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 \[2^n(6-n)=8n.\]

Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10?$

Two circles intersect in points $A$ and $B$. Their common tangent nearer $B$ touches the circles at points $E$ and $F$, and intersects the extension of $AB$ at the point $M$. The point $K$ is chosen on the extention of $AM$ so that $KM = MA$. The line $KE$ intersects the circle containing $E$ again at the point $C$. The line $KF$ intersects the circle containing $F$ again at the point $D$. Prove that the points $A, C$ and $D$ are collinear.

Two distinct numbers $ a$ and $ b$ are chosen randomly from the set $ \{ 2, 2^2, 2^3, \ldots, 2^{25} \}$. What is the probability that $ \log_{a} b$ is an integer? $ \textbf{(A)}\ \frac {2}{25} \qquad \textbf{(B)}\ \frac {31}{300} \qquad \textbf{(C)}\ \frac {13}{100} \qquad \textbf{(D)}\ \frac {7}{50} \qquad \textbf{(E)}\ \frac {1}{2}$

6. The baker has baked a rectangular pancake. He then cut it into $n^{2}$ rectangles by making $n-1$ horizontal and $n-1$ vertical cuts. Being rounded to the closest integer, the areas of resulting rectangles equal to all positive integers from 1 to $n^{2}$ in some order. For which maximal $n$ could this happen? (Half-integers are rounded upwards.) Georgy Karavaev

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?

The number $17^6$ when written out in base $10$ contains $8$ distinct digits from $1,2,\ldots,9,$ with no repeated digits or zeroes. Compute the missing nonzero digit.

In $\triangle ABC$, shown on the right, let $r$ denote the radius of the inscribed circle, and let $r_A$, $r_B$, and $r_C$ denote the radii of the smaller circles tangent to the inscribed circle and to the sides emanating from $A$, $B$, and $C$, respectively. Prove that $r \leq r_A + r_B + r_C$

Find all the real number triples $(x, y, z)$ satisfying the following system of inequalities under the condition $0 < x, y, z < \sqrt{2}$: $$y\sqrt{4-x^2y^2}\ge \frac{2}{\sqrt{xz}}$$ $$x\sqrt{4-x^2z^2}\ge \frac{2}{\sqrt{yz}}$$ $$z\sqrt{4-y^2z^2}\ge \frac{2}{\sqrt{xy}}$$.

Let $C, B, E$ be three points on a straight line $\ell$ in that order. Suppose that $A$ and $D$ are two points on the same side of $\ell$ such that (i) $\angle ACE = \angle CDE = 90^o$ and (ii) $CA = CB = CD$. Let $F$ be the point of intersection of the segment $AB$ and the circumcircle of $\vartriangle ADC$. Prove that $F$ is the incentre of $\vartriangle CDE$.

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

Let the inequality $$Aa(Bb + Cc) + Bb(Aa + Cc) + Cc(Aa + Bb) > \frac{ABc^2 + BCa^2 + CAb^2}{2}$$ with given $a > 0, b > 0, c > 0$ hold for all $A > 0, B > 0, C > 0$. Is it possible to construct a triangle with sides of lengths $a, b, c$?