Consider the $4 \times 4$ “multiplication table” below. The numbers in the first column multiplied by the numbers in the first row give the remaining numbers in the table. For example, the $3$ in the first column times the $4$ in the first row give the $12 (= 3 \cdot 4)$ in the cell that is in the 3rd row and 4th column. [asy] size(3cm); for (int x=0; x<=4; ++x) draw((x, 0) -- (x, 4), linewidth(.5pt)); for (int y=0; y<=4; ++y) draw((0, y) -- (4, y), linewidth(.5pt)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); void foo(int x, int y, string n) { label(n, (x+0.5, y+0.5)); } foo(0, 3, "1"); foo(1, 3, "2"); foo(2, 3, "3"); foo(3, 3, "4"); foo(0, 2, "2"); foo(1, 2, "4"); foo(2, 2, "6"); foo(3, 2, "8"); foo(0, 1, "3"); foo(1, 1, "6"); foo(2, 1, "9"); foo(3, 1, "12"); foo(0, 0, "4"); foo(1, 0, "8"); foo(2, 0, "12"); foo(3, 0, "16"); [/asy] We create a path from the upper-left square to the lower-right square by always moving one cell either to the right or down. For example, here is one such possible path, with all the numbers along the path circled: [asy] import graph; size(3cm); for (int x=0; x<=4; ++x) draw((x, 0) -- (x, 4), linewidth(.5pt)); for (int y=0; y<=4; ++y) draw((0, y) -- (4, y), linewidth(.5pt)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); void foo(int x, int y, string n) { label(n, (x+0.5, y+0.5)); } draw(Circle((0.5,3.5),0.5)); draw(Circle((1.5,3.5),0.5)); draw(Circle((2.5,3.5),0.5)); draw(Circle((2.5,2.5),0.5)); draw(Circle((3.5,2.5),0.5)); draw(Circle((3.5,1.5),0.5)); draw(Circle((3.5,0.5),0.5)); foo(0, 3, "1"); foo(1, 3, "2"); foo(2, 3, "3"); foo(3, 3, "4"); foo(0, 2, "2"); foo(1, 2, "4"); foo(2, 2, "6"); foo(3, 2, "8"); foo(0, 1, "3"); foo(1, 1, "6"); foo(2, 1, "9"); foo(3, 1, "12"); foo(0, 0, "4"); foo(1, 0, "8"); foo(2, 0, "12"); foo(3, 0, "16"); [/asy] If we add up the circled numbers in the example above (including the start and end squares), we get $48$. Considering all such possible paths: (a) What is the smallest sum we can possibly get when we add up the numbers along such a path? Prove your answer is correct. (b) What is the largest sum we can possibly get when we add up the numbers along such a path? Prove your answer is correct.

Prove for every integer $n>0$, there exists an integer $k>0$ such that $2^nk$ can be written in decimal notation using only digits 1 and 2.

Light of a blue laser (wavelength $\lambda=475 \, \text{nm}$) goes through a narrow slit which has width $d$. After the light emerges from the slit, it is visible on a screen that is $ \text {2.013 m} $ away from the slit. The distance between the center of the screen and the first minimum band is $ \text {765 mm} $. Find the width of the slit $d$, in nanometers. [i](Proposed by Ahaan Rungta)[/i]

Let $S$ be the set of lattice points $(x,y) \in \mathbb{Z}^2$ such that $-10\leq x,y \leq 10$. Let the point $(0,0)$ be $O$. Let Scotty the Dog's position be point $P$, where initially $P=(0,1)$. At every second, consider all pairs of points $C,D \in S$ such that neither $C$ nor $D$ lies on line $OP$, and the area of quadrilateral $OCPD$ (with the points going clockwise in that order) is $1$. Scotty finds the pair $C,D$ maximizing the sum of the $y$ coordinates of $C$ and $D$, and randomly jumps to one of them, setting that as the new point $P$. After $50$ such moves, Scotty ends up at point $(1, 1)$. Find the probability that he never returned to the point $(0,1)$ during these $50$ moves. [i]Proposed by David Tang[/i]

Let $n \ge 2$ be an integer. Prove that if $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq \sqrt{\frac{n}{3}}$, then $k^2 +k + n$ is prime for all integers $k$ such that $0 \leq k \leq n - 2$.

Determine the smallest possible value of $| A_{1} \cup A_{2} \cup A_{3} \cup A_{4} \cup A_{5} |$, where $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$ sets simultaneously satisfying the following conditions: $(i)$ $| A_{i}\cap A_{j} | = 1$ for all $1\leq i < j\leq 5$, i.e. any two distinct sets contain exactly one element in common; $(ii)$ $A_{i}\cap A_{j} \cap A_{k}\cap A_{l} =\varnothing$ for all $1\leq i<j<k<l\leq 5$, i.e. any four different sets contain no common element. Where $| S |$ means the number of elements of $S$.

Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$ [i]Proposed by Md. Fuad Al Alam[/i]

A sequence $a_n$ is defined by $a_0 = 0$, and for all $n \ge 1$, $a_n = a_{n-1} + (-1)^n \cdot n^2$. Compute $a_{100}$

Find the formula to express the number of $ n\minus{}$series of letters which contain an even number of vocals (A,I,U,E,O).

