1. Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum $\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}$ is greater than or equal to $\dfrac{2^n-1}{2^n}$.
2. Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.
3. Let $p$ be a fixed odd prime. A $p$-tuple $(a_1,a_2,a_3,\ldots,a_p)$ of integers is said to be good if
• $0\le a_i\le p-1$ for all $i$, and
• $a_1+a_2+a_3+\cdots+a_p$ is not divisible by $p$, and
• $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$ is divisible by $p$.
Determine the number of good $p$-tuples.
4. The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle PAB = \angle PBC = \angle PCD = \angle PDA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.
5. Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.
 MOlympiad.NET là dự án thu thập và phát hành các đề thi tuyển sinh và học sinh giỏi toán. Quý bạn đọc muốn giúp chúng tôi chỉnh sửa đề thi này, xin hãy để lại bình luận facebook (có thể đính kèm hình ảnh) hoặc google (có thể sử dụng $\LaTeX$) bên dưới. BBT rất mong bạn đọc ủng hộ UPLOAD đề thi và đáp án mới hoặc liên hệbbt.molympiad@gmail.comChúng tôi nhận tất cả các định dạng của tài liệu: $\TeX$, PDF, WORD, IMG,...
Anonymous
This comment has been removed by a blog administrator.
You can use $\LaTeX$ in comment