What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation? $ \textbf{(A)}\ 760 \qquad\textbf{(B)}\ 756 \qquad\textbf{(C)}\ 755 \qquad\textbf{(D)}\ 752 \qquad\textbf{(E)}\ 750 $

We call a composite positive integer $n$ nice if it is possible to arrange its factors that are larger than $1$ on a circle such that two neighboring numbers are not coprime. How many of the elements of the set $\{1, 2, 3, ..., 100\}$ are nice?

Angles of a given triangle $ABC$ are all smaller than $120^\circ$. Equilateral triangles $AFB, BDC$ and $CEA$ are constructed in the exterior of $ABC$. (a) Prove that the lines $AD, BE$, and $CF$ pass through one point $S.$ (b) Prove that $SD + SE + SF = 2(SA + SB + SC).$

Prove that for all real numbers $a,b,c$ the inequality $(a^2+b^2-c^2)(b^2+c^2-a^2)(c^2+a^2-b^2) \leq (a+b-c)^2(b+c-a)^2(c+a-b)^2$ holds.

Solve the equation \[ x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x}\] where $ a$ is a real parameter.

If $x, y, z$ satisfy $x+y+z = 12, \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 2$ and $x^3+y^3+z^3 = -480,$ find $$x^2 y + xy^2 + x^2 z + xz^2 + y^2 z + yz^2.$$ [i]Proposed by Mahith Gottipati[/i]

Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students that could have been standing before her third wave? $\textbf{(A) }23\qquad\textbf{(B) }27\qquad\textbf{(C) }29\qquad\textbf{(D) }31\qquad\textbf{(E) }33$

A polynomial $P(x,y)$ with integer coefficients satisfies two following conditions: 1. for every integer $a$ there exists exactly one integer $y$, such that $P(a,y)=0$ 2. for every integer $b$ there exists exactly one integer $x$, such that $P(x,b)=0$ a) Prove that if the degree of $P$ is $2$, then it is divisible by either $x-y+C$ for some integer $C$, or $x+y+C$ for some integer $C$ b) Is there a polynomial $P$ that isn't divisible by any of $x-y+C$ or $x+y+C$ for integers $C$?

The check for a luncheon of 3 sandwiches, 7 cups of coffee and one piece of pie came to $ \$3.15$. The check for a luncheon consisting of 4 sandwiches, 10 cups of coffee and one piece of pie came to $ \$4.20$ at the same place. The cost of a luncheon consisting of one sandwich, one cup of coffee, and one piece of pie at the same place will come to $ \textbf{(A)}\ \$1.70 \qquad \textbf{(B)}\ \$1.65 \qquad \textbf{(C)}\ \$1.20 \qquad \textbf{(D)}\ \$1.05 \qquad \textbf{(E)}\ \$0.95$

An ant begins at a vertex of a convex regular icosahedron (a figure with 20 triangular faces and 12 vertices). The ant moves along one edge at a time. Each time the ant reaches a vertex, it randomly chooses to next walk along any of the edges extending from that vertex (including the edge it just arrived from). Find the probability that after walking along exactly six (not necessarily distinct) edges, the ant finds itself at its starting vertex.

[b]p1.[/b] In a given tetrahedron the sum of the measures of the three face angles at each of the vertices is $180$ degrees. Prove that all faces of the tetrahedron are congruent triangles. [img]https://cdn.artofproblemsolving.com/attachments/c/c/40f03324fd19f6a5e0a5e541153a2b38faac79.png[/img] [b]p2.[/b] The digital sum $D(n)$ of a positive integer $n$ is defined recursively by: $D(n) = n$ if $1 \le n \le 9$ $D(n) = D(a_0 + a_1 + ... + a_m)$ if $n>9$ where $a_0 , a_1 ,..,a_m$ are all the digits of $n$ expressed in base ten. (For example, $D(959) = D(26) = D(8) = 8$.) Prove that $D(n \times 1234)= D(n)$ fcr all positive integers $n$ . [b]p3.[/b] A right triangle has area $A$ and perimeter $P$ . Find the largest possible value for the positive constant $k$ such that for every such triangle, $P^2 \ge kA$ . [b]p4.[/b] In the accompanying diagram, $\overline{AB}$ is tangent at $A$ to a circle of radius $1$ centered at $O$ . The segment $\overline{AP}$ is equal in length to the arc $AB$ . Let $C$ be the point of intersection of the lines $AO$ and $PB$ . Determine the length of segment $\overline{AC}$ in terms of $a$ , where $a$ is the measure of $\angle AOB$ in radians. [img]https://cdn.artofproblemsolving.com/attachments/e/0/596e269a89a896365b405af7bc6ca47a1f7c57.png[/img] [b]p5.[/b] Let $a_1 = a > 0$ and $a_2 = b >a$. Consider the sequence $\{a_1,a_2,a_3,...\}$ of positive numbers defined by: $a_3=\sqrt{a_1a_2}$, $a_4=\sqrt{a_2a_3}$, $...$ and in general, $a_n=\sqrt{a_{n-2}a_{n-1}}$, for $n\ge 3$ . Develop a formula $a_n$ expressing in terms of $a$, $b$ and $n$ , and determine $\lim_{n \to \infty} a_n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Josh and Mike live 13 miles apart. Yesterday, Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?

Prove that in each polyhedron there exist two faces with the same number of edges.

Find the sum of all positive integers $n$ such that $7<n < 100$ and $1573_{n}$ has $6$ factors when written in base $10$. [i]Proposed by Aidan Duncan[/i]

The ratio of the sides of a parallelogram is $\lambda>1$. Given $\lambda$, determine the maximum of the acute angle subtended by the diagonals of the parallelogram.