The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $\gcd (F_{m}, F_{n})=F_{\gcd (m, n)}$ for all $m, n \in \mathbb{N}$.

Let $n$ be a fixed positive integer. Ben is playing a computer game. The computer picks a tree $T$ such that no vertex of $T$ has degree $2$ and such that $T$ has exactly $n$ leaves, labeled $v_1,\ldots, v_n$. The computer then puts an integer weight on each edge of $T$, and shows Ben neither the tree $T$ nor the weights. Ben can ask queries by specifying two integers $1\leq i < j \leq n$, and the computer will return the sum of the weights on the path from $v_i$ to $v_j$. At any point, Ben can guess whether the tree's weights are all zero. He wins the game if he is correct, and loses if he is incorrect. (a) Show that if Ben asks all $\binom n2$ possible queries, then he can guarantee victory. (b) Does Ben have a strategy to guarantee victory in less than $\binom n2$ queries? [i]Brandon Wang[/i]

Let $ p$ and $ q$ be two coprime positive integers, and $ n$ be a non-negative integer. Determine the number of integers that can be written in the form $ ip \plus{} jq$, where $ i$ and $ j$ are non-negative integers with $ i \plus{} j \leq n$.

Let $ABC$ be a triangle. Find the point $D$ on its side $AC$ and the point $E$ on its side $AB$ such that the area of triangle $ADE$ equals to the area of the quadrilateral $DEBC$, and the segment $DE$ has minimum possible length.

Circles $Q_1$ and $Q_2$ are tangent to each other externally at $T$. Points $A$ and $E$ are on $Q_1$, lines $AB$ and $DE$ are tangent to $Q_2$ at $B$ and $D$, respectively, lines $AE$ and $BD$ meet at point $P$. Prove that (1) $\frac{AB}{AT}=\frac{ED}{ET}$; (2) $\angle ATP + \angle ETP = 180^{\circ}$. [asy]import graph; size(5.97cm); real lsf=0.5; pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-6,xmax=5.94,ymin=-3.19,ymax=3.43; pair Q_1=(-2.5,-0.5), T=(-1.5,-0.5), Q_2=(0.5,-0.5), A=(-2.09,0.41), B=(-0.42,1.28), D=(-0.2,-2.37), P=(-0.52,2.96); D(CR(Q_1,1)); D(CR(Q_2,2)); D(A--B); D((-3.13,-1.27)--D); D(P--(-3.13,-1.27)); D(P--D); D(T--(-3.13,-1.27)); D(T--A); D(T--P); D(Q_1); MP("Q_1",(-2.46,-0.44),NE*lsf); D(T); MP("T",(-1.46,-0.44),NE*lsf); D(Q_2); MP("Q_2",(0.54,-0.44),NE*lsf); D(A); MP("A",(-2.22,0.58),NE*lsf); D(B); MP("B",(-0.35,1.45),NE*lsf); D((-3.13,-1.27)); MP("E",(-3.52,-1.62),NE*lsf); D(D); MP("D",(-0.17,-2.31),NE*lsf); D(P); MP("P",(-0.47,3.02),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

On the plane there are two non-intersecting circles with equal radii and with centres $O_1$ and $O_2$, line $s$ going through these centres, and their common tangent $t$. The third circle is tangent to these two circles in points $K$ and $L$ respectively, line $s$ in point $M$ and line $t$ in point $P$. The point of tangency of line $t$ and the first circle is $N$. a) Find the length of the segment $O_1O_2$. b) Prove that the points $M, K$ and $N$ lie on the same line

Water is flowing out through the smaller base of a hollow conical frustum formed by taking a downwards pointing cone of radius $12\text{m}$ and slicing off the tip of the cone in a cut parallel to the base so that the radius of the cross-section of the slice is $6\text{m}$ (meaning the smaller base has a radius of $6\text{m}$). The height of the frustum is $10\text{m}$. If the height of the water level in the frustum is decreasing at $3\text{m/s}$ and the current height is $5\text{m}$, then the volume of the water in the frustum is decreasing at $d\text{ m}^3\text{/s}$. Compute $d$.

On the sides of a parallelogram, squares are constructed outwards. Prove that the centers of these squares are vertices of a square.

Acute triangle $ABP$, where $AB > BP$, has altitudes $BH$, $PQ$, and $AS$. Let $C$ denote the intersection of lines $QS$ and $AP$, and let $L$ denote the intersection of lines $HS$ and $BC$. If $HS = SL$ and $HL$ is perpendicular to $BC$, find the value of $\frac{SL}{SC}$.

Suppose that the natural number $a, b, c, d$ satisfy the equation $a^ab^{a + b} = c^cd^{c + d}$. (a) If gcd $(a, b) = $ gcd $(c, d) = 1$, prove that $a = c$ and $b = d$. (b) Does the conclusion $a = c$ and $b = d$ apply, without the condition gcd $(a, b) = $ gcd $(c, d) = 1$?

