Chủ đề này có 1 trả lời

[Zaraki]

Lớp 9.
P1: Find all numbers $x,y\in\mathbb N$ for which the relation $ x+2y+\frac{3x}{y}=2012$ holds.

Proposed by Bela Kovacs

P2: Let $a_1,b_1,c_1,a_2,b_2,c_2\in\mathbb R \setminus \{0\}$ with $a_1^2+b_1^2+c_1^2=a_2^2+b_2^2+c_2^2.$ Prove that at least one of equations $a_1x^2+2c_2x+b_1=0,$ $b_1x^2+2a_2x+c_1=0,$ and $c_1x^2+2b_2x+a_1=0$ has real solutions.

Proposed by Mihaly Bencze

P3: Solve equation $2^{[x]}=1+2x$ with $x\in\mathbb R,$ where $[x]$ denotes the integer part of $x$.

Proposed by Anna-Maria Darvas

P4: Prove that for every acute angled and not isosceles triangle with the half of the segment determined by a vertex and the orthocenter, with the median from the same vertex, and with the circumradius of the triangle we can construct a triangle.

Proposed by Ferenc Olosz

P5: The measures of two angles of a triangle are of $45^\circ$ and $30^\circ.$ Find the ratio of the longest side of the triangle and the median drawn from the vertex of the angle of $45^\circ.$

Proposed by Ferenc Olosz

P6: Prove that among every seven vertexes of a regular $12$-gon there exist three which are vertexes of a right-angled triangle! Is it also true that among every seven vertexes of a regular $12$-gon there exist three which are vertexes of a right-angled and isosceles triangle?

Proposed by Zoltan Biro

[Zaraki]

P1. Ta có $y|3x$ nên $(x,y)=(uv,v)$ hoặc $(uv,3v)$ vơí $u,v\in\mathbb N.$

• $(x,y)=(uv,v)$ $\implies$ $uv+2v+3u=2012$ $\implies$ $(u+2)(v+3)=2018=2\times 1009$ vô nghiệm.
• $(x,y)=(uv,3v)$ $\implies$ $uv+6v+u=2012$ $\implies$ $(u+6)(v+1)=2018=2\times 1009$, ta tìm được $(u,v)=(1003,1).$

Kết quả. $\boxed{(x,y)=(1003,3)}$