Find all positive integers $k > 1$ for which there exists a positive integer $n$ such that $\tbinom{n}{k}$ is divisible by $n$, and $\tbinom{n}{m}$ is not divisible by $n$ for $2\leq m < k$. [i]Merlijn Staps[/i]

All the points in the plane are colored using three colors.Prove that there exists a triangle with vertices having the same color such that [i]either[/i] it is isosceles [i]or[/i] its angles are in geometric progression.

Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations: \[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\] Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$

Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$, we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.

Consider a polynomial with coefficients of real numbers $ \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3\plus{}9a^2d\minus{}7abc \le 0.\] [i]Hery Susanto, Malang[/i]

Suppose that $p$ and $q$ are positive numbers for which \[ \log_{9}(p) = \log_{12}(q) = \log_{16}(p+q) \] What is the value of $\frac{q}{p}$? $\textbf{(A)}\ \frac{4}{3}\qquad\textbf{(B)}\ \frac{1+\sqrt{3}}{2}\qquad\textbf{(C)}\ \frac{8}{5}\qquad\textbf{(D)}\ \frac{1+\sqrt{5}}{2}\qquad\textbf{(E)}\ \frac{16}{9} $

a) A positive integer is called [i]nice [/i] if it can be represented as an arithmetic mean of some (not necessarily distinct) positive integers each being a nonnegative power of $2$. Prove that all positive integers are nice. b) A positive integer is called [i]ugly [/i] if it can not be represented as an arithmetic mean of some pairwise distinct positive integers each being a nonnegative power of $2$. Prove that there exist infinitely many ugly positive integers. (A. Romanenko, D. Zmeikov)

A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where $[x]$ denotes the smallest integer $\leq x)$

For each real number $x$, let $\lfloor x \rfloor$ denote the largest integer not exceeding $x$. A sequence $\{a_n \}_{n=1}^{\infty}$ is defined by $a_n = \frac{1}{4^{\lfloor -\log_4 n \rfloor}}, \forall n \geq 1.$ Let $b_n = \frac{1}{n^2} \left( \sum_{k=1}^n a_k - \frac{1}{a_1+a_2} \right), \forall n \geq 1.$ a) Find a polynomial $P(x)$ with real coefficients such that $b_n = P \left( \frac{a_n}{n} \right), \forall n \geq 1$. b) Prove that there exists a strictly increasing sequence $\{n_k \}_{k=1}^{\infty}$ of positive integers such that $$\lim_{k \to \infty} b_{n_k} = \frac{2024}{2025}.$$

Let $n\ge3$ be a positive integer. Each edge of a complete graph $K_n$ is assigned a real number satisfying the following conditions: $(i)$ For any three vertices, the numbers assigned to two of the edges among them are equal, and the number on the third edge is strictly greater. $(ii) $ The weight of a vertex is defined as the sum of the numbers assigned to the edges emanating from that vertex. The weights of all vertices are equal. Find all possible values of $n$.

The line passing through the center $I$ of the inscribed circle of a triangle $ABC$, perpendicular to $AI$ and intersects sides $AB$ and $AC$ at points $C'$ and $B'$, respectively. In the triangles $BC'I$ and $CB'I$, the altitudes $C'C_1$ and $B'B_1$ were drawn, respectively. Prove that the midpoint of the segment $B_1C_1$ lies on a straight line passing through point $I$ and perpendicular to $BC$.

Each of three given circles with radii $1$, $r$ and $r$ touches the others from the outside. For what values of $r$ does there exist a triangle “circumscribed” to these circles? (This means the circles lie inside the triangle, each circle touching two sides of the triangle and each side of the triangle touching two circles.) (N.B. Vasiliev, Moscow)

Let $n\geq2$ be a positive integer. Determine all possible values of the sum \[ S=\left\lfloor x_{2}-x_{1}\right\rfloor +\left\lfloor x_{3}-x_{2}\right\rfloor+...+\left\lfloor x_{n}-x_{n-1}\right\rfloor \] where $x_i\in \mathbb{R}$ satisfying $\lfloor{x_i}\rfloor=i$ for $i=1,2,\ldots n$.