Evaluate $2023^2 -2022^2 +2021^2 -2020^2$.

In $\triangle ABC$, let $AD$ be the angle bisector of $\angle BAC$, with $D$ on $BC$. The perpendicular from $B$ to $AD$ intersects the circumcircle of $\triangle ABD$ at $B$ and $E$. Prove that $E$, $A$ and the circumcenter $O$ of $\triangle ABC$ are collinear.

What is the value of \[ (3x \minus{} 2)(4x \plus{} 1) \minus{} (3x \minus{} 2)4x \plus{} 1\]when $ x \equal{} 4$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 12$

Do there exist positive integers $b,n>1$ such that when $n$ is expressed in base $b$, there are more than $n$ distinct permutations of its digits? For example, when $b=4$ and $n=18$, $18 = 102_4$, but $102$ only has $6$ digit arrangements. (Leading zeros are allowed in the permutations.) [i]Lewis Chen.[/i]

Let $ ABC$ be an isosceles triangle with $ AC \equal{} BC.$ Its incircle touches $ AB$ in $ D$ and $ BC$ in $ E.$ A line distinct of $ AE$ goes through $ A$ and intersects the incircle in $ F$ and $ G.$ Line $ AB$ intersects line $ EF$ and $ EG$ in $ K$ and $ L,$ respectively. Prove that $ DK \equal{} DL.$

Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$

The diagonals $AC,BD$ of the quadrilateral $ABCD$ intersect at $E$. Let $M,N$ be the midpoints of $AB,CD$ respectively. Let the perpendicular bisectors of the segments $AB,CD$ meet at $F$. Suppose that $EF$ meets $BC,AD$ at $P,Q$ respectively. If $MF\cdot CD=NF\cdot AB$ and $DQ\cdot BP=AQ\cdot CP$, prove that $PQ\perp BC$.

a) Show that the set $ \mathbb{Q}^{ + }$ of all positive rationals can be partitioned into three disjoint subsets. $ A,B,C$ satisfying the following conditions: \[ BA = B; \& B^2 = C; \& BC = A; \] where $ HK$ stands for the set $ \{hk: h \in H, k \in K\}$ for any two subsets $ H, K$ of $ \mathbb{Q}^{ + }$ and $ H^2$ stands for $ HH.$ b) Show that all positive rational cubes are in $ A$ for such a partition of $ \mathbb{Q}^{ + }.$ c) Find such a partition $ \mathbb{Q}^{ + } = A \cup B \cup C$ with the property that for no positive integer $ n \leq 34,$ both $ n$ and $ n + 1$ are in $ A,$ that is, \[ \text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34. \]

Let us have $6050$ points in the plane, no three collinear. Find the maximum number $k$ of non-overlapping triangles without common vertices in this plane.

Let $A,B,C$ be points in that order along a line, such that $AB=20$ and $BC=18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_1$ and $\ell_2$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_1$ and $\ell_2$. Let $X$ lie on segment $\overline{KA}$ and $Y$ lie on segment $\overline{KC}$ such that $XY\|BC$ and $XY$ is tangent to $\omega$. What is the largest possible integer length for $XY$?

The square $DEAF$ is constructed inside the $30^o-60^o-90^o$ triangle $ABC$, with the hypotenuse $BC = 4$, $D$ on side $BC$, E on side $AC$, and F on side $AB$. What is the side length of the square?

Find all functions $f:\mathbb{Q}^{+} \to \mathbb{Q}^{+}$ such that for all $x\in \mathbb{Q}^+$: [list] [*] $f(x+1)=f(x)+1$, [*] $f(x^2)=f(x)^2$. [/list]

$O$ -circumcenter of $ABCD$. $AC$ and $BD$ intersect in $E$, $AD$ and $BC$ in $F$. $X,Y$ - midpoints of $AD$ and $BC$. $O_1$ -circumcenter of $EXY$. Prove that $OF \parallel O_1E$

The Macedonian Mathematical Olympiad is held in two rooms numbered $1$ and $2$. At the beginning all of the competitors enter room No. $1$. The final arrangement of the competitors to the rooms is obtained in the following way: a list with the names of a few of the competitors is read aloud; after a name is read, the corresponding competitor and all of his/her acquaintances from the rest of the competitors change the room in which they currently are. Hence, to each list of names corresponds one final arrangement of the competitors to the rooms. Show that the total number of possible final arrangements is not equal to $2009$ (acquaintance between competitors is a symmetrical relation).

We say that $(a,b,c)$ form a [i]fantastic triplet[/i] if $a,b,c$ are positive integers, $a,b,c$ form a geometric sequence, and $a,b+1,c$ form an arithmetic sequence. For example, $(2,4,8)$ and $(8,12,18)$ are fantastic triplets. Prove that there exist infinitely many fantastic triplets.