Some marbles in a bag are red and the rest are blue. If one red marble is removed, then one-seventh of the remaining marbles are red. If two blue marbles are removed instead of one red, then one-fifth of the remaining marbles are red. How many marbles were in the bag originally? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 22 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 57 \qquad\textbf{(E)}\ 71 $

Product of two integers is $1$ less than three times of their sum. Find those integers.

In the regular hexagonal prism $ABCDEFA_1B_1C_1D_1E_1F_1$, We construct $, Q$, the projections of point $A$ on the lines $A_1B$ and $A_1C$ repsectilvely. We construct $R,S$, the projections of point $D_1$ on the lines $A_1D$ and $C_1D$ respectively. a) Determine the measure of the angle between the planes $(AQP)$ and $(D_1RS)$. b) Show that $\angle AQP = \angle D_1RS$.

Given a cyclic quadrilateral $ABCD$, let it's diagonals intersect at the point $O$. Take the midpoints of $AD$ and $BC$ as $M$ and $N$ respectively. Take a point $S$ on the arc $AB$ not containing $C$ or $D$ such that $$\angle SMA=\angle SNB$$ Prove that if the diagonals of the quadrilateral made from the lines $SM$, $SN$, $AB$, and $CD$ intersect at the point $T$, then $S$, $O$, and $T$ are collinear.

A permutation of a set is a bijection from the set to itself. For example, if $\sigma$ is the permutation $1 7\mapsto 3$, $2 \mapsto 1$, and $3 \mapsto 2$, and we apply it to the ordered triplet $(1, 2, 3)$, we get the reordered triplet $(3, 1, 2)$. Let $\sigma$ be a permutation of the set $\{1, ... , n\}$. Let $$\theta_k(m) = \begin{cases} m + 1 & \text{for} \,\, m < k\\ 1 & \text{for} \,\, m = k\\ m & \text{for} \,\, m > k\end{cases}$$ Call a finite sequence $\{a_i\}^{j}_{i=1}$ a disentanglement of $\sigma$ if $\theta_{a_j} \circ ...\circ \theta_{a_j} \circ \sigma$ is the identity permutation. For example, when $\sigma = (3, 2, 1)$, then $\{2, 3\}$ is a disentaglement of $\sigma$. Let $f(\sigma)$ denote the minimum number $k$ such that there is a disentanglement of $\sigma$ of length $k$. Let $g(n)$ be the expected value for $f(\sigma)$ if $\sigma$ is a random permutation of $\{1, ... , n\}$. What is $g(6)$?

Let $n \ne 0$ be a natural number and integers $x_1, x_2, ...., x_n, y_1, y_2, ...., y_n$ with the properties: a) $x_1 + x_2 + .... + x_n = y_1 + y_2 + .... + y_n = 0,$ b) $x_1 ^ 2 + y_1 ^ 2 = x_2 ^ 2 + y_2 ^ 2 = .... = x_n ^ 2 + y_n ^ 2$. Show that $n$ is even.

Let $\lambda$ be a line and let $M, N$ be two points on $\lambda$. Circles $\alpha$ and $\beta$ centred at $A$ and $B$ respectively are both tangent to $\lambda$ at $M$, with $A$ and $B$ being on opposite sides of $\lambda$. Circles $\gamma$ and $\delta$ centred at $C$ and $D$ respectively are both tangent to $\lambda$ at $N$, with $C$ and $D$ being on opposite sides of $\lambda$. Moreover $A$ and $C$ are on the same side of $\lambda$. Prove that if there exists a circle tangent to all circles $\alpha, \beta, \gamma, \delta$ containing all of them in its interior, then the lines $AC, BD$ and $\lambda$ are either concurrent or parallel.

During his excursion to the historical park, Pepito set out to collect stones whose weight in kilograms is a power of two. Once the first stone has been collected, Pepito only collects stones strictly heavier than the first. At the end of the excursion, her partner Ana chooses a positive integer $K \ge 2$ and challenges Pepito to divide the stones into $K$ groups of equal weight. a) Can Pepito meet the challenge if all the stones he collected have different weights? b) Can Pepito meet the challenge if some collected stones are allowed to have equal weight?

From the altitudes of an acute-angled triangle, a triangle can be composed. Prove that a triangle can be composed from the bisectors of this triangle.

Let $ x$ and $ y$ be two-digit integers such that $ y$ is obtained by reversing the digits of $ x$. The integers $ x$ and $ y$ satisfy $ x^2 \minus{} y^2 \equal{} m^2$ for some positive integer $ m$. What is $ x \plus{} y \plus{} m$? $ \textbf{(A)}\ 88\qquad \textbf{(B)}\ 112\qquad \textbf{(C)}\ 116\qquad \textbf{(D)}\ 144\qquad \textbf{(E)}\ 154$

Let p be a prime and $a_1 , a_2 , ..., a_k$ pairwise incongruent modulo p . Prove that $[\sqrt {k-1}]$ of the elements can be selected from $a_i$'s such that adding any numbers different from the selected ones will never give a number divisible by p .

