How many multiples of $12$ divide $12!$ and have exactly $12$ divisors? [i]Proposed by Adam Bertelli[/i]

Robert tiles a $420 \times 420$ square grid completely with $1 \times 2$ blocks, then notices that the two diagonals of the grid pass through a total of $n$ blocks. Find the sum of all possible values of $n$.

Let us say that a pair of distinct positive integers is nice if their arithmetic mean and their geometric mean are both integer. Is it true that for each nice pair there is another nice pair with the same arithmetic mean? (The pairs $(a, b)$ and $(b, a)$ are considered to be the same pair.) [i]Boris Frenkin[/i]

Let the altitude of $\vartriangle ABC$ from $A$ intersect the circumcircle of $\vartriangle ABC$ at $D$. Let $E$ be a point on line $AD$ such that $E \ne A$ and $AD = DE$. If $AB = 13$, $BC = 14$, and $AC = 15$, what is the area of quadrilateral $BDCE$?

For a real parameter $a$, solve the equation $x^4-2ax^2+x+a^2-a=0$. Find all $a$ for which all solutions are real.

In the city of $\textrm{Las Cobayas}$, the houses are arranged in a rectangular grid of $3$ rows and $n\geq 2$ columns, as illustrated in the figure. $\textrm{Mich}$ plans to move there and wants to tour the city to visit some of the houses in a way that he visits at least one house from each column and does not visit the same house more than once. During his tour, $\textrm{Mich}$ can move between adjacent houses, that is, after visiting a house, he can continue his journey by visiting one of the neighboring houses to the north, south, east, or west, which are at most four. The figure illustrates one $\textrm{Mich´s}$ position (circle), and the houses to which he can move (triangles). Let $f(n)$ be the number of ways $\textrm{Mich}$ can complete his tour starting from a house in the first column and ending at a house in the last column. Prove that $f(n)$ is odd. [asy]size(200); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((2,0)--(3,0)--(3,1)--(2,1)--cycle); draw((4,0)--(5,0)--(5,1)--(4,1)--cycle); draw((0,2)--(1,2)--(1,3)--(0,3)--cycle); draw((2,2)--(3,2)--(3,3)--(2,3)--cycle); draw((4,2)--(5,2)--(5,3)--(4,3)--cycle); draw((0,4)--(1,4)--(1,5)--(0,5)--cycle); draw((2,4)--(3,4)--(3,5)--(2,5)--cycle); draw((4,4)--(5,4)--(5,5)--(4,5)--cycle); fill(circle((0.5,2.5), 0.4), black); fill((0.1262,4.15)--(0.8738,4.15)--(0.5,4.7974)--cycle, black); fill((0.1262,0.15)--(0.8738,0.15)--(0.5,0.7974)--cycle, black); fill((2.1262,2.15)--(2.8738,2.15)--(2.5,2.7974)--cycle, black); fill(circle((6,0.5), 0.07), black); fill(circle((6.3,0.5), 0.07), black); fill(circle((6.6,0.5), 0.07), black); fill(circle((6,2.5), 0.07), black); fill(circle((6.3,2.5), 0.07), black); fill(circle((6.6,2.5), 0.07), black); fill(circle((6,4.5), 0.07), black); fill(circle((6.3,4.5), 0.07), black); fill(circle((6.6,4.5), 0.07), black); draw((8,0)--(9,0)--(9,1)--(8,1)--cycle); draw((10,0)--(11,0)--(11,1)--(10,1)--cycle); draw((8,2)--(9,2)--(9,3)--(8,3)--cycle); draw((10,2)--(11,2)--(11,3)--(10,3)--cycle); draw((8,4)--(9,4)--(9,5)--(8,5)--cycle); draw((10,4)--(11,4)--(11,5)--(10,5)--cycle); draw((0,-0.2)--(0,-0.5)--(5.5,-0.5)--(5.5,-0.8)--(5.5,-0.5)--(11,-0.5)--(11,-0.5)--(11,-0.2)); label("$n$", (5.22,-1.15), dir(0), fontsize(10)); label("$\textrm{West}$", (-2,2.5), dir(0), fontsize(10)); label("$\textrm{East}$", (11.1,2.5), dir(0), fontsize(10)); label("$\textrm{North}$", (4.5,5.7), dir(0), fontsize(10)); label("$\textrm{South}$", (4.5,-2), dir(0), fontsize(10)); draw((0.5,2.5)--(2,2.5)--(1.8,2.7)--(2,2.5)--(1.8,2.3)); draw((0.5,2.5)--(0.5,4)--(0.3,3.7)--(0.5,4)--(0.7,3.7)); draw((0.5,2.5)--(0.5,1)--(0.3,1.3)--(0.5,1)--(0.7,1.3)); [/asy]

Find all pairs $(p,q)$ of real numbers such that $p+q=1998$ and the solutions of the equation $x^2+px+q=0$ are integers.

