Found problems: 85335
Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n $ be real numbers such that $\sum_{k=1}^nx_k$ be integer . $d_k=\underset{m\in {Z}}{\min}\left|x_k-m\right| $, $1\leq k\leq n$ .Find the maximum value of $\sum_{k=1}^nd_k$.
Let \(ABCDEF\) be a convex cyclic hexagon such that quadrilateral \(ABDF\) is a square, and the incenter of \(\triangle ACE\) lines on \(\overline{BF}\). Diagonal \(CE\) intersects diagonals \(BD\) and \(DF\) at points \(P\) and \(Q\), respectively. Prove that the circumcircle of \(\triangle DPQ\) is tangent to \(\overline{BF}\).
[i]Proposed by Elliott Liu[/i]
Find the least positive integer ending in $7$ with exactly $12$ positive divisors.
Let $m$ and $n$ be positive integers. A child walks the Cartesian plane taking a few steps.
The child begins its journey at the point $(0, n)$ and ends at the point $(m, 0)$ in such a way that:
$\bullet$ Each step has length $1$ and is parallel to either the $X$ or $Y$ axis.
$\bullet$ For each point $(x, y)$ of its path it is true that $x\ge 0$ and $y\ge 0$.
For each step of the child, the distance between the child and the axis to which said step is parallel is calculated. If the step causes the child to be further from the point $(0, 0)$ than before, we consider that distance as positive, otherwise, we consider that distance as negative. Prove that at the end of the boy's journey, the sum of all the distances is $0$.
The integers from $1$ to $1993$ are written in a line in some order. The following operation is performed with this line: if the first number is $k$ then the first $k$ numbers are rewritten in reverse order. Prove that after some finite number of these operations, the first number in the line of numbers will be $1$.
Show that the inequality \[3(1 + a^2 + a^4)\geq (1 + a + a^2)^2\]
holds for all real numbers $a.$
A piece of cardboard has the shape of a pentagon $ABCDE$ in which $BCDE$ is a square and $ABE$ is an isosceles triangle with a right angle at $A$. Prove that the pentagon can be divided in two different ways in three parts that can be rearranged in order to recompose a right isosceles triangle.
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$
[i]Proposed by Michael Ren[/i]
On a dance evening, each of the gentlemen present has sex with at least one of the ladies present danced and each of the ladies present danced with at least one of the gentlemen present. No gentleman has sex with every lady present and no lady has sex with every gentleman present danced.
It must be proven that there were two such ladies and two such gentlemen among those present has that that evening each of the two ladies with exactly one of the two men, and each of the both men danced with exactly one of the two women.
It is assumed that the dance evening did not take place without ladies and gentlemen, i.e. the crowd, which consists of all the ladies and gentlemen present, is not empty.
[hide=original wording]An einem Tanzabend hat jeder der anwesenden Herren mit mindestens einer der anwesenden Damen getanzt und jede der anwesenden Damen mit mindestens einem der anwesenden Herren.
Kein Herr hat mit jeder der anwesenden Damen und keine Dame mit jedem der anwesenden Herren
getanzt.
Es ist zu beweisen, dass es unter den Anwesenden zwei solche Damen und zwei solche Herren gegeben
hat, dass an dem Abend jede der beiden Damen mit genau einem der beiden Herren, und jeder der
beiden Herren mit genau einer der beiden Damen getanzt hat.
Es wird vorausgesetzt, dass der Tanzabend nicht ohne Damen und Herren stattgefunden hat, d.h.,
die Menge, die aus allen anwesenden Damen und Herren besteht, ist nicht leer.[/hide]
There are several counters of various colours and sizes. No two of them have, simultaneously, the same colour and the same size. On each counter $F$ two numbers are written. One of them is the number of counters that have the same colour as $F$ but a different size than $F$. The other number is the number of counters that have the same size as $F$ but a different colour. It is known that each of the $101$ numbers $0,1,\ldots,100$ is written at least once. Determine the smallest number of counters for which this is possible.
Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle?
$ \textbf{(A) }\frac{\sqrt3}3\qquad\textbf{(B) }\frac{\sqrt3}2\qquad\textbf{(C) }1\qquad\textbf{(D) }\sqrt2\qquad\textbf{(E) }2$
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.
Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.
[i]Proposed by Anton Trygub, Ukraine[/i]
Let $ ABC$ be a triangle with centroid $ G$ and $ A_1,B_1,C_1$ midpoints of the sides $ BC,CA,AB$. A paralel through $ A_1$ to $ BB_1$ intersects $ B_1C_1$ at $ F$. Prove that triangles $ ABC$ and $ FA_1A$ are similar if and only if quadrilateral $ AB_1GC_1$ is cyclic.
Let $0<x<y<z<p$ be integers where $p$ is a prime. Prove that the following statements are equivalent:
$(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p$
$(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p$
For each integer $k\ge2$, the decimal expansions of the numbers $1024,1024^2,\dots,1024^k$ are concatenated, in that order, to obtain a number $X_k$. (For example, $X_2 = 10241048576$.) If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n\ge2$ is an integer.
[i]Proposed by Evan Chen[/i]
In acute triangle $ABC, CA \ne BC$. Let $I$ denote the incenter of triangle $ABC$. Points $A_1$ and $B_1$ lie on rays $CB$ and $CA$, respectively, such that $2CA_1 = 2CB_1 = AB + BC + CA$. Line $CI$ intersects the circumcircle of triangle $ABC$ again at $P$ (other than $C$). Point $Q$ lies on line $AB$ such that $PQ \perp CP$. Prove that $QI \perp A_1B_1$.
Let $AD, BE$, and $CF$ denote the altitudes of triangle $\vartriangle ABC$. Points $E'$ and $F'$ are the reflections of $E$ and $F$ over $AD$, respectively. The lines $BF'$ and $CE'$ intersect at $X$, while the lines $BE'$ and $CF'$ intersect at the point $Y$. Prove that if $H$ is the orthocenter of $\vartriangle ABC$, then the lines $AX, YH$, and $BC$ are concurrent.
Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be $100$ pieces of equal weights. Can Vova prevent Pasha's win?
Let $ABC$ be a triangle, and its incenter touches the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $AD$ intersects the incircle of $ABC$ at $M$ distinct from $D$. Let $DF$ intersects the circumcircle of $CDM$ at $N$ distinct from $D$. Let $CN$ intersects $AB$ at $G$. Prove that $EC = 3GF$.
Let $ABC$ be a triangle and $M$ be the midpoint of $AB$. If it is known that $\angle CAM + \angle MCB = 90^o$, show that triangle $ABC$ is isosceles or right.
Find the remainder when $\prod_{n=3}^{33}2n^4 - 25n^3 + 33n^2$ is divided by $2019$.
Let $ \mathbb{Z}[x]$ be the ring of polynomials with integer coefficients, and let $ f(x), g(x) \in\mathbb{Z}[x]$ be nonconstant polynomials such that $ g(x)$ divides $ f(x)$ in $ \mathbb{Z}[x]$. Prove that if the polynomial $ f(x)\minus{}2008$ has at least 81 distinct integer roots, then the degree of $ g(x)$ is greater than 5.
Let $ABCD$ be a cyclic convex quadrilateral and let $r_a,r_b,r_c,r_d$ be the radii of the circles inscribed in the triangles $BCD, ACD, ABD, ABC$, respectively. Prove that $r_a+r_c=r_b+r_d$.
Find the sum of all real values of $A$ such that the equation $Axy+25x^2+25y^2-20x-22y+5=0$ has a unique solution in real numbers $(x,y)$.