Let $a,b$ be real numbers and $k$ a positive integer. Show that $$ \left| \frac{ \cos kb \cos a - \cos ka \cos b}{\cos b -\cos a} \right|<k^2 -1$$ whenever the left side is defined.

A natural number has four natural divisors: $1$, the number itself, and two real divisors. That number plus $9$ is equal to seven times the sum of the true divisors. Determine that number and prove that it is unique.

Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius $30$. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$.

The quadratic polynomial $f(x)$ has the expansion $2x^2 - 3x + r$. What is the largest real value of $r$ for which the ranges of the functions $f(x)$ and $f(f(x))$ are the same set?

Let $a_1, a_2, ...,a_n$ represent an arbitrary arrangement of the numbers $1, 2, ...,n$. Prove that, if $n$ is odd, the product $$(a_1 - 1)(a_2 - 2) ... (a_n -n)$$ is an even number.

Real numbers $a, b, c, d$ satisfy $$a^2+b^2+c^2+d^2=4.$$ Find the greatest possible value of $$E(a,b,c,d)=a^4+b^4+c^4+d^4+4(a+b+c+d)^2 .$$

Positive real numbers $a, b$ are such that $a^3 + b^3 = 2$. Show that that $\frac{1}{a}+\frac{1}{b}\ge 2(a^2 - a + 1)(b^2 - b + 1)$.

Let $k$ be a nonzero natural number and $m$ an odd natural number . Prove that there exist a natural number $n$ such that the number $m^n+n^m$ has at least $k$ distinct prime factors.

We define the sequence $x_n$ so that \[x_1=a, x_2=b, x_n=\frac{{x_{n-1}}^2+{x_{n-2}}^2}{x_{n-1}+x_{n-2}} \quad \forall n \geq 3.\] Where $a,b >1$ are relatively prime numbers. Show that $x_n$ is not an integer for $n \geq 3$.

There is a unique quadruple of positive integers $(a,b,c,k)$ such that $c$ is not a perfect square and $a+\sqrt{b+\sqrt{c}}$ is a root of the polynomial $x^4-20x^3+108x^2-kx+9.$ Compute $c.$

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

A $\text{5.0 kg}$ block with a speed of $\text{8.0 m/s}$ travels $\text{2.0 m}$ along a horizontal surface where it makes a head-on, perfectly elastic collision with a $\text{15.0 kg}$ block which is at rest. The coefficient of kinetic friction between both blocks and the surface is $0.35$. How far does the $\text{15.0 kg}$ block travel before coming to rest? (A) $\text{0.76 m}$ (B) $\text{1.79 m}$ (C) $\text{2.29 m}$ (D) $\text{3.04 m}$ (E) $\text{9.14 m}$

In the parallelogram $CMNP$ extend the bisectors of angles $MCN$ and $PCN$ and intersect with extensions of sides PN and $MN$ at points $A$ and $B$, respectively. Prove that the bisector of the original angle $C$ of the the parallelogram is perpendicular to $AB$. [img]https://cdn.artofproblemsolving.com/attachments/f/3/fde8ef133758e06b1faf8bdd815056173f9233.png[/img]

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Tyler has entered a buffet line in which he chooses one kind of meat, two different vegetables and one dessert. If the order of food items is not important, how many different meals might he choose? - Meat: beef, chicken, pork - Vegetables: baked beans, corn, potatoes, tomatoes - Dessert: brownies, chocolate cake, chocolate pudding, ice cream $ \text{(A)}\ 4\qquad\text{(B)}\ 24\qquad\text{(C)}\ 72\qquad\text{(D)}\ 80\qquad\text{(E)}\ 144 $

In a triangle $ ABC$, take point $ D$ on $ BC$ such that $ DB \equal{} 14, DA \equal{} 13, DC \equal{} 4$, and the circumcircle of $ ADB$ is congruent to the circumcircle of $ ADC$. What is the area of triangle $ ABC$?

