Form $10A$ has $29$ students who are listed in order on its duty roster. Form $10B$ has $32$ students who are listed in order on its duty roster. Every day two students are on duty, one from form $10A$ and one from form $10B$. Each day just one of the students on duty changes and is replaced by the following student on the relevant roster (when the last student on a roster is replaced he is replaced by the first). On two particular days the same two students were on duty. Is it possible that starting on the first of these days and ending the day before the second, every pair of students (one from $10A$ and one from $10B$) shared duty exactly once?

A three-digit number is written $xyz$ in the base $7$ system and $zyx$ in the base $9$ system . What is the number?

From vertex $A$ of an equilateral triangle $ABC$, a ray $Ax$ intersects $BC$ at point $D$. Let $E$ be a point on $Ax$ such that $BA =BE$. Calculate $\angle AEC$.

Rectangle $ABCD$ has $AB=5$ and $BC=4$. Point $E$ lies on $\overline{AB}$ so that $EB=1$, point $G$ lies on $\overline{BC}$ so that $CG=1$. and point $F$ lies on $\overline{CD}$ so that $DF=2$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$? [asy] pair A1=(2,0),A2=(4,4); pair B1=(0,4),B2=(5,1); pair C1=(5,0),C2=(0,4); draw(A1--A2); draw(B1--B2); draw(C1--C2); draw((0,0)--B1--(5,4)--C1--cycle); dot((20/7,12/7)); dot((3.07692307692,2.15384615384)); label("$Q$",(3.07692307692,2.15384615384),N); label("$P$",(20/7,12/7),W); label("$A$",(0,4), NW); label("$B$",(5,4), NE); label("$C$",(5,0),SE); label("$D$",(0,0),SW); label("$F$",(2,0),S); label("$G$",(5,1),E); label("$E$",(4,4),N); dot(A1); dot(A2); dot(B1); dot(B2); dot(C1); dot(C2); dot((0,0)); dot((5,4));[/asy] $\textbf{(A)}~\frac{\sqrt{13}}{16} \qquad \textbf{(B)}~\frac{\sqrt{2}}{13} \qquad \textbf{(C)}~\frac{9}{82} \qquad \textbf{(D)}~\frac{10}{91}\qquad \textbf{(E)}~\frac19$

Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.

Find all functions $f : R_+ \to R_+$ such that $$x^2(f(x)+f(y)) = (x+y)f(f(x)y)$$ for all positive real $x, y$.

$K(x, y), f(x)$ and $g(x)$ are positive and continuous for $x, y \subseteq [0, 1]$. $\int_{0}^{1} f(y) K(x, y) dy = g(x)$ and $\int_{0}^{1} g(y) K(x, y) dy = f(x)$ for all $x \subseteq [0, 1]$. Show that $f = g$ on $[0, 1]$.

Let $a_0=29$, $b_0=1$ and $$a_{n+1} = a_n+a_{n-1}\cdot b_n^{2019}, \qquad b_{n+1}=b_nb_{n-1}$$ for $n\geq 1$. Determine the smallest positive integer $k$ for which $29$ divides $\gcd(a_k, b_k-1)$ whenever $a_1,b_1$ are positive integers and $29$ does not divide $b_1$.

Let $\gamma$ be a circle and $l$ a line in its plane. Let $K$ be a point on $l$, located outside of $\gamma$. Let $KA$ and $KB$ be the tangents from $K$ to $\gamma$, where $A$ and $B$ are distinct points on $\gamma$. Let $P$ and $Q$ be two points on $\gamma$. Lines $PA$ and $PB$ intersect line $l$ in two points $R$ and respectively $S$. Lines $QR$ and $QS$ intersect the second time circle $\gamma$ in points $C$ and $D$. Prove that the tangents from $C$ and $D$ to $\gamma$ are concurrent on line $l$.

