Let $n$ be a positive number. Prove that there exists an integer $N =\overline{m_1m_2...m_n}$ with $m_i \in \{1, 2\}$ which is divisible by $2^n$.

Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357, 89,\text{and } 5$ are all uphill integers, but $32, 1240, \text{and } 466$ are not. How many uphill integers are divisible by $15$? $\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 8$

A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $1/2$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?

Given a triangle $ABC$, with right $\angle A$, we know: the point $T$ of tangency of the circumference inscribed in $ABC$ with the hypotenuse $BC$, the point $D$ of intersection of the angle bisector of $\angle B$ with side AC and the point E of intersection of the angle bisector of $\angle C$ with side $AB$ . Describe a construction with ruler and compass for points $A$, $B$, and $C$. Justify.

All vertices of a convex polyhedron with 100 edges are cut off by some planes. The planes do not intersect either inside or on the surface of the polyhedron. For this new polyhedron find a) the number of vertices; [i](1 point)[/i] b) the number of edges. [i](2 points)[/i]

In a country between each pair of cities there is at most one direct road. There is a connection (using one or more roads) between any two cities even after the elimination of any given city and all roads incident to this city. We say that the city $A$ can be[i] k -directionally[/i] connected to the city $B$, if : we can orient at most $k$ roads such that after[i] arbitrary[/i] orientation of remaining roads for any fixed road $l$ (directly connecting two cities) there is a path passing through roads in the direction of their orientation starting at $A$, passing through $l$ and ending at $B$ and visiting each city at most once. Suppose that in a country with $n$ cities, any two cities can be[i] k - directionally[/i] connected. What is the minimal value of $k$?

Solve the system of equations $$\begin{cases} \dfrac{x+y}{xy}+\dfrac{xy}{x+y}= a+ \dfrac{1}{a}\\ \\\dfrac{x-y}{xy}+\dfrac{xy}{x-y}= c+ \dfrac{1}{c}\end{cases}$$

Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} \plus{} 1$ is composite for all $ n \in \mathbb{N}_{0}$.

The numbers $1, 2, \dots, 1999$ are written on the board. Two players take turn choosing $a,b$ from the board and erasing them then writing one of $ab$, $a+b$, $a-b$. The first player wants the last number on the board to be divisible by $1999$, the second player want to stop him. Determine the winner.

$BL $ is the bisector of an isosceles triangle $ABC $. A point $D $ is chosen on the Base $BC $ and a point $E $ is chosen on the lateral side $AB $ so that $AE=\frac {1}{2}AL=CD $. Prove that $LE=LD $. Tuymaada 2017 Q5 Juniors

A cube with side length $100cm$ is filled with water and has a hole through which the water drains into a cylinder of radius $100cm.$ If the water level in the cube is falling at a rate of $1 \frac{cm}{s} ,$ how fast is the water level in the cylinder rising?

Find all integers $n=2k+1>1$ so that there exists a permutation $a_0, a_1,\ldots,a_{k}$ of $0, 1, \ldots, k$ such that \[a_1^2-a_0^2\equiv a_2^2-a_1^2\equiv \cdots\equiv a_{k}^2-a_{k-1}^2\pmod n.\] [i]Proposed by usjl[/i]

At a party, $2019$ people decide to form teams of three. To do so, each turn, every person not on a team points to one other person at random. If three people point to each other (that is, $A$ points to $B$, $B$ points to $C$, and $C$ points to $A$), then they form a team. What is the probability that after $65, 536$ turns, exactly one person is not on a team

Let n > 3 be a given integer. Find the largest integer d (in terms of n) such that for any set S of n integers, there are four distinct (but not necessarily disjoint) nonempty subsets, the sum of the elements of each of which is divisible by d.

Let $ABC$ be a triangle for which $AB \neq AC$. Points $K$, $L$, $M$ are the midpoints of the sides $BC$, $CA$, $AB$. The incircle of $ABC$ with center $I$ is tangent to $BC$ in $D$. A line passing through the midpoint of $ID$ perpendicular to $IK$ meets the line $LM$ in $P$. Prove that $\angle PIA = 90 ^{\circ}$.

Let $ \triangle{XOY}$ be a right-angled triangle with $ m\angle{XOY}\equal{}90^\circ$. Let $ M$ and $ N$ be the midpoints of legs $ OX$ and $ OY$, respectively. Given that $ XN\equal{}19$ and $ YM\equal{}22$, find $ XY$. $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 26 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 32$

Does there exist an increasing sequence of positive integers $a_1 , a_2 ,\cdots$ with the following two properties? (i) Every positive integer $n$ can be uniquely expressed in the form $n = a_j - a_i$ , (ii) $\frac{a_k}{k^3}$ is bounded.

