Let $n_1 = \overline{abcabc}$ and $n_2= \overline{d00d}$ be numbers represented in the decimal system, with $a\ne 0$ and $d \ne 0$. a) Prove that $\sqrt{n_1}$ cannot be an integer. b) Find all positive integers $n_1$ and $n_2$ such that $\sqrt{n_1+n_2}$ is an integer number. c) From all the pairs $(n_1,n_2)$ such that $\sqrt{n_1 n_2}$ is an integer find those for which $\sqrt{n_1 n_2}$ has the greatest possible value

Let $a \ne 0$ be a natural number. Prove that $a$ is a perfect square if and only if for every $b \in N^*$ there exists $c \in N^*$ such that $a + bc$ is a perfect square.

The triangle $ABC$ has $\angle ACB = 30^o$, $BC = 4$ cm and $AB = 3$ cm . Compute the altitudes of the triangle.

The quadrilateral $ABCD$ has two parallel sides. Let $M$ and $N$ be the midpoints of $[DC]$ and $[BC]$, and $P$ the common point of the lines $AM$ and $DN$. If $\frac{PM}{AP}=\frac{1}{4}$, prove that $ABCD$ is a parallelogram.

Let $k$ be an integer number and $P(X)$ be the polynomial $$P(X) = X^{1997}-X^{1995} +X^2-3kX+3k+1$$Prove that: a) the polynomial has no integer root; β) the numbers $P(n)$ and $P(n) + 3$ are relatively prime, for every integer $n$.

I found this inequality in "Topics in Inequalities" (I 85) For all positive reals $x,y,z$ with $xyz=1$ prove: \[ \frac{x^9+y^9}{x^6+x^3y^3+y^6}+\frac{y^9+z^9}{y^6+y^3z^3+z^6}+\frac{z^9+x^9}{z^6+z^3x^3+x^6}\geq 2 \]

$ABCDA'B'CD'$ is a rectangular parallelepiped with $AA'= 2AB = 8a$ , $E$ is the midpoint of $(AB)$ and $M$ is the point of $(DD')$ for which $DM = a \left( 1 + \frac{AD}{AC}\right)$. a) Find the position of the point. $F$ on the segment $(AA')$ for which the sum $CF + FM$ has the minimum possible value. b) Taking $F$ as above, compute the measure of the angle of the planes $(D, E, F)$ and $(D, B', C')$. c) Knowing that the straight lines $AC'$ and $FD$ are perpendicular, compute the volume of the parallelepiped $ABCDA'B'C'D'$.

Let $S$ be a point outside of the plane of the parallelogram $ABCD$, such that the triangles $SAB$, $SBC$, $SCD$ and $SAD$ are equivalent. a) Prove that $ABCD$ is a rhombus. b) If the distance from $S$ to the plane $(A, B, C, D)$ is $12$, $BD = 30$ and $AC = 40$, compute the distance from the projection of the point $S$ on the plane $(A, B, C, D)$ to the plane $(S,B,C)$ .

Let $C_1,C_2,..., C_n$ , $(n\ge 3)$ be circles having a common point $M$. Three straight lines passing through $M$ intersect again the circles in $A_1, A_2,..., A_n$ ; $B_1,B_2,..., B_n$ and $X_1,X_2,..., X_n$ respectively. Prove that if $$A_1A_2 =A_2A_3 =...=A_{n-1}A_n$$and $$B_1B_2 =B_2B_3 =...=B_{n-1}B_n$$then $$X_1X_2 =X_2X_3 =...=X_{n-1}X_n.$$

Suppose that $a,b,c,d\in\mathbb{R}$ and $f(x)=ax^3+bx^2+cx+d$ such that $f(2)+f(5)<7<f(3)+f(4)$. Prove that there exists $u,v\in\mathbb{R}$ such that $u+v=7 , f(u)+f(v)=7$

function $f:\mathbb{N}^{\star} \times \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star}$ ($\mathbb{N}^{\star}=\mathbb{N}\cup \{0\}$)with these conditon: 1- $f(0,x)=x+1$ 2- $f(x+1,0)=f(x,1)$ 3- $f(x+1,y+1)=f(x,f(x+1,y))$(romania 1997) find $f(3,1997)$

Suppse that $n\geq 3$ be an inetegr number and $x\in \mathbb{R}$ s.t. $x,x^2$ and $x^n$ have a same decimal representation. Prove that $x$ is an integer number.

Let two bijective and continuous functions$f,g: \mathbb{R}\to\mathbb{R}$ such that : $\left(f\circ g^{-1}\right)(x)+\left(g\circ f^{-1}\right)(x)=2x$ for any real $x$. Show that If we have a value $x_{0}\in\mathbb{R}$ such that $f(x_{0})=g(x_{0})$, then $f=g$.

Prove that: $\int_{-1}^1f^2(x)dx\ge \frac 1 2 (\int_{-1}^1f(x)dx)^2 +\frac 3 2(\int_{-1}^1xf(x)dx)^2$ Please give a proof without using even and odd functions. (the oficial proof uses those and seems to be un-natural)

Suppose that $(f_n)_{n\in N}$ be the sequence from all functions $f_n:[0,1]\rightarrow \mathbb{R^+}$ s.t. $f_0$ be the continuous function and $\forall x\in [0,1] , \forall n\in \mathbb {N} , f_{n+1}(x)=\int_0^x \frac {1}{1+f_n (t)}dt$. Prove that for every $x\in [0,1]$ the sequence of $(f_n(x))_{n\in N}$ be the convergent sequence and calculate the limitation.