This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 21

MathLinks Contest 5th, 7.2
Tags: number theory , 5th edition
For any positive integer $n$, let $s(n)$ be the sum of its digits, written in decimal base. Prove that for each integer $n \ge 1$ there exists a positive integer $x$ such that the fraction $\frac{x + k}{s(x + k)}$ is not integral, for each integer $k$ with $0 \le k \le n$.

MathLinks Contest 5th, 3.2
Tags: number theory , algebra , 5th edition
Let $0 < a_1 < a_2 <... < a_{16} < 122$ be $16$ integers. Prove that there exist integers $(p, q, r, s)$, with $1 \le p < r \le s < q \le 16$, such that $a_p + a_q = a_r + a_s$. An additional $2$ points will be awarded for this problem, if you can find a larger bound than $122$ (with proof).

MathLinks Contest 5th, 4.2
Tags: combinatorics , 5th edition
Given is a unit cube in space. Find the maximal integer $n$ such that there are $n$ points, satisfying the following conditions: (a) All points lie on the surface of the cube; (b) No face contains all these points; (c) The $n$ points are the vertices of a polygon.

MathLinks Contest 5th, 4.1
Tags: geometry , 5th edition
Let $ABC$ be an acute angled triangle. Let $M$ be the midpoint of $BC$, and let $BE$ and $CF$ be the altitudes of the triangle. Let $D \ne M$ be a point on the circumcircle of the triangle $EFM$ such that $DE = DF$. Prove that $AD \perp BC$.

MathLinks Contest 5th, 3.1
Tags: algebra , 5th edition
Let $\{x_n\}_n$ be a sequence of positive rational numbers, such that $x_1$ is a positive integer, and for all positive integers $n$. $x_n = \frac{2(n - 1)}{n} x_{n-1}$, if $x_{n_1} \le 1$ $x_n = \frac{(n - 1)x_{n-1} - 1}{n}$ , if $x_{n_1} > 1$. Prove that there exists a constant subsequence of $\{x_n\}_n$.

MathLinks Contest 5th, 4.3
Tags: inequalities , algebra , 5th edition
Let $a_1,..., a_n$ be positive reals and let $x_1, ... , x_n$ be real numbers such that $a_1x_1 +...+ a_nx_n = 0$. Prove that $$\sum_{1\le i<j \le n} x_ix_j |a_i - a_j | \le 0.$$ When does the equality take place?

MathLinks Contest 5th, 5.2
Tags: algebra , inequalities , function , 5th edition
Prove or disprove the existence of a function $f : S \to R$ such that for all $x \ne y \in S$ we have $|f(x) - f(y)| \ge \frac{1}{x^2 + y^2}$, in each of the cases: a) $S = R$ b) $S = Q$.

MathLinks Contest 5th, 7.1
Tags: number theory , 5th edition
Prove that the numbers $${{2^n-1} \choose {i}}, i = 0, 1, . . ., 2^{n-1} - 1,$$ have pairwise different residues modulo $2^n$

MathLinks Contest 5th, 1.2
Tags: number theory , 5th edition
Find all the integers $n \ge 5$ such that the residue of $n$ when divided by each prime number smaller than $\frac{n}{2}$ is odd.

MathLinks Contest 5th, 2.2
Tags: combinatorics , 5th edition
Suppose that $\{D_n\}_{n\ge 1}$ is an finite sequence of disks in the plane whose total area is less than $1$. Prove that it is possible to rearrange the disks so that they are disjoint from each other and all contained inside a disk of area $4$.

MathLinks Contest 5th, 6.3
Tags: algebra , inequalities , 5th edition
Let $x, y, z$ be three positive numbers such that $(x + y-z) \left( \frac{1}{x}+ \frac{1}{y}- \frac{1}{z} \right)=4$. Find the minimal value of the expression $$E(x, y, z) = (x^4 + y^4 + z^4) \left( \frac{1}{x^4}+ \frac{1}{y^4}+ \frac{1}{z^4} \right) .$$

MathLinks Contest 5th, 5.3
Tags: combinatorics , 5th edition
A student wants to make his birthday party special this year. He wants to organize it such that among any groups of $4$ persons at the party there is one that is friends with exactly another person in the group. Find the largest number of his friends that he can possibly invite at the party.

MathLinks Contest 5th, 5.1
Tags: number theory , 5th edition
Find all real numbers $a > 1$ such that there exists an integer $k \ge 1$ such that the sequence $\{x_n\}_{n\ge 1}$ formed with the first $k$ digits of the number $\lfloor a^n\rfloor$ is periodical.

MathLinks Contest 5th, 1.3
Tags: geometry , 5th edition
Let $ABC$ be a triangle and let $A' \in BC$, $B' \in CA$ and $C' \in AB$ be three collinear points. a) Prove that each pair of circles of diameters $AA'$, $BB'$ and $CC'$ has the same radical axis; b) Prove that the circumcenter of the triangle formed by the intersections of the lines $AA' , BB'$ and $CC'$ lies on the common radical axis found above.

MathLinks Contest 5th, 1.1
Tags: number theory , 5th edition
Find all pairs of positive integers $x, y$ such that $x^3 - y^3 = 2005(x^2 - y^2)$.

MathLinks Contest 5th, 6.1
Tags: geometry , 5th edition
Let $ABC$ be a triangle and let $C$ be a circle that intersects the sides $BC, CA$ and $AB$ in the points $A_1, A_2, B_1, B_2$ and $C_1, C_2$ respectively. Prove that if $AA_1, BB_1$ and $CC_1$ are concurrent lines then $AA_2, BB_2$ and $CC_2$ are also concurrent lines.

MathLinks Contest 5th, 6.2
Tags: number theory , 5th edition
We say that a positive integer $n$ is nice if $\frac{4}{n}$ cannot be written as $\frac{1}{x}+\frac{1}{xy}+\frac{1}{z}$ for any positive integers $x, y, z$. Let us denote by $ a_n$ the number of nice numbers smaller than $n$. Prove that the sequence $\frac{n}{a_n}$ is not bounded.

MathLinks Contest 5th, 3.3
Tags: inequalities , 5th edition , algebra
Let $x_1, x_2,... x_n$ be positive numbers such that $S = x_1+x_2+...+x_n =\frac{1}{x_1}+...+\frac{1}{x_n}$ Prove that $$\sum_{i=1}^{n} \frac{1}{n - 1 + x_i} \ge \sum_{i=1}^{n} \frac{1}{1+S - x_i}$$

MathLinks Contest 5th, 2.3
Tags: inequalities , algebra , 5th edition
Let $a, b, c$ be positive numbers such that $abc \le 8$. Prove that $$\frac{1}{a^2 - a + 1} +\frac{1}{b^2 - b + 1}++\frac{1}{c^2 - c + 1} \ge 1$$

MathLinks Contest 5th, 2.1
Tags: functional equation , 5th edition
For what positive integers $k$ there exists a function $f : N \to N$ such that for all $n \in N$ we have $\underbrace{\hbox{f(f(... f(n)....))}}_{\hbox{k times}} = f(n) + 2$ ?

MathLinks Contest 5th, 7.3
Tags: combinatorics , 5th edition
Given is a square of sides $3\sqrt7 \times 3\sqrt7$. Find the minimal positive integer $n$ such that no matter how we put $n$ unit disks inside the given square, without overlapping, there exists a line that intersects $4$ disks.