For some positive integer \( n \), Katya wrote the numbers from \( 1 \) to \( 2^n \) in a row in increasing order. Oleksii rearranged Katya's numbers and wrote the new sequence directly below the first row. Then, they calculated the sum of the two numbers in each column. Katya calculated \( N \), the number of powers of two among the results, while Oleksii calculated \( K \), the number of distinct powers of two among the results. What is the maximum possible value of \( N + K \)? [i]Proposed by Oleksii Masalitin[/i]

Let $A_1,B_1,C_1$ are the midpoints of the sides of the triangle $ABC, I$ is the center of the circle inscribed in it. Let $C_2$ be the intersection point of lines $C_1 I$ and $A_1B_1$. Let $C_3$ be the intersection point of lines $CC_2$ and $AB$. Prove that line $IC_3$ is perpendicular to line $AB$. (A. Zaslavsky)

Let $ f(x)\equal{}x^{2}\plus{}|x|$. Prove that $ \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx$.

Let $m,n$ be positive integer numbers. Prove that there exist infinite many couples of positive integer nubmers $(a,b)$ such that \[a+b| am^a+bn^b , \quad\gcd(a,b)=1.\]

Say that a function $f : \{1, 2, . . . , 1001\} \to Z$ is [i]almost [/i] polynomial if there is a polynomial $p(x) = a_{200}x^{200} +... + a_1x + a_0$ such that each an is an integer with $|a_n| \le 201$, and such that $|f(x) - p(x)| \le 1$ for all $x \in \{1, 2, . . . , 1001\}$. Let $N$ be the number of almost polynomial functions. Compute the remainder upon dividing $N$ by $199$.

The sequence ($a_n$) is defined by $a_1 = a_2 = 1$ and $a_{n+2 }= a_{n+1} +a_n +k$, where $k$ is a positive integer. Find the least $k$ for which $a_{1991}$ and $1991$ are not coprime.

[asy]size(180); draw((1,0)--(2,0)--(2,10)--(1,10)--cycle); draw((3,0)--(4,0)--(4,8)--(3,8)--cycle); draw((5,0)--(6,0)--(6,6)--(5,6)--cycle); draw((7,0)--(8,0)--(8,6)--(7,6)--cycle); draw((9,0)--(10,0)--(10,10)--(9,10)--cycle); draw((0,2)--(-0.5,2)); draw((0,4)--(-0.5,4)); draw((0,6)--(-0.5,6)); draw((0,8)--(-0.5,8)); draw((0,10)--(-0.5,10)); draw((0,10)--(0,0)); draw((0,0)--(10,0)); label("1",(-0.5,2),W); label("2",(-0.5,4),W); label("3",(-0.5,6),W); label("4",(-0.5,8),W); label("5",(-0.5,10),W); label("A",(1.5,-0.5),S); label("B",(3.5,-0.5),S); label("C",(5.5,-0.5),S); label("D",(7.5,-0.5),S); label("F",(9.5,-0.5),S); label("Grade",(5,-3),S); label("$\#$ of Students",(-4,5),W);[/asy] The bar graph shows the grades in a mathematics class for the last grading period. If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory? \[ \textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{2}{3} \qquad \textbf{(C)}\ \frac{3}{4} \qquad \textbf{(D)}\ \frac{4}{5} \qquad \textbf{(E)}\ \frac{9}{10} \]

Each edge and each diagonal of the convex $ n$-gon $ (n\geq 3)$ is colored in red or blue. Prove that the vertices of the $ n$-gon can be labeled as $ A_1,A_2,...,A_n$ in such a way that one of the following two conditions is satisfied: (a) all segments $ A_1A_2,A_2A_3,...,A_{n\minus{}1}A_n,A_nA_1$ are of the same colour. (b) there exists a number $ k, 1<k<n$ such that the segments $ A_1A_2,A_2A_3,...,A_{k\minus{}1}A_k$ are blue, and the segments $ A_kA_{k\plus{}1},...,A_{n\minus{}1}A_n,A_nA_1$ are red.

Find the sum of all integers $n$ that fulfill the equation \[2^n(6-n)=8n.\]