Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying \[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\] for all $x, y\in \mathbb{R}^{+}$.

If $x,y,z$ satisfies $x^2+y^2+z^2=1$, find the maximum possible value of $$(x^2-yz)(y^2-zx)(z^2-xy)$$

Let $S$ be the set of all positive integers between 1 and 2017, inclusive. Suppose that the least common multiple of all elements in $S$ is $L$. Find the number of elements in $S$ that do not divide $\frac{L}{2016}$. [i]Proposed by Yannick Yao[/i]

Find the smallest positive integer $n$ that satisfies the following: We can color each positive integer with one of $n$ colors such that the equation $w + 6x = 2y + 3z$ has no solutions in positive integers with all of $w, x, y$ and $z$ having the same color. (Note that $w, x, y$ and $z$ need not be distinct.)

The real numbers $a,b,c$ satisfy the condition: for all $x$, such that for $ -1 \le x \le 1$, the inequality $$| ax^2 + bx + c | \le 1$$ is held. Prove that for the same $x$ , $$| cx^2 + bx + a | \le 2$$

We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$. Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$. Is there always a number $x$ that satisfies all the equations? [i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]

Let $ABC$ be a triangle with $\angle A = 90^o$ and let $D$ be an orthogonal projection of $A$ onto $BC$. The midpoints of $AD$ and $AC$ are called $E$ and $F$, respectively. Let $M$ be the circumcentre of $\vartriangle BEF$. Prove that $AC\parallel BM$.

A complete graph is in a plane such that no three of its vertices are collinear. The edges of the graph, which are straight segments connecting the vertices, are colored with two colors. Prove that there is a non-self-intersecting spanning tree consisting of edges of the same color.

Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\overline{AC}$ and points $D$ and $F$ are on $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have the same perimeter. What is $DE+FG$? [asy] size(180); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); real s=1/2,m=5/6,l=1; pair A=origin,B=(l,0),C=rotate(60)*l,D=(s,0),E=rotate(60)*s,F=m,G=rotate(60)*m; draw(A--B--C--cycle^^D--E^^F--G); dot(A^^B^^C^^D^^E^^F^^G); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,S); label("$E$",E,NW); label("$F$",F,S); label("$G$",G,NW); [/asy] $\textbf{(A) }1\qquad \textbf{(B) }\dfrac{3}{2}\qquad \textbf{(C) }\dfrac{21}{13}\qquad \textbf{(D) }\dfrac{13}{8}\qquad \textbf{(E) }\dfrac{5}{3}\qquad$

The five numbers $17$, $98$, $39$, $54$, and $n$ have a mean equal to $n$. Find $n$.

Find $26$ elements of $\{1, 2, 3, ... , 40\}$ such that the product of two of them is never a square. Show that one cannot find $27$ such elements.

A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$. [i]Albania[/i]

Let $X$ be an interior point of a rectangle $ABCD$. Let the bisectors of $\angle DAX$ and $\angle CBX$ intersect in $P$. A point $Q$ satisfies $\angle QAP=\angle QBP=90^\circ$. Show that $PX=QX$.

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

Nestor ordered Juan to do the following work: draw a circle, draw one of its diameters and mark the extreme points of the diameter with the numbers 1 and 2 respectively. Place 100 points in each of the semicircles that determines the diameter layout (different from the ends of the diameter) and mark these points randomly with the numbers $1$ and $2$. To finish, paint red all small segments that have different markings on their extremes. After a certain amount of time passed, Juan finished the work and told Nestor that “he painted 47 segments red.” Prove that if Juan made no mistakes, what he said is false.

Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.

How many positive perfect squares less than $2023$ are divisible by $5$? $\textbf{(A) } 8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad\textbf{(E) } 12$

Find all pairs of positive integers $(a, b)$ such that $f(x)=x$ is the only function $f:\mathbb{R}\to \mathbb{R}$ that satisfies $$f^a(x)f^b(y)+f^b(x)f^a(y)=2xy$$ for all $x, y\in \mathbb{R}$.

Find the smallest possible distance of points $ P$ and $ Q$ on a $ xy$-plane, if $ P$ lies on the line $ y \equal{} x$ and $ Q$ lies on the curve $ y \equal{} 2^x$.

A bus route consists of a circular road of circumference 10 miles and a straight road of length 1 mile which runs from a terminus to the point $Q$ on the circular road (see diagram). [img]6763[/img] It is served by two buses, each of which requires 20 minutes for the round trip. Bus No. 1, upon leaving the terminus, travels along the straight road, once around the circle clockwise and returns along the straight road to the terminus. Bus No. 2, reaching the terminus 10 minutes after Bus No. 1, has a similar route except that it proceeds counterclockwise around the circle. Both buses run continuously and do not wait at any point on the route except for a negligible amount of time to pick up and discharge passengers. A man plans to wait at a point $P$ which is $x$ miles ($0\le x < 12$) from the terminus along the route of Bus No. 1 and travel to the terminus on one of the buses. Assuming that he chooses to board that bus which will bring him to his destination at the earliest moment, there is a maximum time $w(x)$ that his journey (waiting plus travel time) could take. Find $w(2)$; find $w(4)$. For what value of $x$ will the time $w(x)$ be the longest? Sketch a graph of $y = w(x)$ for $0\le x < 12$.

Let us denote by $b(n)$ the number of ways to represent $n$ in the form $$n = a_0+a_1 \cdot 2 +a_2 \cdot 2^2+...+ a_k \cdot 2^k,$$ where the coefficients at, $r = 1$,$2$,$...$, $k$ can be equal to $0$, $1$ or $2$. Find $b(1996)$.

Let $ABC$ be a triangle with $BA=BC$ and $\angle ABC=90^{\circ}$. Let $D$ and $E$ be the midpoints of $CA$ and $BA$ respectively. The point $F$ is inside of $\triangle ABC$ such that $\triangle DEF$ is equilateral. Let $X=BF\cap AC$ and $Y=AF\cap DB$. Prove that $DX=YD$.

Find the number of ordered pairs of integers $(a, b)$ such that $a, b$ are divisors of 720 but $ab$ is not.

Let $A$ be a set of seven positive numbers. Determine the maximal number of triples $(x, y, z)$ of elements of $A$ satisfying $x < y$ and $x + y = z$.