Found problems: 85335
Let $ m,n\in \mathbb{N}^*$. Find the least $ n$ for which exists $ m$, such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$, $ n \minus{} 1$ of length $ 2$, $ ...$, $ 1$ square of length $ n$. For the found value of $ n$ give the example of covering.
The product of $10$ natural numbers is equal to $10^{10}$. What is the largest possible sum of these numbers?
A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?
We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.
The vertices of a triangle are lattice points. There are no lattice points on the sides (apart from the vertices) and $n$ lattice points inside the triangle. Show that its area is $n + \frac12$. Find the formula for the general case where there are also $m$ lattice points on the sides (apart from the vertices).
The points $D, E, F$ lie respectively on the sides $BC$, $CA$, $AB$ of the triangle ABC such that $F B = BD$, $DC = CE$, and the lines $EF$ and $BC$ are parallel. Tangent to the circumscribed circle of triangle $DEF$ at point $F$ intersects line $AD$ at point $P$. Perpendicular bisector of segment $EF$ intersects the segment $AC$ at $Q$. Prove that the lines $P Q$ and $BC$ are parallel.
The points $A_1$,$B_1$,$C_1$ are middle points of the arcs $\widehat{BC}, \widehat{CA}, \widehat{AB}$ of the circumscribed circle of $\Delta ABC$, respectively. The points $I_a,I_b,I_c$ are the reflections in the middle points of $BC,CA,AB$ of the center $I$ of the inscribed circle in the triangle. Prove that $I_a A_1,I_b B_1$, and $I_c C_1$ are concurrent.
$k$ and $n$ are positive integers that are greater than $1$. $N$ is the set of positive integers. $A_1, A_2, \cdots A_k$ are pairwise not-intersecting subsets of $N$ and $A_1 \cup A_2 \cup \cdots \cup A_k = N$.
Prove that for some $i \in \{ 1,2,\cdots,k \}$, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in $A_i$.
Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.
Let $M, \alpha, \beta \in \mathbb{R} $ with $M > 0$ and $\alpha, \beta \in (0,1)$. If $R>1$ is a real number, we say that a sequence of positive real numbers $\{ C_n \}_{n\geq 0}$ is $R$-[i]inoceronte[/i] if $ \sum_{i=1}^n R^{n-i}C_i \leq R^n \cdot M$ for all $n \geq 1$. Determine the smallest real $R>1$ for which exists a $R$-[i]inoceronte[/i] sequence $ \{ C_n \}_{n\geq 0}$ such that $\sum_{n=1}^{\infty} \beta ^n C_n^{\alpha}$ diverges.
For every natural number $a$ and $b$, define the notation $[a,b]$ as the least common multiple of $a $ and $b$ and the notation $(a,b)$ as the greatest common divisor of $a$ and $b$. Find all $n \in \mathbb{N}$ that satisfies
\[
4 \sum_{k=1}^{n} [n,k] = 1 + \sum_{k=1}^{n} (n,k) + 2n^2 \sum_{k=1}^{n} \frac{1}{(n,k)}
\]
Triangle $ABC$ lies in a regular decagon as shown in the figure.
What is the ratio of the area of the triangle to the area of the entire decagon?
Write the answer as a fraction of integers.
[img]https://1.bp.blogspot.com/-Ld_-4u-VQ5o/Xzb-KxPX0wI/AAAAAAAAMWg/-qPtaI_04CQ3vvVc1wDTj3SoonocpAzBQCLcBGAsYHQ/s0/2007%2BMohr%2Bp1.png[/img]
Let $\Gamma_1$ and $\Gamma_2$ be circles internally tangent at point $A$, with $\Gamma_1$ inside $\Gamma_2$. Let $BC$ be a chord of $\Gamma_2$ which is tangent to $\Gamma_1$ at point $D$. Prove that line $AD$ is the angle bisector of $\angle BAC$.
In the plane, three distinct non-collinear points $A,B,C$ are marked. In each step, Ege can do one of the following:
[list]
[*] For marked points $X,Y$, mark the reflection of $X$ across $Y$.
[*]For distinct marked points $X,Y,Z,T$ which do not form a parallelogram, mark the center of spiral similarity which takes segment $XY$ to $ZT$.
