Found problems: 85335
. Find all triplets $(x, y, z)$, $x > y > z$ of positive integers such that $\frac{1}{x}+\frac{2}{y}+\frac{3}{z}= 1$
Find the orders of the subgroups of the group of rotations of the cube, and find its normal subgroups.
In a triangle $ABC$ with a right angle at $A$, $H$ is the foot of the altitude from $A$. Prove that the sum of the inradii of the triangles $ABC$, $ABH$, and $AHC$ is equal to $AH$.
$\textbf{Juan's Old Stamping Grounds}$
Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.)
[asy]
/* AMC8 2002 #8, 9, 10 Problem */
size(3inch, 1.5inch);
for ( int y = 0; y <= 5; ++y )
{
draw((0,y)--(18,y));
}
draw((0,0)--(0,5));
draw((6,0)--(6,5));
draw((9,0)--(9,5));
draw((12,0)--(12,5));
draw((15,0)--(15,5));
draw((18,0)--(18,5));
draw(scale(0.8)*"50s", (7.5,4.5));
draw(scale(0.8)*"4", (7.5,3.5));
draw(scale(0.8)*"8", (7.5,2.5));
draw(scale(0.8)*"6", (7.5,1.5));
draw(scale(0.8)*"3", (7.5,0.5));
draw(scale(0.8)*"60s", (10.5,4.5));
draw(scale(0.8)*"7", (10.5,3.5));
draw(scale(0.8)*"4", (10.5,2.5));
draw(scale(0.8)*"4", (10.5,1.5));
draw(scale(0.8)*"9", (10.5,0.5));
draw(scale(0.8)*"70s", (13.5,4.5));
draw(scale(0.8)*"12", (13.5,3.5));
draw(scale(0.8)*"12", (13.5,2.5));
draw(scale(0.8)*"6", (13.5,1.5));
draw(scale(0.8)*"13", (13.5,0.5));
draw(scale(0.8)*"80s", (16.5,4.5));
draw(scale(0.8)*"8", (16.5,3.5));
draw(scale(0.8)*"15", (16.5,2.5));
draw(scale(0.8)*"10", (16.5,1.5));
draw(scale(0.8)*"9", (16.5,0.5));
label(scale(0.8)*"Country", (3,4.5));
label(scale(0.8)*"Brazil", (3,3.5));
label(scale(0.8)*"France", (3,2.5));
label(scale(0.8)*"Peru", (3,1.5));
label(scale(0.8)*"Spain", (3,0.5));
label(scale(0.9)*"Juan's Stamp Collection", (9,0), S);
label(scale(0.9)*"Number of Stamps by Decade", (9,5), N);
[/asy]
In dollars and cents, how much did his South American stampes issued before the '70s cost him?
$ \text{(A)}\ \textdollar 0.40\qquad\text{(B)}\ \textdollar 1.06\qquad\text{(C)}\ \textdollar 1.80\qquad\text{(D)}\ \textdollar 2.38\qquad\text{(E)}\ \textdollar 2.64 $
If $r_1$, $r_2$, and $r_3$ are the solutions to the equation $x^3 - 5x^2 + 6x - 1 = 0$, then what is the value of $r_1^2 + r_2^2 + r_3^2$?
[i]Proposed by Eugene Chen[/i]
Three pairwairs distinct positive integers $a,b,c,$ with $gcd(a,b,c)=1$, satisfy \[a|(b-c)^2 ,b|(a-c)^2 , c|(a-b)^2\] Prove that there doesnt exist a non-degenerate triangle with side lengths $a,b,c.$
An integer $n\ge 1$ is called balanced if it has an even number of distinct prime divisors. Prove that there exist infinitely many positive integers $n$ such that there are exactly two balanced numbers among $n,n+1,n+2$ and $n+3$.
For positive numbers $ a $, $ b $ and $ c $ satisfying the equality $ a + b + c = 1 $, prove the inequality
$$
\frac {a ^ 7 + b ^ 7} {a ^ 5 + b ^ 5} + \frac {b ^ 7 + c ^ 7} {b ^ 5 + c ^ 5} + \frac {c ^ 7 + a ^ 7} {c ^ 5 + a ^ 5} \geq \frac {1} {3}.
$$
Let there be an integer $n\geq2$. In a chess tournament $n$ players play between each other one game. No game ended in a draw. Show that after the end of the tournament the players can be arranged in a list: $P_1, P_2, P_3,\ldots,P_n$ such that for every $i (1\leq i\leq n-1)$ the player $P_i$ won against player $P_{i+1}$.
