## Juniors

1. Quadratic trinomials $F$ and $G$ satisfy $$F(F(x))>F(G(x))>G(G(x))$$ for all real $x$. Prove that $F(x)>G(x)$ for all real $x$.
2. The bisector of angle $B$ of a parallelogram $A B C D$ meets its diagonal $A C$ at $E$, and the external bisector of angle $B$ meets line $A D$ at $F . M$ is the midpoint of $B E$. Prove that $C M \| E F$.
3. For $n$ distinct positive integers all their $n(n-1) / 2$ pairwise sums are considered. For each of these sums Ivan has written on the board the number of original integers which are less than that sum and divide it. What is the maximum possible sum of the numbers written by Ivan?
4. Some manors of Lipshire county are connected by roads. The inhabitants of manors connected by a road are called neighbours. Is it always possible to settle in each manor a knight (who always tells truth) or a liar (who always lies) so that every inhabitant can say "The number of liars among my neighbours is at least twice the number of knights"?
5. In a $100 \times 100$ table 110 unit squares are marked. Is it always possible to rearrange rows and columns so that all the marked unit squares are above the main diagonal or on it?
6. Given are real $y>1$ and positive integer $n \leqslant y^{50}$ such that all prime divisors of $n$ do not exceed $y$. Prove that $n$ is a product of 99 positive integer factors (not necessarily primes) not exceeding $y$.
7. A pile contains $2021^{2021}$ stones. In a move any pile can be divided into two piles so that the numbers of stones in them differ by a power of 2 with non-negative integer exponent. After some move it turned out that the number of stones in each pile is a power of 2 with non-negative integer exponent. Prove that the number of moves performed was even.
8. An acute triangle $A B C$ is given, $A C \neq B C$. The altitudes drawn from $A$ and $B$ meet at $H$ and intersect the external bisector of the angle $C$ at $Y$ and $X$ respectively. The external bisector of the angle $A H B$ meets the segments $A X$ and $B Y$ at $P$ and $Q$ respectively. If $P X=Q Y$, prove that $A P+B Q \geqslant 2 C H$.

## Seniors

1. Polynomials $F$ and $G$ satisfy $$F(F(x))>F(G(x))>G(G(x))$$ for all real $x$. Prove that $F(x)>G(x)$ for all real $x$.
2. In trapezoid $A B C D$, $M$ is the midpoint of base $A D$. Point $E$ lies on the segment $B M$. It is known that $\angle A D B=\angle M A E=\angle B M C$. Prove that the triangle $B C E$ is isosceles.
3. Positive real numbers $a_{1}, \ldots, a_{k}, b_{1}, \ldots, b_{k}$ are given. Let $A=\sum_{i=1}^{k} a_{i}, B=\sum_{i=1}^{k} b_{i}$. Prove the inequality $$\left(\sum_{i=1}^{k} \frac{a_{i} b_{i}}{a_{i} B+b_{i} A}-1\right)^{2} \geqslant \sum_{i=1}^{k} \frac{a_{i}^{2}}{a_{i} B+b_{i} A} \cdot \sum_{i=1}^{k} \frac{b_{i}^{2}}{a_{i} B+b_{i} A} .$$
4. An $n \times n$ square ( $n$ is a positive integer) consists of $n^{2}$ unit squares. A monotonous path in this square is a path of length $2 n$ beginning in the left lower corner of the square, ending in its right upper corner and going along the sides of unit squares. For each $k, 0 \leq k<2 n-1$, let $S_{k}$ be the set of all the monotonous paths such that the number of unit squares lying below the path leaves remainder $k$ upon division by $2 n-1$. Prove that all $S_{k}$ contain equal number of elements.
5. Sines of three acute angles form an arithmetic progression, while the cosines of these angles form a geometric progression. Prove that all three angles are equal.
6. In a $n \times n$ table $(n>1) k$ unit squares are marked. One wants to rearrange rows and columns so that all the marked unit squares are above the main diagonal or on it. For what maximum $k$ is it always possible?
7. An acute triangle $A B C$ is given, $A C \neq B C$. The altitudes drawn from $A$ and $B$ meet at $H$ and intersect the external bisector of the angle $C$ at $Y$ and $X$ respectively. The external bisector of the angle $A H B$ meets the segments $A X$ and $B Y$ at $P$ and $Q$ respectively. If $P X=Q Y$, prove that $A P+B Q \geqslant 2 C H$.
8. In a sequence $P_{n}$ of quadratic trinomials each trinomial, starting with the third, is the sum of the two preceding trinomials. The first two trinomials do not have common roots. Is it possible that $P_{n}$ has an integral root for each $n$ ?
 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,...