 1000$ (*) and the number of operations that Mazo can subsequently perform with the triple $x, y, z$ is (a) maximal (i.e. there is no triple of positive integers satisfying (*) that would allow him to do more operations); (b) minimal (i.e. every triple of positive integers satisfying (*) allows him to perform at least so many operations).

Some cells of a $10\times10$ board are marked so that each cell has an even number of neighboring (i.e. sharing a side) marked cells. Find the maximum possible number of marked cells.