Found problems: 85335
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?
How many positive integers less than or equal to $150$ have exactly three distinct prime factors?
Does there exist a convex polygon such that all its sidelengths are equal and all triangle formed by its vertices are obtuse-angled?
For which $n$ is $n^4+6n^3+11n^2+3n+31$ a perfect square?
A cube of edge length $3$ consists of $27$ unit cubes. Find the number of lines passing through exactly three centers of these $27$ cubes, as well as the number of those passing through exactly two such centers.
A large collection of congruent right triangles is given, each with side length 3,4,5. Find the maximal number of such triangles you can place inside a 20x20 square, with no two triangles intersecting (in their interiors).
Let $ Y_n$ be a binomial random variable with parameters $ n$ and $ p$. Assume that a certain set $ H$ of positive integers has a density and that this density is equal to $ d$. Prove the following statements:
(a) $ \lim _{n \rightarrow \infty}P(Y_n\in H)\equal{}d$ if $ H$ is an arithmetic progression.
(b) The previous limit relation is not valid for arbitrary $ H$.
(c) If $ H$ is such that $ P(Y_n \in H)$ is convergent, then the limit must be equal to $ d$.
[i]L. Posa[/i]
Prove that all real roots of the polynomial
$$ P=X^{1985}-X^{1984}+1983\cdot X^{1983}+1994\cdot X^{992} -884064 $$
are positive.
In country there are two capitals ($A$ and $B$) and finite number of towns.
Some towns (or town with one of capital) connected with roads (one-way). (between every two towns or capital and town there are arbitrary number of roads) such that exist at least one way from $A$ to $B$.
Given, that any two ways from $A$ to $B$ have at least one common road.
Prove, that exist one road, such that all ways from $A$ to $B$ pass through this road.
Find the least positive integer such that the product of its digits is $8! = 8 \cdot 7 \cdot6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$.
The line $\ell$ passes through vertex$ B$ and the interior of regular hexagon $ABCDEF$. If the distances from $\ell$ to the vertices $A$ and $C$ are $7$ and $4$, respectively, compute the area of hexagon $ABCDEF$.
A $(2n + 1) \times (2n + 1)$ grid, with $n> 0$, is colored in such a way that each of the cell is white or black. A cell is called [i]special[/i] if there are at least $n$ other cells of the same color in its row, and at least another $n$ cells of the same color in its column.
(a) Prove that there are at least $2n + 1$ special boxes.
(b) Provide an example where there are at most $4n$ special cells.
(c) Determine, as a function of $n$, the minimum possible number of special cells.
Given an acute triangle $ABC$. The inscribed circle of triangle $ABC$ is tangent to $AB$ and $AC$ at $X$ and $Y$ respectively. Let $CH$ be the altitude. The perpendicular bisector of the segment $CH$ intersects the line $XY$ at $Z$. Prove that $\angle BZC = 90^o.$
([b]5[/b]) Find $ p$ so that $ \lim_{x\rightarrow\infty}x^p\left(\sqrt[3]{x\plus{}1}\plus{}\sqrt[3]{x\minus{}1}\minus{}2\sqrt[3]{x}\right)$ is some non-zero real number.
Find all pairs $(p,q)$ of prime numbers which $p>q$ and
$$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$
is an integer.
Let $a_1, a_2, \cdots, a_n$ be a sequence of real numbers with $a_1+a_2+\cdots+a_n=0$. Define the score $S(\sigma)$ of a permutation $\sigma=(b_1, \cdots b_n)$ of $(a_1, \cdots a_n)$ to be the minima of the sum $$(x_1-b_1)^2+\cdots+(x_n-b_n)^2$$ over all real numbers $x_1\le \cdots \le x_n$.
Prove that $S(\sigma)$ attains the maxima over all permutations $\sigma$, if and only if for all $1\le k\le n$, $$b_1+b_2+\cdots+b_k\ge 0.$$
[i]Proposed by Anzo Teh Zhao Yang[/i]
Let $ABC$ and $XYZ$ be two triangles. Define
\[A_1=BC\cap ZX, A_2=BC\cap XY,\]\[B_1=CA\cap XY, B_2=CA\cap YZ,\]\[C_1=AB\cap YZ, C_2=AB\cap ZX.\]
Hereby, the abbreviation $g\cap h$ means the point of intersection of two lines $g$ and $h$.
Prove that $\frac{C_1C_2}{AB}=\frac{A_1A_2}{BC}=\frac{B_1B_2}{CA}$ holds if and only if $\frac{A_1C_2}{XZ}=\frac{C_1B_2}{ZY}=\frac{B_1A_2}{YX}$.
Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$.
[i]Proposed by Jozsef Pelikan, Hungary[/i]
Let $A_1A_2A_3 \ldots A_{21}$ be a 21-sided regular polygon inscribed in a circle with centre $O$. How many triangles $A_iA_jA_k$, $1 \leq i < j < k \leq 21$, contain the centre point $O$ in their interior?
There are several rational numbers written on a board. If the numbers $x{}$ and $y{}$ are present on the board, we can add the number $(x+y)/(1-xy)$ to it. Prove that there is a rational number that cannot ever appear on the board.
Let $ ABC$ be acute triangle. The circle with diameter $ AB$ intersects $ CA,\, CB$ at $ M,\, N,$ respectively. Draw $ CT\perp AB$ and intersects above circle at $ T$, where $ C$ and $ T$ lie on the same side of $ AB$. $ S$ is a point on $ AN$ such that $ BT = BS$. Prove that $ BS\perp SC$.
Prove that every positive rational number can be represented in the form $\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}$ where a,b,c,d are positive integers.
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
Let $a,b,c$ be three integers for which the sum
\[ \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}\]
is integer.
Prove that each of the three numbers
\[ \frac{ab}{c}, \quad \frac{ac}{b},\quad \frac{bc}{a}\]
is integer.
(Proposed by Gerhard J. Woeginger)
Let $k$ be a given integer, $3 < k \leq n$. Consider a graph $G$ with $n$ vertices satisfying the condition: for any two non-adjacent vertices $x$ and $y$ in graph $G$, the sum of their degrees must satisfy $d(x) + d(y) \geq k$. Please answer the following questions and prove your conclusions.
(1) Suppose the length of the longest path in graph $G$ is $l$ satisfying the inequality $3 \leq l < k$, does graph $G$ necessarily contain a cycle of length $l+1$? (The length of a path or cycle refers to the number of edges that make up the path or cycle.)
(2) For the case where $3 < k \leq n-1$ and graph $G$ is connected, can we determine that the length of the longest path in graph $G$, $l \geq k$?
(3) For the case where $3 < k = n-1$, is it necessary for graph $G$ to have a path of length $n-1$ (i.e., a Hamiltonian path)?