Let $a, b$ be positive real numbers satisfying $a + b = 1$. Show that if $x_1,x_2,...,x_5$ are positive real numbers such that $x_1x_2...x_5 = 1$, then $(ax_1+b)(ax_2+b)...(ax_5+b)>1$

Let $a$, $b$, $c$ be real numbers. Prove that \[ab + bc + ca + \max\{|a - b|, |b - c|, |c - a|\} \le 1 + \frac{1}{3} (a + b + c)^2.\]

Let $f(x) = x-\tfrac1{x}$, and define $f^1(x) = f(x)$ and $f^n(x) = f(f^{n-1}(x))$ for $n\ge2$. For each $n$, there is a minimal degree $d_n$ such that there exist polynomials $p$ and $q$ with $f^n(x) = \tfrac{p(x)}{q(x)}$ and the degree of $q$ is equal to $d_n$. Find $d_n$.

The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled? $ \textbf{(A)}\ 80\qquad\textbf{(B)}\ 96\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108\qquad\textbf{(E)}\ 192 $

Let $f, g$ be functions from the positive integers to the integers. Vlad the impala is jumping around the integer grid. His initial position is $x_0 = (0, 0)$, and for every $n \ge 1$, his jump is $x_n - x_{n - 1} = (\pm f(n), \pm g(n))$ or $(\pm g(n), \pm f(n)),$ with eight possibilities in total. Is it always possible that Vlad can choose his jumps to return to his initial location $(0, 0)$ infinitely many times when (a) $f, g$ are polynomials with integer coefficients? (b) $f, g$ are any pair of functions from the positive integers to the integers?

Solve in real numbers the equation \[ x+\log_{2}\left( 1+\sqrt{\frac{5^{x}}{3^{x}+4^{x}}}\right) =4+\log_{1/2}\left(1+\sqrt{\frac{25^{x}}{7^{x}+24^{x}}}\right) \]

In triangle $ABC$, points $D$ and $E$ lie on the interior of segments $AB$ and $AC$, respectively,such that $AD = 1$, $DB = 2$, $BC = 4$, $CE = 2$ and $EA = 3$. Let $DE$ intersect $BC$ at $F$. Determine the length of $CF$.

Below is a net consisting of $3$ squares, $4$ equilateral triangles, and $1$ regular hexagon. Each polygon has side length $1$. When we fold this net to form a polyhedron, what is the volume of the polyhedron? (This figure is called a "triangular cupola".) Net: [asy] pair A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P; A = origin; B = (0,3); C = 3*dir(150); D = (0,1); E = (0,2); F = C+2*dir(30); G = C+dir(30); H = 2*dir(150); I = dir(150); J = (1,1); K = J+dir(30); L = (1,2); M = F+dir(120); N = G+dir(120); O = H+dir(240); P = I+dir(240); draw(A--B--C--cycle); draw(D--E--F--G--H--I--cycle); draw(D--E--L--J--cycle); draw(F--G--N--M--cycle); draw(H--I--P--O--cycle); draw(J--K--L--cycle); [/asy] Resulting polyhedron: [img]https://upload.wikimedia.org/wikipedia/commons/9/93/Triangular_cupola.png[/img]

The sign shown below consists of two uniform legs attached by a frictionless hinge. The coefficient of friction between the ground and the legs is $\mu$. Which of the following gives the maximum value of $\theta$ such that the sign will not collapse? $\textbf{(A) } \sin \theta = 2 \mu \\ \textbf{(B) } \sin \theta /2 = \mu / 2\\ \textbf{(C) } \tan \theta / 2 = \mu\\ \textbf{(D) } \tan \theta = 2 \mu \\ \textbf{(E) } \tan \theta / 2 = 2 \mu$

Cut an arbitrary triangle into $2019$ pieces so that one of them turns out to be a triangle, one is a quadrilateral, ... one is a $2019$-gon and one is a $2020$-gon. Polygons do not have to be convex.

In any $\vartriangle ABC, \ell$ is any line through $C$ and points $P, Q$. If $BP, AQ$ are perpendicular to the line $\ell$ and $M$ is the midpoint of the line segment $AB$, then prove that $MP = MQ$

Let $ f:[0,\infty )\longrightarrow\mathbb{R} $ a bounded and periodic function with the property that $$ |f(x)-f(y)|\le |\sin x-\sin y|,\quad\forall x,y\in[0,\infty ) . $$ Show that the function $ [0,\infty ) \ni x\mapsto x+f(x) $ is monotone.

Let $p$ be an odd prime number and suppose that $2^h \not \equiv 1 \text{ (mod } p\text{)}$ for all integer $1 \leq h \leq p-2$. Let $a$ be an even number such that $\frac{p}{2} < a < p$. Define the sequence $a_0, a_1, a_2, \ldots$ as $$a_0 = a, \qquad a_{n+1} = p -b_n, \qquad n = 0,1,2, \ldots,$$ where $b_n$ is the greatest odd divisor of $a_n$. Show that the sequence is periodic and determine its period.

Let $P (x)$ be a polynomial with integer coefficients and $P (n) =m$ for some integers $n, m$ ($m \ne 10$). Prove that $P (n + km)$ is divisible by $m$ for any integer $k$.

Are there positive integers $a, b$ with $b \ge 2$ such that $2^a + 1$ is divisible by $2^b - 1$?

Alice needs to replace a light bulb located $10$ centimeters below the ceiling of her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters? $ \textbf{(A)}\ 32 \qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 38\qquad\textbf{(E)}\ 40 $

Sokal da tries to find out the largest positive integer n such that if n transforms to base-7, then it looks like twice of base-10. $156$ is such a number because $(156)_{10}$ = $(312)_7$ and 312 = 2$\times$156. Find out Sokal da's number.

