Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds

Let the four real solutions to the equation $x^2 + \frac{144}{x^2} = 25$ be $r_1, r_2, r_3$, and $r_4$. Find $|r_1| +|r_2| +|r_3| +|r_4|$.

We are given twelve coins, one of which is a fake with a different mass from the other eleven. Determine that coin with three weighings and whether it is heavier or lighter than the others.

In triangle $ABC$ with centroid $G$ and circumcircle $\omega$, line $\overline{AG}$ intersects $BC$ at $D$ and $\omega$ at $P$. Given that $GD =DP = 3$, and $GC = 4$, find $AB^2$. [i]Proposed by Muztaba Syed[/i]

Suppose $n$ coins are available that their mass is unknown. We have a pair of balances and every time we can choose an even number of coins and put half of them on one side of the balance and put another half on the other side, therefore a [i]comparison[/i] will be done. Our aim is determining that the mass of all coins is equal or not. Show that at least $n-1$ [i]comparisons[/i] are required.

Solve in integers $x$ and $y$ the equation $x^3-y^3=2xy+8$.

Prove that if the point $ P $ runs through a circle inscribed in the triangle $ ABC $, then the value of the expression $ a \cdot PA^2 + b \cdot PB^2 + c \cdot PC^2 $ is constant ($ a, b, c $ are the lengths of the sides opposite the vertices $ A, B, C $, respectively).

The probability that event $A$ occurs is $\frac{3}{4}$; the probability that event $B$ occurs is $\frac{2}{3}$. Let $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval $ \textbf{(A)}\ \Big[\frac{1}{12},\frac{1}{2}\Big]\qquad\textbf{(B)}\ \Big[\frac{5}{12},\frac{1}{2}\Big]\qquad\textbf{(C)}\ \Big[\frac{1}{2},\frac{2}{3}\Big]\qquad\textbf{(D)}\ \Big[\frac{5}{12},\frac{2}{3}\Big]\qquad\textbf{(E)}\ \Big[\frac{1}{12},\frac{2}{3}\Big]$

Find all numbers $\alpha$ for which the equation \[x^2 - 2x[x] + x -\alpha = 0\] has two nonnegative roots. ($[x]$ denotes the largest integer less than or equal to x.)

Find all functions $f: N_0 \to N_0$, (where $N_0$ is the set of all non-negative integers) such that $f(f(n))=f(n)+1$ for all $n \in N_0$ and the minimum of the set $\{ f(0), f(1), f(2) \cdots \}$ is $1$.

Let be a natural number $ n $ and $ 2n $ real numbers $ b_1,b_2,\ldots ,b_n,a_1<a_2<\cdots <a_n. $ Show that [b]a)[/b] if $ b_1,b_2,\ldots ,b_n>0, $ then there exists a polynomial $ f\in\mathbb{R}[X] $ irreducible in $ \mathbb{R}[X] $ such that $$ f\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$ [b]b)[/b] there exists a polynom $ g\in\mathbb{R} [X] $ of degree at least $ 1 $ which has only real roots and such that $$ g\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$

A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a 256 by 1 strip of quadruple thickness. This process is repeated 8 more times. After the last fold, the strip has become a stack of 1024 unit squares. How many of these squares lie below the square that was originally the 942nd square counting from the left?

Find an integer $n$, where $100 \leq n \leq 1997$, such that \[ \frac{2^n+2}{n} \] is also an integer.

Let $n$ be a positive integer. There are $n$ ants walking along a line at constant nonzero speeds. Different ants need not walk at the same speed or walk in the same direction. Whenever two or more ants collide, all the ants involved in this collision instantly change directions. (Different ants need not be moving in opposite directions when they collide, since a faster ant may catch up with a slower one that is moving in the same direction.) The ants keep walking indefinitely. Assuming that the total number of collisions is finite, determine the largest possible number of collisions in terms of $n$.

