Let $a,b,c\in\mathbb R^{n}, a+b+c=0$ and $\lambda>0$. Prove that \[\prod_{cycle}\frac{|a|+|b|+(2\lambda+1)|c|}{|a|+|b|+|c|}\geq(2\lambda+3)^{3}\]

$f: \mathbb R\longrightarrow\mathbb R^{+}$ is a non-decreasing function. Prove that there is a point $a\in\mathbb R$ that \[f(a+\frac1{f(a)})<2f(a)\]

$a_{n}$ is a sequence that $a_{1}=1,a_{2}=2,a_{3}=3$, and \[a_{n+1}=a_{n}-a_{n-1}+\frac{a_{n}^{2}}{a_{n-2}}\] Prove that for each natural $n$, $a_{n}$ is integer.

$a_{n}$ ($n$ is integer) is a sequence from positive reals that \[a_{n}\geq \frac{a_{n+2}+a_{n+1}+a_{n-1}+a_{n-2}}4\] Prove $a_{n}$ is constant.

$\omega$ is circumcirlce of triangle $ABC$. We draw a line parallel to $BC$ that intersects $AB,AC$ at $E,F$ and intersects $\omega$ at $U,V$. Assume that $M$ is midpoint of $BC$. Let $\omega'$ be circumcircle of $UMV$. We know that $R(ABC)=R(UMV)$. $ME$ and $\omega'$ intersect at $T$, and $FT$ intersects $\omega'$ at $S$. Prove that $EF$ is tangent to circumcircle of $MCS$.

$M$ is midpoint of $BC$.$P$ is an arbitary point on $BC$. $C_{1}$ is tangent to big circle.Suppose radius of $C_{1}$ is $r_{1}$ Radius of $C_{4}$ is equal to radius of $C_{1}$ and $C_{4}$ is tangent to $BC$ at P. $C_{2}$ and $C_{3}$ are tangent to big circle and line $BC$ and circle $C_{4}$. Invalid image file Prove : \[r_{1}+r_{2}+r_{3}=R\] ($R$ radius of big circle)

In triangle $ABC$, $AD$ is angle bisector ($D$ is on $BC$) if $AB+AD=CD$ and $AC+AD=BC$, what are the angles of $ABC$?

Circles $C_{1}$ and $C_{2}$ are tangent to each other at $K$ and are tangent to circle $C$ at $M$ and $N$. External tangent of $C_{1}$ and $C_{2}$ intersect $C$ at $A$ and $B$. $AK$ and $BK$ intersect with circle $C$ at $E$ and $F$ respectively. If AB is diameter of $C$, prove that $EF$ and $MN$ and $OK$ are concurrent. ($O$ is center of circle $C$.)

Let $ M$ and $ N$ be points on the side $ BC$ of triangle $ ABC$, with the point $ M$ lying on the segment $ BN$, such that $ BM = CN$. Let $ P$ and $ Q$ be points on the segments $ AN$ and $ AM$, respectively, such that $ \measuredangle PMC =\measuredangle MAB$ and $ \measuredangle QNB =\measuredangle NAC$. Prove that $ \measuredangle QBC =\measuredangle PCB$.

Click for solution Let's go for a nonstandard proof: The reflection with respect to the perpendicular bisector of the segment BC maps the point B to the point C and vice versa; also, it maps the point M to the point N and vice versa (in fact, the points M and N lie on the segment BC and satisfy BM = CN; this means that they are symmetric to each other with respect to the midpoint of the segment BC, and thus with respect to the perpendicular bisector of the segment BC as well). Let D be the image of the point A under this reflection, and let R be the image of the point P under this reflection. Altogether, the reflection with respect to the perpendicular bisector of the segment BC maps the points B, C, M, N, A, P to the points C, B, N, M, D, R, respectively. Hence, as the point P lies on the segment AN, its image R lies on the segment DM, and we have < RNB = < PMC, < MDB = < NAC and < RBC = < PCB (since reflections are congruence transformations and keep angles invariant - non-directed angles at least). Since the points N and D are the reflections of the points M and A in the perpendicular bisector of the segment BC, the quadrilateral AMND is an isosceles trapezoid; thus, it has a circumcircle. Let the line NQ meet the line BD at a point U, and let the line NR meet the line AB at a point V. Since < UNM = < QNB = < NAC = < MDB = < UDM, the points U, M, N, D lie on one circle, i. e. the point U lies on the circumcircle of the quadrilateral AMND. Since < VNM = < RNB = < PMC = < MAB = < VAM, the points V, M, N, A lie on one circle, i. e. the point V also lies on the circumcircle of the quadrilateral AMND. Altogether, we see that both points U and V lie on the circumcircle of the quadrilateral AMND, so the hexagon AMDUNV is cyclic; thus, according to the Pascal theorem, the points $AM\cap UN$, $MD\cap NV$ and $DU\cap VA$ are collinear. Since $AM\cap UN=Q$, $MD\cap NV=R$ and $DU\cap VA=B$, this means that the points Q, R and B are collinear. Consequently, < QBC = < RBC. Since < RBC = < PCB, this yields < QBC = < PCB, and the problem is solved. Darij

$H,I,O,N$ are orthogonal center, incenter, circumcenter, and Nagelian point of triangle $ABC$. $I_{a},I_{b},I_{c}$ are excenters of $ABC$ corresponding vertices $A,B,C$. $S$ is point that $O$ is midpoint of $HS$. Prove that centroid of triangles $I_{a}I_{b}I_{c}$ and $SIN$ concide.

