Prove that there exist nonconstant polynomials $f, g$ with integer coefficients, such that for infinitely many primes $p$, $p \nmid f(x)-g(y)$ for any integers $x, y$.

In a language$,$ an alphabet with $25$ letters is used$;$ words are exactly all sequences of $($ not necessarily different $)$ letters of length $17.$ Two ends of a paper strip are glued so that the strip forms a ring$;$ the strip bears a sequence of $5^{18}$ letters$.$ Say that a word is singular if one can cut a piece bearing exactly that word from the strip$,$ but one cannot cut out two such non-overlapping pieces$.$ It is known that one can cut out $5^{16}$ non-overlapping pieces each containing the same word$.$ Determine the largest possible number of singular words$.$ [i](Bogdanov I.)[/i]

Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that $$k\le2\lfloor\sqrt n\rfloor.$$

Let $n$ be an integer number greater than $2$, let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that \[\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1\] and let $k$ be a real number greater than $1$. Show that: \[\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}\] and determine the cases of equality.

Find all non-negative integers $n$, $a$, and $b$ satisfying \[2^a + 5^b + 1 = n!.\]

Let be a function $ f:(0,\infty )\longrightarrow\mathbb{R} $ satisfying the following two properties: $ \text{(i) } 2\lfloor x \rfloor \le f(x) \le 2 \lfloor x \rfloor +2,\quad\forall x\in (0,\infty ) $ $ \text{(ii) } f\circ f $ is monotone Can $ f $ be non-monotone? Justify.

Let $a, b, c$ be positive real numbers such that $$a^2 + b^2 + c^2 = \frac{1}{4}.$$ Prove that $$\frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}} + \frac{1}{\sqrt{a^2 + b^2}} \le \frac{\sqrt{2}}{(a + b)(b + c)(c + a)}.$$ [i]Proposed by Petar Filipovski, Macedonia[/i]

Two right triangles, $T_1$ and $T_2$, have areas of 1 and 2, respectively. One side length of one triangle is congruent to a different side length in the other, and another side length of the first triangle is congruent to yet another side length in the other. What is the square of the product of the third side lengths of $T_1$ and $T_2$? $\textbf{(A) }\frac{28}3\qquad\textbf{(B) }10\qquad\textbf{(C) }\frac{32}3\qquad\textbf{(D) }\frac{34}3\qquad\textbf{(E) }12$

Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying (i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$. (ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.

In a national park there is a group of sequoia trees, all of which have a positive integer age. Their average age is $41$ years. After a $2010$ year old building was destroyed by lightning, the average age drops to $40$ years. How many trees were originally in the group? At most, how many of them were exactly $2010$ years old? (W. Janous, WRG Ursulinen, Innsbruck)

Let $AD, BE$, and $CF$ denote the altitudes of triangle $\vartriangle ABC$. Points $E'$ and $F'$ are the reflections of $E$ and $F$ over $AD$, respectively. The lines $BF'$ and $CE'$ intersect at $X$, while the lines $BE'$ and $CF'$ intersect at the point $Y$. Prove that if $H$ is the orthocenter of $\vartriangle ABC$, then the lines $AX, YH$, and $BC$ are concurrent.

[b]p1.[/b] (See the diagram below.) $ABCD$ is a square. Points $G$, $H$, $I$, and $J$ are chosen in the interior of $ABCD$ so that: (i) $H$ is on $\overline{AG}$, $I$ is on $\overline{BH}$, $J$ is on $\overline{CI}$, and $G$ is on $\overline{DJ}$ (ii) $\vartriangle ABH \sim \vartriangle BCI \sim \vartriangle CDJ \sim \vartriangle DAG$ and (iii) the radii of the inscribed circles of $\vartriangle ABH$, $\vartriangle BCI$, $\vartriangle CDJ$, $\vartriangle DAK$, and $GHIJ$ are all the same. What is the ratio of $\overline{AB}$ to $\overline{GH}$? [img]https://cdn.artofproblemsolving.com/attachments/f/b/47e8b9c1288874bc48462605ecd06ddf0f251d.png[/img] [b]p2.[/b] The three solutions $r_1$, $r_2$, and $r_3$ of the equation $$x^3 + x^2 - 2x - 1 = 0$$ can be written in the form $2 \cos (k_1 \pi)$, $2 \cos (k_2 \pi)$, and $2 \cos (k_3 \pi)$ where $0 \le k_1 < k_2 < k_3 \le 1$. What is the ordered triple $(k_1, k_2, k_3)$? [b]p3.[/b] $P$ is a convex polyhedron, all of whose faces are either triangles or decagons ($10$-sided polygon), though not necessarily regular. Furthermore, at each vertex of $P$ exactly three faces meet. If $P$ has $20$ triangular faces, how many decagonal faces does P have? [b]p4.[/b] $P_1$ is a parabola whose line of symmetry is parallel to the $x$-axis, has $(0, 1)$ as its vertex, and passes through $(2, 2)$. $P_2$ is a parabola whose line of symmetry is parallel to the $y$-axis, has $(1, 0)$ as its vertex, and passes through $(2, 2)$. Find all four points of intersection between $P_1$ and $P_2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for all odd $ n > 1$, we have $ 8n \plus{} 4|C^{4n}_{2n}$.

