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Moldova, Third Team Selection Test for IMO and BMO 2008, March 30.

Posted by aimingbeyond on April 1, 2008

Moldova, Third Team Selection Test for IMO-BMO 2008, March 30

Determine a subset $A\subset \mathbb N^*$ having exactly $5$ distinct elements, whose sum of squares equals their product.

Problem 2

Let $p$ be a prime number and $k,n$ positive integers so that $\gcd(p,n)=1$ . Prove that $\displaystyle\binom{n\cdot p^k}{p^k}$ and $p$ are coprime.

Problem 3

In triangle $ABC$ the bisector of $\angle{ACB}$ intersects $AB$ at $D$ . Consider an arbitrary circle $O$ passing through $C$ and $D$ and not tangent to $BC$ or $CA$ . Let $O\cap BC=\{M\}$ and $O\cap CA=\{N\}$ .

Prove that there is a circle $S$ so that $DM$ and $DN$ are tangent to $S$ in $M$ and $N$ respectively.
Circle $S$ intersects $BC$ and $CA$ in $P$ and $Q$ respectively. Prove that the lengths of $MP$ and $NQ$ do not depend on the choice of circle $O$ .

Problem 4

A non-empty set $S$ of positive integers is said to be good if there is a coloring with $2008$ colors of all positive integers so that no number in $S$ is the sum of two different positive integers (not necessarily in $S$ ) of the same color. Find the largest value $t$ can take so that the set $S=\{a+1,a+2,\ldots,a+t\}$ is good, for any positive integer $a$ .

Posted in Elementary Mathematics, Olympiads | Tagged: BMO, Elementary Mathematics, IMO, Moldova, Olympiad, Team Selection Test | No Comments »

Moldova, Second Team Selection Test for IMO-BMO 2008

Posted by aimingbeyond on March 29, 2008

Here is the pdf file. Time allowed - 4 hours, 30 minutes. Each problem is worth 7 points.

Moldova, Second Team Selection Test for IMO-BMO 2008, March 29

Problem 1

Find all solutions $(x,y)\in\mathbb{R}\times\mathbb{R}$ of the system:

$\begin{cases} x^3+3xy^2=49,\\ x^2+8xy+y^2=8y+17x.\end{cases}$

Problem 2

Let $a_1,a_2,\ldots,a_n$ be positive reals so that $a_1+a_2+\ldots+a_n\le\dfrac n2$ . Find the minimal value of

$\displaystyle\sqrt{a_1^2+\frac1{a_2^2}}+\sqrt{a_2^2+\frac1{a_3^2}}+\ldots+\sqrt{a_n^2+\frac1{a_1^2}}$

Problem 3

Let $\omega$ be the circumcircle of $ABC$ and let $D$ be a fixed point on $(BC)$ . $X$ is a variable point on $(BC)$ , $X\neq D$ . Denote by $Y$ the second intersection of $AX$ and $\omega$ . Prove that the circumcircle of triangle $XYD$ passes through a fixed point.

Problem 4

Find the number of even permutations of $\{1,2,\ldots,n\}$ which have no fixed points.

Posted in Elementary Mathematics, Olympiads | Tagged: BMO, Elementary Mathematics, IMO, Moldova, Olympiad, Team Selection Test | No Comments »

Moldova, First Team Selection Test for IMO and BMO, 2008, March 3.

Posted by aimingbeyond on March 3, 2008

Here goes the pdf file. Time allowed - 4 hours 30 mintues. Each problem is worth 7 points.

Moldova, First Team Selection Test for IMO-BMO 2008, March 3

Problem 1

Let $p$ be a prime number. Solve in $\mathbb{N}_0\times\mathbb{N}_0$ the equation $x^3+y^3-3xy=p-1$ .

Problem 2

We say the set $\{1,2,\ldots,3k\}$ has property $D$ if it can be split into disjoint triples, so that in each such triple, one number is the sum of the other two. Prove that

The set $\{1,2,\ldots,3324\}$ has property $D$ .
The set $\{1,2,\ldots,3309\}$ hasn’t property $D$ .

Problem 3

Let $\Gamma(I,r)$ and $\Gamma(O,R)$ be the incircle and circumcircle, respectively, of triangle $ABC$ . Consider all triangles $A_iB_iC_i$ which are simultaneously inscribed in $\Gamma(O,R)$ and circumscribed to $\Gamma(I,r)$ . Prove that the centroids of the triangles $A_iB_iC_i$ lie on a circle.

Problem 4

A non-zero polynomial $S\in\mathbb{R}[X,Y]$ is called homogeneous of degre $d$ if there is a positive integer $d$ so that $S(\lambda x,\lambda y)=\lambda^dS(x,y)$ for any $\lambda\in\mathbb{R}$ . Let $P,Q\in\mathbb{R}[X,Y]$ so that $Q$ is homogeneous and $P$ divides $Q$ (that is $P|Q$ ). Prove that $P$ is homogeneous too.

