Consider the following increasing sequence $1,3,5,7,9,…$ of all positive integers consisting only of odd digits. Find the $2017$ -th term of the above sequence.

Let $n\ge 1$ be an integer. In town $X$ there are $n$ girls and $n$ boys, and each girl knows each boy. In town $Y$ there are $n$ girls, $g_1,g_2,\ldots ,g_n$, and $2n-1$ boys, $b_1,b_2,\ldots ,b_{2n-1}$. For $i=1,2,\ldots ,n$, girl $g_i$ knows boys $b_1,b_2,\ldots ,b_{2i-1}$ and no other boys. Let $r$ be an integer with $1\le r\le n$. In each of the towns a party will be held where $r$ girls from that town and $r$ boys from the same town are supposed to dance with each other in $r$ dancing pairs. However, every girl only wants to dance with a boy she knows. Denote by $X(r)$ the number of ways in which we can choose $r$ dancing pairs from town $X$, and by $Y(r)$ the number of ways in which we can choose $r$ dancing pairs from town $Y$. Prove that $X(r)=Y(r)$ for $r=1,2,\ldots ,n$.

Let $p$ and $n$ be positive integers, with $p\geq2$, and let $a$ be a real number such that $1\leq a<a+n\leq p$. Prove that the set \[ \mathcal {S}=\left\{\left\lfloor \log_{2}x\right\rfloor +\left\lfloor \log_{3}x\right\rfloor +\cdots+\left\lfloor \log_{p}x\right\rfloor\mid x\in\mathbb{R},a\leq x\leq a+n\right\} \] has exactly $n+1$ elements.

Prove that if $ a_1 < a_2 < \ldots < a_n $ and $ b_1 < b_2 < \ldots < b_n $, where $ n \geq 2 $, then $$\qquad (a_1 + a_2 + \ldots + a_n)(b_1 + b_2 + \ldots + b_n) < n(a_1b_1 + a_2b_2 + \ldots + a_nb_n).$$

There are $76$ cards with different numbers written on them. These cards are laid out on the table in a circle, number down. Try to find some three cards in a row such that the number written on the middle of these three cards is greater than on each of the two neighboring ones. You can turn over no more than $10$ cards in succession. How should one proceed to be sure to find three cardboard boxes for which the specified condition is met?

What is the smallest positive integer that when raised to the $6^{th}$ power, it can be represented by a sum of the $6^{th}$ powers of distinct smaller positive integers?

Let $K,L,M$ be points on sides $AB,BC,CA$, respectively, of a triangle $ABC$ such that $AK/AB = BL/BC = CM/CA = 1/3$. Show that if the circumcircles of the triangles $AKM, BLK, CML$ are equal, then so are the incircles of these triangles.

Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

Anne-Marie has a deck of $16$ cards, each with a distinct positive factor of $2002$ written on it. She shuffles the deck and begins to draw cards from the deck without replacement. She stops when there exists a nonempty subset of the cards in her hand whose numbers multiply to a perfect square. What is the expected number of cards in her hand when she stops? [i]Proposed by Michael Ren.[/i]

Let $f(x)=x^2+bx+c$, where $b,c \in \mathbb{R}$ and $b>0$ Do there exist disjoint sets $A$ and $B$, whose union is $[0,1]$ and $f(A)=B$, where $f(X)=\{f(x), x \in X\}$ [i]D. Zmiaikou[/i]

Let $n \geq 2$ be a positive integer. For a positive integer $a$, let $Q_a(x)=x^n+ax$. Let $p$ be a prime and let $S_a=\{b | 0 \leq b \leq p-1, \exists c \in \mathbb {Z}, Q_a(c) \equiv b \pmod p \}$. Show that $\frac{1}{p-1}\sum_{a=1}^{p-1}|S_a|$ is an integer.

Keiko tosses one penny and Ephraim tosses two pennies. The probability that Ephraim gets the same number of heads that Keiko gets is $\text{(A)}\ \dfrac{1}{4} \qquad \text{(B)}\ \dfrac{3}{8} \qquad \text{(C)}\ \dfrac{1}{2} \qquad \text{(D)}\ \dfrac{2}{3} \qquad \text{(E)}\ \dfrac{3}{4}$

For any angle $0 < \theta < \pi/2$, show that \[0 < \sin \theta + \cos \theta + \tan \theta + \cot \theta - \sec \theta - \csc \theta < 1.\]

Eight poles stand along the road. A sparrow starts at the first pole and once in a minute flies to a neighboring pole. Let $a(n)$ be the number of ways to reach the last pole in $2n + 1$ flights (we assume $a(m) = 0$ for $m < 3$). Prove that for all $n \ge 4$ $$a(n) - 7a(n-1)+ 15a(n-2) - 10a(n-3) +a(n-4)=0.$$ [i](T. Amdeberhan, F. Petrov )[/i]

The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at points $A_1$, $B_1$, and $C_1$. The line through the midpoints of segments $AB_1$ and $AC_1$ intersects the tangent at $A$ to the circumcircle of triangle $ABC$ at point $A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that points $A_2$, $B_2$, and $C_2$ lie on a line.

Let \(n\) be a positive integer. For every positive integer $1 \leq k \leq n$ the sequence ${\displaystyle {\{ a_{i}+ki\}}_{i=1}^{n }}$ is defined, where $a_1,a_2, \dots ,a_n$ are integers. Among these \(n\) sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by \(n\)?

Chords $AB$ and $CD$ of a circle $\omega$ meet at point $E$ in such a way that $AD = AE = EB$. Let $F$ be a point of segment $CE$ such that $ED = CF$. The bisector of angle $AFC$ meets an arc $DAC$ at point $P$. Prove that $A$, $E$, $F$, and $P$ are concyclic.

Positive integers $m$ and $n$ satisfy $$(m + n)(24mn + 1) = 2023.$$ Find $m + n + 12mn$.

Prove that a ring that has a prime characteristic admits nonzero nilpotent elements if and only if its characteristic divides the number of its units.

Let $ABCD$ be a cyclic quadrilateral and $\omega_1, \omega_2$ the incircles of triangles $ABC$ and $BCD$. Show that the common external tangent line of $\omega_1$ and $\omega_2$, the other one than $BC$, is parallel with $AD$

Let $ABCD$ be a convex quadrilateral, and write $\alpha=\angle DAB$; $\beta=\angle ADB$; $\gamma=\angle ACB$; $\delta= \angle DBC$; and $\epsilon=\angle DBA$. Assuming that $\alpha<\pi/2$, $\beta+\gamma=\pi /2$, and $\delta+2\epsilon=\pi$, prove that \[(DB+BC)^2=AD^2+AC^2\] [color=red][Moderator edit: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=30569 .][/color]

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

Let $I{}$ be the incenter of triangle $ABC{}.$ Let $N{}$ be the foot of the bisector of angle $B{}.$ The tangent line to the circumcircle of triangle $AIN$ at $A{}$ and the tangent line to the circumcircle of triangle $CIN{}$ at $C{}$ intersect at $D{}.$ Prove that lines $AC{}$ and $DI$ are perpendicular. [i]Mikhail Evdokimov[/i]