Let $ N$ be the number of ordered pairs of nonempty sets $ \mathcal{A}$ and $ \mathcal{B}$ that have the following properties: • $ \mathcal{A} \cup \mathcal{B} \equal{} \{1,2,3,4,5,6,7,8,9,10,11,12\}$, • $ \mathcal{A} \cap \mathcal{B} \equal{} \emptyset$, • The number of elements of $ \mathcal{A}$ is not an element of $ \mathcal{A}$, • The number of elements of $ \mathcal{B}$ is not an element of $ \mathcal{B}$. Find $ N$.

How many ordered pairs $(a, b)$ such that $a$ is a real positive number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017} = \log_b (a^{2017})$? $ \textbf{(A) \ }198\qquad \textbf{(B) \ } 199 \qquad \textbf{(C) \ } 398 \qquad \textbf{(D) \ }399\qquad \textbf{(E) \ } 597$

What is the sum of all the integers $n$ such that $\left|n-1\right|<\pi$? [i]2016 CCA Math Bonanza Lightning #1.1[/i]

Find the maximum number of colors used in coloring integers $n$ from $49$ to $94$ such that if $a, b$ (not necessarily different) have the same color but $c$ has a different color, then $c$ does not divide $a+b$.

Suppose $n\geq 4$ and let $S$ be a finite set of points in the space ($\mathbb{R}^3$), no four of which lie in a plane. Assume that the points in $S$ can be colored with red and blue such that any sphere which intersects $S$ in at least 4 points has the property that exactly half of the points in the intersection of $S$ and the sphere are blue. Prove that all the points of $S$ lie on a sphere.

The $260$ volumes of the [i]Encyclopedia Galactica[/i] are out of order in the library. Fortunately for the librarian, the books are numbered. Due to his religion, which holds both encyclopedias and the concept of parity in high esteem, the librarian can only rearrange the books two at a time, and then only by switching the position of an even numbered volume with that of an odd numbered volume. Find the minimum number of such transpositions sufficient to get the books back into ordinary sequential order, regardless of the starting positions of the books. (Find the minimum number of transpositions in the worst-case scenario.)

In a Greek village of $100$ inhabitants in the beginning there lived $12$ Olympians and $88$ humans, they were the first generation. The Olympians are $100\%$ gods while humans are $0\%$ gods. In each generation they formed $50$ couples and each couple had $2$ children, who form the next generation (also consisting of $100$ members). From the second generation onwards, every person’s percentage of godness is the average of the percentages of his/her parents’ percentages. (For example the children of $25\%$ and $12.5\% $gods are $18.75\%$ gods.) a) Which is the earliest generation in which it is possible that there are equally many $100\%$ gods as $ 0\%$ gods? b) At most how many members of the fifth generation are at least 25% gods?

A man walks $x$ miles due west, turns $150^\circ$ to his left and walks 3 miles in the new direction. If he finishes a a point $\sqrt{3}$ from his starting point, then $x$ is $\text{(A)} \ \sqrt 3 \qquad \text{(B)} \ 2\sqrt{5} \qquad \text{(C)} \ \frac 32 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$

Given a circle and a chord $AB$, different from the diameter. Point $C$ moves along the large arc $AB$. The circle passing through passing through points $A, C$ and point $H$ of intersection of altitudes of of the triangle $ABC$, re-intersects the line $BC$ at point $P$. Prove that line $PH$ passes through a fixed point independent of the position of point $C$.

A nonempty set $S$ is called $\emph{well-filled}$ if for every $m \in S$, there are fewer than $\frac 12 m$ elements of $S$ which are less than $m$. Determine the number of well-filled subsets of $\left\{ 1,2,\dots,42 \right\}$.

Let $a,b,c$ be real numbers with $0 < a< 1$, $0 < b < 1$, $0 < c < 1$, and $a+b + c = 2$. Prove that $\dfrac{a}{1-a} \cdot \dfrac{b}{1-b} \cdot \dfrac{c}{1-c} \geq 8$.

For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s. (a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$. (b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.