Chủ đề này có 4 trả lời

[Crystal]

The competition took place on the 19th of April, 2012. The best 44 students qualified for part one of the Federal Competition for advanced students.


Problem 1:
Prove that the inequality \[ a + a^3 - a^4 - a^6 < 1\] holds for all real numbers $a$.

Problem 2:
Determine all integer solutions $(x, y)$ of the equation \[(x - 1)x(x + 1) + (y - 1)y(y + 1) = 24 - 9xy\mbox{.}\]
Problem 3:
In an arithmetic sequence, the difference of consecutive terms in constant. We consider sequences of integers in which the difference of consecutive terms equals the sum of the differences of all preceding consecutive terms. Which of these sequences with $a_0 = 2012$ and $1\leqslant d = a_1-a_0 \leqslant 43$ contain square numbers?

Problem 4:
In a triangle $ABC$, let $H_a$, $H_b$ and $H_c$ denote the base points of the altitudes on the sides $BC$, $CA$ and $AB$, respectively. Determine for which triangles $ABC$ two of the lengths $H_aH_b$, $H_bH_c$ and $H_aH_c$ are equal.

Time: 4.5 hours.

[Crystal]

The competition took place on the 17th of May, 2012. The best 24 out of 44 students qualified for part two of the Federal Competition for advanced students (taking place on the 6th and 7th of June).

Problem 1:
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying the following property: For each pair of integers $m$ and $n$ (not necessarily distinct), $\mathrm{gcd}(m, n)$ divides $f(m) + f(n)$.

Note: If $n\in\mathbb{Z}$, $\mathrm{gcd}(m, n)=\mathrm{gcd}(|m|, |n|)$ and $\mathrm{gcd}(n, 0)=n$.

Problem 2:
Determine all solutions $(n, k)$ of the equation $n!+A
n = n^k$ with $n, k \in\mathbb{N}$ for $A = 7$ and for $A = 2012$.

Problem 3:
Consider a stripe of $n$ fieds, numbered from left to right with the integers $1$ to $n$ in ascending order. Each of the fields is colored with one of the colors $1$, $2$ or $3$. Even-numbered fields can be colored with any color. Odd-numbered fields are only allowed to be colored with the odd colors $1$ and $3$. How many such colorings are there such that any two neighboring fields have different colors?

Problem 4:
Let $ABC$ be a scalene (i.e. non-isosceles) triangle. Let $U$ be the center of the circumcircle of this triangle and $I$ the center of the incircle. Assume that the intersection of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$ lies on the perpendicular bisector of $UI$. Show that $\gamma$ is the second-largest angle in the triangle $ABC$.

Time: 4.5 hours.

AoPS

[Zaraki]

Problem 1:
Prove that the inequality \[ a + a^3 - a^4 - a^6 < 1\] holds for all real numbers $a$.

$\fbox{1}.$ Chứng minh rằng bất đẳng thức \[ a + a^3 - a^4 - a^6 < 1\] đúng với mọi số thực $a$.

Problem 2:
Determine all integer solutions $(x, y)$ of the equation \[(x - 1)x(x + 1) + (y - 1)y(y + 1) = 24 - 9xy\mbox{.}\]

Giải phương trình nghiệm nguyên \[(x - 1)x(x + 1) + (y - 1)y(y + 1) = 24 - 9xy\mbox{.}\]

Problem 1:
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying the following property: For each pair of integers $m$ and $n$ (not necessarily distinct), $\mathrm{gcd}(m, n)$ divides $f(m) + f(n)$.

Note: If $n\in\mathbb{Z}$, $\mathrm{gcd}(m, n)=\mathrm{gcd}(|m|, |n|)$ and $\mathrm{gcd}(n, 0)=n$.

$\fbox{1}.$ Xác định mọi hàm số $f: \mathbb{Z}\to\mathbb{Z}$ thỏa mãn: Với mỗi cặp số nguyên $m$ và $n$ (không nhất thiết phải khác nhau) $\mathrm{gcd}(m, n)$ là ước của $f(m) + f(n)$
Chú ý: Nếu $n\in\mathbb{Z}$, $\mathrm{gcd}(m, n)=\mathrm{gcd}(|m|, |n|)$ và $\mathrm{gcd}(n, 0)=n$.

Problem 2:
Determine all solutions $(n, k)$ of the equation $n!+A
n = n^k$ with $n, k \in\mathbb{N}$ for $A = 7$ and for $A = 2012$.

Tìm nghiệm $(n, k)$ của phương trình $n!+An = n^k$ với $n, k \in\mathbb{N}$ nếu $A = 7$ và nếu $A = 2012$.

Problem 4:
Let $ABC$ be a scalene (i.e. non-isosceles) triangle. Let $U$ be the center of the circumcircle of this triangle and $I$ the center of the incircle. Assume that the intersection of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$ lies on the perpendicular bisector of $UI$. Show that $\gamma$ is the second-largest angle in the triangle $ABC$.

Cho $ABC$ là tam giác không cân. Lấy $U$ là tâm đường tròn ngoại tiếp của tam giác này và $I$ là tâm đường tròn nội tiếp. Giả sử rằng các giao điểm của phân giác góc $\gamma = \angle ACB$ với đường tròn ngoại tiếp $ABC$ nằm trên đường vuông góc với phân giác của $UI$. Hãy chỉ ra rằng $\gamma$ là góc lớn thứ hai trong tam giác $ABC$.

Bài viết đã được chỉnh sửa nội dung bởi Phạm Quang Toàn: 24-05-2012 - 07:30

[tranghieu95]

$\fbox{1}.$ Chứng minh rằng bất đẳng thức \[ a + a^3 - a^4 - a^6 < 1\] đúng với mọi số thực $a$.

$a^6-a^4-a^3-a+1=(a^3-\dfrac{1}{2})^2+(a^2-\dfrac{1}{2})^2+(a-\dfrac{1}{2})^2+\dfrac{1}{4} >0$

[tranghieu95]

Giải phương trình nghiệm nguyên \[(x - 1)x(x + 1) + (y - 1)y(y + 1) = 24 - 9xy\mbox{.}\]

$(x-1)x(x+1)+(y-1)y(y+1)=24-9xy$
$\Rightarrow x^3+y^3-x-y+9xy-24=0$
$\Rightarrow (x+y-3)(x^2-4xy+y^2+3x+3y+8)=0$
$\Rightarrow (x+y-3)(4x^2-4xy+4y^2+12x+12y+32)=0$
$\Rightarrow (x+y-3)[(2x-y)^2+3(y-3)^2-4]=0$
$\begin{bmatrix}
x+y=3\\
(2x-y)^2+3(y-3)^2=4
\end{bmatrix}$
Ta có: $4=1^2+3.1^2 \Rightarrow \left\{\begin{matrix}
(2x-y)^2=1\\
(y-3)^2=1
\end{matrix}\right.$
Do đó pt có nghiệm $(x; y)$ là $(n; 3-n), (-2,-2),(-3,-4),(-3,-2),(-4;-4),(-2,-3),(-4,-3)$