In an $m\times n$ table there is a policeman in cell $(1,1)$, and there is a thief in cell $(i,j)$. A move is going from a cell to a neighbor (each cell has at most four neighbors). Thief makes the first move, then the policeman moves and ... For which $(i,j)$ the policeman can catch the thief?

We have a bipartite graph $G$ (with parts $X$ and $Y$). We orient each edge arbitrarily. Hessam chooses a vertex at each turn and reverse the orientation of all edges that $v$ is one of their endpoint. Prove that with these steps we can reach to a graph that for each vertex $v$ in part $X$, $\deg^{+}(v)\geq \deg^{-}(v)$ and for each vertex in part $Y$, $\deg^{+}v\leq \deg^{-}v$

$f,g$ are two permutations of set $X=\{1,\dots,n\}$. We say $f,g$ have common points iff there is a $k\in X$ that $f(k)=g(k)$. a) If $m>\frac{n}{2}$, prove that there are $m$ permutations $f_{1},f_{2},\dots,f_{m}$ from $X$ that for each permutation $f\in X$, there is an index $i$ that $f,f_{i}$ have common points. b) Prove that if $m\leq\frac{n}{2}$, we can not find permutations $f_{1},f_{2},\dots,f_{m}$ satisfying the above condition.

A subset $S$ of $\mathbb N$ is eventually linear iff there are $k,N\in\mathbb N$ that for $n>N,n\in S\Longleftrightarrow k|n$. Let $S$ be a subset of $\mathbb N$ that is closed under addition. Prove that $S$ is eventually linear.

Click for solution This is equivalent to: Characterize all submonoids of the monoid $\mathbb{N}_{0}$. Proof: Let $a_{1}, a_{2}\in S$. Then all big enough multiples $m$ of $c_{1}= \gcd(a_{1}, a_{2})$ are of type $m=xa_{1}+ya_{2}$ with $x,y \geq 0$, thus are in $S$. Now if there is an $a_{3}$ that is not a multiple of $\gcd(a_{1}, a_{2})$, we procede to same way to get that all big enough multiples of $\gcd(a_{1},a_{2},a_{3})$ are in $S$. We do this again and again. It stops at some point since the number of different residues $\mod \gcd(a_{1},a_{2},a_{3},..., a_{i})$ strictly decreases.

Let A be be a point outside the circle C, and AB and AC be the two tangents from A to this circle C. Let L be an arbitrary tangent to C that cuts AB and AC in P and Q. A line through P parallel to AC cuts BC in R. Prove that while L varies, QR passes through a fixed point.

Click for solution Suppose $T$ is intersection of $QR$ and $AB$ If the tangent from $T$ to circle (not $AB$ the other) is' not parallel to $AC$ by Brianchon theorem $PR$ , $AC$ and tangent from $T$ to circle are concurrent. But $PR$ is parallel to $AC$ contradiction. So $T$ is constant.

For positive $a,b,c$, \[a^{2}+b^{2}+c^{2}+abc=4\] Prove $a+b+c \leq3$

Find the smallest natural number $n$ that the following statement holds : Let $A$ be a finite subset of $\mathbb R^{2}$. For each $n$ points in $A$ there are two lines including these $n$ points. All of the points lie on two lines.

Find all continious $f: \mathbb R\longrightarrow\mathbb R$ that for any $x,y$ \[f(x)+f(y)+f(xy)=f(x+y+xy)\]

$I$ is incenter of triangle $ABC$. Incircle of $ABC$ touches $AB,AC$ at $X,Y$. $XI$ intersects incircle at $M$. Let $CM\cap AB=X'$. $L$ is a point on the segment $X'C$ that $X'L=CM$. Prove that $A,L,I$ are collinear iff $AB=AC$.

$a_{0}=2,a_{1}=1$ and for $n\geq 1$ we know that : $a_{n+1}=a_{n}+a_{n-1}$ $m$ is an even number and $p$ is prime number such that $p$ divides $a_{m}-2$. Prove that $p$ divides $a_{m+1}-1$.

Excircle of triangle $ABC$ corresponding vertex $A$, is tangent to $BC$ at $P$. $AP$ intersects circumcircle of $ABC$ at $D$. Prove \[r(PCD)=r(PBD)\] whcih $r(PCD)$ and $r(PBD)$ are inradii of triangles $PCD$ and $PBD$.

15000 years ago Tilif ministry in Persia decided to define a code for $n\geq2$ cities. Each code is a sequence of $0,1$ such that no code start with another code. We know that from $2^{m}$ calls from foreign countries to Persia $2^{m-a_{i}}$ of them where from the $i$-th city (So $\sum_{i=1}^{n}\frac1{2^{a_{i}}}=1$). Let $l_{i}$ be length of code assigned to $i$-th city. Prove that $\sum_{i=1}^{n}\frac{l_{i}}{2^{i}}$ is minimum iff $\forall i,\ l_{i}=a_{i}$

Find all polynomials $p$ with real coefficients that if for a real $a$,$p(a)$ is integer then $a$ is integer.

$A,B,C$ are on circle $\mathcal C$. $I$ is incenter of $ABC$ , $D$ is midpoint of arc $BAC$. $W$ is a circle that is tangent to $AB$ and $AC$ and tangent to $\mathcal C$ at $P$. ($W$ is in $\mathcal C$) Prove that $P$ and $I$ and $D$ are on a line.

An ant walks on the interior surface of a cube, he moves on a straight line. If ant reaches to an edge the he moves on a straight line on cube's net. Also if he reaches to a vertex he will return his path. a) Prove that for each beginning point ant can has infinitely many choices for his direction that its path becomes periodic. b) Prove that if if the ant starts from point $A$ and its path is periodic, then for each point $B$ if ant starts with this direction, then his path becomes periodic.