Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline? $ \textbf{(A) }10\qquad\textbf{(B) }16\qquad\textbf{(C) }25\qquad\textbf{(D) }30\qquad\textbf{(E) }40 $

The plan of a house has the shape of a capital $L$, obtained by suitably placing side-by-side four squares whose sides are $10$ metres long. The external walls of the house are $10$ metres high. The roof of the house has six faces, starting at the top of the six external walls, and each face forms an angle of $30^\circ$ with respect to a horizontal plane. Determine the volume of the house (that is, of the solid delimited by the six external walls, the six faces of the roof, and the base of the house).

A set of $61$ coins that look alike is given. Two coins (whose weights are equal) are counterfeit. The other $59$ (genuine) coins also have the same weight, but a different weight from that of the counterfeit coins. However it is not known whether it is the genuine coins or the counterfeit coins which are heavier. How can this question be resolved by three weighings on the one balance? (It is not required to separate the counterfeit coins from the genuine ones.) (D. Fomin, Leningrad)

Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.

Find all monic polynomials $P(x)$ such that the polynomial $P(x)^2-1$ is divisible by the polynomial $P(x+1)$.

Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition : 1) $f(-1)\geq f(1).$ 2) $x+f(x)$ is non decreasing function. 3) $\int_{-1}^ 1 f(x)\ dx=0.$ Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

Triangle $ABC$ has $AB = 13$, $BC = 14$, and $CA = 15$. The internal angle bisector of $\angle ABC$ intersects side $CA$ at $X$. The circumcircles of triangles $AXB$ and $BXC$ intersect sides $BC$ and $AB$ at $M$ and $N$, respectively. The value of $MN^2$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m+n$ is divided by $1000$. [i]Proposed by [b] Th3Numb3rThr33[/b][/i]

Let $A = {1, 2, 3, 4, 5}$. Find the number of functions $f$ from the nonempty subsets of $A$ to $A$, such that $f(B) \in B$ for any $B \subset A$, and $f(B \cup C)$ is either $f(B)$ or $f(C)$ for any $B$, $C \subset A$

Find all functions $f : R \to R$ that satisfy $f(x^2 + f(y) - y) = (f(x))^2$ for all $x,y \in R$.

Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker $ \$ \underline{1}$ $ \underline{A}$ $ \underline{2}$. What is the missing digit $A$ of this $3$-digit number? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4$

Several black and white checkers (tokens?) are standing along the circumference. Two men remove checkers in turn. The first removes all the black ones that had at least one white neighbour, and the second -- all the white ones that had at least one black neighbour. They stop when all the checkers are of the same colour. a) Let there be $40$ checkers initially. Is it possible that after two moves of each man there will remain only one (checker)? b) Let there be $1000$ checkers initially. What is the minimal possible number of moves to reach the position when there will remain only one (checker)?

Let $ABC$ be a triangle with $AB \ne AC$. The incirle of triangle $ABC$ is tangent to $BC, CA, AB$ at $D, E, F$, respectively. The perpendicular line from $D$ to $EF$ intersects $AB$ at $X$. The second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX \perp T F$

There is a row of $100$ cells each containing a token. For $1$ dollar it is allowed to interchange two neighbouring tokens. Also it is allowed to interchange with no charge any two tokens such that there are exactly $3$ tokens between them. What is the minimum price for arranging all the tokens in the reverse order? (Egor Bakaev)

Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.

Find a four-digit number whose division by two given distinct one-digit numbers goes along the following lines: [img]https://cdn.artofproblemsolving.com/attachments/2/a/e1d3c68ec52e11ad59de755c3dbdc2cf54a81f.png[/img]

Compute the number of monic polynomials $q(x)$ with integer coefficients of degree $12$ such that there exists an integer polynomial $p(x)$ satisfying $q(x)p(x) = q(x^2).$ [i]Proposed by Yang Liu[/i]

Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?

Determine all positive integers $n$ such that $\frac{a^2+n^2}{b^2-n^2}$ is a positive integer for some $a,b\in \mathbb{N}$. $Turkey$

