Let $\tau (n)$ denote the number of positive integer divisors of $n$. Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$.

$ABCD$ is a convex quadrilateral; half-lines $DA$ and $CB$ meet at point $Q$; half-lines $BA$ and $CD$ meet at point $P$. It is known that $\angle AQB=\angle APD$. The bisector of angle $\angle AQB$ meets the sides $AB$ and $CD$ of the quadrilateral at points $X$ and $Y$, respectively; the bisector of angle $\angle APD$ meets the sides $AD$ and $BC$ at points $Z$ and $T$, respectively. The circumcircles of triangles $ZQT$ and $XPY$ meet at point $K$ inside the quadrilateral. Prove that $K$ lies on the diagonal $AC$. [i]Proposed by S. Berlov[/i]

[b]Q[/b] Let $p,q,r$ be non negative reals such that $pqr=1$. Find the maximum value for the expression $$\sum_{cyc} p[r^{4}+q^{4}-p^{4}-p]$$ [i]Proposed by Aritra12[/i]

Let $ABC$ a triangle and $P$ a point such that the triangles $PAB, PBC, PCA$ have the same area and the same perimeter. Prove that if: a) $P$ is in the interior of the triangle $ABC$ then $ABC$ is equilateral. b) $P$ is in the exterior of the triangle $ABC$ then $ABC$ is right angled triangle.

We draw $n$ convex quadrilaterals in the plane. They divide the plane into regions (one of the regions is infinite). Determine the maximal possible number of these regions.

The incircle of a non-isosceles triangle $ABC$ has centre $I$ and is tangent to $BC$ and $CA$ in $D$ and $E$, respectively. Let $H$ be the orthocentre of $ABI$, let $K$ be the intersection of $AI$ and $BH$ and let $L$ be the intersection of $BI$ and $AH$. Show that the circumcircles of $DKH$ and $ELH$ intersect on the incircle of $ABC$.

Let $ABC$ be an acute triangle with $AB<AC$ and $D,E,F$ be the contact points of the incircle $(I)$ with $BC,AC,AB$. Let $M,N$ be on $EF$ such that $MB \perp BC$ and $NC \perp BC$. $MD$ and $ND$ intersect the $(I)$ in $D$ and $Q$. Prove that $DP=DQ$.

Show that there is no infinite sequence of primes $p_1, p_2, p_3, . . .$ there any for each $ k$: $p_{k+1} = 2p_k - 1$ or $p_{k+1} = 2p_k + 1$ is fulfilled. Note that not the same formula for every $k$.

It is given that $x$ varies directly as $y$ and inversely as the square of $z$, and that $x=10$ when $y=4$ and $z=14$. Then, when $y=16$ and $z=7$, $x$ equals: $ \textbf{(A)}\ 180\qquad\textbf{(B)}\ 160\qquad\textbf{(C)}\ 154\qquad\textbf{(D)}\ 140\qquad\textbf{(E)}\ 120 $

An equilateral triangle of side$ 1$ is rotated around its center, yielding another equilareral triangle. Find the area of the intersection of these two triangles.

Let $l, m, n$ be positive integers and $p$ be prime. If $p^{2l-1}m(mn+1)^2 + m^2$ is a perfect square, prove that $m$ is also a perfect square.

Solve the equation \[ 3^x \minus{} 5^y \equal{} z^2.\] in positive integers. [i]Greece[/i]

On the sides $BC$ and $AC$ of the isosceles triangle $ABC$ ($AB = BC$), points $E$ and $D$ are marked, respectively, so that $DE \parallel AB$. On the extendsion of side $CB$ beyond the point $B$, point $K$ was arbitrarily marked. Let $P$ be the intersection point of the lines $AB$ and $KD$. Let $Q$ be the intersection point of the lines $AK$ and $DE$. Prove that $CA$ is the bisector of angle $\angle PCQ$.

A piecewise linear function is defined by \[f(x) = \begin{cases} x & \text{if } x \in [-1, 1) \\ 2 - x & \text{if } x \in [1, 3)\end{cases}\] and $f(x + 4) = f(x)$ for all real numbers $x.$ The graph of $f(x)$ has the sawtooth pattern depicted below. [color=transparent]Diagram from RandomMath.[/color] [center][img width=45]https://i.ibb.co/JW8jH2Dr/image.png[/img][/center] The parabola $x = 34y^2$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$-coordinates of these intersection points can be expressed in the form $\tfrac{a + b\sqrt c}d,$ where $a, b, c$ and $d$ are positive integers, $a, b,$ and $d$ has greatest common divisor equal to $1,$ and $c$ is not divisible by the square of any prime. Find $a + b + c + d.$

For how many positive integers $n \le 1000$ is $$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$$ not divisible by $3$? (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) $\textbf{(A) } 22 \qquad\textbf{(B) } 23 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 26$

How many sigma $(\sigma)$ and pi $(\pi)$ bonds are in a molecule of ethyne (acetylene), $\ce{HCCH}?$ $ \textbf{(A) } 1 \sigma \text{ and } 1 \pi \qquad\textbf{(B) }2 \sigma \text{ and } 1 \pi \qquad\textbf{(C) }2 \sigma \text{ and } 3\pi \qquad\textbf{(D) }3 \sigma \text{ and } 2 \pi\qquad$

A [i]hook[/i] consists of three segments of longitude $1$ forming two right angles as demonstrated in the figure. |_| We have a square of side length $n$ divided into $n^2$ squares of side length $1$ by lines parallel to its sides. Hooks are placed on this square in such a way that each segment of the hook covers one side of a little square. Two segements of a hook cannot overlap. Determine all possible values of n for which it is possible to cover the sides of the $n^2$ small squares.

