Found problems: 85335
In how many ways can the set of integers $\{1,2,\ldots ,1995\}$ be partitioned into three non-empty sets so that none of these sets contains any pair of consecutive integers?
Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$.
(Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)
A group of 6 students decided to make [i]study groups[/i] and [i]service activity groups[/i] according to the following principle:
Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.
Supposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.
The average of two positive real numbers is equal to their difference. What is the ratio of the larger number to the smaller one?
[i]Author: Ray Li[/i]
Determine all the injective functions $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, such that for each pair of integers $(m, n)$ the following conditions hold:
$a)$ $f(mn) = f(m)f(n)$
$b)$ $f(m^2 + n^2) \mid f(m^2) + f(n^2).$
In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
Let $S$ be the set of real numbers $k$ with the following property: for all set of real numbers $(a,b,c)$ satisfying $ab+bc+ca=1$, we always have the inequality:$$\frac{a}{{\sqrt {{a^2} + ab + {b^2} + k} }} + \frac{b}{{\sqrt {{b^2} + bc + {c^2} + k} }} + \frac{c}{{\sqrt {{c^2} + ca + {a^2} + k} }} \ge \sqrt {\frac{3}{{k + 1}}} .$$
a) Assume that $k\in S$. Prove that: $k\ge 2$.
b) Prove that: $2\in S$.
The convex pentagon $ABC DE$ is given, such that $AB,BC,CD$ and $DE$ are parallel to one of the diagonals. Prove that this also applies to $EA$.
Let $ABCD$ be a square with $AB=6.$ A point $P$ in the interior is $2$ units away from side $BC$ and $3$ units away from side $CD.$ What is the distance from $P$ to $A?$
Let $a_1, a_2, a_3, \dots, a_{20}$ be a permutation of the numbers $1, 2, \dots, 20$. How many different values can the expression $a_1-a_2+a_3-\dots - a_{20}$ have?
Find all pairs $(x, y)$ of positive integers such that $(x + y)(x^2 + 9y)$ is the cube of a prime number.
Elfs and trolls are seated at a round table, 60 creatures in total. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake.
Everybody claims to sit between an elf and a troll, but exactly two elfs made a mistake! How many trolls are there at this table?
Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$.
Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$.
[i]Author: Farzan Barekat, Canada[/i]
Let $ABCD$ be a convex quadrilateral. In the triangle $ABC$ let $I$ and $J$ be the incenter and the excenter opposite to vertex $A$, respectively. In the triangle $ACD$ let $K$ and $L$ be the incenter and the excenter opposite to vertex $A$, respectively. Show that the lines $IL$ and $JK$, and the bisector of the angle $BCD$ are concurrent.
A net for hexagonal pyramid is constructed by placing a triangle with side lengths $x$, $x$, and $y$ on each side of a regular hexagon with side length $y$. What is the maximum volume of the pyramid formed by the net if $x+y=20$?
We have $2021$ colors and $2021$ chips of each color. We place the $2021^2$ chips in a row. We say that a chip $F$ is [i]bad[/i] if there is an odd number of chips that have a different color to $F$ both to the left and to the right of $F$.
(a) Determine the minimum possible number of bad chips.
(b) If we impose the additional condition that each chip must have at least one adjacent chip of the same color, determine the minimum possible number of bad chips.
If $ x^{ \minus{} 1} \minus{} 1$ is divided by $ x \minus{} 1$ the quotient is:
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac {1}{x \minus{} 1} \qquad\textbf{(C)}\ \frac { \minus{} 1}{x \minus{} 1} \qquad\textbf{(D)}\ \frac {1}{x} \qquad\textbf{(E)}\ \minus{} \frac {1}{x}$
Find the minimum value of $xy$, given that $x^2+y^2+z^2=7$, $xz+xy+yz=4$, and $x, y, z$ are real numbers
If $x\neq y$ and the sequences $x,a_1,a_2,y$ and $x,b_1,b_2,b_3,y$ each are in arithmetic progression, then $(a_2-a_1)/(b_2-b_1)$ equals
$\textbf{(A) }\frac{2}{3}\qquad\textbf{(B) }\frac{3}{4}\qquad\textbf{(C) }1\qquad\textbf{(D) }\frac{4}{3}\qquad \textbf{(E) }\frac{3}{2}$
Let $a,b,c,d,e$ be non-negative real numbers such that $a+b+c+d+e>0$. What is the least real number $t$ such that $a+c=tb$, $b+d=tc$, $c+e=td$?
$ \textbf{(A)}\ \frac{\sqrt 2}2
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ \sqrt 2
\qquad\textbf{(D)}\ \frac32
\qquad\textbf{(E)}\ 2
$
Find the remainder when $$\left \lfloor \frac{149^{151} + 151^{149}}{22499}\right \rfloor$$ is divided by $10^4$.
[i]Proposed by Vijay Srinivasan[/i]
Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The [i]distance[/i] between two cities is the least number of roads on any path between the two cities.)
For which $n$ is it possible for Mark to achieve this?
[i]Proposed by Michael Ren[/i]
$a,b,c$ are positive numbers such that $a+b+c=3$, show that: \[\frac{a^{2}+9}{2a^{2}+(b+c)^{2}}+\frac{b^{2}+9}{2b^{2}+(a+c)^{2}}+\frac{c^{2}+9}{2c^{2}+(a+b)^{2}}\leq 5\]
Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius $r$, there exist one (or more) with maximum area. If so, determine their shape and area.