Let $N=1890*1930*1970$, find the number of divisors of N which are not divisors of $45$
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$
with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$
Suppose $a_1$, $a_2$, $a_3$, $\dots$ is an arithmetic sequence such that \[a_1+a_2+a_3+\cdots+a_{48}+a_{49}=1421.\] Find the value of $a_1+a_4+a_7+a_{10}+\cdots+a_{49}$.
[i]Proposed by Tony Kim[/i]
An equilateral triangle with side length $1$ has the same area as a square with side length $s$. Find $s$.
$\textbf{(A) }\dfrac{\sqrt[4]3}2\hspace{14em}\textbf{(B) }\dfrac{\sqrt[4]3}{\sqrt2}\hspace{14em}\textbf{(C) }1$
$\textbf{(D) }\dfrac34\hspace{14.9em}\textbf{(E) }\dfrac43\hspace{14.9em}\textbf{(F) }\sqrt3$
$\textbf{(G) }\dfrac{\sqrt6}2$
Given that ${a_n}$ and ${b_n}$ are two sequences of integers defined by
\begin{align*}
a_1=1, a_2=10, a_{n+1}=2a_n+3a_{n-1} & ~~~\text{for }n=2,3,4,\ldots, \\
b_1=1, b_2=8, b_{n+1}=3b_n+4b_{n-1} & ~~~\text{for }n=2,3,4,\ldots.
\end{align*}
Prove that, besides the number $1$, no two numbers in the sequences are identical.
Let $s_1, s_2, s_3$ be the three roots of $x^3 + x^2 +\frac92x + 9$.
$$\prod_{i=1}^{3}(4s^4_i + 81)$$ can be written as $2^a3^b5^c$. Find $a + b + c$.
We define a sequence ${a_n}$:
$$a_1=1,a_{n+1}=\sqrt{a_n+n^2},n=1,2,...$$
(1)Find $\lfloor a_{2019}\rfloor$
(2)Find $\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor$
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
Find the maximal value of \[ \left( x^3+1 \right) \left( y^3 + 1\right) , \] where $x,y \in \mathbb R$, $x+y=1$.
[i]Dan Schwarz[/i]
This is a really easy one for Junior level :p
$a^2+b^2+c^2+ab+bc+ac=6$
a,b,c>0
Find max{a+b+c}
Let $ABCDE$ be a convex pentagon such that $AE\parallel BC$ and $\angle ADE = \angle BDC$. The diagonals $AC$ and $BE$ intersect at point $F$. Prove that $\angle CBD= \angle ADF$.
Let $n$ be a positive integer. We have $n$ boxes where each box contains a non-negative number of pebbles. In each move we are allowed to take two pebbles from a box we choose, throw away one of the pebbles and put the other pebble in another box we choose. An initial configuration of pebbles is called [i]solvable[/i] if it is possible to reach a configuration with no empty box, in a finite (possibly zero) number of moves. Determine all initial configurations of pebbles which are not solvable, but become solvable when an additional pebble is added to a box, no matter which box is chosen.
In a pentagon $\Pi$ in the plane, $M_1,...M_5$ are the midpoints of the consecutive sides. $Z_i$ is the centroid of the triangle $M_{i} M_{i+1} M_{i+3}$, where $i=1,2...5$ and it is understood that $M_{j\cdot 5}=M_j$ Given pentagon $Z_{1}Z_{2}Z_{3}Z_{4}Z_{5}$, determine the original pentagon $\Pi$.
Let an be a sequence such that $a_0 = 0$ and:
$a_{3n+1} = a_{3n} + 1 = a_n + 1$
$a_{3n+2} = a_{3n} + 2 = a_n + 2$
for all natural numbers $n$. How many $n$ less than $2012$ have the property that $a_n = 7$?
For some integers $b$ and $c$, neither of the equations below have real solutions:
\begin{align*}
2x^2+bx+c&=0\\
2x^2+cx+b&=0.
\end{align*}
What is the largest possible value of $b+c$?
Let $n \geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$) such that there exists a convex $n$-gon $A_{1}A_{2}\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$.)