Let $f(x) = ax^2 + bx+ c$ and $g(x) = cx^2 + bx + a$. If $|f(0)| \leq 1, |f(1)| \leq 1, |f(-1)| \leq 1$, prove that for $|x| \leq 1$, [b](a)[/b] $|f(x)| \leq 5/4$, [b](b)[/b] $|g(x)| \leq 2$.

Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all positive real $x, y$ holds $$xf(x)+yf(y)=(x+y)f\left(\frac{x^2+y^2}{x+y}\right)$$. [i]Fedir Yudin[/i]

Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

Let $B, M$, and $T$ be the three roots of the equation $x^3 + 20x^2 -18x-19 = 0$. What is the value of $|(B + 1)(M + 1)(T + 1)|$?

[b]p1.[/b] Solve for $x$ and $y$ if $\frac{1}{x^2}+\frac{1}{xy}=\frac{1}{9}$ and $\frac{1}{y^2}+\frac{1}{xy}=\frac{1}{16}$ [b]p2.[/b] Find positive integers $p$ and $q$, with $q$ as small as possible, such that $\frac{7}{10} <\frac{p}{q} <\frac{11}{15}$. [b]p3.[/b] Define $a_1 = 2$ and $a_{n+1} = a^2_n -a_n + 1$ for all positive integers $n$. If $i > j$, prove that $a_i$ and $a_j$ have no common prime factor. [b]p4.[/b] A number of points are given in the interior of a triangle. Connect these points, as well as the vertices of the triangle, by segments that do not cross each other until the interior is subdivided into smaller disjoint regions that are all triangles. It is required that each of the givien points is always a vertex of any triangle containing it. Prove that the number of these smaller triangular regions is always odd. [b]p5.[/b] In triangle $ABC$, let $\angle ABC=\angle ACB=40^o$ is extended to $D$ such that $AD=BC$. Prove that $\angle BCD=10^o$. [img]https://cdn.artofproblemsolving.com/attachments/6/c/8abfbf0dc38b76f017b12fa3ec040849e7b2cd.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ M $ be a finite set of integers, and let be a function $ \varphi :\mathbb{Z}\longrightarrow\mathbb{Z} $ whose restriction to $ \mathbb{Z}\setminus M $ evaluates to a constant $ c, $ such that $$ 2\le |\varphi (M)|=|M|\neq \frac{1}{c}\cdot \sum_{\iota \in \varphi (M) } \iota . $$ Prove that $ \varphi $ is not a sum between an injective function and a surjective function. [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

[u]Round 1 [/u] [b]p1.[/b] In the convex quadrilateral $ABCD$ with diagonals $AC$ and $BD$, you know that angle $BAC$ is congruent to angle $CBD$, and that angle $ACD$ is congruent to angle $ADB$. Show that angle $ABC$ is congruent to angle $ADC$. [img]https://cdn.artofproblemsolving.com/attachments/5/d/41cd120813d5541dc73c5d4a6c86cc82747fcc.png[/img] [b]p2.[/b] In how many different ways can you place $12$ chips in the squares of a $4 \times 4$ chessboard so that (a) there is at most one chip in each square, and (b) every row and every column contains exactly three chips. [b]p3.[/b] Students from Hufflepuff and Ravenclaw were split into pairs consisting of one student from each house. The pairs of students were sent to Honeydukes to get candy for Father's Day. For each pair of students, either the Hufflepuff student brought back twice as many pieces of candy as the Ravenclaw student or the Ravenclaw student brought back twice as many pieces of candy as the Hufflepuff student. When they returned, Professor Trelawney determined that the students had brought back a total of $1000$ pieces of candy. Could she have possibly been right? Why or why not? Assume that candy only comes in whole pieces (cannot be divided into parts). [b]p4.[/b] While you are on a hike across Deception Pass, you encounter an evil troll, who will not let you across the bridge until you solve the following puzzle. There are six stones, two colored red, two colored yellow, and two colored green. Aside from their colors, all six stones look and feel exactly the same. Unfortunately, in each colored pair, one stone is slightly heavier than the other. Each of the lighter stones has the same weight, and each of the heavier stones has the same weight. Using a balance scale to make TWO measurements, decide which stone of each color is the lighter one. [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2 [/u] [b]p6.[/b] Consider a set of finitely many points on the plane such that if we choose any three points $A,B,C$ from the set, then the area of the triangle $ABC$ is less than $1$. Show that all of these points can be covered by a triangle whose area is less than $4$. [b]p7.[/b] A palindrome is a number that is the same when read forward and backward. For example, $1771$ and $23903030932$ are palindromes. Can the number obtained by writing the numbers from $1$ to $n$ in order be a palindrome for some $n > 1$ ? (For example, if $n = 11$, the number obtained is $1234567891011$, which is not a palindrome.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that for polynomials \( P, Q, R \) with positive integer coefficients, the following two conditions hold: \(\bullet\) The constant terms of \( P, Q, R \) are equal. \(\bullet\) For all real numbers \( x \), the following relations hold: \[ P(Q(R(x))) = Q(R(P(x))) = R(P(Q(x))) = P(R(Q(x))) = Q(P(R(x))) = R(Q(P(x))). \] Prove that for every real number \( x \), \( P(x) = Q(x) = R(x) \). Proposed by Soroush Behroozifar & Ali Nazarboland

Prove that, if $\tan (a/2)$ is rational (or else, if $ a$ is an odd multiple of $\pi$ so that $\tan (a/2)$ is not defined), then $\cos a$ and $\sin a$ are rational. And, conversely, if $\cos a$ and $\sin a$ are rational, then $\tan (a/2)$ is rational unless $a$ is an odd multiple of $\pi$ so that $\tan (a/2)$ is not defined.

Given a convex polygon $ A_1A_2 \ldots A_n$ with area $ S$ and a point $ M$ in the same plane, determine the area of polygon $ M_1M_2 \ldots M_n,$ where $ M_i$ is the image of $ M$ under rotation $ R^{\alpha}_{A_i}$ around $ A_i$ by $ \alpha_i, i \equal{} 1, 2, \ldots, n.$

Suppose the polynomial $x^3 - x^2 + bx + c$ has real roots $a, b, c$. What is the square of the minimum value of $abc$?

For a certain complex number $c$, the polynomial \[ P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\] has exactly 4 distinct roots. What is $|c|$? $\textbf{(A) } 2 \qquad \textbf{(B) } \sqrt{6} \qquad \textbf{(C) } 2\sqrt{2} \qquad \textbf{(D) } 3 \qquad \textbf{(E) } \sqrt{10}$

Prove the inequality: \[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\] for positive reals $ a_{1},a_{2},\ldots,a_{n}$. [i]Proposed by Dusan Dukic, Serbia[/i]

Show that there are infinitely many integers $m \ge 2$ such that $m \choose 2$ $= 3$ $n \choose 4$ holds for some integer $n \ge 4$. Give the general form of all such $m$.

Let $\mathbb{Z}_+=\{1,2,3,4...\}$ be the set of all positive integers. Find, with proof, all functions $f : \mathbb{Z}_+ \mapsto \mathbb{Z}_+$ with the property that $$f(x+f(y)+f(f(z)))=z+f(y)+f(f(x))$$ for all positive integers $x,y,z$.

For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$

We consider the real sequence $(x_n)$ defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2x_n$ for $n=0,1,...$ We define the sequence $(y_n)$ by $y_n=x_n^2+2^{n+2}$ for every non negative integer $n$. Prove that for every $n>0$, $y_n$ is the square of an odd integer

Find all functions $\mathbb{R}^+\rightarrow\mathbb{R}^+$ for which we have for all $x,y\in \mathbb{R}^+$ that $f(x)f(yf(x))=f(x+y)$.

Show that there is no function $f:{{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$ such that $f(x+y)>f(x)(1+yf(x))$ for all $x,y\in {{\mathbb{R}}^{+}}$.

Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.

Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$. [i]Evan O' Dorney.[/i]

$x \geq 0$ and $y$ are real numbers for which $y^2 \geq x(x + 1)$. Prove that: $(y - 1)^2 \geq x(x-1)$.

