Two sums, each consisting of $n$ addends , are shown below: $S = 1 + 2 + 3 + 4 + ...$ $T = 100 + 98 + 96 + 94 +...$ . For what value of $n$ is it true that $S = T$ ?

Show that the number $ 3^{{4}^{5}} \plus{} 4^{{5}^{6}}$ can be expresed as the product of two integers greater than $ 10^{2009}$

The sum of the numerical coefficients in the complete expansion of $ (x^2 \minus{} 2xy \plus{} y^2)^7$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 128 \qquad \textbf{(E)}\ 128^2$

Consider the sequence $(x_n)_{n\in\mathbb{N^*}}$ such that $$x_0=0,\quad x_1=2024,\quad x_n=x_{n-1}+x_{n-2}, \forall n\geq2.$$ Prove that there is an infinity of terms in this sequence that end with $2024.$

Show that for every convex $ n$-gon $ ( n \ge 4)$, the arithmetic mean of the lengths of its sides is less than the arithmetic mean of the lengths of all its diagonals.

Let $p_{1}, p_{2}, \cdots, p_{n}$ be distinct primes greater than $3$. Show that \[2^{p_{1}p_{2}\cdots p_{n}}+1\] has at least $4^{n}$ divisors.

Let $N$ denote the number of ordered pairs of sets $(A, B)$ such that $A \cup B$ is a size-$999$ subset of $\{1,2,\dots,1997\}$ and $(A \cap B) \cap \{1,2\} = \{1\}$. If $m$ and $k$ are integers such that $3^m5^k$ divides $N$, compute the the largest possible value of $m+k$. [i]Proposed by Michael Tang[/i]

Seller reduced price of one shirt for $20\%$,and they raised it by $10\%$. Does he needs to reduce or raise the price and how many, so that price of shirt will be reduced by $10\%$ from the original price

Let $S_n=\int_0^{\pi} \sin ^ n x\ dx\ (n=1,\ 2,\ ,\ \cdots).$ Find $\lim_{n\to\infty} nS_nS_{n+1}.$

If $x,y,z$ are real numbers and \begin{align*} 2x+y+z\leq66\\ x+2y+z\leq60\\ x+y+2z\leq70\\ x+2y+3z\leq110\\ 3x+y+2z\leq98\\ 2x+3y+z\leq89\\ \end{align*} What is the maximum possible value of $x+y+z$?

It is given positive real number $a$ such that: $$\left\{\frac{1}{a}\right\}=\{a^2\}$$ $$ 2<a^2<3$$ Find the value of $$a^{12}-\frac{144}{a}$$

How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$? $\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad$

Let $n$ be a nonnegative integer and $M =\{n^3, n^3+1, n^3+2, ..., n^3+n\}$. Consider $A$ and $B$ two nonempty, disjoint subsets of $M$ such that the sum of elements of the set $A$ divides the sum of elements of the set $B$. Prove that the number of elements of the set $A$ divides the number of elements of the set $B$.

Let $x,y$ be positive integers such that $3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect square.

Find all pairs of natural numbers $(m,n)$, for which $m\mid 2^{\varphi(n)} +1$ and $n\mid 2^{\varphi (m)} +1$.

Let $n \in \mathbb{N}$ such that $p=17^{2n}+4$ is a prime. Show \begin{align*} p \mid 7^{\tfrac{p-1}{2}} +1 \end{align*}

A quadrilateral is circumscribed around a circle. Prove that the straight lines connecting neighboring tangent points either meet on the extension of a diagonal of the quadrilateral or are parallel to it.

Let $ ABC$ be a triangle with $ AB \equal{} 5$, $ BC \equal{} 4$ and $ AC \equal{} 3$. Let $ \mathcal P$ and $ \mathcal Q$ be squares inside $ ABC$ with disjoint interiors such that they both have one side lying on $ AB$. Also, the two squares each have an edge lying on a common line perpendicular to $ AB$, and $ \mathcal P$ has one vertex on $ AC$ and $ \mathcal Q$ has one vertex on $ BC$. Determine the minimum value of the sum of the areas of the two squares. [asy]import olympiad; import math; import graph; unitsize(1.5cm); pair A, B, C; A = origin; B = A + 5 * right; C = (9/5, 12/5); pair X = .7 * A + .3 * B; pair Xa = X + dir(135); pair Xb = X + dir(45); pair Ya = extension(X, Xa, A, C); pair Yb = extension(X, Xb, B, C); pair Oa = (X + Ya)/2; pair Ob = (X + Yb)/2; pair Ya1 = (X.x, Ya.y); pair Ya2 = (Ya.x, X.y); pair Yb1 = (Yb.x, X.y); pair Yb2 = (X.x, Yb.y); draw(A--B--C--cycle); draw(Ya--Ya1--X--Ya2--cycle); draw(Yb--Yb1--X--Yb2--cycle); label("$A$", A, W); label("$B$", B, E); label("$C$", C, N); label("$\mathcal P$", Oa, origin); label("$\mathcal Q$", Ob, origin);[/asy]

Consider the sequence $a_1, a_2, a_3, ...$ defined by $a_1 = 9$ and $a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}$ for $n \ge 1$. Find all natural numbers $n$ for which $a_n$ is a perfect square of an integer.

Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \] for every $x, y \in \mathbb{Z}$.

Find the rightmost nonzero digit of $ \frac{100!}{5^{20}}$ (here $ n!\equal{}1\cdot2\cdot3\cdot\ldots\cdot n$).

Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is [i]monotonically bounded[/i] if $1\leq a_i\leq i$ for all $1\leq i\leq 2^{16}$. We say that a term $a_k$ in the sequence with $2\leq k\leq 2^{16}-1$ is a [i]mountain[/i] if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$. Evan writes out all possible monotonically bounded sequences. Let $N$ be the total number of mountain terms over all such sequences he writes. Find the remainder when $N$ is divided by $p$. [i]Proposed by Michael Ren[/i]

In an $n$-element set $S$, several subsets $A_1, A_2, \ldots , A_k$ are distinguished, each consists of at least two, but not all elements of $S$. What is the largest $k$ that it’s possible to write down the elements of $S$ in a row in the order such that we don’t find all of the element of an $A_i$ set in the consecutive elements of the row?

Let $n\ge 16$ be an integer, and consider the set of $n^2$ points in the plane: \[ G=\big\{(x,y)\mid x,y\in\{1,2,\ldots,n\}\big\}.\] Let $A$ be a subset of $G$ with at least $4n\sqrt{n}$ elements. Prove that there are at least $n^2$ convex quadrilaterals whose vertices are in $A$ and all of whose diagonals pass through a fixed point.

Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$. It is known that the symmetric point of $M$with respect to the line $GI$ lies on $AC$. Find the measure of $\measuredangle B$.