Find all functions $f: \mathbb{Q}\to \mathbb{R}$ such that for all $x,y\in \mathbb{Q}$: \[f(xy)=f(x)f(y)-f(x+y)+1.\]

A rectangle $ABCD$ is given. On the side $AB$, $n$ different points are chosen strictly between $A$ and $B$. Similarly, $m$ different points are chosen on the side $AD$. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way? (One possibility is shown in the figure.) [img]https://cdn.artofproblemsolving.com/attachments/0/1/dcf48e4ce318fdcb8c7088a34fac226e26e246.png[/img]

For a positive integer $n$ we denote $D_2(n)$ to the number of divisors of $n$ which are perfect squares and $D_3(n)$ to the number of divisors of $n$ which are perfect cubes. Prove that there exists such that $D_2(n)=999D_3(n).$ Note. The perfect squares are $1^2,2^2,3^2,4^2,…$ , the perfect cubes are $1^3,2^3,3^3,4^3,…$ .

Prove that in any triangle the length of the shortest bisector does not exceed three times the radius of the incircle.

Let $A_1A_2 \dots A_n$ be a polygon (not necessarily regular) with $n$ sides. Suppose there is a translation that maps each point $A_i$ to a point $B_i$ in the same plane. For convenience, define $A_0 = A_n$ and $B_0 = B_n$. Prove that \[ \sum_{i=1}^{n} (A_{i-1} B_{i})^2 = \sum_{i=1}^{n} (B_{i-1} A_{i})^2 \, . \]

What is the greatest number of regions into which four planes can divide three-dimensional space?

If $ x$ and $ y$ are positive integers for which $ 2^x3^y \equal{} 1296$, what is the value of $ x\plus{}y$? $ \textbf{(A)}\ 8\qquad \textbf{(B)}\ 9\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 11\qquad \textbf{(E)}\ 12$

Line segment $\overline{AE}$ of length $17$ bisects $\overline{DB}$ at a point $C$. If $\overline{AB} = 5$, $\overline{BC} = 6$ and $\angle BAC = 78^o$ degrees, calculate $\angle CDE$.

Consider a circumscribed pentagon ${ABCDE}$ . Its incenter lies on the diagonal ${AC}$ . Prove that $$ {AB} + {BC} > {CD} + {DE} + {EA}. $$ Egor Bakaev

For every positive integer $n$, determine the biggest positive integer $k$ so that $2^k |\ 3^n+1$

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

suppose that $A$ is the set of all Closed intervals $[a,b] \subset \mathbb{R}$. Find all functions $f:\mathbb{R} \rightarrow A$ such that $\bullet$ $x \in f(y) \Leftrightarrow y \in f(x)$ $\bullet$ $|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing$ $\bullet$ For all real numbers $0\leq r\leq 1$, $f(r)=[r^2-1,r^2+1]$ Proposed by Matin Yousefi

Do there exist integers $a$ and $b$ such that : (a) the equation $x^2 + ax + b = 0$ has no real roots, and the equation $\lfloor x^2 \rfloor + ax + b = 0$ has at least one real root? [i](2 points)[/i] (b) the equation $x^2 + 2ax + b$ = 0 has no real roots, and the equation $\lfloor x^2 \rfloor + 2ax + b = 0$ has at least one real root? [i]3 points[/i] (By $\lfloor k \rfloor$ we denote the integer part of $k$, that is, the greatest integer not exceeding $k$.) [i]Alexandr Khrabrov[/i]

Let $f:[1,\infty)\to(0,\infty)$ be a continuous function. Assume that for every $a>0$, the equation $f(x)=ax$ has at least one solution in the interval $[1,\infty)$. (a) Prove that for every $a>0$, the equation $f(x)=ax$ has infinitely many solutions. (b) Give an example of a strictly increasing continuous function $f$ with these properties.

A [i]smooth number[/i] is a positive integer of the form $2^m 3^n$, where $m$ and $n$ are nonnegative integers. Let $S$ be the set of all triples $(a, b, c)$ where $a$, $b$, and $c$ are smooth numbers such that $\gcd(a, b)$, $\gcd(b, c)$, and $\gcd(c, a)$ are all distinct. Evaluate the infinite sum $\sum_{(a,b,c) \in S} \frac{1}{abc}$. Recall that $\gcd(x, y)$ is the greatest common divisor of $x$ and $y$.