[b]p1.[/b] It’s currently $6:00$ on a $12$ hour clock. What time will be shown on the clock $100$ hours from now? Express your answer in the form hh : mm. [b]p2.[/b] A tub originally contains $10$ gallons of water. Alex adds some water, increasing the amount of water by 20%. Barbara, unhappy with Alex’s decision, decides to remove $20\%$ of the water currently in the tub. How much water, in gallons, is left in the tub? Express your answer as an exact decimal. [b]p3.[/b] There are $2000$ math students and $4000$ CS students at Berkeley. If $5580$ students are either math students or CS students, then how many of them are studying both math and CS? [b]p4.[/b] Determine the smallest integer $x$ greater than $1$ such that $x^2$ is one more than a multiple of $7$. [b]p5.[/b] Find two positive integers $x, y$ greater than $1$ whose product equals the following sum: $$9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29.$$ Express your answer as an ordered pair $(x, y)$ with $x \le y$. [b]p6.[/b] The average walking speed of a cow is $5$ meters per hour. If it takes the cow an entire day to walk around the edges of a perfect square, then determine the area (in square meters) of this square. [b]p7.[/b] Consider the cube below. If the length of the diagonal $AB$ is $3\sqrt3$, determine the volume of the cube. [img]https://cdn.artofproblemsolving.com/attachments/4/d/3a6fdf587c12f2e4637a029f38444914e161ac.png[/img] [b]p8.[/b] I have $18$ socks in my drawer, $6$ colored red, $8$ colored blue and $4$ colored green. If I close my eyes and grab a bunch of socks, how many socks must I grab to guarantee there will be two pairs of matching socks? [b]p9.[/b] Define the operation $a @ b$ to be $3 + ab + a + 2b$. There exists a number $x$ such that $x @ b = 1$ for all $b$. Find $x$. [b]p10.[/b] Compute the units digit of $2017^{(2017^2)}$. [b]p11.[/b] The distinct rational numbers $-\sqrt{-x}$, $x$, and $-x$ form an arithmetic sequence in that order. Determine the value of $x$. [b]p12.[/b] Let $y = x^2 + bx + c$ be a quadratic function that has only one root. If $b$ is positive, find $\frac{b+2}{\sqrt{c}+1}$. [b]p13.[/b] Alice, Bob, and four other people sit themselves around a circular table. What is the probability that Alice does not sit to the left or right of Bob? [b]p14.[/b] Let $f(x) = |x - 8|$. Let $p$ be the sum of all the values of $x$ such that $f(f(f(x))) = 2$ and $q$ be the minimum solution to $f(f(f(x))) = 2$. Compute $p \cdot q$. [b]p15.[/b] Determine the total number of rectangles ($1 \times 1$, $1 \times 2$, $2 \times 2$, etc.) formed by the lines in the figure below: $ \begin{tabular}{ | l | c | c | r| } \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline \end{tabular} $ [b]p16.[/b] Take a square $ABCD$ of side length $1$, and let $P$ be the midpoint of $AB$. Fold the square so that point $D$ touches $P$, and let the intersection of the bottom edge $DC$ with the right edge be $Q$. What is $BQ$? [img]https://cdn.artofproblemsolving.com/attachments/1/1/aeed2c501e34a40a8a786f6bb60922b614a36d.png[/img] [b]p17.[/b] Let $A$, $B$, and $k$ be integers, where $k$ is positive and the greatest common divisor of $A$, $B$, and $k$ is $1$. Define $x\# y$ by the formula $x\# y = \frac{Ax+By}{kxy}$ . If $8\# 4 = \frac12$ and $3\# 1 = \frac{13}{6}$ , determine the sum $A + B + k$. [b]p18.[/b] There are $20$ indistinguishable balls to be placed into bins $A$, $B$, $C$, $D$, and $E$. Each bin must have at least $2$ balls inside of it. How many ways can the balls be placed into the bins, if each ball must be placed in a bin? [b]p19.[/b] Let $T_i$ be a sequence of equilateral triangles such that (a) $T_1$ is an equilateral triangle with side length 1. (b) $T_{i+1}$ is inscribed in the circle inscribed in triangle $T_i$ for $i \ge 1$. Find $$\sum^{\infty}_{i=1} Area (T_i).$$ [b]p20.[/b] A [i]gorgeous [/i] sequence is a sequence of $1$’s and $0$’s such that there are no consecutive $1$’s. For instance, the set of all gorgeous sequences of length $3$ is $\{[1, 0, 0]$,$ [1, 0, 1]$, $[0, 1, 0]$, $[0, 0, 1]$, $[0, 0, 0]\}$. Determine the number of gorgeous sequences of length $7$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n>1$ be an integer. Find all $r\in \mathbb{R}$ so that the system of equations in real variables $x_1, x_2, \dots, x_n$: \begin{align*} &(r\cdot x_1-x_2)(r\cdot x_1-x_3)\dots (r\cdot x_1-x_n)=\\ =&(r\cdot x_2-x_1)(r\cdot x_2-x_3)\dots (r\cdot x_2-x_n)=\\ &\qquad \qquad \qquad \qquad \vdots \\ =&(r\cdot x_n-x_1)(r\cdot x_n-x_2)\dots (r\cdot x_n-x_{n-1}) \end{align*} has a solution where the numbers $x_1, x_2, \dots, x_n$ are pairwise distinct.

Given $n$ a positive integer, define $T_n$ the number of quadruples of positive integers $(a,b,x,y)$ such that $a>b$ and $n= ax+by$. Prove that $T_{2023}$ is odd.