MathNet 2023-part-04 — 191 problems (EN–VI)

Let $\Gamma$ and $\omega$ be two circles, intersecting at points $C$ and $D$. The center of circle $\omega$ is denoted as $P$, and it lies on circle $\Gamma$. A line passing through point $D$ intersects circle $\Gamma$ at point $A$ and circle $\omega$ at point $B$, with $D$ being located between points $A$ and $B$. Line $PB$ intersects circle $\Gamma$ at a point $E$ (distinct from $P$). Prove that $AE = AB$.

Let $p$, $q$, $r$ be prime numbers such that $p \lt q \lt r$ and at least two of $p+n$, $q+n$, $r+n$ are relatively prime for any positive integer $n$. Find all possible values of the pair $(p, q)$.

Let $n$ be a positive integer and suppose that $n = a_1 + a_2 + \cdots + a_m$ for $m \ge 2$ and $a_1 \gt a_2 \gt \cdots \gt a_m$ distinct positive integers satisfying
$$
\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_m} = 1.
$$
Prove that the inequality $n + 2m + 1 \le a_1 a_2$ holds and determine the conditions under which equality holds.

Let $n$, $m$, $k$ be integers such that $\frac{mk+n}{n^2+1}$ and $\frac{nk+m}{m^2+1}$ are integers. Show that $k^2 - 1$ is divisible by the greatest common divisor of $n^2+1$ and $m^2+1$.

Let $ABCD$ be a quadrilateral with $\angle DAB = \angle ABC$. The circumcircle of triangle $ABC$ intersects segment $AD$ at point $K$ and segment $CD$ at point $L$. Segment $AL$ intersects segment $CK$ at point $P$. If $\angle ADB = \angle PDC$, prove that $CP = CL$.

Let $X$ be a finite set of positive integers such that no two subsets have the same sum. Prove that the sum of the inverses of the elements of $X$ satisfy
$$
\sum_{x \in X} \frac{1}{x} \lt \frac{5}{2}.
$$
For example, for $X = \{1, 2, 5\}$ the sums $1, 2, 5, 1+2, 1+5, 2+5, 1+2+5$ are all distinct and $1/1 + 1/2 + 1/5 = 1.7 \lt 2.5$ holds.

Let $0 \le a \le 1$. Prove that $m^2 + 2amn + an^2 \ge m + an$ holds for all integers $m$ and $n$, and determine the condition under which equality holds.

Let $p \ge 5$ be a prime number and $f(x) = (x-1)(x^3-1)\dots(x^{p-4}-1)(x^{p-2}-1)$. Show that $f(n^k) - f(n)$ is divisible by $p$ for any positive integer $k$ relatively prime to $p-1$.

Show that every positive integer $n$ can be expressed as a sum of positive integers, where the sum of their reciprocals is less than or equal to $4$. For instance, as $5 = 2 + 2 + 1$, we find that $1/2 + 1/2 + 1/1 = 2 \le 4$.

Show that the sum or the product of $a, b, c$ is equal to $1$ if $a, b, c$ are real numbers satisfying
$$
(ab - c)(bc - a) + (bc - a)(ca - b) + (ca - b)(ab - c) = 4abc.
$$

Let $ABC$ be a triangle with points $D$ and $E$ taken on side $BC$ such that $D$ is between $B$ and $E$. A point $F$ lies on segment $AD$ and satisfies the condition $\frac{AF}{FD} = \frac{BE}{EC}$. The circumcircle of triangle $ABD$ intersects segment $AE$ at a point $G$ (distinct from $A$). If lines $EF$ and $GC$ are parallel, prove that $GE = EC$.

Suppose that there are $n$ students standing in a line, and each student randomly raises either their left or right hand, but not both. Let $P_n$ denote the probability that in every group of three students standing in a row, at least one student raised their right hand. Prove
$$
P_n \lt \left(\frac{12}{13}\right)^{n-2}.
$$

There are four pipes installed in the pool. The first two pipes are used to fill the pool with water, while the remaining two pipes are responsible for draining water from the pool. If all four pipes are operational simultaneously, the reservoir fills up in $2.5$ hours. When only pipes $1$, $2$, and $3$ are in use, the pool fills up in $1.5$ hours. Similarly, when only pipes $1$, $3$, and $4$ are active, it takes $15$ hours to fill the pool. How long does it take for the pool to be full when only pipes $1$ and $3$ are operational?

Are there ten positive integers such that no number in the set is divisible by another number, and the product of any two numbers is divisible by the remaining numbers in the set?

Bill wants to write a set of pairwise distinct positive integers on the board, ensuring that each number is divisible by at most one other number on the board.
(1)
Prove that Bill can write 14 numbers on the board.
(2)
Prove that Bill cannot write 15 numbers on the board.

Let $a$, $b$, and $c$ be real numbers.
(1)
Show that if $a^2 + b^2 = c^2$ and $a^3 + b^3 = c^3$, then either $a = 0$ or $b = 0$.

(2)
Show that if $a^3 + b^3 = c^3$ and $a^4 + b^4 = c^4$, then either $a = 0$ or $b = 0$.

Find all three-digit numbers that are divisible by $37$ and can be expressed as the sum of the cubes of their digits.

Prove that there exists a sequence of 2023 consecutive positive integers in which exactly 2000 numbers are composite.

Prove that, when a crooked dice is thrown twice, the probability of obtaining the same parity faces on both rolls is not less than the probability of obtaining different parity faces on the two rolls.

Let $\omega$ be the circumcircle of a triangle $ABC$. Let $P$ be a point on the arc $AB$ of $\omega$, not containing $C$. The circle inscribed in triangle $ABC$ has center $I$, and the line $AI$ intersects $\omega$ at a point $M$, distinct from $A$. The circumcircle of triangle $API$ intersects segment $AB$ at a point $E$ (other than $A$), and segments $PM$ and $BC$ intersect at point $D$. Prove that $BE = BD$.