What is the least integer a greater than $14$ so that the triangle with side lengths $a - 1$, $a$, and $a + 1$ has integer area?

a) In the decimal expression of a positive number, $a$, all decimals beginning with the third after the decimal point, are deleted (i.e., we take an approximation of $a$ with accuracy to $0.01$ with deficiency). The number obtained is divided by $a$ and the quotient is similarly approximated with the same accuracy by a number $b$. What numbers $b$ can be thus obtained? Write all their possible values. b) same as (a) but with accuracy to $0.001$ c) same as (a) but with accuracy to $0.0001$

For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)+y)+xf(y)=f(xy+y)+f(x)$$ for reals $x, y$.

There are 7 different numbers on the board, their sum is $10$. For each number on the board, Petya wrote the product of this number and the sum of the remaining 6 numbers in his notebook. It turns out that the notebook only has 4 distinct numbers in it. Determine one of the numbers that is written on the board.

In $\triangle ABC$, $\angle A=55^{\circ}$, $\angle C=75^{\circ}$, $D$ is on side $\overline{AB}$ and $E$ is on side $\overline{BC}$. If $DB=BE$, then $\angle BED=$ [asy] size((100)); draw((0,0)--(10,0)--(8,10)--cycle); draw((4,5)--(9.2,4)); dot((0,0)); dot((10,0)); dot((8,10)); dot((4,5)); dot((9.2,4)); label("A", (0,0), SW); label("B", (8,10), N); label("C", (10,0), SE); label("D", (4,5), NW); label("E", (9.2,4), E); label("$55^{\circ}$", (.5,0), NE); label("$75^{\circ}$", (9.8,0), NW); [/asy] $ \textbf{(A)}\ 50^{\circ} \qquad\textbf{(B)}\ 55^{\circ} \qquad\textbf{(C)}\ 60^{\circ} \qquad\textbf{(D)}\ 65^{\circ} \qquad\textbf{(E)}\ 70^{\circ} $

Consider a $2n\times 2n$ chessboard with all the $4n^2$ cells being white to start with. The following operation is allowed to be performed any number of times: "Three consecutive cells (in a row or column) are recolored (a white cell is colored black and a black cell is colored white)." Find all possible values of $n\ge 2$ for which using the above operation one can obtain the normal chess coloring of the given board.

All ordinary proper irreducible fractions whose numerators are two-digit numbers were ordered in ascending order. Between what two consecutive fractions is the number $\frac58$ located?

In a chess tournament, each player played at most one game against each other, and the number of games played by each player is not less than the set natural number $ n $. Prove that it is possible to divide players into two groups $ A $ and $ B $ in such a way that the number of games played by each player of group $ A $ with players of group $ B $ is not less than $ n/2 $ and at the same time the number of games played by each player of the $ B $ group with players of the $ A $ group was not less than $ n/2 $.

Let $n$ be a positive integer. Find as many as possible zeros as last digits the following expression: $1^n + 2^n + 3^n + 4^n$.

Let a convex polyhedron be given with volume $V$ and full surface $S$. Prove that inside a polyhedron it is possible to arrange a ball of radius $\frac{V}{S}$.

Let $0\leq p_i \leq 1$ for $i=1,2, \dots, n.$ Show that $$\sum_{i=1}^{n} \frac{1}{|x-p_i|} \leq 8n(1+1/3+1/5+\dots +\frac{1}{2n-1})$$ for some $x$ satisfying $0\leq x \leq 1.$

Let $n\geqslant 3$ be an integer. Ion draws a regular $n$-gon and all its diagonals. On every diagonal and edge, Ion writes a positive integer, such that for any triangle formed with the vertices of the $n$-gon, one of the numbers on its edges is the sum of the two other numbers on its edges. Determine the smallest possible number of distinct values that Ion can write.

Find the value of $$\frac{1}{\sqrt{2}^1} + \frac{4}{\sqrt{2}^2} + \frac{9}{\sqrt{2}^3} + \cdots$$

Find all primes $p$ such that for any integer $k$, there exist two integers $x$ and $y$ such that $$x^3+2023xy+y^3 \equiv k \pmod p$$ [i]Proposed by Tristan Chaang Tze Shen[/i]

In a room, there are $2005$ boxes, each of them containing one or several sorts of fruits, and of course an integer amount of each fruit. [b]a)[/b] Show that we can find $669$ boxes, which altogether contain at least a third of all apples and at least a third of all bananas. [b]b)[/b] Can we always find $669$ boxes, which altogether contain at least a third of all apples, at least a third of all bananas and at least a third of all pears?

Find the total number of primes $p<100$ such that $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.

Let $n$ be a integer and $n \ge 3$, and $T_1T_2...T_n$ is a regular n-gon. Distinct $3$ points $T_i , T_j , T_k$ are chosen randomly. Determine the probability of triangle $T_iT_jT_k$ being an acute triangle.