A sequence of integers $a_0, a_1 …$ is called [i]kawaii[/i] if $a_0 =0, a_1=1,$ and $$(a_{n+2}-3a_{n+1}+2a_n)(a_{n+2}-4a_{n+1}+3a_n)=0$$ for all integers $n \geq 0$. An integer is called [i]kawaii[/i] if it belongs to some kawaii sequence. Suppose that two consecutive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that $m$ is divisible by $3,$ and that $m/3$ is also kawaii.

In $\triangle ABC$, $AB>AC$, bisector of outer angle $\angle A$ intersects circumcircle of $\triangle ABC$ at $E$. Projection of $E$ on $AB$ is $F$. Prove that $2AF=AB-AC$.

Without tables and such, prove that $\frac{1}{\log_2 \pi}+\frac{1}{\log_5 \pi} >2$

Five lilypads lie in a line on a pond. At first, a frog sits on the third lilypad. Then, each minute there is a $\tfrac{1}{2}$ probability that the frog jumps to the lilypad to its left and $\tfrac{1}{2}$ probability that it jumps to its right. If the frog jumps to the left from the leftmost lilypad or right from the rightmost lilypad, it will fall in the pond and stay there forever. Compute the probability that the frog is not in the pond after $14$ minutes have passed.

Let $n\in {{\mathbb{N}}^{*}}$ and ${{a}_{1}},{{a}_{2}},...,{{a}_{n}}\in \mathbb{R}$ so ${{a}_{1}}+{{a}_{2}}+...+{{a}_{k}}\le k,\left( \forall \right)k\in \left\{ 1,2,...,n \right\}.$Prove that $\frac{{{a}_{1}}}{1}+\frac{{{a}_{2}}}{2}+...+\frac{{{a}_{n}}}{n}\le \frac{1}{1}+\frac{1}{2}+...+\frac{1}{n}$

Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is $\dfrac{7}{2}$ times the original one.

Unit square $ABCD$ is divided into $10^{12}$ smaller squares (not necessarily equal). Prove that the sum of perimeters of all the smaller squares having common points with diagonal $AC$ does not exceed 1500. [i]Proposed by A. Kanel-Belov[/i]

The diameters of two circles are $ 8$ inches and $ 12$ inches respectively. The ratio of the area of the smaller to the area of the larger circle is: $ \textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{4}{9} \qquad\textbf{(C)}\ \frac{9}{4} \qquad\textbf{(D)}\ \frac{1}{2} \qquad\textbf{(E)}\ \text{none of these}$

Let $ABC$ be an acute triangle and let $A_1$, $B_1$, $C_1$ be points on the sides $BC, CA$ and $AB$, respectively. Show that the triangles $ABC$ and $A_1B_1C_1$ are similar ($\angle A = \angle A_1, \angle B = \angle B_1,\angle C = \angle C_1$) if and only if the orthocentre of the triangle $A_1B_1C_1$ and the circumcentre of the triangle $ABC$ coincide.

[b]p1.[/b] A permutation on $\{1, 2,..., n\}$ is an ordered arrangement of the numbers. For example, $32154$ is a permutation of $\{1, 2, 3, 4, 5\}$. Does there exist a permutation $a_1a_2... a_n$ of $\{1, 2,..., n\}$ such that $i+a_i$ is a perfect square for every $1 \le i \le n$ when a) $n = 6$ ? b) $n = 13$ ? c) $n = 86$ ? Justify your answers. [b]p2.[/b] Circle $C$ and circle $D$ are tangent at point $P$. Line $L$ is tangent to $C$ at point $Q$ and to $D$ at point $R$ where $Q$ and $R$ are distinct from $P$. Circle $E$ is tangent to $C, D$, and $L$, and lies inside triangle $PQR$. $C$ and $D$ both have radius $8$. Find the radius of $E$, and justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/f/b/4b98367ea64e965369345247fead3456d3d18a.png[/img] [b]p3.[/b] (a) Prove that $\sin 3x = 4 \cos^2 x \sin x - \sin x$ for all real $x$. (b) Prove that $$(4 \cos^2 9^o - 1)(4 \cos^2 27^o - 1)(4 cos^2 81^o - 1)(4 cos^2 243^o - 1)$$ is an integer. [b]p4.[/b] Consider a $3\times 3\times 3$ stack of small cubes making up a large cube (as with the small cubes in a Rubik's cube). An ant crawls on the surface of the large cube to go from one corner of the large cube to the opposite corner. The ant walks only along the edges of the small cubes and covers exactly nine of these edges. How many different paths can the ant take to reach its goal? [b]p5.[/b] Let $m$ and $n$ be positive integers, and consider the rectangular array of points $(i, j)$ with $1 \le i \le m$, $1 \le j \le n$. For what pairs m; n of positive integers does there exist a polygon for which the $mn$ points $(i, j)$ are its vertices, such that each edge is either horizontal or vertical? The figure below depicts such a polygon with $m = 10$, $n = 22$. Thus $10$, $22$ is one such pair. [img]https://cdn.artofproblemsolving.com/attachments/4/5/c76c0fe197a8d1ebef543df8e39114fe9d2078.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle, let ${A}'$, ${B}'$, ${C}'$ be the orthogonal projections of the vertices $A$ ,$B$ ,$C$ on the lines $BC$, $CA$ and $AB$, respectively, and let $X$ be a point on the line $A{A}'$.Let $\gamma_{B}$ be the circle through $B$ and $X$, centred on the line $BC$, and let $\gamma_{C}$ be the circle through $C$ and $X$, centred on the line $BC$.The circle $\gamma_{B}$ meets the lines $AB$ and $B{B}'$ again at $M$ and ${M}'$, respectively, and the circle $\gamma_{C}$ meets the lines $AC$ and $C{C}'$ again at $N$ and ${N}'$, respectively.Show that the points $M$, ${M}'$, $N$ and ${N}'$ are collinear.