If $ a$, $ b$, and $ c$ are positive real numbers such that $ a(b \plus{} c) \equal{} 152$, $ b(c \plus{} a) \equal{} 162$, and $ c(a \plus{} b) \equal{} 170$, then abc is $ \textbf{(A)}\ 672 \qquad \textbf{(B)}\ 688 \qquad \textbf{(C)}\ 704 \qquad \textbf{(D)}\ 720 \qquad \textbf{(E)}\ 750$

A plane, flying at a height of $3000$ meters above the level ground below, receives a signal from the airport where the pilot intends to land. Using a vertical dipole antenna, the airport's air traffic control system is capable of transmitting 110-watt, 24 MHz signals. When the plane's horizontal position is 5 kilometers from the airport, what is the intensity of the signal at the plane's receiving antenna, in $\text{W}/\text{m}^2$? (The height of the transmitting antenna is negligible.) [i]Problem proposed by Kimberly Geddes[/i]

Let $G$ be the intersection of medians of $\triangle ABC$ and $I$ be the incenter of $\triangle ABC$. If $|AB|=c$, $|AC|=b$ and $GI \perp BC$, what is $|BC|$? $ \textbf{(A)}\ \dfrac{b+c}2 \qquad\textbf{(B)}\ \dfrac{b+c}{3} \qquad\textbf{(C)}\ \dfrac{\sqrt{b^2+c^2}}{2} \qquad\textbf{(D)}\ \dfrac{\sqrt{b^2+c^2}}{3\sqrt 2} \qquad\textbf{(E)}\ \text{None of the preceding} $

For odd positive integers $n$, define $f(n)$ to be the smallest odd integer greater than $n$ that is not relatively prime to $n$. Compute the smallest $n$ such that $f(f(n))$ is not divisible by $3$.

Let $A,B,C$ denote intersection points of diagonals $A_1A_4$ and $A_2A_5$, $A_1A_6$ and $A_2A_7$, $A_1A_9$ and $A_2A_{10}$ of the regular decagon $A_1A_2...A_{10}$ respectively Find the angles of the triangle $ABC$

Let $a$ and $b$ be perfect squares whose product exceeds their sum by $4844$. Compute the value of \[\left(\sqrt a + 1\right)\left(\sqrt b + 1\right)\left(\sqrt a - 1\right)\left(\sqrt b - 1\right) - \left(\sqrt{68} + 1\right)\left(\sqrt{63} + 1\right)\left(\sqrt{68} - 1\right)\left(\sqrt{63} - 1\right).\]

Show that the equation ${n \choose k}=m^{l}$ has no integral solution with $l \ge 2$ and $4 \le k \le n-4$.

Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e. a uniformly random bearing $\theta\in [0,2\pi]$) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$.

Let $K_1, K_2, K_3, K_4, K_5$ be 5 distinguishable keys, and let $D_1, D_2, D_3, D_4, D_5$ be $5$ distinguishable doors. For $1 \leq i \leq 5$, key $K_i$ opens doors $D_{i}$ and $D_{i+1}$ (where $D_6 = D_1$) and can only be used once. The keys and doors are placed in some order along a hallway. Key\$ha walks into the hallway, picks a key and opens a door with it, such that she never obtains a key before all the doors in front of it are unlocked. In how many orders can the keys and doors be placed such that Key\$ha can open all of the doors? [i]Author: Mitchell Lee[/i] [hide="Clarifications"] [list=1][*]The doors and keys are in series. In other words, the doors aren't lined up along the side of the hallway. They are blocking Key\$ha's path to the end, and the only way she can get past them is by getting the appropriate keys along the hallway. [*]The doors and keys appear consecutively along the hallway. For example, she might find $K_1 D_1 K_2 D_2 K_3 D_3 K_4 D_4 K_5 D_5$ down the hallway in that order. Also, by "she never obtains a key before all the doors in front of it are unlocked," we mean that she cannot obtain a key before all the doors appearing before the key are unlocked. In essence, it merely states that locked doors cannot be passed. [*]The doors and keys do not need to alternate down the hallway.[/list][/hide]

Given the set $S$ whose elements are zero and the even integers, positive and negative. Of the five operations applied to any pair of elements: (1) addition (2) subtraction (3) multiplication (4) division (5) finding the arithmetic mean (average), those elements that only yield elements of $S$ are: $ \textbf{(A)}\ \text{all} \qquad\textbf{(B)}\ 1,2,3,4\qquad\textbf{(C)}\ 1,2,3,5\qquad\textbf{(D)}\ 1,2,3\qquad\textbf{(E)}\ 1,3,5 $

Consider all nine-digit numbers, consisting of non-repeating digits from $1$ to $9$ in an arbitrary order. A pair of such numbers is called “conditional” if their sum is equal to $987654321$. (a) Prove that there exist at least two conditional pairs (noting that ($a,b$) and ($b,a$) is considered to be one pair). (b) Prove that the number of conditional pairs is odd. (G Galperin, Moscow)