A number $ x$ is uniformly chosen on the interval $ [0,1]$, and $ y$ is uniformly randomly chosen on $ [\minus{}1,1]$. Find the probability that $ x>y$.

Let $ G=(V,E)$ be a simple graph. a) Let $ A,B$ be a subsets of $ E$, and spanning subgraphs of $ G$ with edges $ A,B,A\cup B$ and $ A\cap B$ have $ a,b,c$ and $ d$ connected components respectively. Prove that $ a+b\leq c+d$. We say that subsets $ A_1,A_2,\dots,A_m$ of $ E$ have $ (R)$ property if and only if for each $ I\subset\{1,2,\dots,m\}$ the spanning subgraph of $ G$ with edges $ \cup_{i\in I}A_i$ has at most $ n-|I|$ connected components. b) Prove that when $ A_1,\dots,A_m,B$ have $ (R)$ property, and $ |B|\geq2$, there exists an $ x\in B$ such that $ A_1,A_2,\dots,A_m,B\backslash\{x\}$ also have property $ (R)$. Suppose that edges of $ G$ are colored arbitrarily. A spanning subtree in $ G$ is called colorful if and only if it does not have any two edges with the same color. c) Prove that $ G$ has a colorful subtree if and only if for each partition of $ V$ to $ k$ non-empty subsets such as $ V_1,\dots,V_k$, there are at least $ k\minus{}1$ edges with distinct colors that each of these edges has its two ends in two different $ V_i$s. d) Assume that edges of $ K_n$ has been colored such that each color is repeated $ \left[\frac n2\right]$ times. Prove that there exists a colorful subtree. e) Prove that in part d) if $ n\geq5$ there is a colorful subtree that is non-isomorphic to $ K_{1,n-1}$. f) Prove that in part e) there are at least two non-intersecting colorful subtrees.

An observer stands on the side of the front of a stationary train. When the train starts moving with constant acceleration, it takes $5$ seconds for the first car to pass the observer. How long will it take for the $10\text{th}$ car to pass? $\textbf{(A)} \hspace{1mm} 1.07s\\ \textbf{(B)} \hspace{1mm } 0.98s\\ \textbf{(C)}\hspace{1mm} 0.91s\\ \textbf{(D)}\hspace{1mm} 0.86s\\ \textbf{(E)}\hspace{1mm} 0.81s$

Given triangle $ABC$ and the points$D,E\in \left( BC \right)$, $F,G\in \left( CA \right)$, $H,I\in \left( AB \right)$ so that $BD=CE$, $CF=AG$ and $AH=BI$. Note with $M,N,P$ the midpoints of $\left[ GH \right]$, $\left[ DI \right]$ and $\left[ EF \right]$ and with ${M}'$ the intersection of the segments $AM$and $BC$. a) Prove that $\frac{B{M}'}{C{M}'}=\frac{AG}{AH}\cdot \frac{AB}{AC}$. b) Prove that the segments$AM$, $BN$ and $CP$ are concurrent.

Let $a<b<c<d$ be integers. If one of the roots of the equation $(x-a)(x-b)(x-c)(x-d)-9$ is $x=7$, what is $a+b+c+d$? $ \textbf{(A)}\ 14 \qquad\textbf{(B)}\ 21 \qquad\textbf{(C)}\ 28 \qquad\textbf{(D)}\ 42 \qquad\textbf{(E)}\ 63 $

Let be four real numbers $ x,y,z,t $ satisfying the following system: $$ \left\{ \begin{matrix} \sin x+\sin y+\sin z +\sin t =0 \\ \cos x+\cos y+\cos z+\cos t=0 \end{matrix} \right. $$ Prove that $$ \sin ((1+2k)x) +\sin ((1+2k)y) +\sin ((1+2k)z) +\sin ((1+2k)t) =0, $$ for any integer $ k. $ [i]Aurel Bârsan[/i]

Call a positive integer $k$ $\textit{pretty}$ if for every positive integer $a$, there exists an integer $n$ such that $n^2+n+k$ is divisible by $2^a$ but not $2^{a+1}$. Find the remainder when the $2021$st pretty number is divided by $1000$. [i]Proposed by i3435[/i]

We say that a positive integer $n$ is $m$[i]-expressible[/i] if it is possible to get $n$ from some $m$ digits and the six operations $+,-,\times,\div$, exponentiation $^\wedge$, and concatenation $\oplus$. For example, $5625$ is $3$-expressible (in two ways): both $5\oplus (5^\wedge 4)$ and $(7\oplus 5)^\wedge 2$ yield $5625$. Does there exist a positive integer $N$ such that all positive integers with $N$ digits are $(N-1)$-expressible? [i]Proposed by Krit Boonsiriseth[/i]

Let $ABC$ be a triangle with $\angle BAC = 90^o$ and $D$ be the point on the side $BC$ such that $AD \perp BC$. Let$ r, r_1$, and $r_2$ be the inradii of triangles $ABC, ABD$, and $ACD$, respectively. If $r, r_1$, and $r_2$ are positive integers and one of them is $5$, find the largest possible value of $r+r_1+ r_2$.