Posted in Olympiads | Tagged: BMO, Elementary Mathematics, IMO, Moldova, Olympiad, Team Selection Test | No Comments »

Moldova National Mathematical Olympiad 2008.

Posted by aimingbeyond on March 2, 2008

Yesterday was the first day and today the second day. Here are the problems for 12th grade.

Check here for the problems in pdf format. Next week I have holiday, so I will try to upload problems for other grades, eventually with solutions.

Day 1

Problem 1

Consider the equation $x^4-4x^3+4x^2+ax+b=0$ , where $a,b\in\mathbb{R}$ . Determine the largest value $a+b$ can take, so that the given equation has two distinct positive roots $x_1,x_2$ so that $x_1+x_2=2x_1x_2$ .

Problem 2

Evaluate

$\displaystyle E=\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x$ .

Problem 3

In the usual coordinate system $xOy$ , line $d$ intersects circles $C_1:(x+1)^2+y^2=1$ and $C_2:(x-2)^2+y^2=4$ in the points $A,B,C,D$ (in this order), all having positive $Oy$ coordinates. Given that $A\left(-\frac32,\frac{\sqrt3}2\right)$ and $m(\angle{BOC})=60^\circ$ find the slope of $d$ .

Problem 4

Define the sequence $(a_p)_{pge0}$ as follows:

$a_p=\displaystyle\frac{\binom p0}{2\cdot 4}-\frac{\binom p1}{3\cdot5}+\frac{\binom p2}{4\cdot6}-\ldots+(-1)^p\cdot\frac{\binom pp}{(p+2)(p+4)}$

Compute $\displaystyle\lim_{n\to\infty}(a_0+a_1+\ldots+a_n)$ .

Day 2

Problem 5

Find the least positive integer $n$ so that the polynomial $P(X)=\sqrt3X^{n+1}-X^n-1$ has at least one root of modulus $1$ .

Problem 6

For $n\ge1$ , let

$\displaystyle a_n=\frac1{\sqrt{n^2+8n-1}}+\frac1{\sqrt{n^2+16n-1}}+\frac1{\sqrt{n^2+24n-1}}+\ldots+\frac1{\sqrt{9n^2-1}}$

Find $\displaystyle\lim_{n\to\infty}a_n$ .

Problem 7

Vertices $B,C$ of triangle are fixed and $BC=2$ , while $A$ is variable. Denote by $H$ and $G$ the orthocenter and centroid respectively of triangle $ABC$ . Let $F\in (HG)$ so that $HF/FG=3$ . Find the locus of the point $A$ so that $F\in BC$ .

Problem 8

Evaluate

$\displaystyle I = \int_0^{\frac\pi4}\left(\sin^62x + \cos^62x\right)\cdot \ln(1 + \tan x)\text{d}x$

Posted in Olympiads | Tagged: National Mathematical Olympiad | No Comments »

Moldova City Olympiad 2008.

Posted by aimingbeyond on February 24, 2008

Here are the problems for the 12th grade. I will try to post the problems for other grades soon. The problems are rather easy, for beginners mainly. However when I find time, I will try to provide their solutions.

Moldova, Chisinau, City Olympiad 2008, February 23.

Grade 12.

Problem 1.

The polynomial $P(X)=aX^3+bX^2+cX+d$ takes integer values for $X=-1,0,1,2$ . Prove that $P(X)$ is an integer for any integer $X$ .

Problem 2.

Prove that $\displaystyle 1-e^{-\frac\pi2}<\int_0^{\frac\pi2}e^{-\sin x}\textrm{d}x<\frac\pi2(1-e^{-1})$ .

Problem 3.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ , $f(x)=x^{2007}e^x\textrm{d}x$ . Prove that if $F(x)$ is a primitive for $f(x)$ then there exist $2008$ unique integers $a_0,a_1,ldots,a_{2007}$ so that $F(x)=[a_0+a_1(x-1)+a_2(x-1)^2+\ldots+a_{2007}(x-1)^{2007}]e^x+C$ .

Problem 4.

Prove that the volume of any regular pyramid is less than $\displaystyle\frac7{17}$ of the cube of the lateral edge.

The pdf file is available here.

Posted in Olympiads | Tagged: Mathematical Olympiad, Moldova City Olympiad | No Comments »

PEN: Problem of the bi-week

Posted by aimingbeyond on January 5, 2008

Inside PEN (Problems in Elementary Number Theory), the Problem of the bi-week project has been launched.

For an idea of what all this is about, here’s what Hojoo Lee, creator of this project, said about “Problem of the bi-week“:

“First week, you meet a problem (from PEN book) and then, next week, I’ll upload the solution file so that everyone can see the solutions, and so on.“

For details, check Problem of the bi-week on Facebook.

Or you may go directly to the project.

Happy Problem Solving!