Let $n$ be square with 4 digits, such that all its digits are less than 6. If we add 1 to each digit the resulting number is another square. Find $n$

Let $ABCD$ an inscribed quadrilateral and $r$ and $s$ the reflections of the straight line through $A$ and $B$ over the inner angle bisectors of angles $\angle{CAD}$ and $\angle{CBD}$, respectively. Let $P$ the point of intersection of $r$ and $s$ and let $O$ the circumcentre of $ABCD$. Prove that $OP \perp CD$.

Let $n$ and $k$ be two positive integers. Prove that if $n$ is relatively prime with $30$, then there exist two integers $a$ and $b$, each relatively prime with $n$, such that $\frac{a^2 - b^2 + k}{n}$ is an integer. Malik Talbi

[u]Round 1[/u] [b]p1.[/b] How many powers of $2$ are greater than $3$ but less than $2013$? [b]p2.[/b] What number is equal to six greater than three times the answer to this question? [b]p3.[/b] Surya Cup-a-tea-raju goes to Starbucks Coffee to sip coffee out of a styrofoam cup. The cup is a cylinder, open on one end, with base radius $3$ centimeters and height $10$ centimeters. What is the exterior surface area of the styrofoam cup? [u]Round 2[/u] [b]p4.[/b] Andrew has two $6$-foot-length sticks that he wishes to make into two of the sides of the entrance to his fort, with the ground being the third side. If he wants to make his entrance in the shape of a triangle, what is the largest area that he can make the entrance? [b]p5.[/b] Ethan and Devin met a fairy who told them “if you have less than $15$ dollars, I will give you cake”. If both had integral amounts of dollars, and Devin had 5 more dollars than Ethan, but only Ethan got cake, how many different amounts of money could Ethan have had? [b]p6.[/b] If $2012^x = 2013$, for what value of $a$, in terms of $x$, is it true that $2012^a = 2013^2$? [u]Round 3[/u] [b]p7.[/b] Find the ordered triple $(L, M, T)$ of positive integers that makes the following equation true: $$1 + \dfrac{1}{L + \dfrac{1}{M+\dfrac{1}{T}}}=\frac{79}{43}.$$ [b]p8.[/b] Jonathan would like to start a banana plantation so he is saving up to buy an acre of land, which costs $\$600,000$. He deposits $\$300,000$ in the bank, which gives $20\%$ interest compounded at the end of each year. At this rate, how many years will Jonathan have to wait until he can buy the acre of land? [b]p9.[/b] Arul and Ethan went swimming at their town pool and started to swim laps to see who was in better shape. After one hour of swimming at their own paces, Ethan completed $32$ more laps than Arul. However, after that, Ethan got tired and swam at half his original speed while Arul’s speed didn’t change. After one more hour, Arul swam a total of $320$ laps. How many laps did Ethan swim after two hours? [u]Round 4[/u] [b]p10.[/b] A right triangle with a side length of $6$ and a hypotenuse of 10 has circles of radius $1$ centered at each vertex. What is the area of the space inside the triangle but outside all three circles? [b]p11.[/b] In isosceles trapezoid $ABCD$, $\overline{AB} \parallel\overline{CD}$ and the lengths of $\overline{AB}$ and $\overline{CD}$ are $2$ and $6$, respectively. Let the diagonals of the trapezoid intersect at point $E$. If the distance from $E$ to $\overline{CD}$ is $9$, what is the area of triangle $ABE$? [b]p12.[/b] If $144$ unit cubes are glued together to form a rectangular prism and the perimeter of the base is $54$ units, what is the height? PS. You should use hide for answers. Rounds 6-8 are [url=https://artofproblemsolving.com/community/c3h3136014p28427163]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3137069p28442224]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A prime $p$ and a positive integer $n$ are given. The product $$(1^3+1)(2^3+1)...((n-1)^3+1)(n^3+1)$$ is divisible by $p^3$. Prove that $p \leq n+1$. [i]Proposed by Z. Luria[/i]

[b]p1.[/b] Prove that the equation $x^6 - 143x^5 - 917x^4 + 51x^3 + 77x^2 + 291x + 1575 = 0$ has no integer solutions. [b]p2.[/b] There are $81$ wheels in a storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that it can be detected with certainty after four measurements on a balance scale. [b]p3.[/b] Rob and Ann multiplied the numbers from $1$ to $100$ and calculated the sum of digits of this product. For this sum, Rob calculated the sum of its digits as well. Then Ann kept repeating this operation until he got a one-digit number. What was this number? [b]p4.[/b] Rui and Jui take turns placing bishops on the squares of the $ 8\times 8$ chessboard in such a way that bishops cannot attack one another. (In this game, the color of the rooks is irrelevant.) The player who cannot place a rook loses the game. Rui takes the first turn. Who has a winning strategy, and what is it? [b]p5.[/b] The following figure can be cut along sides of small squares into several (more than one) identical shapes. What is the smallest number of such identical shapes you can get? [img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for all positive $a_1. a_2, ..., a_n$ the inequality $$\left( 1+\frac{a_1^2}{a_2}\right) \left( 1+\frac{a_2^2}{a_3}\right) ...\left( 1+\frac{a_n^2}{a_1}\right) \ge (1+a_1)(1+a_2)...(1+a_n)$$ holds. (LD Kurliandchik)

If $(1.0025)^{10}$ is evaluated correct to $5$ decimal places, then the digit in the fifth decimal place is $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }5\qquad \textbf{(E) }8$