Let $ BC$ of right triangle $ ABC$ be the diameter of a circle intersecting hypotenuse $ AB$ in $ D$. At $ D$ a tangent is drawn cutting leg $ CA$ in $ F$. This information is [u]not[/u] sufficient to prove that $ \textbf{(A)}\ DF \text{ bisects }CA \qquad \textbf{(B)}\ DF \text{ bisects }\angle CDA$ $ \textbf{(C)}\ DF \equal{} FA \qquad \textbf{(D)}\ \angle A \equal{} \angle BCD \qquad \textbf{(E)}\ \angle CFD \equal{} 2\angle A$

Find all positive integers $n$ that are quadratic residues modulo all primes greater than $n$.

The results of a cross-country team's training run are graphed below. Which student has the greatest average speed? [asy] for ( int i = 1; i <= 7; ++i ) { draw((i,0)--(i,6)); } for ( int i = 1; i <= 5; ++i ) { draw((0,i)--(8,i)); } draw((-0.5,0)--(8,0), linewidth(1)); draw((0,-0.5)--(0,6), linewidth(1)); label("$O$", (0,0), SW); label(scale(.85)*rotate(90)*"distance", (0, 3), W); label(scale(.85)*"time", (4, 0), S); dot((1.25, 4.5)); label(scale(.85)*"Evelyn", (1.25, 4.8), N); dot((2.5, 2.2)); label(scale(.85)*"Briana", (2.5, 2.2), S); dot((4.25,5.2)); label(scale(.85)*"Carla", (4.25, 5.2), SE); dot((5.6, 2.8)); label(scale(.85)*"Debra", (5.6, 2.8), N); dot((6.8, 1.4)); label(scale(.85)*"Angela", (6.8, 1.4), E); [/asy] $ \textbf{(A)}\ \text{Angela}\qquad\textbf{(B)}\ \text{Briana}\qquad\textbf{(C)}\ \text{Carla}\qquad\textbf{(D)}\ \text{Debra}\qquad\textbf{(E)}\ \text{Evelyn} $

Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\ ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]

The circle $\omega_2$ passing through the center $O$ of the circle $\omega_1$, is tangent to the circle $\omega_2$ at the point $A$. On the circle $\omega_2$, the point $C$ is taken so that the ray $AC$ intersects the circle $\omega_1$ for second time at point $D$, the ray $OC$ intersects the circle $\omega_1$ at point $E$ and the lines $DE$ and $AO$ are parallel. Find the size of the angle $DAE$.

Let $n$ be an even natural number and let $A$ be the set of all non-zero sequences of length $n$, consisting of numbers $0$ and $1$ (length $n$ binary sequences, except the zero sequence $(0,0,\ldots,0)$). Prove that $A$ can be partitioned into groups of three elements, so that for every triad $\{(a_1,a_2,\ldots,a_n), (b_1,b_2,\ldots,b_n), (c_1,c_2,\ldots,c_n)\}$, and for every $i = 1, 2,\ldots,n$, exactly zero or two of the numbers $a_i, b_i, c_i$ are equal to $1$.

Find the minimum $k>2$ for which there are $k$ consecutive integers such that the sum of their squares is a square.

A board of $6\times 6$ is totally covered by $18$ dominoes (of $2\times 1$), that is, there are no overlaps, gaps, and the tiles do not come off the board. Prove that, regardless of the arrangement of the tiles, there is always a line that divides the board into two non-empty parts, and without cutting tiles.

A pair $(m, n)$ of positive integers is called “[i]linked[/i]” if $m$ divides $3n + 1$ and $n$ divides $3m + 1$. If $a, b, c$ are distinct positive integers such that $(a, b)$ and $( b, c)$ are linked pairs, prove that the number $1$ belongs to the set $\{a, b, c\}$

The quadrilateral $ABCD$ is circumscribed about the circle, touches the sides $AB, BC, CD, DA$ in the points $K, L, M, N,$ respectively. Let $P, Q, R, S$ midpoints of the sides $KL, LM, MN, NK$. Prove that $PR = QS$ if and only if $ABCD$ is inscribed.

Is it possible to paint the cells of a $5\times 5$ board in $4$ colors so that the cells standing at the intersection of any two rows and any two columns were painted in at least $ 3$ colors?

Luke and Carissa are finding the sum of the first $20$ positive integers by adding them one at a time. Luke forgets to add a number and gets an answer of $207$. Carissa adds a number twice by mistake and gets an answer of $225$. What is the sum of the number that Luke forgot and the number that Carissa added twice?

Let $x+y=a$ and $xy=b$. The expression $x^6+y^6$ can be written as a polynomial in terms of $a$ and $b$. What is this polynomial?

The points $D$ and $E$ lie on the side $(BC)$ of the triangle $ABC$ such that $D$ is between $B$ and $E.$ A point $R$ on the segment $(AE)$ is called [i]remarkable[/i] if the lines $PQ$ and $BC$ are parallel, where $\{P\}=DR \cap AC$ and $\{Q\}=CR \cap AB.$ A point $R'$ on the segment $(AD)$ is called [i]remarkable[/i] if the lines $P'Q'$ and $BC$ are parallel, where $\{P'\}=BR' \cap AC$ and $\{Q'\}=ER' \cap AB.$ a) If there exists a remarkable point on the segment $(AE),$ prove that any point of the segment $(AE)$ is remarkable. b) If each of the segments $(AD)$ and $(AE)$ contains a remarkable point, prove that $BD=CE=\varphi \cdot DE,$ where $\varphi= \frac{1+\sqrt{5}}{2}$ is the golden ratio.