Is it possible to partition $3$-dimensional Euclidean space into $1979$ mutually isometric subsets?

How many ways are there to color the vertices of a cube red, blue, or green such that no edge connects two vertices of the same color? Rotations and reflections are considered distinct colorings.

A sequence of integers $a_1,a_2,a_3,\ldots$ is called [i]exact[/i] if $a_n^2-a_m^2=a_{n-m}a_{n+m}$ for any $n>m$. Prove that there exists an exact sequence with $a_1=1,a_2=0$ and determine $a_{2007}$.

The smaller angle between the hands of a clock at $ 12: 25$ p.m. is: $ \textbf{(A)}\ 132^\circ 30' \qquad \textbf{(B)}\ 137^\circ 30' \qquad \textbf{(C)}\ 150^\circ \qquad \textbf{(D)}\ 137^\circ 32' \qquad \textbf{(E)}\ 137^\circ$

Prove that among any $1001$ numbers taken from the numbers $1,2,\ldots,1997$ there exist two with the difference $4$.

Find all possible values of integer $n > 3$ such that there is a convex $n$-gon in which, each diagonal is the perpendicular bisector of at least one other diagonal. Proposed by Mahdi Etesamifard

Find $A$ so that the ratio of $3\frac23$ to $22$ is the same as the ratio of $7\frac56$ to $A$

[u]Round 5[/u] [i]Each of the three problems in this round depends on the answer to two of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. [/i] [b]p13.[/b] Let $B$ be the answer to problem $14$, and let $C$ be the answer to problem $15$. A quadratic function $f(x)$ has two real roots that sum to $2^{10} + 4$. After translating the graph of $f(x)$ left by $B$ units and down by $C$ units, the new quadratic function also has two real roots. Find the sum of the two real roots of the new quadratic function. [b]p14.[/b] Let $A$ be the answer to problem $13$, and let $C$ be the answer to problem $15$. In the interior of angle $\angle NOM = 45^o$, there is a point $P$ such that $\angle MOP = A^o$ and $OP = C$. Let $X$ and $Y$ be the reflections of $P$ over $MO$ and $NO$, respectively. Find $(XY)^2$. [b]p15.[/b] Let $A$ be the answer to problem $13$, and let $B$ be the answer to problem $14$. Totoro hides a guava at point $X$ in a flat field and a mango at point $Y$ different from $X$ such that the length $XY$ is $B$. He wants to hide a papaya at point $Z$ such that $Y Z$ has length $A$ and the distance $ZX$ is a nonnegative integer. In how many different locations can he hide the papaya? [u]Round 6[/u] [b]p16.[/b] Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $AB = 4$, $CD = 8$, $BC = 5$, and $AD = 6$. Given that point $E$ is on segment $CD$ and that $AE$ is parallel to $BC$, find the ratio between the area of trapezoid $ABCD$ and the area of triangle $ABE$. [b]p17.[/b] Find the maximum possible value of the greatest common divisor of $\overline{MOO}$ and $\overline{MOOSE}$, given that $S$, $O$, $M$, and $E$ are some nonzero digits. (The digits $S$, $O$, $M$, and $E$ are not necessarily pairwise distinct.) [b]p18.[/b] Suppose that $125$ politicians sit around a conference table. Each politician either always tells the truth or always lies. (Statements of a liar are never completely true, but can be partially true.) Each politician now claims that the two people beside them are both liars. Suppose that the greatest possible number of liars is $M$ and that the least possible number of liars is $N$. Determine the ordered pair $(M,N)$. [u]Round 7[/u] [b]p19.[/b] Define a [i]lucky [/i] number as a number that only contains $4$s and $7$s in its decimal representation. Find the sum of all three-digit lucky numbers. [b]p20.[/b] Let line segment $AB$ have length $25$ and let points $C$ and $D$ lie on the same side of line $AB$ such that $AC = 15$, $AD = 24$, $BC = 20$, and $BD = 7$. Given that rays $AC$ and $BD$ intersect at point $E$, compute $EA + EB$. [b]p21.[/b] A $3\times 3$ grid is filled with positive integers and has the property that each integer divides both the integer directly above it and directly to the right of it. Given that the number in the top-right corner is $30$, how many distinct grids are possible? [u]Round 8[/u] [b]p22.[/b] Define a sequence of positive integers $s_1, s_2, ... , s_{10}$ to be [i]terrible [/i] if the following conditions are satisfied for any pair of positive integers $i$ and $j$ satisfying $1 \le i < j \le 10$: $\bullet$ $s_i > s_j $ $\bullet$ $j - i + 1$ divides the quantity $s_i + s_{i+1} + ... + s_j$ Determine the minimum possible value of $s_1 + s_2 + ...+ s_{10}$ over all terrible sequences. [b]p23.[/b] The four points $(x, y)$ that satisfy $x = y^2 - 37$ and $y = x^2 - 37$ form a convex quadrilateral in the coordinate plane. Given that the diagonals of this quadrilateral intersect at point $P$, find the coordinates of $P$ as an ordered pair. [b]p24.[/b] Consider a non-empty set of segments of length $1$ in the plane which do not intersect except at their endpoints. (In other words, if point $P$ lies on distinct segments $a$ and $b$, then $P$ is an endpoint of both $a$ and $b$.) This set is called $3$-[i]amazing [/i] if each endpoint of a segment is the endpoint of exactly three segments in the set. Find the smallest possible size of a $3$-amazing set of segments. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2934024p26255963]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].