Prove that no Fibonacci number can be factored into a product of two smaller Fibonacci numbers, each greater than 1.

There is a shape like this (Attachment down below) Find the number of triangles made by choosing 3 vertex from the 8 vertex in the attachment.

What is the largest prime factor of 8091?

A circle of radius 2 has center at (2,0). A circle of radius 1 has center at (5,0). A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the $y$-intercept of the line? $\text{(A)} \ \sqrt{2}/4 \qquad \text{(B)} \ 8/3 \qquad \text{(C)} \ 1 + \sqrt 3 \qquad \text{(D)} \ 2 \sqrt 2 \qquad \text{(E)} \ 3$

Four siblings ordered an extra large pizza. Alex ate $\frac15$, Beth $\frac13$, and Cyril $\frac14$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed? $\textbf{(A) } \text{Alex, Beth, Cyril, Dan}$ $\textbf{(B) } \text{Beth, Cyril, Alex, Dan}$ $\textbf{(C) } \text{Beth, Cyril, Dan, Alex}$ $\textbf{(D) } \text{Beth, Dan, Cyril, Alex}$ $\textbf{(E) } \text{Dan, Beth, Cyril, Alex}$

The two diagonals of a quadrilateral have lengths $12$ and $9$, and the two diagonals are perpendicular to each other. Find the area of the quadrilateral.

What is the value in simplest form of the following expression? \[\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}\] $\textbf{(A) }5 \qquad \textbf{(B) }4 + \sqrt{7} + \sqrt{10} \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 15 \qquad \textbf{(E) } 4 + 3\sqrt{3} + 2\sqrt{5} + \sqrt{7}$

Let $2=p_1<p_2<\ldots<p_n<\ldots$ be all prime numbers. Prove that for any positive integer $n \geq 3$ there exist at least $p_n+n-1$ prime numbers, that do not exceed $p_1p_2\ldots p_n$ [i]I. Voronovich[/i]

Suppose $JHMT$ is a convex quadrilateral with perimeter $68$ and satisfies $\angle HJT = 120^\circ,$ $HM = 20,$ and $JH + JT = JM > HM.$ Furthermore, $\overrightarrow{JM}$ bisects $\angle HJT.$ Compute $JM.$

A chord $AB$ is drawn in a certain circle. The smaller of the two arcs $AB$ corresponds to a central angle of $120^o$. A tangent $p$ to this arc is drawn. Two circles with radii $R$ and $r$ are constructed, touching this smaller arc $AB$ and straight lines $AB$ and $p$. Find the radius of the original circle.

In a soccer tournament each team plays exactly one game with all others. The winner gets $3$ points, the loser gets $0$ and each team gets $1$ point in case of a draw. It is known that $n$ teams ($n \geq 3$) participated in the tournament and the final classification is given by the arithmetical progression of the points, the last team having only 1 point. [list=a] [*] Prove that this configuration is unattainable when $n=12$ [*] Find all values of $n$ and all configurations when this is possible [/list]

Determine all nonnegative integer solutions of the equation $2^x-2^y = 1$

Find all functions $f : Z_{>0} \to Z_{>0}$ for which $f(n) | f(m) - n$ if and only if $n | m$ for all natural numbers $m$ and $n$.

In trapezoid $ABCD$ holds $AD \mid \mid BC$, $\angle ABC = 30^{\circ}$, $\angle BCD = 60^{\circ}$ and $BC=7$. Let $E$, $M$, $F$ and $N$ be midpoints of sides $AB$, $BC$, $CD$ and $DA$, respectively. If $MN=3$, find $EF$

In the plane, Point $ O$ is the center of the equilateral triangle $ABC$ , Points $P,Q$ such that $\overrightarrow{OQ}=2\overrightarrow{PO}$. Prove that\[|PA|+|PB|+|PC|\le |QA|+|QB|+|QC|.\]

Let $n$ be a fixed positive integer. - Show that there exist real polynomials $p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]$ such that \[(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)\] - Find the least natural number $k$, depending on $n$, such that the above polynomials $p_1, p_2, \cdots, p_k$ exist.

Let $ABC$ be an acute-angled triangle. The external bisector of $\angle BAC$ meets the line $BC$ at point $N$. Let $M$ be the midpoint of $BC$. $P$ and $Q$ are two points on line $AN$ such that, $\angle PMN=\angle MQN=90^{\circ}$. If $PN=5$ and $BC=3$, then the length $QA$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are co-prime positive integers. What is the value of $(a+b)$?

Let $D$ be the foot of perpendicular from $A$ to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle $ABC$. A circle $\omega$ with centre $S$ passes through $A$ and $D$, and it intersects sides $AB$ and $AC$ at $X$ and $Y$ respectively. Let $P$ be the foot of altitude from $A$ to $BC$, and let $M$ be the midpoint of $BC$. Prove that the circumcentre of triangle $XSY$ is equidistant from $P$ and $M$.

Is it true that in any convex $n$-gon with $n > 3$, there exists a vertex and a diagonal passing through this vertex such that the angles of this diagonal with both sides adjacent to this vertex are acute? [i]Proposed by Boris Frenkin - Russia[/i]

Denote $f(x) = x^4 + 2x^3 - 2x^2 - 4x+4$. Prove that there are infinitely many primes $p$ that satisfies the following. For all positive integers $m$, $f(m)$ is not a multiple of $p$.

The quadrilateral $ABCD$ is inscribed in the circle ω. The diagonals $AC$ and $BD$ intersect at the point $O$. On the segments $AO$ and $DO$, the points $E$ and $F$ are chosen, respectively. The straight line $EF$ intersects ω at the points $E_1$ and $F_1$. The circumscribed circles of the triangles $ADE$ and $BCF$ intersect the segment $EF$ at the points $E_2$ and $F_2$ respectively (assume that all the points $E, F, E_1, F_1, E_2$ and $F_2$ are different). Prove that $E_1E_2 = F_1F_2$. $(N. Sedrakyan)$

Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: [list] [*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin. [*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line. [*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line. [/list] Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

$n$ coins lies in the circle. If two neighbour coins lies both head up or both tail up, then we can flip both. How many variants of coins are available that can not be obtained from each other by applying such operations?

Call an arrangement of n not necessarily distinct nonnegative integers in a circle [i]wholesome[/i] when, for any subset of the integers such that no pair of them is adjacent in the circle, their average is an integer. Over all wholesome arrangements of $n$ integers where at least two of them are distinct, let $M(n)$ denote the smallest possible value for the maximum of the integers in the arrangement. What is the largest integer $n < 2023$ such that $M(n+1)$ is strictly greater than $M(n)$?

Find all pairs of primes $(p, q)$ such that $p^3 - q^5 = (p + q)^2$ .