What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?

A convex quadrilateral of area $1$ is given. Prove that there exist four points in the interior or on the sides of the quadrilateral such that each triangle with the vertices in three of these four points has an area greater than or equal to $1/4$.

Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.

let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the $$\max( [x]y + [y]z + [z]x ) $$ ( $[a]$ is the biggest integer not exceeding $a$)

Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

A finite number of chords is drawn in a circle $1$ cm in diameter so that any diameter of the circle intersects at most $N$ of these chords. Prove that the sum of the lengths of all chords is less than $3.15 \cdot N$ cm.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] The number $123456789$ is written on the blackboard. At each step it is allowed to choose its digits $a$ and $b$ of the same parity and to replace each of them by $\frac{a+b}{2}.$ Is it possible to obtain a number larger then a)$800000000$; b)$880000000$ by such replacements?

Say that an $n$-by-$n$ matrix $A=(a_{ij})_{1\le i,j \le n}$ with integer entries is very odd if, for every nonempty subset $S$ of $\{1,2,\dots,n \}$, the $|S|$-by-$|S|$ submatrix $(a_{ij})_{i,j \in S}$ has odd determinant. Prove that if $A$ is very odd, then $A^k$ is very odd for every $k \ge 1$.

A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A' $ at points $A_o, B_o, C_o$ respectively. It is known that $\angle A_oBB' = \angle B_oCC' =\angle C_oAA'$. a) Find all possible values of these angles. b) Prove that segments $AA_o, BB_o, CC_o$ concur. c) Prove that the projections of the incenter to $A'B', B'C', C'A'$ are the vertices of a regular triangle.

Find the integer closest to \[\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}\]

Let $ \clubsuit(x)$ denote the sum of the digits of the positive integer $ x$. For example, $ \clubsuit(8)\equal{}8$ and $ \clubsuit(123)\equal{}1\plus{}2\plus{}3\equal{}6$. For how many two-digit values of $ x$ is $ \clubsuit(\clubsuit(x))\equal{}3$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

In trapezoid $ABCD$, the bisectors of angles $A$ and $D$ intersect at point $E$ lying on the side of $BC$. These bisectors divide the trapezoid into three triangles into which the circles are inscribed. One of these circles touches the base $AB$ at the point $K$, and two others touch the bisector $DE$ at points $M$ and $N$. Prove that $BK = MN$.

Find all real numbers $a$ such that there exists a function $f:\mathbb R\to \mathbb R$ such that the following conditions are simultaneously satisfied: (a) $f(f(x))=xf(x)-ax,\;\forall x\in\mathbb{R};$ (b) $f$ is not a constant function; (c) $f$ takes the value $a$.

Let $ABC$ be a triangle with $\angle C=90^{\circ}$, and $K, L $ be the midpoints of the minor arcs AC and BC of its circumcircle. Segment $KL$ meets $AC$,at point $N$. Find angle $NIC$ where $I$is the incenter of $ABC$.

As usual, let ${\mathbb Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : {\mathbb Z}[x] \to {\mathbb Z}$ such that for any polynomials $p,q \in {\mathbb Z}[x]$, [list] [*]$\theta(p+1) = \theta(p)+1$, and [*]if $\theta(p) \neq 0$ then $\theta(p)$ divides $\theta(p \cdot q)$. [/list] [i]Evan Chen and Yang Liu[/i]

If the base $8$ representation of a perfect square is $ab3c$, where $a\ne 0$, then $c$ equals $\textbf{(A) } 0\qquad \textbf{(B) }1 \qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } \text{not uniquely determined}$

Find all natural solutions of $3^{x}= 2^{x}y+1.$

Find all integers $x$ for which $2x^2-x-36$ is the square of a prime number.

We consider permutations $f$ of the set $\mathbb{N}$ of non-negative integers, i.e. bijective maps $f$ from $\mathbb{N}$ to $\mathbb{N}$, with the following additional properties: \[f(f(x)) = x \quad \mbox{and}\quad \left| f(x)-x\right| \leqslant 3\quad\mbox{for all } x \in\mathbb{N}\mbox{.}\] Further, for all integers $n > 42$, \[\left.M(n)=\frac{1}{n+1}\sum_{j=0}^n \left|f(j)-j\right|<2,011\mbox{.}\right.\] Show that there are infinitely many natural numbers $K$ such that $f$ maps the set \[\left\{ n\mid 0\leqslant n\leqslant K\right\}\] onto itself.

Let be a group and two coprime natural numbers $ m,n. $ Show that if the applications $ G\ni x\mapsto x^{m+1},x^{n+1} $ are surjective endomorphisms, then the group is commutative.

Let $r$, $s$ be real numbers, find maximum $t$ so that if $a_1, a_2, \ldots$ is a sequence of positive real numbers satisfying \[ a_1^r + a_2^r + \cdots + a_n^r \le 2023 \cdot n^t \] for all $n \ge 2023$ then the sum \[ b_n = \frac 1{a_1^s} + \cdots + \frac 1{a_n^s} \] is unbounded, i.e for all positive reals $M$ there is an $n$ such that $b_n > M$.

Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that: i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}. ii) The same as above replacing "area" for "perimeter".

Let $ABC$ be a triangle and $M$ be the midpoint of $BC$, and let $AM$ meet the circumcircle of $ABC$ again at $R$. A line passing through $R$ and parallel to $BC$ meet the circumcircle of $ABC$ again at $S$. Let $U$ be the foot from $R$ to $BC$, and $T$ be the reflection of $U$ in $R$. $D$ lies in $BC$ such that $AD$ is an altitude. $N$ is the midpoint of $AD$. Finally let $AS$ and $MN$ meets at $K$. Prove that $AT$ bisector $MK$.

$\textbf{(Self-Isogonal Cubics)}$ Let $ABC$ be a triangle with $AB = 2$, $AC = 3$, $BC = 4$. The $\emph{isogonal conjugate}$ of a point $P$, denoted $P^\ast$, is the point obtained by intersecting the reflection of lines $PA$, $PB$, $PC$ across the angle bisectors of $\angle A$, $\angle B$, and $\angle C$, respectively. Given a point $Q$, let $\mathfrak K(Q)$ denote the unique cubic plane curve which passes through all points $P$ such that line $PP^\ast$ contains $Q$. Consider: [list] [*] the M'Cay cubic $\mathfrak K(O)$, where $O$ is the circumcenter of $\triangle ABC$, [*] the Thomson cubic $\mathfrak K(G)$, where $G$ is the centroid of $\triangle ABC$, [*] the Napoleon-Feurerbach cubic $\mathfrak K(N)$, where $N$ is the nine-point center of $\triangle ABC$, [*] the Darboux cubic $\mathfrak K(L)$, where $L$ is the de Longchamps point (the reflection of the orthocenter across point $O$), [*] the Neuberg cubic $\mathfrak K(X_{30})$, where $X_{30}$ is the point at infinity along line $OG$, [*] the nine-point circle of $\triangle ABC$, [*] the incircle of $\triangle ABC$, and [*] the circumcircle of $\triangle ABC$. [/list] Estimate $N$, the number of points lying on at least two of these eight curves. An estimate of $E$ earns $\left\lfloor 20 \cdot 2^{-|N-E|/6} \right\rfloor$ points.

Let $(a_1, a_2, ..., a_8)$ be a permutation of $(1, 2, ... , 8)$. Find, with proof, the maximum possible number of elements of the set $$\{a_1, a_1 + a_2, ... , a_1 + a_2 + ... + a_8\}$$ that can be perfect squares.