[*] For distinct marked points $X,Y,Z,T$, mark the intersection of lines $XY$ and $ZT$.
[/list]
No matter how the points $A,B,C$ are marked in the beginning, can Ege always mark, after finitely many moves,
a) The circumcenter of $\triangle ABC$.
b) The incenter of $\triangle ABC$.
Proposed by [i]Deniz Can Karaçelebi[/i]
A quadratic polynomial $p(x)$ with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)?$
$\textbf{(A) }\dfrac5{16} \qquad \textbf{(B) }\dfrac12 \qquad \textbf{(C) }\dfrac58 \qquad \textbf{(D) }1 \qquad \textbf{(E) }\dfrac98$
In an island shaped like a regular polygon of $n$ sides, there are airports at each vertex of the island. The island would like to add $k$ new airports into the interior of the island, but it must follow the following rules:\\
$1$. It must be in the interior of the island (none on borders).\\
$2$. No two airports can be at the exact same location.\\
$3$. Every triple of $1$ new and $2$ old airports must form an isoceles triangle.\\
$4$. No three airports can be collinear.\\
Find the maximum value of $k$ for each $n$
[hide=Harder Version]Replace $1$ new and $2$ old with $1$ old and $2$ new.[/hide]
Let $b_1 = 1$ and $ b_{n+1} = 1 + \frac{1}{n(n+1)b_1b_2...b_n}$ for $n \ge 1$. Find $b_12$.
Let $ABCD$ be a convex quadrilateral with $\angle ABC = 120^{\circ}$ and $\angle BCD = 90^{\circ}$, and let $M$ and $N$ denote the midpoints of $\overline{BC}$ and $\overline{CD}$. Suppose there exists a point $P$ on the circumcircle of $\triangle CMN$ such that ray $MP$ bisects $\overline{AD}$ and ray $NP$ bisects $\overline{AB}$. If $AB + BC = 444$, $CD = 256$ and $BC = \frac mn$ for some relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Michael Ren[/i]
Each real number greater than 1 is colored red or blue with both colors being used. Prove that there exist real numbers $a$ and $b$ such that the numbers $a+\frac1b$ and $b+\frac1a$ are different colors.
Let $n$ is a positive integers ,$a_1,a_2,\cdots,a_n\in\{0,1,\cdots,n\}$ . For the integer $j$ $(1\le j\le n)$ ,define $b_j$ is the number of elements in the set $\{i|i\in\{1,\cdots,n\},a_i\ge j\}$ .For example ：When $n=3$ ,if $a_1=1,a_2=2,a_3=1$ ,then $b_1=3,b_2=1,b_3=0$ .
$(1)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^2\ge \sum_{i=1}^{n}(i+b_i)^2.$$
$(2)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^k\ge \sum_{i=1}^{n}(i+b_i)^k,$$
for the integer $k\ge 3.$
A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S}\cup\{1\}$ is 13 less than the mean of $\mathcal{S}$, and the mean of $\mathcal{S}\cup\{2001\}$ is 27 more than the mean of $\mathcal{S}.$ Find the mean of $\mathcal{S}.$
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.
Real numbers$ x$ and $y$ it is know that $\frac{x+y}{x-y}+\frac{x-y}{x+y}= 1999$. Find the value of the expression $$\frac{x^4+y^4}{x^4-y^4}+\frac{x^4-y^4}{x^4+y^4}.$$
[b]a)[/b] prove that for every compressed set $K$ in the space $\mathbb R^3$, the function $f:\mathbb R^3 \longrightarrow \mathbb R$ that $f(p)=inf\{|p-k|,k\in K\}$ is continuous.
[b]b)[/b] prove that we cannot cover the sphere $S^2\subseteq \mathbb R^3$ with it's three closed sets, such that none of them contain two antipodal points.
A strongman Bambula can carry several weights at the same time, if their total weight does not exceed $200$ kg, and these weights are no more than three. On the way to work, he injured his finger and found that he could now carry no more than two weights (and still no more than $200$ kg). At what minimum $k$ is the statement true: [i]any set of $100$ weights that Bambula could previously carry in $50$ runs, with a sore finger, he will be able to carry in no more than $k$ runs?[/i]