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]

Determine all positive real solutions of the following system of equations: $$a =\ max \{ \frac{1}{b} , \frac{1}{c}\} \,\,\,\,\,\, b = \max \{ \frac{1}{c} , \frac{1}{d}\} \,\,\,\,\,\, c = \max \{ \frac{1}{d}, \frac{1}{e}\} $$ $$d = \max \{ \frac{1}{e} , \frac{1}{f }\} \,\,\,\,\,\, e = \max \{ \frac{1}{f} , \frac{1}{a}\} \,\,\,\,\,\, f = \max \{ \frac{1}{a} , \frac{1}{b}\}$$

A circular disc with diameter $D$ is placed on an $8\times 8$ checkerboard with width $D$ so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is $\textbf{(A) }48\qquad\textbf{(B) }44\qquad\textbf{(C) }40\qquad\textbf{(D) }36\qquad \textbf{(E) }32$

Let's write down a segment of a series of integers from $0$ to $1995$. Among the numbers written out, two have been crossed out. Let's consider the longest arithmetic progression contained among the remaining $1994$ numbers. Let $K$ be the length of the progression. Which two numbers must be crossed out so that the value of $K$ is the smallest?

Spaceman Fred's spaceship (which has negligible mass) is in an elliptical orbit about Planet Bob. The minimum distance between the spaceship and the planet is $R$; the maximum distance between the spaceship and the planet is $2R$. At the point of maximum distance, Spaceman Fred is traveling at speed $v_\text{0}$. He then fires his thrusters so that he enters a circular orbit of radius $2R$. What is his new speed? [asy] size(300); // Shape draw(circle((0,0),25),dashed+gray); draw(circle((0,0),3.5),linewidth(2)); draw(ellipse((5,0),20,15)); // Dashed Lines draw((25,13)--(25,-35),dotted); draw((0,-35)--(0,-3.3),dotted); draw((0,3.3)--(0,13),dotted); draw((-15,13)--(-15,-35),dotted); // Labels draw((-14,-35)--(-1,-35),Arrows(size=6,SimpleHead)); label(scale(1.2)*"$R$",(-7.5,-35),N); draw((24,-35)--(1,-35),Arrows(size=6,SimpleHead)); label(scale(1.2)*"$2R$",(10,-35),N); // Blobs on Earth path A=(-1.433, 2.667)-- (-1.433, 2.573)-- (-1.360, 2.478)-- (-1.408, 2.360)-- (-1.493, 2.207)-- (-1.554, 2.160)-- (-1.614, 2.113)-- (-1.675, 2.065)-- (-1.735, 1.959)-- (-1.772, 1.877)-- (-1.723, 1.759)-- (-1.748, 1.676)-- (-1.748, 1.523)-- (-1.772, 1.369)-- (-1.760, 1.240)-- (-1.857, 1.145)-- (-1.941, 1.098)-- (-2.050, 1.122)-- (-2.111, 1.086)-- (-2.244, 1.039)-- (-2.390, 1.004)-- (-2.511, 0.909)-- (-2.486, 0.697)-- (-2.499, 0.555)-- (-2.535, 0.414)-- (-2.668, 0.308)-- (-2.765, 0.237)-- (-2.910, 0.131)-- (-3.068, 0.036)-- (-3.250, 0.024)-- (-3.310, 0.154)-- (-3.274, 0.272)-- (-3.286, 0.402)-- (-3.298, 0.532)-- (-3.250, 0.650)-- (-3.165, 0.768)-- (-3.128, 0.933)-- (-3.068, 1.074)-- (-3.032, 1.204)-- (-2.971, 1.310)-- (-2.886, 1.452)-- (-2.801, 1.558)-- (-2.729, 1.652)-- (-2.656, 1.770)-- (-2.583, 1.912)-- (-2.486, 1.995)-- (-2.365, 2.089)-- (-2.244, 2.207)-- (-2.123, 2.313)-- (-2.014, 2.419)-- (-1.905, 2.478)-- (-1.832, 2.573)-- (-1.687, 2.643)-- (-1.578, 2.714)--cycle; filldraw(A,gray); path B=(-0.397, 2.527)-- (-0.468, 2.321)-- (-0.538, 2.154)-- (-0.639, 2.065)-- (-0.760, 2.085)-- (-0.922, 2.085)-- (-0.993, 2.016)-- (-0.770, 