The largest one of numbers $ p_1^{\alpha_1}, p_2^{\alpha_2}, \cdots, p_t^{\alpha_t}$ is called a $ \textbf{Good Number}$ of positive integer $ n$, if $ \displaystyle n\equal{} p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdots p_t^{\alpha_t}$, where $ p_1$, $ p_2$, $ \cdots$, $ p_t$ are pairwisely different primes and $ \alpha_1, \alpha_2, \cdots, \alpha_t$ are positive integers. Let $ n_1, n_2, \cdots, n_{10000}$ be $ 10000$ distinct positive integers such that the $ \textbf{Good Numbers}$ of $ n_1, n_2, \cdots, n_{10000}$ are all equal. Prove that there exist integers $ a_1, a_2, \cdots, a_{10000}$ such that any two of the following $ 10000$ arithmetical progressions $ \{ a_i, a_i \plus{} n_i, a_i \plus{} 2n_i, a_i \plus{} 3n_i, \cdots \}$($ i\equal{}1,2, \cdots 10000$) have no common terms.

Let $ABCD$ be a cyclic quadrilateral with $O$ is circumcenter and $AC$ meets $BD$ at $I$ Suppose that rays $DA,CD$ meet at $E$ and rays $BA,CD$ meet at $F$. The Gauss line of $ABCD$ meets $AB,BC,CD,DA$ at points $M,N,P,Q$ respectively. Prove that the circle of diameter $OI$ is tangent to two circles $(ENQ), (FMP)$

Let $p > 3$ be a prime number and $ x$ an integer, denote by $r ( x )\in \{ 0 , 1 , ... , p - 1 \}$ to the rest of $x$ modulo $p$ . Let $x_1, x_2, ... , x_k$ ( $2 < k < p$) different integers modulo $p$ and not divisible by $p$. We say that a number $a \in \{ 1 , 2 ,..., p -1 \}$ is [i]good [/i] if $r ( a x_1) < r ( a x_2) <...< r ( a x_k)$. Show that there are at most $\frac{2 p}{k + 1}-{ 1}$ [i]good [/i] numbers.

Given five circles of radii $1, 2, 3, 4,$ and $5$, what is the maximum number of points of intersections possible (every distinct point where two circles intersect counts).

Let $a_0=-2,b_0=1$, and for $n\geq 0$, let \begin{align*}a_{n+1}&=a_n+b_n+\sqrt{a_n^2+b_n^2},\\b_{n+1}&=a_n+b_n-\sqrt{a_n^2+b_n^2}.\end{align*} Find $a_{2012}$.

The sides of a triangle are equal to $\sqrt2, \sqrt3, \sqrt4$ and its angles are $\alpha, \beta, \gamma$, respectively. Prove that the equation $x\sin \alpha + y\sin \beta + z\sin \gamma = 0$ has exactly one solution in integers $x, y, z$.

Let $D,E,F$ be the midpoints of $BC$ ,$CA$, $AB$ in $\vartriangle ABC$ such that $AD= 9$, $BE= 12$, $CF= 15$. Calculate the area of $\vartriangle ABC$

Sasha has a bag that holds $6$ red marbles and $7$ green marbles. How many ways can Sasha pick a handful of (zero or more) marbles from the bag such that her handful contains at least as many red marbles as green marbles (any two marbles are distinguishable, even if they have the same color)?

A quadrilateral $KLMN$ is given. A circle with center $O$ meets its side $KL$ at points $A$ and $A_1$, side $LM$ at points $B$ and $B_1$, etc. Prove that if the circumcircles of triangles $KDA, LAB, MBC$ and $NCD$ concur at point $P$, then a) the circumcircles of triangles $KD_1A_1, LA_1B_1, MB_1C_1$ and $NC1D1$ also concur at some point $Q$; b) point $O$ lies on the perpendicular bisector to $PQ$.

Zandrew Hao has $n^2$ dollars, where $n$ is an integer. He is a massive fan of the singer Pachary Zerry, and he wants to buy many copies of his $3$ albums, which cost $\$8$, $\$623$, and $\$835$ (two of them are very rare). Find the sum of the $3$ greatest values of $n$ such that Zandrew can't spend all of his money on albums.

A country has 1001 cities, and each two cities are connected by a one-way street. From each city exactly 500 roads begin, and in each city 500 roads end. Now an independent republic splits itself off the country, which contains 668 of the 1001 cities. Prove that one can reach every other city of the republic from each city of this republic without being forced to leave the republic.

Prove that in any group of people, the number of those who know an odd number of the others in the group is even. Assume that knowing is a symmetric relation.

Let $m$ and $n$ two given integers. Ana thinks of a pair of real numbers $x$, $y$ and then she tells Beto the values of $x^m+y^m$ and $x^n+y^n$, in this order. Beto's goal is to determine the value of $xy$ using that information. Find all values of $m$ and $n$ for which it is possible for Beto to fulfill his wish, whatever numbers that Ana had chosen.

A set of points on the plane is [i]antiparallelogram [/i] if any four points of the set are [b]not[/b] vertices of a parallelogram. Prove that for any set of $2023$ points on the plane, [b]no[/b] three of them are collinears, there exists a subset of $17$ points, such that this subset is antiparallelogram.