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[Solutions] Romanian Mathematical Competitions 2001

Romania National Olympiad 2001

Grade level 7

  1. Show that there exist no integers $a$ and $b$ such that $$a^3+a^2b+ab^2+b^3=2001.$$
  2. Let $a$ and $b$ be real, positive and distinct numbers. We consider the set \[M=\{ ax+by\mid x,y\in\mathbb{R},\ x>0,\ y>0,\ x+y=1\}.\] Prove that
    a) $\dfrac{2ab}{a+b}\in M;$
    b) $\sqrt{ab}\in M.$
  3. We consider a right trapezoid $ABCD$, in which $AB||CD$, $AB>CD$, $AD\perp AB$ and $AD>CD$. The diagonals $AC$ and $BD$ intersect at $O$. The parallel through $O$ to $AB$ intersects $AD$ in $E$ and $BE$ intersects $CD$ in $F$. Prove that $CE\perp AF$ if and only if $AB\cdot CD=AD^2-CD^2$ .
  4. Consider the acute angle $ABC$. On the half-line $BC$ we consider the distinct points $P$ and $Q$ whose projections onto the line $AB$ are the points $M$ and $N$. Knowing that $AP=AQ$ and $AM^2-AN^2=BN^2-BM^2$, find the angle $ABC$.

Grade level 8

  1. Determine all real numbers $a$ and $b$ such that $a+b\in\mathbb{Z}$ and $a^2+b^2=2$.
  2. For every rational number $m>0$ we consider the function $f_m:\mathbb{R}\rightarrow\mathbb{R}$, $f_m(x)=\dfrac{1}{m}x+m$. Denote by $G_m$ the graph of the function $f_m$. Let $p,q,r$ be positive rational numbers.
    a) Show that if $p$ and $q$ are distinct then $G_p\cap G_q$ is non-empty.
    b) Show that if $G_p\cap G_q$ is a point with integer coordinates, then $p$ and $q$ are integer numbers.
    c) Show that if $p,q,r$ are consecutive natural numbers, then the area of the triangle determined by intersections of $G_p,G_q$ and $G_r$ is equal to $1$.
  3. We consider the points $A$, $B$, $C$, $D$, not in the same plane, such that $AB\perp CD$ and $AB^2+CD^2=AD^2+BC^2$.
    a) Prove that $AC\perp BD$.
    b) Prove that if $CD<BC<BD$, then the angle between the planes $(ABC)$ and $(ADC)$ is greater than $60^{\circ}$.
  4. In the cube $ABCDA'B'C'D'$, with side $a$, the plane $(AB'D')$ intersects the planes $(A'BC)$, $(A'CD)$, $(A'DB)$ after the lines $d_1$, $d_2$ and $d_3$ respectively.
    a) Show that the lines $d_1$, $d_2$, $d_3$ intersect pairwise.
    b) Determine the area of the triangle formed by these three lines.

Grade level 9

  1. Let $A$ be a set of real numbers which verifies:
    a) $ 1 \in A$
    b) $x\in A\implies x^2\in A$
    c) $x^2-4x+4\in A\implies x\in A$
    Show that $2000+\sqrt{2001}\in A$.
  2. Let $ABC$ be a triangle $(A=90^{\circ})$ and $D\in (AC)$ such that $BD$ is the bisector of $B$. Prove that $BC-BD=2AB$ if and only if
    \[\frac{1}{BD}-\frac{1}{BC}=\frac{1}{2AB} \]
  3. Let $n\in\mathbb{N}^*$ and $v_1,v_2,\ldots ,v_n$ be vectors in the plane with lengths less than or equal to $1$. Prove that there exists $\xi_1,\xi_2,\ldots ,\xi_n\in\{-1,1\}$ such that
    \[ | \xi_1v_1+\xi_2v_2+\ldots +\xi_nv_n|\le\sqrt{2}\]
  4. Determine the ordered systems $(x,y,z)$ of positive rational numbers for which $x+\dfrac{1}{y}$, $y+\dfrac{1}{z}$ and $z+\dfrac{1}{x}$ are integers.

