In an $n$ by $n$ grid, each cell is filled with an integer between $1$ and $6$. The outmost cells all contain the number $1$, and any two cells that share a vertex has difference not equal to $3$. For any vertex $P$ inside the grid (not including the boundary), there are $4$ cells that have $P$ has a vertex. If these four cells have exactly three distinct numbers $i$, $j$, $k$ (two cells have the same number), and the two cells with the same number have a common side, we call $P$ an $ijk$-type vertex. Let there be $A_{ijk}$ vertices that are $ijk$-type. Prove that $A_{123}\equiv A_{246} \pmod 2$.

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$s than $n^2$.

The side lengths of a scalene triangle are roots of the polynomial $$x^3-20x^2+131x-281.3.$$ Find the square of the area of the triangle.

Of the following sets of data the only one that does not determine the shape of a triangle is: $ \textbf{(A)}\ \text{the ratio of two sides and the included angle} \\ \qquad\textbf{(B)}\ \text{the ratios of the three altitudes} \\ \qquad\textbf{(C)}\ \text{the ratios of the three medians} \\ \qquad\textbf{(D)}\ \text{the ratio of the altitude to the corresponding base} \\ \qquad\textbf{(E)}\ \text{two angles}$

Let $P_i(x) = x^2 + b_i x + c_i , i = 1,2, ..., n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \le i < j \le n , i, j \in N$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.

From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. It is found that $t_1$, the length of the tangent, is the mean proportional between $m$ and $n$. If $m$ and $t$ are integers, then $t$ may have the following number of values: $ \textbf{(A)}\ \text{zero} \qquad\textbf{(B)}\ \text{one} \qquad\textbf{(C)}\ \text{two} \qquad\textbf{(D)}\ \text{three} \qquad\textbf{(E)}\ \text{infinitely many} $

Show that given any 9 points inside a square of side 1 we can always find 3 which form a triangle with area less than $\frac 18$. [i]Bulgaria[/i]

An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ passes point $(2,1)$, then all points $(x,y)$ on the ellipse that $|y|>1$ are (shown as shadow) [img]https://graph.baidu.com/resource/122481219e60931bb707101582696834.jpg[/img]

The price of $175$ Humpties is more than the price of $125$ Dumpties but less than that of $126$ Dumpties. Prove that you cannot buy three Humpties and one Dumpty for (a) $80$ cents. (b) $1$ dollar. (S Fomin, Leningrad) PS. (a) for Juniors , (a),(b) for Seniors

Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)

A straight road consists of green and red segments in alternating colours, the first and last segment being green. Suppose that the lengths of all segments are more than a centimeter and less than a meter, and that the length of each subsequent segment is larger than the previous one. A grasshopper wants to jump forward along the road along these segments, stepping on each green segment at least once an without stepping on any red segment (or the border between neighboring segments). Prove that the grasshopper can do this in such a way that among the lengths of his jumps no more than $8$ different values occur. [i]Proposed by T. Korotchenko[/i]

Suppose that $ a,b,$ and $ \sqrt[3]{a}\plus{}\sqrt[3]{b}$ are rational numbers. Prove that $ \sqrt[3]{a}$ and $ \sqrt[3]{b}$ are also rational.

Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.

Let $ ABC $ be an obtuse triangle with $ \angle A > 90^{\circ} $. Let circle $ O $ be the circumcircle of $ ABC $. $ D $ is a point lying on segment $ AB $ such that $ AD = AC $. Let $ AK $ be the diameter of circle $ O $. Two lines $ AK $ and $ CD $ meet at $ L $. A circle passing through $ D, K, L $ meets with circle $ O $ at $ P ( \ne K ) $ . Given that $ AK = 2, \angle BCD = \angle BAP = 10^{\circ} $, prove that $ DP = \sin ( \frac{ \angle A}{2} )$.

Jody has $6$ distinguishable balls and $6$ distinguishable sticks, all of the same length. How many ways are there to use the sticks to connect the balls so that two disjoint non-interlocking triangles are formed? Consider rotations and reflections of the same arrangement to be indistinguishable.

At a tourist camp, each person has at least $50$ and at most $100$ friends among the other persons at the camp. Show that one can hand out a t-shirt to every person such that the t-shirts have (at most) $1331$ different colors, and any person has $20$ friends whose t-shirts all have pairwisely different colors.

Call a three-digit number $\overline{ABC}$ $\textit{spicy}$ if it satisfies $\overline{ABC}=A^3+B^3+C^3$. Compute the unique $n$ for which both $n$ and $n+1$ are $\textit{spicy}$.

In the quadrilateral $ABCD$, shown in fig. , the equations are true: $\angle ABC = \angle BCD$ and $2AB = CD$. On the side $BC$, a point $X$ is selected such that $\angle BAX = \angle CDA$. Prove that $AX = AD$. [img]https://cdn.artofproblemsolving.com/attachments/2/9/0884eb311d1e40300c1e5980fd53eaadfa7a25.png[/img]

For a non-empty finite complex number set $A$, define the "[i]Tao root[/i]" of $A$ as $\left|\sum_{z\in A} z \right|$. