Let $a,b,c$ be positive real numbers with $a \leq c$ and $b \leq c$. Prove that $$ (a +10b)(b +22c)(c +7a) \geq 2024 abc.$$

Determine all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that \[ f(2x+f(y))=x+y +f(f(x)) , \ \ \ \forall x,y \in \mathbb{R}^+.\]

Let $ABC$ be a right-angled triangle and $CH$ be the altitude from its right angle $C$. Points $O_1$ and $O_2$ are the incenters of triangles $ACH$ and $BCH$ respectively, $P_1$ and $P_2$ are the touching points of their incircles with $AC$ and $BC$. Prove that lines $O_1P_1$ and $O_2P_2$ meet on $AB$.

A polynomial $ Q (x) = k_n x ^ n + k_ {n-1} x ^ {n-1} + \ldots + k_1 x + k_0 $ with real coefficients is called [i]powerful[/i] if the equality $ | k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n | $, and [i]non-increasing[/i] , if $ k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n $. Let for the polynomial $ P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0 $ with nonzero real coefficients, where $ a_d> 0 $, the polynomial $ P (x) (x-1) ^ t (x + 1) ^ s $ is [i]powerful[/i] for some non-negative integers $ s $ and $ t $ ($ s + t> 0 $). Prove that at least one of the polynomials $ P (x) $ and $ (- 1) ^ d P (-x) $ is [i]nonincreasing[/i].

Let $S$ denote the set of fractions $\dfrac mn$ for relatively prime positive integers $m$ and $n$ with $m+n\le 10000$. The least fraction in $S$ that is strictly greater than \[\prod_{i=0}^\infty \left(1-\dfrac{1}{10^{2i+1}}\right)\] can be expressed in the form $\dfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $1000p+q$. [i]Proposed by James Lin[/i]

A cyclic quadrilateral $ABCD$ has circumcircle $\Gamma$, and $AB+BC=AD+DC$. Let $E$ be the midpoint of arc $BCD$, and $F (\neq C)$ be the antipode of $A$ [i]wrt[/i] $\Gamma$. Let $I,J,K$ be the incenter of $\triangle ABC$, the $A$-excenter of $\triangle ABC$, the incenter of $\triangle BCD$, respectively. Suppose that a point $P$ satisfies $\triangle BIC \stackrel{+}{\sim} \triangle KPJ$. Prove that $EK$ and $PF$ intersect on $\Gamma.$

Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 14$

Find the greatest possible number of lines in space that all pass through a single point and the angle between any two of them is the same.

A square is partitioned in $n^2\geq 4$ rectanles using $2(n-1)$ lines,$n-1$ of which,are parallel to the one side of the square,$n-1$ are parallel to the other side.Prove that we can choose $2n$ rectangles of the partition,such that,for each two of them,we can place the one inside the other (possibly with rotation).

Find all pairs $(m, n)$ of positive integers such that $m^2 + n^2 =(m + 1)(n + 1).$

Find the number of integers $a$ with $1\le a\le 2012$ for which there exist nonnegative integers $x,y,z$ satisfying the equation \[x^2(x^2+2z) - y^2(y^2+2z)=a.\] [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]$x,y,z$ are not necessarily distinct.[/list][/hide]

The extensions of the sides $AB$ and $CD$ of the trapezoid $ABCD$ intersect at point $E$. Denote by $H$ and $G$ the midpoints of $BD$ and $AC$. Find the ratio of the area $AEGH$ to the area $ABCD$.

[b]a)[/b] Let be real numbers $ s,t\ge 0 $ and $ a,b\ge 1. $ Show that for any real $ x, $ it holds: $$ a^{s\sin x+t\cos x}b^{s\cos x+t\sin x}\le 10^{(s+t)\sqrt{\text{tg}^2 a+\text{tg}^2 b}} $$ [b]b)[/b] For $ a,b>0 $ is the above inequality still true? [i]Ilie Diaconu[/i]

[b]p1.[/b] Fill in each box with an integer from $1$ to $9$. Each number in the right column is the product of the numbers in its row, and each number in the bottom row is the product of the numbers in its column. Some numbers may be used more than once, and not every number from $1$ to $9$ is required to be used. [img]https://cdn.artofproblemsolving.com/attachments/c/0/0212181d87f89aac374f75f1f0bde6d0600037.png[/img] [b]p2.[/b] A set $X$ is called [b]prime-difference free [/b] (henceforth pdf) if for all $x, y \in X$, $|x - y|$ is not prime. Find the number n such that the following both hold. $\bullet$ There is a pdf set of size $n$ that is a subset of $\{1,..., 2016\}$, and $\bullet$ There is no pdf set of size $n + 1$ that is a subset of $\{1,..., 2016\}$. [b]p3.[/b] Let $X_1,...,X_{15}$ be a sequence of points in the $xy$-plane such that $X_1 = (10, 0)$ and $X_{15} = (0, 10)$. Prove that for some $i \in \{1, 2,..., 14\}$, the midpoint of $X_iX_{i+1}$ is of distance greater than $1/2$ from the origin. [b]p4.[/b] Suppose that $s_1, s_2,..., s_{84}$ is a sequence of letters from the set $\{A,B,C\}$ such that every four-letter sequence from $\{A,B,C\}$ occurs exactly once as a consecutive subsequence $s_k$, $s_{k+1}$, $s_{k+2}$, $s_{k+3}$. Suppose that $$(s_1, s_2, s_3, s_4, s_5) = (A,B,B,C,A).$$ What is $s_{84}$? Prove your answer. [b]p5.[/b] Determine (with proof) whether or not there exists a sequence of positive real numbers $a_1, a_2, a_3,...$ with both of the following properties: $\bullet$ $\sum^n_{i=1} a_i \le n^2$, for all $n \ge 1$, and $\bullet$ $\sum^n_{i=1} \frac{1}{a_i} \le 2016$, for all $n \ge 1$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$

For given positive integer $a$, find every $(x_1, x_2, …, x_{2002})$ that satisfies the following: (1) $x_1 \geq x_2 \geq … \geq x_{2002} \geq 0$ (2) $0< x_1+x_2+…+x_{2003}<a+1$ (3) $ x^2_1+x^2_2+…+x^2_{2003}+9=a^2$