Grade level 10

  1. Let $a$ and $b$ be complex non-zero numbers and $z_1,z_2$ the roots of the polynomials $X^2+aX+b$. Show that $|z_1+z_2|=|z_1|+|z_2|$ if and only if there exists a real number $\lambda\ge 4$ such that $a^2=\lambda b$.
  2. In the tetrahedron $OABC$ we denote by $\alpha,\beta,\gamma$ the measures of the angles $\angle BOC,\angle COA,$ and $\angle AOB$, respectively. Prove the inequality
    \[\cos^2\alpha+\cos^2\beta+\cos^2\gamma<1+2\cos\alpha\cos\beta\cos\gamma \]
  3. Let $m,k$ be positive integers, $k<m$ and $M$ a set with $m$ elements. Prove that the maximal number of subsets $A_1,A_2,\ldots ,A_p$ of $M$ for which $A_i\cap A_j$ has at most $k$ elements, for every $1\le i<j\le p$, equals \[ p_{\max}=\binom{m}{0}+\binom{m}{1}+\binom{m}{2}+\ldots+\binom{m}{k+1}\]
  4. Let $n\ge 2$ be an even integer and $a,b$ real numbers such that $b^n=3a+1$. Show that the polynomial $P(X)=(X^2+X+1)^n-X^n-a$ is divisible by $Q(X)=X^3+X^2+X+b$ if and only if $b=1$.

Grade level 11

  1. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function, derivable on $R\backslash\{x_0\}$, having finite side derivatives in $x_0$. Show that there exists a derivable function $g:\mathbb{R}\rightarrow\mathbb{R}$, a linear function $h:\mathbb{R}\rightarrow\mathbb{R}$ and $\alpha\in\{-1,0,1\}$ such that \[ f(x)=g(x)+\alpha |h(x)|,\ \forall x\in\mathbb{R} \]
  2. We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$. a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command. B=%Error. "rank" is a bad command. C = r$, such that $A=BC$. b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.
  3. Let $f:\mathbb{R}\rightarrow[0,\infty )$ be a function with the property that $$|f(x)-f(y)|\le |x-y|$$ for every $x,y\in\mathbb{R}$. Show that:
    a) If $\lim_{n\rightarrow \infty} f(x+n)=\infty$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\infty$.
    b) If $\lim_{n\rightarrow \infty} f(x+n)=\alpha ,\alpha\in[0,\infty )$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\alpha$.
  4. The continuous function $f:[0,1]\rightarrow\mathbb{R}$ has the property: \[\lim_{x\rightarrow\infty}\ n\left(f\left(x+\frac{1}{n}\right)-f(x)\right)=0 \] for every $x\in [0,1)$. Show that
    a) For every $\epsilon >0$ and $\lambda\in (0,1)$, we have: \[ \sup\ \{x\in[0,\lambda )\mid |f(x)-f(0)|\le \epsilon x \}=\lambda \] b) $f$ is a constant function.

Grade level 12

  1. a) Consider the polynomial $P(X)=X^5\in \mathbb{R}[X]$. Show that for every $\alpha\in\mathbb{R}^*$, the polynomial $P(X+\alpha )-P(X)$ has no real roots.
    b) Let $P(X)\in\mathbb{R}[X]$ be a polynomial of degree $n\ge 2$, with real and distinct roots. Show that there exists $\alpha\in\mathbb{Q}^*$ such that the polynomial $P(X+\alpha )-P(X)$ has only real roots.
  2. Let $A$ be a finite ring. Show that there exists two natural numbers $m,p$ where $m> p\ge 1$, such that $a^m=a^p$ for all $a\in A$.
  3. Let $f:[-1,1]\rightarrow\mathbb{R}$ be a continuous function. Show that:
    a) if $\int_0^1 f(\sin (x+\alpha ))\, dx=0$, for every $\alpha\in\mathbb{R}$, then $f(x)=0$, $\forall x\in [-1,1]$.
    b) if $\int_0^1 f(\sin (nx))\, dx=0$, for every $n\in\mathbb{Z}$, then $f(x)=0$, $\forall x\in [-1,1]$.
  4. Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$.
    a) Show that \[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\]
    b) Show that \[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]

