Found problems: 85335
A set $P$ of $2002$ persons is given. The family of subsets of $P$ containing exactly $1001$ persons has the property that the number of acquaintance pairs in each such subset is the same. (It is assumed that the acquaintance relation is symmetric). Find the best lower estimation of the acquaintance pairs in the set $P$.
A circle is circumscribed around an isosceles triangle $ABC$ with base $BC$. The bisector of the angle $C$ and the bisector of the angles $A$ intersect the circle at the points $E$ and $D$, respectively, and the segment $DE$ intersects the sides $BC$ and $AB$ at the points $P$ and $Q$, respectively. Reconstruct $\vartriangle ABC$ given points $D, P, Q$, if it is known in which half-plane relative to the line $DQ$ lies the vertex $A$.
(Maria Rozhkova)
For a positive integer $ n>2$, let $ \left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a sequence of $ n$ positive reals which is either non-decreasing (this means, we have $ a_{1}\leq a_{2}\leq \ldots \leq a_{n}$) or non-increasing (this means, we have $ a_{1}\geq a_{2}\geq \ldots \geq a_{n}$), and which satisfies $ a_{1}\neq a_{n}$. Let $ x$ and $ y$ be positive reals satisfying $ \frac{x}{y}\geq \frac{a_{1}-a_{2}}{a_{1}-a_{n}}$. Show that:
\[ \frac{a_{1}}{a_{2}\cdot x+a_{3}\cdot y}+\frac{a_{2}}{a_{3}\cdot x+a_{4}\cdot y}+\ldots+\frac{a_{n-1}}{a_{n}\cdot x+a_{1}\cdot y}+\frac{a_{n}}{a_{1}\cdot x+a_{2}\cdot y}\geq \frac{n}{x+y}. \]
For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p\plus{}q}{2} \right) \leq \frac{f(p) \plus{} f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always:
\[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]
How many 5 digit positive integers are there such that each of its digits, except for the last one, is greater than or equal to the next digit?
Inscribe the equilateral triangle of the largest perimeter in a given semicircle.
Given is a convex pentagon $ABCDE$ in which $\angle A = 60^o$, $\angle B = 100^o$, $\angle C = 140^o$.
Show that this pentagon can be placed in a circle with a radius of $\frac23 AD$.
Let $f(n)$ be the integer closest to $\sqrt{n}$. Compute the largest $N$ less than or equal to $2018$ such that $\sum_{i=1}^N\frac{1}{f(i)}$ is integral.
Find all pairs of positive integers $(n, k)$ that satisfy the equation
$$n!+n=n^k$$
Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$,
\[\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),\]where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$.
[i]Calvin Deng.[/i]
If $a,b,c$ are the roots of $x^3+20x^2+1x+5$, compute $(a^2+1)(b^2+1)(c^2+1)$.
[i]2015 CCA Math Bonanza Tiebreaker Round #2[/i]
Find all integers $n\geq 3$ for which every convex equilateral $n$-gon of side length $1$ contains an equilateral triangle of side length $1$. (Here, polygons contain their boundaries.)
A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?
$
\textbf{(A) }2\sqrt{3}\qquad
\textbf{(B) }4\qquad
\textbf{(C) }3\sqrt{2}\qquad
\textbf{(D) }2\sqrt{5}\qquad
\textbf{(E) }5\qquad
$
Prove that the product of the sides of a quadrilateral inscribed in a circle with radius $1$ does not exceed $4$.
\[\int_0^\infty e^{-\lfloor x \rfloor(1+\{x\})}\,\mathrm dx\]
[i]Proposed by Vlad Oleksenko[/i]
Determine all solutions $(n, k)$ of the equation $n!+An = n^k$ with $n, k \in\mathbb{N}$ for $A = 7$ and for $A = 2012$.
Determine all complex numbers $z$ such that \[\Bigl|z-\bigl|z+|z|\bigr|\Bigr|-|z|\sqrt3\ge0\] and draw the set of all such $z$ in complex plane.
Let $d,p,q$ be fixed positive integers, and $d$ is not a perfect square. $\mathbb{N}$ is the set of all positive integers, and $S=\{m+n\sqrt{d}|m,n \in \mathbb{N}\} \cup \{0\}$. Suppose the function $f: S \to S$ satisfies the following conditions for all $x,y \in S$:
(i) $f((xy)^p)=(f(x)f(y))^p$
(ii)$f((x+y)^q)=(f(x)+f(y))^q$
Find the function $f$.
Quadrilateral $ABCD$ has $AB=BC=CD$, $\angle ABC=70^\circ$, and $\angle BCD=170^\circ$. What is the degree measure of $\angle BAD$?
$ \textbf{(A)}\ 75\qquad
\textbf{(B)}\ 80\qquad
\textbf{(C)}\ 85\qquad
\textbf{(D)}\ 90\qquad
\textbf{(E)}\ 95$
There are $100$ students who want to sign up for the class Introduction to Acting. There are three class sections for Introduction to Acting, each of which will fit exactly $20$ students. The $100$ students, including Alex and Zhu, are put in a lottery, and 60 of them are randomly selected to fill up the classes. What is the probability that Alex and Zhu end up getting into the same section for the class?
Determine all positive rational numbers $r \neq 1$ such that $\sqrt[r-1]{r}$ is rational.
Let $p$ be a prime number. A set of $p + 2$ positive integers, not necessarily distinct, is called [i]interesting [/i] if the sum of any $p$ of them is divisible by each of the other two. Determine all interesting sets.
Two points $J$ and $H$ lie $26$ units apart on a given plane. Let $M$ be the locus of points $T$ on this plane such that $JT^2 + HT^2 = 2020$. Then, M encloses a region on the plane with area $a$ and perimeter $p$. If $q$ and $r$ are coprime positive integers and $\frac{a}{p} = \frac{q}{r}$ , then compute $q + r$.
A random selector can only select one of the nine integers $ 1,2,\ldots,9$, and it makes these selections with equal probability. Determine the probability that after $ n$ selections ($ n>1$), the product of the $ n$ numbers selected will be divisible by 10.
Distinct positive integers $a, b, c, d$ satisfy
$$\begin{cases} a \mid b^2 + c^2 + d^2,\\
b\mid a^2 + c^2 + d^2,\\
c \mid a^2 + b^2 + d^2,\\
d \mid a^2 + b^2 + c^2,\end{cases}$$
and none of them is larger than the product of the three others. What is the largest possible number of primes among them?