Given an acute triangle $ABC$. Let $A'$ be a point symmetric to $A$ with respect to $BC, O_A$ is the center of the circle passing through $A$ and the midpoints of the segments $A'B$ and $A'C. O_B$ and $O_C$ points are defined similarly. Find the ratio of the radii of the circles circumscribed around the triangles $ABC$ and $O_AO_BO_C$.

Suppose $2016$ points of the circumference of a circle are colored red and the remaining points are colored blue . Given any natural number $n\ge 3$, prove that there is a regular $n$-sided polygon all of whose vertices are blue.

Let $(x,y)$ be the coordinates of a point chosen uniformly at random within the unit square with vertices at $(0,0), (0,1), (1,0),$ and $(1,1).$ The probability that $|x - \tfrac{1}{2}| + |y - \tfrac{1}{2}| < \tfrac{1}{2}$ is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p + q.$

Prove that the equation \[ x^{2008}\plus{} 2008!\equal{} 21^{y}\] doesn't have solutions in integers.

Solve the system of equations: $ y^2\equal{}(x\plus{}8)(x^2\plus{}2),$ $ y^2\minus{}(8\plus{}4x)y\plus{}(16\plus{}16x\minus{}5x^2)\equal{}0.$

For how many positive integers $ m$ does there exist at least one positive integer $ n$ such that $ m\cdot n \le m \plus{} n$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}$ infinitely many

Consider a triangle $ABC$ and $I$ its incenter. The line $(AI)$ meets the circumcircle of $ABC$ in $D$. Let $E$ and $F$ be the orthogonal projections of $I$ on $(BD)$ and $(CD)$ respectively. Assume that $IE+IF=\frac{1}{2}AD$. Calculate $\angle{BAC}$. [color=red][Moderator edited: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=5088 .][/color]

Several positive integers $a_0 , a_1 , a_2 , ... , a_n$ are written on a board. On a second board, we write the amount $b_0$ of numbers written on the first board, the amount $b_1$ of numbers on the first board exceeding $1$, the amount $b_2$ of numbers greater than $2$, and so on as long as the $b$s are still positive. Then we stop, so that we do not write any zeros. On a third board we write the numbers $c_0 , c_1 , c_2 , ...$. using the same rules as before, but applied to the numbers $b_0 , b_1 , b_2 , ...$ of the second board. Prove that the same numbers are written on the first and the third boards. (H. Lebesgue - A Kanel)

Let $ABC$ be a triangle and $P$ a point on the side $BC$. Let $S_1$ be the circumference with center $B$ and radius $BP$ that cuts the side $AB$ at $D$ such that $D$ lies between $A$ and $B$. Let $S_2$ be the circumference with center $C$ and radius $CP$ that cuts the side $AC$ at $E$ such that $E$ lies between $A$ and $C$. Line $AP$ cuts $S_1$ and $S_2$ at $X$ and $Y$ different from $P$, respectively. We call $T$ the point of intersection of $DX$ and $EY$. Prove that $\angle BAC+ 2 \angle DTE=180$

Let $\ell$ be a line with negative slope passing through the point $(20,16)$. What is the minimum possible area of a triangle that is bounded by the $x$-axis, $y$-axis, and $\ell$? [i] Proposed by James Lin [/i]

Find real roots $x_1$, $x_2$ of equation $x^5-55x+21=0$, if we know $x_1\cdot x_2=1$

[b]6.[/b] Let $f(x)$ be an arbitrary function, differentiable infinitely many times. Then the $n$th derivative of $f(e^{x})$ has the form $\frac{d^{n}}{dx^{n}}f(e^{x})= \sum_{k=0}^{n} a_{kn}e^{kx}f^{(k)}(e^{x})$ ($n=0,1,2,\dots$). From the coefficients $a_{kn}$ compose the sequence of polynomials $P_{n}(x)= \sum_{k=0}^{n} a_{kn}x^{k}$ ($n=0,1,2,\dots$) and find a closed form for the function $F(t,x) = \sum_{n=0}^{\infty} \frac{P_{n}(x)}{n!}t^{n}.$ [b](S. 22)[/b]

Let $ABCD$ be a trapezium with $AD||BC$; and let $X$ and $Y$ be the midpoints of $AC$ and $BD$ respectively. Prove that if $\angle DAY=\angle CAB$ then the internal bisectors of $\angle XAY$ and $\angle XBY$ meet on $XY$. Proposed by Belur Jana Venkatachala

Determine all composite integers $n>1$ that satisfy the following property: if $d_1$, $d_2$, $\ldots$, $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$.

Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.

The median $ BM $ of a triangle $ ABC $ intersects the circumscribed circle at point $ K $. The circumcircle of the triangle $ KMC $ intersects the segment $ BC $ at point $ P $, and the circumcircle of $ AMK $ intersects the extension of $ BA $ at $ Q $. Prove that $ PQ> AC $.