Romania Team Selection Test 2001

  1. Show that if $a,b,c$ are complex numbers that such that
    \[(a+b)(a+c)=b, \quad (b+c)(b+a)=c, \quad (c+a)(c+b)=a\] then $a,b,c$ are real numbers.
  2. a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function.
    b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.
  3. The sides of a triangle have lengths $a,b,c$. Prove that:
    \begin{align*}(-a+b+c)(a-b+c)\, +\, & (a-b+c)(a+b-c)+(a+b-c)(-a+b+c)\\ &\le\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c})\end{align*}
  4. Three schools have $200$ students each. Every student has at least one friend in each school (if the student $a$ is a friend of the student $b$ then $b$ is a friend of $a$). It is known that there exists a set $E$ of $300$ students (among the $600$) such that for any school $S$ and any two students $x,y\in E$ but not in $S$, the number of friends in $S$ of $x$ and $y$ are different. Show that one can find a student in each school such that they are friends with each other.
  5. Find all polynomials with real coefficients $P$ such that \[ P(x)P(2x^2-1)=P(x^2)P(2x-1)\] for every $x\in\mathbb{R}$.
  6. The vertices $A,B,C$ and $D$ of a square lie outside a circle centred at $M$. Let $AA',BB',CC',DD'$ be tangents to the circle. Assume that the segments $AA',BB',CC',DD'$ are the consecutive sides of a quadrilateral $p$ in which a circle is inscribed. Prove that $p$ has an axis of symmetry.
  7. Find the least $n\in N$ such that among any $n$ rays in space sharing a common origin there exist two which form an acute angle.
  8. Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.
  9. Let $n$ be a positive integer and $f(x)=a_mx^m+\ldots + a_1X+a_0$, with $m\ge 2$, a polynomial with integer coefficients such that:
    a) $a_2,a_3\ldots a_m$ are divisible by all prime factors of $n$,
    b) $a_1$ and $n$ are relatively prime.
    Prove that for any positive integer $k$, there exists a positive integer $c$, such that $f(c)$ is divisible by $n^k$.
  10. Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called ideal if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n + p$ and $ n + q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$
  11. Find all pairs $\left(m,n\right)$ of positive integers, with $m,n\geq2$, such that $a^n-1$ is divisible by $m$ for each $a\in \left\{1,2,3,\ldots,n\right\}$.
  12. Prove that there is no function $f:(0,\infty )\rightarrow (0,\infty)$ such that \[f(x+y)\ge f(x)+yf(f(x)) \] for every $x,y\in (0,\infty )$.
  13. The tangents at $A$ and $B$ to the circumcircle of the acute triangle $ABC$ intersect the tangent at $C$ at the points $D$ and $E$, respectively. The line $AE$ intersects $BC$ at $P$ and the line $BD$ intersects $AC$ at $R$. Let $Q$ and $S$ be the midpoints of the segments $AP$ and $BR$ respectively. Prove that $\angle ABQ=\angle BAS$.
  14. Consider a convex polyhedron $P$ with vertices $V_1,\ldots ,V_p$. The distinct vertices $V_i$ and $V_j$ are called neighbours if they belong to the same face of the polyhedron. To each vertex $V_k$ we assign a number $v_k(0)$, and construct inductively the sequence $v_k(n)\ (n\ge 0)$ as follows: $v_k(n+1)$ is the average of the $v_j(n)$ for all neighbours $V_j$ of $V_k$ . If all numbers $v_k(n)$ are integers, prove that there exists the positive integer $N$ such that all $v_k(n)$ are equal for $n\ge N$ .

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MOlympiad: [Solutions] Romanian Mathematical Competitions 2001
[Solutions] Romanian Mathematical Competitions 2001
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