$ABCD$ is a convex quadrilateral with perimeter $p$. Prove that \[ \dfrac{1}{2}p < AC + BD < p. \] (A polygon is convex if all of its interior angles are less than $180^\circ$.)

Evaluate $ \int_0^\frac{\pi}{4} \frac{x\plus{}\sin x}{1\plus{}\cos x}\ dx$.

$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$

Let a positive integer $n$.Consider square table $3*3$.One use $n$ colors to color all cell of table such that each cell is colored by exactly one color. Two colored table is same if we can receive them from other by a rotation through center of $3*3$ table How many way to color this square table satifies above conditions.

Let $(a,b,c)$ be a Pythagorean triple, i.e. a triplet of positive integers with $ a^2\plus{}b^2\equal{}c^2$. $a)$ Prove that $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2>8$. $b)$ Prove that there are no integer $n$ and Pythagorean triple $(a,b,c)$ satisfying $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2\equal{}n$.

In a convex quadrilateral $ABCD$, We have $\angle{B} = \angle{D} = 120^{\circ}$. Points $A'$, $B'$ and $C'$ are symmetric to $D$ relative to $BC$, $CA$ and $AB$, respectively. Prove that lines $AA'$, $BB'$ and $CC'$ are concurrent.

Let $a,b,c$ be reals and \[f(a, b, c) = \left| \frac{ |b-a|}{|ab|} +\frac{b+a}{ab} -\frac 2c \right| +\frac{ |b-a|}{|ab|} +\frac{b+a}{ab} +\frac 2c\] Prove that $f(a, b, c) = 4 \max \{\frac 1a, \frac 1b,\frac 1c \}.$

Two disjoint circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively lie on the same side of a line $p$ and touch the line at $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects $k_1$ at $B_1$ and $k_2$ at $B_2$. Prove that $A_1B_1\perp A_2B_2$.

Triangle $ABC$ has $AB = 40$, $AC = 31$, and $\sin A = \tfrac15$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$. Find the maximum possible area of $AQRS$.

A square with side $10^6$, with a corner square with side $10^{-3}$ cut off, is partitioned into $10$ rectangles. Prove that at least one of these rectangles has the ratio of the greater side to the smaller one at least $9$.

The numbers $a_1,a_2,...,a_n$ ( $n\ge 3$) satisfy the relations $$a_1=a_n = 0, a_{k-1}+ a_{k+1}\le 2a_k \,\,\, (k = 2, 3,..., n-1)$$ Prove that the numbers $a_1,a_2,...,a_n$ are non-negative.

If $x$ is real and positive and grows beyond all bounds, then $\log_3{(6x-5)}-\log_3{(2x+1)}$ approaches: $\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ \text{no finite number}$

Let $ABCD$ be a cyclic quadrilateral with circumcentre $O,$ and the pair of opposite sides not parallel with each other. Let $M=AB\cap CD$ and $N=AD\cap BC.$ Denote, by $P,Q,S,T;$ the intersection of the internal angle bisectors of $\angle MAN$ and $\angle MBN;$ $\angle MBN$ and $\angle MCN;$ $\angle MDN$ and $\angle MAN;$ $\angle MCN$ and $\angle MDN.$ Suppose that the four points $P,Q,S,T$ are distinct. (a) Show that the four points $P,Q,S,T$ are concyclic. Find the centre of this circle, and denote it as $I.$ (b) Let $E=AC\cap BD.$ Prove that $E,O,I$ are collinear.

Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD > BC$. Let $E$ be a point on the side $AC$ such that $\frac{AE}{EC} = \frac{BD}{AD-BC}$. Show that $AD > BE$.

Define a "ternary sequence" is a sequence that every number is $0,1$ or $2$. ternary sequence $(x_1,x_2,x_3,\cdots,x_n)$, define its difference to be $$(|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|)$$ A difference will make the length of the sequence decrease by $1$, so we define the "feature value" of a ternary sequence with length $n$ is the number left after $n-1$ differences. How many ternary sequences has length $2023$ and feature value $0$? [i]Proposed by CSJL[/i]

Prove that the only morphisms from a finite symmetric group to the multiplicative group of rational numbers are the identity and the signature.

Let $ABC$ be a triangle with orthocenter $H$; suppose $AB=13$, $BC=14$, $CA=15$. Let $G_A$ be the centroid of triangle $HBC$, and define $G_B$, $G_C$ similarly. Determine the area of triangle $G_AG_BG_C$.

Let A be a symmetric matrix such that the sum of elements of any row is zero. Show that all elements in the main diagonal of cofator matrix of A are equal.

We will call a positive integer [i]good [/i] if all its digits are nonzero. A good integer will be called [i]special [/i] if it has at least $k$ digits and their values strictly increase from left to right. Let a good integer be given. At each move, one may either add some special integer to its digital expression from the left or from the right, or insert a special integer between any two its digits, or remove a special number from its digital expression.What is the largest $k$ such that any good integer can be turned into any other good integer by such moves?

$n$ is a fixed natural number. Find the least $k$ such that for every set $A$ of $k$ natural numbers, there exists a subset of $A$ with an even number of elements which the sum of it's members is divisible by $n$.

In a class among 80 students number of boys is 40 and number of girls is 40. 50 of the students use spectacles. Which of the following is correct? [list=1] [*] Only 10 boys use spectacles [*] Only 20 girls use spectacles [*] At most 25 boys do not use spectacles [*] At most 30 girls do not use spectacles [/list]

Each of the numbers $x_1, x_2, \dots, x_n$ equals $1$ or $-1$ and \[\sum_{i=1}^n x_i x_{i+1} x_{i+2} x_{i+3} =0.\] where $x_{n+i}=x_i $ for all $i$. Prove that $4\mid n$.

Let $ABC$ be an acute-angled triangle, and let $F$ be the foot of its altitude from the vertex $C$. Let $M$ be the midpoint of the segment $CA$. Assume that $CF=BM$. Then the angle $MBC$ is equal to angle $FCA$ if and only if the triangle $ABC$ is equilateral.

A chocolate bar has five lengthwise dents and eight crosswise ones, which can be used to break up the bar into sections (one can get a total of $ 9 \times 6 = 54$ cells). Two players play the following game with such a bar. At each move (the two players move alternatively) one player breaks off a section of width one from the bar along a single dent and eats it, the other player does the same with what’s left of the bar, and so on. When one of the players breaks up a section of width two into two strips of width one, he eats one of the strips and the other player eats the other strip. Prove that the player who has the first move can play so as to eat at least $6$ cells more than his opponent (no matter how his opponent plays). (R Fedorov)

Let $ABC$ be a triangle and $A', B', C'$ the points in which its incircle touches the sides $BC, CA, AB$, respectively. We denote by $I$ the incenter and by $P$ its projection onto $AA' $. Let $M$ be the midpoint of the line segment $[A'B']$ and $N$ be the intersection point of the lines $MP$ and $AC$. Prove that $A'N $is parallel to $B'C'$