MathNet 2017-part-04 — 176 problems (EN–VI)

In a right trapezoid $ABCD$ with right angles at the vertices $A$ and $B$ there are two circles. One of them is tangent to the legs and to the larger base $AD$, and the other is tangent to the legs, to the smaller base $BC$ and to the first circle.

1) The line through the centers of the circles meets $AD$ at the point $P$. Prove that $\frac{AP}{PD} = \sin (\angle D)$.

2) Find the area of the trapezoid if the radii of the circles are $\frac{4}{3}$ and $\frac{1}{3}$.

Prove that $8 \cdot \cos^{2} x \cdot (1-\operatorname{ctg} x) \lt 1$, $\forall x \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right]$.

Compute
$$
\lim _{n \rightarrow \infty}\left(\sum_{k=0}^{n} C_{2 n}^{2 k} 61^{k}\right) /\left(\sum_{k=0}^{n-1} C_{2 n}^{2 k+1} 61^{k}\right).
$$

Find all values of the real parameter $a$ for which the system
$$
\left\{
\begin{array}{c}
x^{2}+5x+y^{2}-y-|x-5y+5|=51 \\
a x - y - 5a + 2 = 0
\end{array}
\right.
$$
has exactly 2 distinct real solutions.

A student had to multiply a 3-digit number by a 4-digit number (the numbers do not start with 0). Instead of doing so, he simply wrote the 4-digit number to the right of the 3-digit number, obtaining as a result a 7-digit number which is $N$ times ($N \in \mathbb{N}$) larger than the correct result.

a) Can $N$ be equal to 2 or 10?

b) Find the largest possible value of $N$.

Let $a_1 \lt a_2 \lt \dots \lt a_{53}$ be positive integers such that the sum of any 27 integers is greater than the sum of the remaining 26 integers.

a) Find the minimum value of $a_1$.

b) Find all possible values of $a_2, \dots, a_{53}$, when $a_1$ is at the minimum value.

Let $D$ and $E$ be points on sides $AB$ and $AC$ of triangle $ABC$, respectively, such that $BD = CE$. Let $M$ and $N$ be midpoints of $BC$ and $DE$, respectively. Prove that $MN$ is parallel to the bisector of $\angle BAC$.

Let $a_1 \lt a_2 \lt \dots$ be the positive divisors of a positive integer $a$ and let $b_1 \lt b_2 \lt \dots$ be the positive divisors of a positive integer $b$. Find all $a, b$ such that
$$
\begin{cases} a_{10} + b_{10} = a \\ a_{11} + b_{11} = b \end{cases}
$$

Let $a$ and $b$ be real numbers such that $a + b = 1$. Prove the following inequality.
$$
\sqrt{1 + 5a^2} + 5\sqrt{2 + b^2} \ge 9
$$

Let $ABCD$ be an isosceles trapezoid with $AD = BC$ and $AB \parallel CD$. Let $O$ be the intersection of the diagonals and let $M$ be the midpoint of $AD$. The circumcircle of $BCM$ intersects $AD$ again at $K$. Prove that $OK$ is parallel to $AB$.

(Proposed by B. Bat-Od)

A $10 \times 10$ square is divided into $1 \times 1$ squares. A *point light* at a vertex of a $1 \times 1$ square lights all the $1 \times 1$ squares to which the vertex belongs. (A point light can be positioned on a vertex on the edge of the big square.) Find the minimum number of point lights required such that all the squares are lit even if one of the point lights is not functioning.
(Proposed by B. Battsengel)

$n$ girls are standing in a circle, each holding exactly 1 card. One girl gives her card to the girl on her left, who in turn gives 2 cards to the girl on her left. The girl who got the cards gives 1 card to the girl on her left. The girl who got the card gives 2 cards to the girl on her left. Continuing this way, each girl gives alternately 1 or 2 cards to the girl on her left. Anyone who has no card leaves the game immediately. Find all values of $n$ such that all cards are collected by one girl.

Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line through $B$ intersects $\omega_1$ at the point $C$ and $\omega_2$ at the point $D$. The line $AD$ intersects $\omega_1$ at a point $E$ different from $A$ and the line $AC$ intersects $\omega_2$ at a point $F$ different from $A$. If $O$ is the circumcenter of triangle $AEF$, prove that $OB \perp CD$.

(Proposed by B. Bat-Od)

Let $p$ be a prime number which is greater than $5$. Prove that there exist natural numbers $m, n$ such that $m + n \lt p$ and $2^m \cdot 3^n - 1$ is divisible by $p$.

Let $\omega$ be the circumcircle of triangle $ABC$ and let $AD$ and $BE$ be altitudes. The line $DE$ intersects the circle $\omega$ at points $P$ and $Q$, in the order $P$, $E$, $D$, $Q$ on the line. Let the bisectors of angles $APQ$ and $BQP$ intersect the circle $\omega$ at points $K$ and $L$, respectively. Prove that the line $KL$ is perpendicular to the bisector of angle $ACB$.
(Proposed by B. Ulziinasan)

For a positive integer $n$, let $\sigma(n)$ be the sum of all divisors of $n$. Show that there exist infinitely many positive integers $n$ such that $n$ divides $2^{\sigma(n)}-1$.

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$, and let $E$ be the intersection of the diagonals $AC$ and $BD$. A line passing through $E$ intersects the lines $AB$, $BC$ at $P, Q$, respectively. Let $R$ ($R \neq D$) be the intersection point of $\omega$ and the circle that passes through $D, E$ and is tangent to the line $PQ$ at $E$. Prove that $B, P, Q, R$ are concyclic.
(Proposed by B. Khoroldagva)

Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying
$$
(a-b)f(a+b) + (b-c)f(b+c) + (c-a)f(c+a) = 0
$$
for all $a, b, c \in \mathbb{R}$.

Let $\omega$ be the circumcircle of a scalene triangle $ABC$. The tangents to $\omega$ at $A$ and $C$ meet at $P$, and the line $BP$ intersects $\omega$ at $D$. Let $BB'$ be a diameter of $\omega$. The exterior angle bisector of $\angle ABC$ intersects the lines $B'A$ and $B'C$ at $A'$ and $C'$, respectively. Prove that $A', B', C', D$ are concyclic.

Do there exist positive integers $a_1, a_2, \dots, a_{2017}$ such that the product
$$
(a_1^{2017} + a_2)(a_2^{2017} + a_3) \dots (a_{2016}^{2017} + a_{2017})(a_{2017}^{2017} + a_1)
$$
is a power of a prime with exponent
a) $2017 \cdot 2018,$
b) $2017 \cdot 2023.$