Found problems: 85335
A meeting was attended by $n$ people. They are welcome to occupy the $k$ table provided $\left( k \le \frac{n}{2} \right)$. Each table is occupied by at least two people. When the meeting begins, the moderator selects two people from each table as representatives for talk to. Suppose that $A$ is the number of ways to choose representatives to speak.
Determine the maximum value of $A$ that is possible.
Does there exist a rational $x_1$, such that all members of the sequence $x_1, x_2, \ldots, x_{2024}$ defined by $x_{n+1}=x_n+\sqrt{x_n^2-1}$ for $n=1, 2, \ldots, 2023$ are greater than $1$ and rational?
[b]Problem Section #2
c). Denote by $\mathbb{Q^+}$ the set of all positive rational numbers. Determine all functions $f:\mathbb{Q^+}\to\mathbb{Q^+}$ which satisfy the following equation for all
$x,y \in \mathbb{Q^+} : f(f(x)^2.y)=x^3.f(xy)$.
Given a point $M$ inside a right tetrahedron. Prove that at least one tetrahedron edge is seen from the $M$ in an angle, that has a cosine not greater than $-1/3$. (e.g. if $A$ and $B$ are the vertices, corresponding to that edge, $cos(\widehat{AMB}) \le -1/3$)
The diagonals of a convex quadrilateral $ABCD$ intersect at $E$. The triangles $ABE, BCE, CDE$ and $DAE$ have centroids $K,L,M$ and $N$, and orthocentres $Q,R,S$ and $T$. Show that the quadrilaterals $QRST$ and $LMNK$ are similar.
Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17$. Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2$, find the product $n_1\cdot n_2$.
Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$.
[i](Proposed by Oleksii Masalitin)[/i]
Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if
\[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\]
Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$.
[i]Proposed by Andrey Badzyan, Russia[/i]
Solve the equation $p+\sqrt{q^{2}+r}=\sqrt{s^{2}+t}$ in prime numbers.
Find the number of real solutions $(x,y)$ to the system of equations:
$$\begin{cases}
\sin(x^2-y) = 0 \\
|x|+|y|=2\pi
\end{cases}$$
Let $n$ be given, $n\ge 4,$ and suppose that $P_1,P_2,\dots,P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i.$ What is the probability that at least one of the vertex angles of this polygon is acute.?
Let $p$ be a prime number. Find all integers $k$ for which $\sqrt{k^2 -pk}$ is a positive integer.
Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$
A positive real number $k$, a triangle $ABC$ with circumcircle $\omega$, and a point $M$ on its side $AB$ are fixed. The point $P$ moves along $\omega$, and $Q$ on segment $CP$ is such that $CQ : QP = k$. The line through $P$, parallel to $CM$, intersects the line $MQ$ at point $N$. Prove that $N$ lies on a constant circle, independent of the choice of $P$.
A circle is inscribed in triangle $ABC$. $M$ and $N$ are the points of its tangency with the sides $BC$ and $CA$, respectively. The segment $AM$ intersects $BN$ at point $P$ and the inscribed circle at point $Q$. It is known that $MP = a$, $PQ = b$. Find $AQ$.
Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.
Let $P_i(x) = x^2 + b_i x + c_i , i = 1,2, ..., n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \le i < j \le n , i, j \in N$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.
Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$,
[*]$x+y=0$,
[*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanferminera colorings are there?
A fancy bed and breakfast inn has $5$ rooms, each with a distinctive color-coded decor. One day $5$ friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than $2$ friends per room. In how many ways can the innkeeper assign the guests to the rooms?
$\textbf{(A) }2100\qquad
\textbf{(B) }2220\qquad
\textbf{(C) }3000\qquad
\textbf{(D) }3120\qquad
\textbf{(E) }3125\qquad$
A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?
If $a, b, c$ are side lengths of a triangle, prove that
\[(a + b)(b + c)(c + a) \geq 8(a + b - c)(b + c - a)(c + a - b).\]
[b]p4.[/b] What is gcd $(2^6 - 1, 2^9 - 1)$?
[b]p5.[/b] Sarah is walking along a sidewalk at a leisurely speed of $\frac12$ m/s. Annie is some distance behind her, walking in the same direction at a faster speed of $s$ m/s. What is the minimum value of $s$ such that Sarah and Annie spend no more than one second within one meter of each other?
[b]p6.[/b] You have a choice to play one of two games. In both games, a coin is flipped four times. In game $1$, if (at least) two flips land heads, you win. In game $2$, if (at least) two consecutive flips land heads, you win. Let $N$ be the number of the game that gives you a better chance of winning, and let $p$ be the absolute difference in the probabilities of winning each game. Find $N + p$.
PS. You should use hide for answers.
Given acute triangle $ABC$ with circumcenter $O$ and the center of nine-point circle $N$. Point $N_1$ are given such that $\angle NAB = \angle N_1AC$ and $\angle NBC = \angle N_1BA$. Perpendicular bisector of segment $OA$ intersects the line $BC$ at $A_1$. Analogously define $B_1$ and $C_1$. Show that all three points $A_1,B_1,C_1$ are collinear at a line that is perpendicular to $ON_1$.
Find six distinct positive integers such that the product of any two of them is divisible by their sum.
(D. Fomin, Leningrad)
In a triangle $ABC$, $\angle B=90^o$ and $\angle A=60^o$, $I$ is the point of intersection of its angle bisectors. A line passing through the point $I$ parallel to the line $AC$, intersects the sides $AB$ and $BC$ at the points $P$ and $T$ respectively. Prove that $3PI+IT=AC$ .
(Anton Trygub)