At the $ABC$ triangle the midpoints of $BC, AC, AB$ are respectively $D, E, F$ and the triangle tangent to the incircle at $G$, $H$ and $I$ in the same order.The midpoint of $AD$ is $J$. $BJ$ and $AG$ intersect at point $K$. The $C-$centered circle passing through $A$ cuts the $[CB$ ray at point $X$. The line passing through $K$ and parallel to the $BC$ and $AX$ meet at $U$. $IU$ and $BC$ intersect at the $P$ point. There is $Y$ point chosen at incircle. $PY$ is tangent to incircle at point $Y$. Prove that $D, E, F, Y$ are cyclic.

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

For any $k \in \mathbb{Z},$ define $$F_k=X^4+2(1-k)X^2+(1+k)^2.$$ Find all values $k \in \mathbb{Z}$ such that $F_k$ is irreducible over $\mathbb{Z}$ and reducible over $\mathbb{Z}_p,$ for any prime $p.$ [i]Marius Vladoiu[/i]

A convex $ABCDE$ is inscribed in a unit circle, $AE$ being its diameter. If $AB = a$, $BC = b$, $CD = c$, $DE = d$ and $ab = cd =\frac{1}{4}$, compute $AC + CE$ in terms of $a, b, c, d.$

Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that \[0<|xy-zw|<C\alpha ^{-4}\] where $\alpha =\frac{|X|}{n}$.

Find the product of all values of $d$ such that $x^{3} +2x^{2} +3x +4 = 0$ and $x^{2} +dx +3 = 0$ have a common root.

Let $N \geq 2$ be an integer, and let $\mathbf a$ $= (a_1, \ldots, a_N)$ and $\mathbf b$ $= (b_1, \ldots b_N)$ be sequences of non-negative integers. For each integer $i \not \in \{1, \ldots, N\}$, let $a_i = a_k$ and $b_i = b_k$, where $k \in \{1, \ldots, N\}$ is the integer such that $i-k$ is divisible by $n$. We say $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] if each $a_i$ equals the following arithmetic mean: \[a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.\] Suppose that neither $\mathbf a $ nor $\mathbf b$ is a constant sequence, and that both $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] and $\mathbf b$ is $\mathbf a$-[i]harmonic[/i]. Prove that at least $N+1$ of the numbers $a_1, \ldots, a_N,b_1, \ldots, b_N$ are zero.

Determine the minimum value of the expression $2x^4 - 2x^2y^2 + y^4 - 8x^2 + 18$ where $x, y \in R$.

Let $n = 2k + 1$ be an odd positive integer, and $m$ be an integer realtively prime to $n{}$. For each $j =1,2,\ldots,k$ we define $p_j$ as the unique integer from the interval $[-k, k]$ congruent to $m\cdot j$ modulo $n{}$. Prove that there are equally many pairs $(i,j)$ for which $1\leqslant i<j\leqslant k$ which satisfy $|p_i|>|p_j|$ as those which satisfy $p_ip_j<0$.

Find all $x, y, z \in \left (0, \frac{1}{2}\right )$ such that $$ \begin{cases} (3 x^{2}+y^{2}) \sqrt{1-4 z^{2}} \geq z; \\ (3 y^{2}+z^{2}) \sqrt{1-4 x^{2}} \geq x; \\ (3 z^{2}+x^{2}) \sqrt{1-4 y^{2}} \geq y. \end{cases} $$ [i]Proposed by Ngo Van Trang, Vietnam[/i]

In a right-angled triangle $ABC$ ($\angle A = 90^o$), the perpendicular bisector of $BC$ intersects the line $AC$ in $K$ and the perpendicular bisector of $BK$ intersects the line $AB$ in $L$. If the line $CL$ be the internal bisector of angle $C$, find all possible values for angles $B$ and $C$. by Mahdi Etesami Fard

Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has $4$ seats: $1$ Driver seat, $1$ front passenger seat, and $2$ back passenger seat. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there? $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 24$

All points in the plane are colored in $n$ colors. In each line, there are point of no more than two colors. What is the maximum number of colors?

Let a rad number be a palindrome such that the square root of the sum of its digits is irrational. Find the number of $4$-digit rad numbers. [i]Lightning 2.2[/i]

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.

Let $a$ be a real number. Determine, for all $a$, all pairs $(x,y)$ of real numbers such that $(x-y^2)(y-x^2)+x^3+y^3=a $.