Found problems: 85335
Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.
Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends when no player can make a move, and the last player to make a move wins. Determine the number of values of $n$ between $1$ and $2013$ (inclusive) for which the first player can guarantee a win, regardless of the moves that the second player makes.
[i]Ray Li[/i]
Show that for any natural number $n$, the number $A = [\frac{n + 3}{4}] + [ \frac{n + 5}{4} ] + [\frac{n}{2} ] +n^2 + 3n + 3$ is a perfect square. ($[x]$ denotes the integer part of the real number x.)
How many distinct four-digit numbers are divisible by $ 3$ and have $ 23$ as their last two digits?
$ \textbf{(A)}\ 27 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 33 \qquad
\textbf{(D)}\ 81 \qquad
\textbf{(E)}\ 90$
Let $S$ be the sum of all distinct real solutions of the equation
\[
\sqrt{x + 2015} = x^2 - 2015.
\]
Compute $\lfloor 1/S \rfloor$. Recall that if $r$ is a real number, then $\lfloor r \rfloor$ (the [i]floor[/i] of $r$) is the greatest integer that is less than or equal to $r$.
Let $n \geq 2$ be a positive integer and define $k$ to be the number of primes $\leq n$. Let $A$ be a subset of $S = \{2,...,n\}$ such that $|A| \leq k$ and no two elements in $A$ divide each other. Show that one can find a set $B$ such that $|B| = k$, $A \subseteq B \subseteq S$ and no two elements in $B$ divide each other.
Let $k$ be a constant such that exactly three real values of $x$ satisfy $$x-|x^2-4x+3| = k$$ The sum of all possible values of $k$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
[i]Proposed by Andy Xu[/i]
a)Is it true that for every non-empty set $A$ and every associative operation $*:A \times A \to A$ the conditions $$x*x*y=y \;\;\; \text{and}\; \;\; y*x*x=y$$ for every $x,y\in A$ imply commutativity of $*$?
b)a)Is it true that for every non-empty set $A$ and every associative operation $*:A \times A \to A$ the condition$$x*x*y=y $$ for every $x,y\in A$ implies commutativity of $*$?
[i]Proposed by Paulius Drungilas, Arturas Dubickas (Vilnius University).
[/i]
Prove that for any real numbers $ a_1, a_2, \dots, a_ {100} $ there exists a real number $ b $ such that all numbers $ a_i + b $ ($ 1 \leq i \leq 100 $) are irrational.
For which $k$ the number $N = 101 ... 0101$ with $k$ ones is a prime?
Let $P(x,y)$ be a polynomial with real coefficients in the variables $x,y$ that is not identically zero. Suppose that $P(\lfloor 2a \rfloor, \lfloor 3a\rfloor) = 0$ for all real numbers $a.$ If $P$ has the minimum possible degree and the coefficient of the monomial $y$ is $4,$ find the coefficient of $x^2y^2$ in $P.$
(The [i]degree[/i] of a monomial $x^my^n$ is $m + n.$ The [i]degree[/i] of a polynomial $P(x,y)$ is then the maximum degree of any of its monomials.)
Is there a quadrilateral in which the position of any vertex can be changed, leaving the other three in place, so that the resulting four points serve as the vertices of a quadrilateral equal to the original one?
A positive integer is [i]dapper[/i] if at least one of its multiples begins with $ 2008$. For example, $ 7$ is dapper because $ 200858$ is a multiple of $ 7$ and begins with $ 2008$. Observe that $ 200858 \equal{} 28694\times 7$.
Prove that every positive integer is dapper.
Let $p(x)$ be an $n$-degree $(n \ge 2)$ polynomial with integer coefficients. If there are infinitely many positive integers $m$, such that $p(m)$ at most $n -1$ different prime factors $f$, prove that $p(x)$ has at most $n-1$ different rational roots .
[color=#f00]a help in translation is welcome[/color]
Let $ABCD$ be a square and let $M$ be the midpoint of $BC$. Let $X$ and $Y$ be points on the segments $AB$ and $CD$, respectively. Prove that $\angle XMY = 90^\circ$ if and only if $BX + CY = XY$.
[i]Note: In the competition, students were only asked to prove the 'only if' direction.[/i]
$AH$ is the altitude of triangle $ABC$ and $H^\prime$ is the reflection of $H$ trough the midpoint of $BC$. If the tangent lines to the circumcircle of $ABC$ at $B$ and $C$, intersect each other at $X$ and the perpendicular line to $XH^\prime$ at $H^\prime$, intersects $AB$ and $AC$ at $Y$ and $Z$ respectively, prove that $\angle ZXC=\angle YXB$.
A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies
\[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \]
for all $k=1,2,\ldots 9$.
Find the number of interesting numbers.
Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$.
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
Determine the largest integer $n\geq 3$ for which the edges of the complete graph on $n$ vertices
can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.
A cube is divided into $8$ smaller cubes of the same size, as shown in the figure. Then, each of these small cubes is divided again into $8$ smaller cubes of the same size. This process is done $4$ more times to each resulting cube. What is the ratio between the sum of the total areas of all the small cubes resulting from the last division and the total area of the initial cube?
For the consumer, a single discount of $n\%$ is more advantageous than any of the following discounts:
$(1)$ two successive $15\%$ discounts
$(2)$ three successive $10\%$ discounts
$(3)$ a $25\%$ discount followed by a $5\%$ discount
What is the smallest possible positive integer value of $n$?
${ \textbf{(A)}\ \ 27\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 29\qquad\textbf{(D)}}\ 31\qquad\textbf{(E)}\ 33$
At a party there were some couples attending. As they arrive each person gets to talk with all the other persons which are [i]already [/i] in the room. During the party, after all the guests arrive, groups of persons form, such that no two persons forming a couple belong to the same group, and for each two persons that do not form a couple, there is one and only one group to which both belong. Find the number of couples attending the party, knowing that there are less groups than persons at the party.
Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.
Determine all real $x$ such that \[\sqrt{\tan(x)-1}\,\Bigl(\log_{\tan(x)}\bigl(2+4\cos^2(x)-2\bigr)\Bigr)\ge0.\]
Let $I$ be the point of intersection of the angle bisectors of the $\vartriangle ABC$, $W_1,W_2,W_3$ be point of intersection of lines $AI, BI, CI$ with the circle circumscribed around the triangle, $r$ and $R$ be the radii of inscribed and circumscribed circles respectively. Prove the inequality $$IW_1+ IW_2 + IW_3\ge 2R + \sqrt{2Rr.}$$