## Search found 108 matches

Sun Feb 19, 2017 11:12 am
Topic: IMO Marathon
Replies: 184
Views: 64044

### Re: IMO Marathon

$\text{Problem } 53$ Let $n$ be a positive integer and let $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$ be two nondecreasing sequences of real numbers such that $$a_1 + \dots + a_i \le b_1 + \dots + b_i \text{ for every } i = 1, \dots, n$$ and  a_1 + \dots + a_n = b_1 + \do...
Sun Feb 19, 2017 11:02 am
Forum: Combinatorics
Topic: Combi Marathon
Replies: 48
Views: 27826

### Combi Marathon

This thread is dedicated for a marathon on combinatorics problems. The rules are similar to the Geo Marathon . A poster must post both the solution of the previous problem and the next problem of the marathon. If no one solves the next problem within $2$ days, he or she must provide a solution or a ...
Sun Feb 19, 2017 10:45 am
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 146
Views: 63365

### Re: Geometry Marathon : Season 3

$\text{Problem 29}$ In acute triangle $ABC$ , segments $AD; BE$ , and $CF$ are its altitudes, and $H$ is its orthocenter. Circle $\omega$, centered at $O$, passes through $A$ and $H$ and intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. The circumcircle of triangle ...
Wed Feb 15, 2017 10:48 pm
Forum: Algebra
Topic: F.E. (You run out of these names fast)
Replies: 1
Views: 1141

A brief sketch of the solution. Plugging in $x=1$ in the second equality gives $f(1)=1$. By induction, it follows that $f(x)=x$ for all positive integers $x$. Note that $f(x+k)=f(x)+k$ for positive integer $k$. So, $f(m+\frac{k}{m})^2 = (f(\frac{k}{m})+m)^2$. By the second F.E condition, $f(m+\frac{... Sat Feb 04, 2017 6:48 pm Forum: Geometry Topic: Circle is tangent to circumcircle and incircle Replies: 3 Views: 2066 ### Re: Circle is tangent to circumcircle and incircle Without loss of generality, let$Y$be closer to$A$than$X$. The crux move is to show that$AD$bisects$\angle SAT$. Let$L$be the midpoint of$XY$. Clearly$X,Y,S,T$all lie on a circle centered at$L$. Denote this circle by$\ell$. Lemma :$CY\perp AX$and$BX\perp AX$Proof : Well known. Chas... Fri Feb 03, 2017 8:21 pm Forum: Number Theory Topic: IMO SL 2005 N5 Replies: 0 Views: 1026 ### IMO SL 2005 N5 Denote by$d(n)$the number of divisors of the positive integer$n$. A positive integer$n$is called highly divisible if$d(n) > d(m)$for all positive integers$m < n$. Two highly divisible integers$m$and$n$with$m < n$are called consecutive if there exists no highly divisible integer$s$sat... Fri Feb 03, 2017 5:30 pm Forum: Algebra Topic: A beautiful FE Replies: 2 Views: 1388 ### Re: A beautiful FE We will show that$f(x)=x+1$. Check to see that this indeed verifies the original equation.$P(0,x)$implies$f(f(x))=f(0)((f(x)-1)+2$whilst$P(1,x)$implies$f(f(x))=\dfrac{f(x)f(1)+2}{2}$. Combining the two gives us$f(x)=\dfrac{2f(0)-2}{2f(0)-f(1)}$implying$f$is a constant, which is readily f... Fri Feb 03, 2017 4:10 pm Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 146 Views: 63365 ### Re: Geometry Marathon : Season 3$\text{Solution of Problem } 20$This problem is readily bary-able. Line$CP$has the equation$x\cdot (b^2+c^2-a^2)-y\cdot (a^2-b^2+c^2)=0$and line$AP$has the equation$b^2z+c^2y=0$. We therefore proceed to find$K$. Redefine$K$to be the point such that$KH$is perpendicular to$AD$. It suffic... Wed Feb 01, 2017 10:34 pm Forum: International Mathematical Olympiad (IMO) Topic: IMO Marathon Replies: 184 Views: 64044 ### Re: IMO Marathon$\text{Solution to Problem } 50$A solution can be found following this link : https://artofproblemsolving.com/community/c6h348658p1871421$\text{Problem } 51$Let$A$be a set of$N$residues$\pmod{N^{2}}$. Prove that there exists a set$B$of of$N$residues$\pmod{N^{2}}$such that$A + B = \{a+...
Wed Feb 01, 2017 7:44 pm
Forum: Algebra
Topic: 2009 IMO SL A3
Replies: 2
Views: 1545

### 2009 IMO SL A3

Determine all functions $f$ from the set of positive integers to the set of positive integers such that, for all positive integers $a$ and $b$, there exists a non-degenerate triangle with sides of lengths $a, f(b) \text{ and } f(b + f(a) - 1).$ (A triangle is non-degenerate if its vertices are...