Source: Romania 1999 7.1 Determine the side lengths of a right trianlge if they are intgers and the product of the leg lengths is equal to three times the perimeter.

Let $a, b, c$ be non zero integers,$ a\ne c$ such that $$\frac{a}{c}=\frac{a^2+b^2}{c^2+b^2}$$Prove that $a^2 +b^2 +c^2$ cannot be a prime number.

Let $ABCD$ be a convex quadrilateral with $\angle BAC = \angle CAD$, $\angle ABC =\angle ACD$, $(AD \cap (BC =\{E\}$, $(AB \cap (DC = \{F\}$. Prove that: a) $AB\cdot DE = BC \cdot CE$ b) $AC^2 < \frac12 (AD \cdot AF + AB \cdot AE).$

In the triangle $ABC$, let $D \in (BC)$, $E \in (AB)$, $EF \parallel BC$, $F \in (AC)$, $EG\parallel AD$, $G\in (BC)$ and $M,N$ be the midpoints of $(AD)$ and $(BC)$, respectively. Prove that: a) $\frac{EF}{BC}+\frac{EG}{AD}=1$ b) the midpoint of $[FG]$ lies on the line $ MN$.

Let $P(x) = 2x^3-3x^2+2$, and the sets: $$A =\{ P(n) | n \in N, n \le 1999\}, B=\{p^2+1 |p \in N\}, C=\{ q^2+2 | q \in N\}$$Prove that the sets $A \cap B$ and $A\cap C$ have the same number of elements

let $x_i,y_i 1 \le i \le n$ be positive numbers such that : $\displaystyle \sum_{i=1}^n x_i \ge \sum_{i=1}^n x_iy_i$ Prove : $\displaystyle \sum_{i=1}^n x_i \le \sum _{i=1}^n \frac{x_i}{y_i}$

Let $a, b, c$ be positive real numbers such that $ab +be + ba \le 3abc$. Prove that $$a^3+b^3+c^3 \ge a+b+c.$$

Let $ABCDA'B'C'D'$ be a right parallelepiped, $E$ and $F$ the projections of $A$ on the lines $A'D$, $A'C$, respectively, and $P, Q$ the projections of $B'$ on the lines $A'C'$ and $A'C$ Prove that a) the planes $(AEF)$ and $(B'PQ)$ are parallel b) the triangles $AEF$ and $B'PQ$ are similar.

Let $SABC$ be a regular pyramid, $O$ the center of basis $ABC$, and $M$ the midpoint of $[BC]$. If $N \in [SA]$ such that $SA = 25 \cdot NS$ and $SO \cap MN=\{P\}$, $AM=2\cdot SO$, prove that the planes $(ABP)$ and $(SBC)$ are perpendicular.

Let $AD$ be the bisector of angle $A$ of the triangle $ABC$. One considers the points M, N on the half-lines $(AB$ and $(AC$, respectively, such that $\angle MDA = \angle B$ and $\angle NDA = \angle C$. Let $AD \cap MN=\{P\}$. Prove that: $$AD^3 = AB \cdot AC\cdot AP$$

For $a, b > 0$, denote by $t(a,b)$ the positive root of the equation $$(a+b)x^2-2(ab-1)x-(a+b) = 0.$$Let $M = \{ (a.b) | \, a \ne b \,\,\, and \,\,\,t(a,b) \le \sqrt{ab} \}$ Determine, for $(a, b)\in M$, the mmimum value of $t(a,b)$.

In the convex quadrilateral $ABCD$, the bisectors of angles $A$ and $C$ intersect in $I$. Prove that $ABCD$ is circumscriptible if and only if $$S[AIB] + S[CID] =S[AID]+S[BIC]$$( $S[XYZ]$ denotes the area of the triangle $XYZ$)

a) Let $a,b\in R$, $a <b$. Prove that $x \in (a,b)$ if and only if there exists $\lambda \in (0,1)$ such that $x=\lambda a +(1-\lambda)b$. b) If the function $f: R \to R$ has the property: $$f (\lambda x+(1-\lambda) y) < \lambda f(x) + (1-\lambda)f(y), \forall x,y \in R, x\ne y, \forall \lambda \in (0,1), $$prove that one cannot find four points on the function’s graph that are the vertices of a parallelogram

On the sides $(AB)$, $(BC)$, $(CD)$ and $(DA)$ of the regular tetrahedron $ABCD$, one considers the points $M$, $N$, $P$, $Q$, respectively Prove that $$MN \cdot NP \cdot PQ \cdot QM \ge AM \cdot BN \cdot CP \cdot DQ.$$