Posted in Number Theory | Tagged: Number Theory, PEN | No Comments »

2 beautiful inequalities with beautiful solutions :)

Posted by aimingbeyond on November 29, 2007

In the book ”Old and New Inequalities” by Titu Andreescu, Gabriel Dospinescu, Vasile Cîrtoaje and Mircea Lascu the following inequality appears as problem 19, b)

[Marian Tetiva] Let $x,y,z$ be positive real numbers satisfying the condition $x^2+y^2+z^2+2xyz=1$

Prove that $x+y+z\le\displaystyle\frac32$

Solution 1:

Clearly $x,y,z\le1$ . The given relation rewrites as $(x+yz)^2=(1-y^2)(1-z^2)$ , or $x=-yz+\sqrt{(1-y^2)(1-z^2)}\le-yz+\displaystyle\frac{1-y^2+1-z^2}2=1-\frac{(y+z)^2}2$ .

Let $s=y+z$ . Then for the inequality $x+y+z\le\displaystyle\frac32$ to be true, it’s enough to have $1-\displaystyle\frac{s^2}2+s\le\frac32$ . But the last one is $-(s-1)^2\le0$ , clearly true $\blacksquare$

Solution 2:

Recall the following identity for an acute triangle: $\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C=1$ . Hence we can substitute $x=\cos A$ and similar for $y,z$ . Now, $\cos$ is a concave function on $\displaystyle\left[0;\frac{\pi}2\right]$ , so, using Jensen’s Inequality we have $x+y+z=\cos A+\cos B+\cos C\le3\cos\displaystyle\frac\pi3=\frac32\blacksquare$

A beautiful solution similar in manner to the Solution 1 above, can be given for Problem 26 d) of the same book:

[Marian Tetiva] Consider positive real numbers $x,y,z$ so that $x^2+y^2+z^2=xyz$ . Prove that $xy+yz+zx\ge2(x+y+z)+9$ .

Solution 1:

By applying the AM-GM inequality, we easily deduce $\displaystyle\frac{x+y+z}3\ge\sqrt[3]{xyz}\ge3$ , hence $x+y+z\ge9$ .

We will prove the stronger inequality $xy+yz+zx\ge3(x+y+z)$ .

Write $x^2+y^2+z^2$ as $-x^2+(x^2+y^2)+(x^2+z^2)$ and deduce $xyz=x^2+y^2+z^2=-x^2+(x^2+y^2)+(x^2+z^2)\ge-x^2+2xy+2xz$ , hence $yz\ge -x+2y+2z$ . In a similar way $xz\ge -y+2x+2z$ and $xy\ge -z+2x+2y$ . Summing up these 3 inequalities we get $xy+yz+zx \ge 3(x+y+z) \blacksquare$ . Isn’t it beautiful?

None of these solutions appears in the book Here’s the solution from the book for the second problem:

Solution 2:

Since $x^2<xyz$ we get $x<yz$ and the likes. Then $xy<yz\cdot xz$ , so $1<z$ and the likes. Set $a=x-1$ , $b=y-1$ and $c=z-1$ . Clearly $a,b,c>0$ . The given condition then becomes, after expanding, $a^2+b^2+c^2+a+b+c+2=abc+ab+bc+ca$ . Set $q=ab+bc+ca$ . It is well-known that $q\le a^2+b^2+c^2$ and $\sqrt{3q}\le a+b+c$ and $abc\le\displaystyle\left(\frac q3\right)^{\frac32}=\frac{(3q)^{\frac32}}{27}$ . Then $q+\sqrt{3q}+2\le a^2+b^2+c^2+a+b+c+2=abc+ba+bc+ca\le\frac{(3q)^{\frac32}}{27}+q$ . Setting $t=\sqrt{3q}$ , we get $t+2\le\frac{t^3}{27}$ , or $(t+3)^2(t-6)\ge0$ . So $\sqrt{3q}\ge6$ , or $ab+bc+ca\ge12$ . Substituting back $x=a+1$ ,…, and expanding we get $xy+yz+zx\ge 2(x+y+z)+9\blacksquare$

The book mentioned at the beginning of this note is a strong one on Inequalities. Check www.gil.ro for it.

Posted in Elementary Mathematics, Inequalities | 1 Comment »

Intro

Posted by aimingbeyond on November 29, 2007

What happened to my older blog?

Well - nothing. It’s still there: http://freeyourminder.spaces.live.com/

But a Live Space is a bit too rigid and, in my opinion, intended to serve more for ‘Social linking’ than what i was looking for. Actually, I can’t exactly define what I’m looking for…

Posted in General | 2 Comments »

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Moldova, Third Team Selection Test for IMO and BMO 2008, March 30.

Moldova, Second Team Selection Test for IMO-BMO 2008

Moldova, First Team Selection Test for IMO and BMO, 2008, March 3.

Moldova National Mathematical Olympiad 2008.

Moldova City Olympiad 2008.

PEN: Problem of the bi-week

2 beautiful inequalities with beautiful solutions :)

Intro