1.918)-- (-0.649, 1.829)-- (-0.498, 1.780)-- (-0.367, 1.770)-- (-0.205, 1.751)-- (-0.084, 1.761)-- (-0.104, 1.613)-- (-0.114, 1.495)-- (-0.094, 1.358)-- (0.007, 1.220)-- (0.067, 1.131)-- (0.108, 1.013)-- (0.188, 0.905)-- (0.239, 0.787)-- (0.330, 0.650)-- (0.461, 0.620)-- (0.622, 0.620)-- (0.794, 0.591)-- (0.905, 0.610)-- (0.956, 0.689)-- (1.026, 0.591)-- (1.097, 0.483)-- (1.198, 0.374)-- (1.258, 0.276)-- (1.339, 0.188)-- (1.319, -0.009)-- (1.309, -0.166)-- (1.198, -0.343)-- (1.077, -0.432)-- (0.935, -0.520)-- (0.814, -0.589)-- (0.633, -0.677)-- (0.481, -0.727)-- (0.350, -0.776)-- (0.229, -0.894)-- (0.229, -1.041)-- (0.229, -1.228)-- (0.340, -1.346)-- (0.522, -1.415)-- (0.643, -1.513)-- (0.693, -1.651)-- (0.784, -1.798)-- (0.723, -1.936)-- (0.612, -2.044)-- (0.471, -2.123)-- (0.350, -2.201)-- (0.249, -2.270)-- (0.108, -2.339)-- (-0.013, -2.418)-- (-0.124, -2.535)-- (-0.135, -2.673)-- (-0.175, -2.811)-- (-0.084, -2.840)-- (0.067, -2.840)-- (0.209, -2.830)-- (0.350, -2.742)-- (0.522, -2.653)-- (0.582, -2.604)-- (0.713, -2.545)-- (0.845, -2.457)-- (0.935, -2.408)-- (1.057, -2.388)-- (1.228, -2.280)-- (1.329, -2.191)-- (1.460, -2.132)-- (1.581, -2.093)-- (1.692, -2.044)-- (1.793, -2.005)-- (1.844, -1.906)-- (1.844, -1.828)-- (1.904, -1.749)-- (2.005, -1.621)-- (1.955, -1.454)-- (1.894, -1.287)-- (1.773, -1.189)-- (1.632, -0.992)-- (1.592, -0.874)-- (1.491, -0.736)-- (1.410, -0.569)-- (1.460, -0.412)-- (1.561, -0.274)-- (1.592, -0.078)-- (1.622, 0.168)-- (1.551, 0.306)-- (1.440, 0.404)-- (1.420, 0.561)-- (1.551, 0.620)-- (1.703, 0.630)-- (1.824, 0.532)-- (1.955, 0.365)-- (2.046, 0.453)-- (2.116, 0.551)-- (2.167, 0.689)-- (2.096, 0.807)-- (1.965, 0.905)-- (1.834, 0.935)-- (1.743, 0.994)-- (1.622, 1.131)-- (1.531, 1.249)-- (1.430, 1.348)-- (1.359, 1.515)-- (1.420, 1.702)-- (1.511, 1.839)-- (1.571, 2.016)-- (1.672, 2.134)-- (1.592, 2.232)-- (1.440, 2.291)-- (1.289, 2.350)-- (1.178, 2.252)-- (1.127, 2.134)-- (1.067, 1.997)-- (0.986, 1.898)-- (0.845, 1.839)-- (0.693, 1.839)-- (0.522, 1.859)-- (0.471, 1.977)-- (0.380, 2.124)-- (0.289, 2.203)-- (0.188, 2.291)-- (0.047, 2.311)-- (-0.074, 2.370)-- (-0.195, 2.508)--cycle; filldraw(B,gray); [/asy] (A) $\sqrt{3/2}v_\text{0}$ (B) $\sqrt{5}v_\text{0}$ (C) $\sqrt{3/5}v_\text{0}$ (D) $\sqrt{2}v_\text{0}$ (E) $2v_\text{0}$

Determine all 4-tuples ($a, b,c, d$) of real numbers satisfying the following four equations: $\begin{cases} ab + c + d = 3 \\ bc + d + a = 5 \\ cd + a + b = 2 \\ da + b + c = 6 \end{cases}$

An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. [quote]For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1[/quote] The number of partitions of n is given by the partition function $p\left ( n \right )$. So $p\left ( 4 \right ) = 5$ . Determine all the positive integers so that $p\left ( n \right )+p\left ( n+4 \right )=p\left ( n+2 \right )+p\left ( n+3 \right )$.

We reflected each vertex of a triangle on the opposite side. Prove that the area of the triangle formed by these three reflection points is smaller than the area of the initial triangle multiplied by five.