Three equal mass satellites $A$, $B$, and $C$ are in coplanar orbits around a planet as shown in the figure. The magnitudes of the angular momenta of the satellites as measured about the planet are $L_A$, $L_B$, and $L_C$. Which of the following statements is correct? [asy] // Code created by riben size(250); dotfactor=12; draw(circle((0,0),1.5),linewidth(2)); draw(circle((0,0),6),dashdotted); draw(circle((0,0),14),dashed); draw(ellipse((4,0),10,8),linewidth(1)); pair A,B,C; A=(-7,12.12); B=(5,7.9); C=(5.7,-1.87); dot(A); dot(B); dot(C); label("A",A,NW*1.5); label("B",B,NW*1.5); label("C",C,E*1.5); filldraw((-1.500, 0.078)-- (-1.428, 0.080)-- (-1.337, 0.094)-- (-1.295, 0.157)-- (-1.246, 0.209)-- (-1.186, 0.227)-- (-1.143, 0.290)-- (-1.148, 0.357)-- (-1.135, 0.469)-- (-1.057, 0.505)-- (-0.996, 0.563)-- (-0.936, 0.526)-- (-0.852, 0.557)-- (-0.773, 0.587)-- (-0.772, 0.716)-- (-0.765, 0.828)-- (-0.781, 0.955)-- (-0.732, 1.035)-- (-0.648, 1.083)-- (-0.605, 1.162)-- (-0.604, 1.246)-- (-0.645, 1.295)-- (-0.736, 1.270)-- (-0.796, 1.229)-- (-0.851, 1.193)-- (-0.941, 1.135)-- (-1.014, 1.076)-- (-1.105, 0.995)-- (-1.154, 0.921)-- (-1.227, 0.841)-- (-1.288, 0.760)-- (-1.349, 0.669)-- (-1.398, 0.556)-- (-1.453, 0.465)-- (-1.485, 0.357)-- (-1.510, 0.239)--cycle,gray); filldraw((-0.119, 1.245)-- (-0.130, 1.193)-- (-0.146, 1.095)-- (-0.202, 1.056)-- (-0.327, 1.033)-- (-0.262, 1.031)-- (-0.278, 0.979)-- (-0.193, 0.949)-- (-0.108, 0.943)-- (-0.013, 0.941)-- (0.032, 0.915)-- (0.026, 0.840)-- (0.015, 0.779)-- (0.019, 0.705)-- (0.074, 0.646)-- (0.113, 0.582)-- (0.162, 0.533)-- (0.167, 0.463)-- (0.241, 0.400)-- (0.311, 0.412)-- (0.416, 0.410)-- (0.465, 0.342)-- (0.541, 0.410)-- (0.611, 0.347)-- (0.679, 0.242)-- (0.728, 0.132)-- (0.732, 0.048)-- (0.671, -0.037)-- (0.615, -0.104)-- (0.540, -0.172)-- (0.409, -0.209)-- (0.324, -0.244)-- (0.253, -0.293)-- (0.188, -0.314)-- (0.162, -0.389)-- (0.181, -0.486)-- (0.270, -0.534)-- (0.340, -0.537)-- (0.380, -0.596)-- (0.424, -0.688)-- (0.418, -0.772)-- (0.352, -0.825)-- (0.281, -0.883)-- (0.241, -0.926)-- (0.145, -0.981)-- (0.044, -1.044)-- (-0.006, -1.107)-- (-0.007, -1.190)-- (0.077, -1.216)-- (0.162, -1.213)-- (0.253, -1.163)-- (0.323, -1.128)-- (0.404, -1.075)-- (0.510, -1.015)-- (0.605, -0.980)-- (0.671, -0.931)-- (0.731, -0.920)-- (0.817, -0.852)-- (0.898, -0.798)-- (0.963, -0.777)-- (0.964, -0.708)-- (1.024, -0.645)-- (1.025, -0.571)-- (0.976, -0.488)-- (0.912, -0.425)-- (0.878, -0.347)-- (0.823, -0.289)-- (0.779, -0.225)-- (0.744, -0.193)-- (0.756, -0.100)-- (0.816, -0.033)-- (0.837, 0.047)-- (0.838, 0.122)-- (0.824, 0.200)-- (0.800, 0.307)-- (0.796, 0.381)-- (0.872, 0.416)-- (0.967, 0.414)-- (1.016, 0.360)-- (1.096, 0.381)-- (1.117, 0.428)-- (1.058, 0.506)-- (0.998, 0.564)-- (0.954, 0.591)-- (0.914, 0.617)-- (0.860, 0.676)-- (0.800, 0.716)-- (0.751, 0.775)-- (0.757, 0.859)-- (0.797, 0.921)-- (0.823, 0.987)-- (0.889, 1.096)-- (0.850, 1.160)-- (0.780, 1.176)-- (0.700, 1.183)-- (0.645, 1.125)-- (0.579, 1.039)-- (0.518, 0.986)-- (0.438, 0.956)-- (0.343, 0.967)-- (0.289, 1.049)-- (0.249, 1.117)-- (0.195, 1.176)-- (0.125, 1.192)-- (0.030, 1.208)-- (-0.040, 1.220)--cycle,gray); [/asy] (A) $L_\text{A} > L_\text{B} > L_\text{C}$ (B) $L_\text{C} > L_\text{B} > L_\text{A}$ (C) $L_\text{B} > L_\text{C} > L_\text{A}$ (D) $L_\text{B} > L_\text{A} > L_\text{C}$ (E) The relationship between the magnitudes is different at various instants in time.

Construct a convex quadrilateral given two opposite angles and sides.

Without calculating the value of the determinant $$ \begin{vmatrix}1 &1 &3& 1\\1& 2& 3 &5\\ 3& 0& 5& 5\\ 0& a& -11a& a^{13}+9a\end{vmatrix} , $$ show that it is divisible by $ 26, $ for any integer $ a. $

Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Point $P$ lies on the segment $AB$, and points $S_1$ and $S_2$ are the centers of the circumcircles of $APC$ and $BPC$, respectively. Show that the midpoint of segment $S_1S_2$ lies on the perpendicular bisector of segment $CM$.

$m,n$ are given integer numbers such that $m+n$ is an odd number. Edges of a complete bipartie graph $K_{m,n}$ are labeled by ${-1,1}$ such that the sum of all labels is $0$. Prove that there exists a spanning tree such that the sum of the labels of its edges is equal to $0$.

Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that $$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$

Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit.