Found problems: 85335
Let $n \geq2$ be an integer number and $D_n$ the set of all the points $(x,y)$ in the plane such that its coordinates are integer numbers with: $-n \le x \le n$ and $-n \le y \le n$.
(a) There are three possible colors in which the points of $D_n$ are painted with (each point has a unique color). Show that with
any distribution of the colors, there are always two points of $D_n$ with the same color such that the line that contains them does not go through any other point of $D_n$.
(b) Find a way to paint the points of $D_n$ with 4 colors such that if a line contains exactly two points of $D_n$, then, this points have different colors.
Let \[ f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3 \] be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$.
Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.
For some positive integers $m>n$, the quantities $a=\text{lcm}(m,n)$ and $b=\gcd(m,n)$ satisfy $a=30b$. If $m-n$ divides $a$, then what is the value of $\frac{m+n}{b}$?
[i]Proposed by Andrew Wu[/i]
Find all real $\alpha,\beta$ such that the following limit exists and is finite: \[\lim_{x,y\rightarrow 0^{+}}\frac{x^{2\alpha}y^{2\beta}}{x^{2\alpha}+y^{3\beta}}\]
Pentagon $ABCDE$ is given in the plane. Let the perpendicular from $A$ to line $CD$ be $F$, the perpendicular from $B$ to $DE$ be $G$, from $C$ to $EA$ be $H$, from $D$ to $AB$ be $I$,and from $E$ to $BC$ be $J$. Given that lines $AF,BG,CH$, and $DI$ concur, show that they also concur with line $EJ$.
Suppose $a_3x^3 - x^2 + a_1x - 7 = 0$ is a cubic polynomial in x whose roots $\alpha,\beta, \gamma$ are positive real numbers satisfying $$\frac{225\alpha^2}{\alpha^2 +7}=\frac{144\beta^2}{\beta^2 +7}=\frac{100\gamma^2}{\gamma^2 +7}.$$ Find $a_1$.
[b]p1.[/b] Find the area of a right triangle with legs of lengths $20$ and $18$.
[b]p2.[/b] How many $4$-digit numbers (without leading zeros) contain only $2,0,1,8$ as digits? Digits can be used more than once.
[b]p3.[/b] A rectangle has perimeter $24$. Compute the largest possible area of the rectangle.
[b]p4.[/b] Find the smallest positive integer with $12$ positive factors, including one and itself.
[b]p5.[/b] Sammy can buy $3$ pencils and $6$ shoes for $9$ dollars, and Ben can buy $4$ pencils and $4$ shoes for $10$ dollars at the same store. How much more money does a pencil cost than a shoe?
[b]p6.[/b] What is the radius of the circle inscribed in a right triangle with legs of length $3$ and $4$?
[b]p7.[/b] Find the angle between the minute and hour hands of a clock at $12 : 30$.
[b]p8.[/b] Three distinct numbers are selected at random fromthe set $\{1,2,3, ... ,101\}$. Find the probability that $20$ and $18$ are two of those numbers.
[b]p9.[/b] If it takes $6$ builders $4$ days to build $6$ houses, find the number of houses $8$ builders can build in $9$ days.
[b]p10.[/b] A six sided die is rolled three times. Find the probability that each consecutive roll is less than the roll before it.
[b]p11.[/b] Find the positive integer $n$ so that $\frac{8-6\sqrt{n}}{n}$ is the reciprocal of $\frac{80+6\sqrt{n}}{n}$.
[b]p12.[/b] Find the number of all positive integers less than $511$ whose binary representations differ from that of $511$ in exactly two places.
[b]p13.[/b] Find the largest number of diagonals that can be drawn within a regular $2018$-gon so that no two intersect.
[b]p14.[/b] Let $a$ and $b$ be positive real numbers with $a > b $ such that $ab = a +b = 2018$. Find $\lfloor 1000a \rfloor$. Here $\lfloor x \rfloor$ is equal to the greatest integer less than or equal to $x$.
[b]p15.[/b] Let $r_1$ and $r_2$ be the roots of $x^2 +4x +5 = 0$. Find $r^2_1+r^2_2$ .
[b]p16.[/b] Let $\vartriangle ABC$ with $AB = 5$, $BC = 4$, $C A = 3$ be inscribed in a circle $\Omega$. Let the tangent to $\Omega$ at $A$ intersect $BC$ at $D$ and let the tangent to $\Omega$ at $B$ intersect $AC$ at $E$. Let $AB$ intersect $DE$ at $F$. Find the length $BF$.
[b]p17.[/b] A standard $6$-sided die and a $4$-sided die numbered $1, 2, 3$, and $4$ are rolled and summed. What is the probability that the sum is $5$?
[b]p18.[/b] Let $A$ and $B$ be the points $(2,0)$ and $(4,1)$ respectively. The point $P$ is on the line $y = 2x +1$ such that $AP +BP$ is minimized. Find the coordinates of $P$.
[b]p19.[/b] Rectangle $ABCD$ has points $E$ and $F$ on sides $AB$ and $BC$, respectively. Given that $\frac{AE}{BE}=\frac{BF}{FC}= \frac12$, $\angle ADE = 30^o$, and $[DEF] = 25$, find the area of rectangle $ABCD$.
[b]p20.[/b] Find the sum of the coefficients in the expansion of $(x^2 -x +1)^{2018}$.
[b]p21.[/b] If $p,q$ and $r$ are primes with $pqr = 19(p+q+r)$, find $p +q +r$ .
[b]p22.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B$ is acute and $AB < AC$. Let $D$ be the foot of altitude from $A$ to $BC$ and $F$ be the foot of altitude from $E$, the midpoint of $BC$, to $AB$. If $AD = 16$, $BD = 12$, $AF = 5$, find the value of $AC^2$.
[b]p23.[/b] Let $a,b,c$ be positive real numbers such that
(i) $c > a$
(ii) $10c = 7a +4b +2024$
(iii) $2024 = \frac{(a+c)^2}{a}+ \frac{(c+a)^2}{b}$.
Find $a +b +c$.
[b]p24.[/b] Let $f^1(x) = x^2 -2x +2$, and for $n > 1$ define $f^n(x) = f ( f^{n-1}(x))$. Find the greatest prime factor of $f^{2018}(2019)-1$.
[b]p25.[/b] Let $I$ be the incenter of $\vartriangle ABC$ and $D$ be the intersection of line that passes through $I$ that is perpendicular to $AI$ and $BC$. If $AB = 60$, $C A =120$, and $CD = 100$, find the length of $BC$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Determine all pairs of prime numbers $(p, q)$ which satisfy the equation
\[
p^3+q^3+1=p^2q^2
\]
A set $S$ of four distinct points is given in the plane. It is known that for any point $X\in S$ the remaining points can be denoted by $Y,Z$ and $W$ so that
$|XY|=|XZ|+|XW|$
Prove that all four points lie on a line.
If $ P(x)$ denotes a polynomial of degree $ n$ such that $ P(k)\equal{}\frac{k}{k\plus{}1}$ for $ k\equal{}0,1,2,\ldots,n$, determine $ P(n\plus{}1)$.
Find all integers $x,y,z$ such that $x^2(x^2+y)=y^{z+1}$.
Each of the $9$ cells in a $3\times 3$ grid is colored either blue or white with equal probability. The expected value of the area of the largest square of blue cells contained within the grid is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Find the minimum value of the expression $ \frac{a\plus{}1}{a(a\plus{}2)}\plus{}\frac{b\plus{}1}{b(b\plus{}2)}\plus{}\frac{c\plus{}1}{c(c\plus{}2)}$, where $ a,b,c$ are positive real numbers with $ a\plus{}b\plus{}c \le 3$.
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.
Prove that Sisyphus cannot reach the aim in less than
\[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]
turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )
A function $f:[0,\infty)\to\mathbb R\setminus\{0\}$ is called [i]slowly changing[/i] if for any $t>1$ the limit $\lim_{x\to\infty}\frac{f(tx)}{f(x)}$ exists and is equal to $1$. Is it true that every slowly changing function has for sufficiently large $x$ a constant sign (i.e., is it true that for every slowly changing $f$ there exists an $N$ such that for every $x,y>N$ we have $f(x)f(y)>0$?)
Find all prime numbers $p$ and $q$ such that
$$1 + \frac{p^q - q^p}{p + q}$$
is a prime number.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Let $ABC$ be an isosceles triangle with vertex $A$ and $AB> BC$. Let $k$ be the circle with center $A$ passsing through $B$ and $C$. Let $H$ be the second intersection of $k$ with the altitude of the triangle $ABC$ through $B$. Further let $G$ be the second intersection of $k$ with the median through $B$ in triangle $ABC$. Let $X$ be the intersection of the lines $AC$ and $GH$. Show that $C$ is the midpoint of $AX$.
Let $L$ be the answer to problem $9$. Find the solution to the equation $4x+\sqrt{L}=0$.
Let $M$ be the answer to problem $7$. Let $f(x)=x^4+4x^3+6x^2+1$. Find $f(M)$.
Let $T$ be the answer to problem $8$. Find the area of a square with side length $T$.
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.
Prove that if $g \ge 2$ is an integer, then two series \[\sum_{n=0}^{\infty}\frac{1}{g^{n^{2}}}\;\; \text{and}\;\; \sum_{n=0}^{\infty}\frac{1}{g^{n!}}\] both converge to irrational numbers.
The point $P$, from the plane in which $\Delta ABC$ lies, is such that if $A_1,B_1$, and $C_1$ are the orthogonal projections of $P$ on the respective altitudes of $ABC$, then $AA_1=BB_1=CC_1=t$. Determine the locus of $P$ and length of $t$.
For positive integers $n,a,b$, if $n=a^2 +b^2$, and $a$ and $b$ are coprime, then the number pair $(a,b)$ is called a [i]square split[/i] of $n$ (the order of $a, b$ does not count). Prove that for any positive $k$, there are only two square splits of the integer $13^k$.
A function $f:\mathbb{R} \to \mathbb{R}$ has the following property:
$$\text{For every } x,y \in \mathbb{R} \text{ such that }(f(x)+y)(f(y)+x) > 0, \text{ we have } f(x)+y = f(y)+x.$$
Prove that $f(x)+y \leq f(y)+x$ whenever $x>y$.
Let $ABC$ be a triangle with $AB\neq AC$. Let the excircle $\omega$ opposite $A$ touch $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Suppose $X$ and $Y$ are points on the segments $AC$ and $AB$, respectively, such that $XY$ and $BC$ are parallel, and let $\Gamma$ be a circle through $X$ and $Y$ which is externally tangent to $\omega$ at $Z$. Prove that the lines $EF$, $DZ$, and $XY$ are concurrent.