Set $S$ contains exactly $36$ elements in the form of $2^m \cdot 5^n$ for integers $ 0 \le m,n \le 5$. Two distinct elements of $S$ are randomly chosen. Given that the probability that their product is divisible by $10^7$ is $a/b$, where $a$ and $b$ are relatively prime positive integers, find $a +b$. [i]Proposed by Ada Tsui[/i]

Let \(ABCD\) be a convex quadrilateral in the plane and let \(O_{A}, O_{B}, O_{C}\) and \(O_{D}\) be the circumcenters of the triangles \(BCD, CDA, DAB\) and \(ABC\), respectively. Suppose these four circumcenters are distinct points. Prove that these points are not on a same circle.

Prove that the arithmetic mean $a$ of $x_1,\ldots ,x_n$ satisfies \[ (x_1-a)^2+\ldots +(x_n-a)^2\le \frac{1}{2}(|x_1-a|+\ldots +|x_n-a|)^2\]

Find all positive integers $m$ and $n$ for which \[1!+2!+3!+\cdots+n!=m^{2}\]

Show that the sum of the first $n$ terms in the binomial expansion of $(2-1)^{-n}$ is $\frac{1}{2},$ where $n$ is a positive integer.

Let $n_{1}, \cdots, n_{k}$ and $a$ be positive integers which satify the following conditions:[list][*] for any $i \neq j$, $(n_{i}, n_{j})=1$, [*] for any $i$, $a^{n_{i}} \equiv 1 \pmod{n_i}$, [*] for any $i$, $n_{i}$ does not divide $a-1$. [/list] Show that there exist at least $2^{k+1}-2$ integers $x>1$ with $a^{x} \equiv 1 \pmod{x}$.

The base of a pyramid is a regular triangle having side of size $1$. Two of three angles at the vertex of the pyramid are right. Find the maximum value of the volume of the pyramid.

I define a "good day" as a day when both the day and the month evenly divide the concatenation of the two. For example, today (March $15$) is a good day since $3$ and $15$ both divide $315.$ However, March $9$ is not a good day since $9$ does not divide $39.$ How many good days are in March, April, and May combined?

Let $N=\overline{abc}$ be a three-digit number. It is known that we can construct an isoceles triangle with $a,b$ and $c$ as the length of sides. Determine how many possible three-digit number $N$ there are. ($N=\overline{abc}$ means that $a,b$ and $c$ are digits of $N$, and not $N=a\times b\times c$.)

Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

A terrain ( $ABCD$ ) has a rectangular trapezoidal shape. The angle in $A$ measures $90^o$. $AB$ measures $30$ m, $AD$ measures $20$ m and $DC$ measures 45 m. This land must be divided into two areas of the same area, drawing a parallel to the $AD$ side . At what distance from $D$ do we have to draw the parallel? [img]https://1.bp.blogspot.com/-DnyNY3x4XKE/XNYvRUrLVTI/AAAAAAAAKLE/gohd7_S9OeIi-CVUVw-iM63uXE5u-WmGwCK4BGAYYCw/s400/image002.gif[/img]

A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral? [asy] unitsize(1.5cm); defaultpen(.8); pair A = (0,0), B = (3,0), C = (1.4, 2), D = B + 0.4*(C-B), Ep = A + 0.3*(C-A); pair F = intersectionpoint( A--D, B--Ep ); draw( A -- B -- C -- cycle ); draw( A -- D ); draw( B -- Ep ); filldraw( D -- F -- Ep -- C -- cycle, mediumgray, black ); label("$7$",(1.25,0.2)); label("$7$",(2.2,0.45)); label("$3$",(0.45,0.35));[/asy] $ \textbf{(A) }15\qquad\textbf{(B) }17\qquad\textbf{(C) }\frac{35}{2}\qquad\textbf{(D) }18\qquad\textbf{(E) }\frac{55}{3} $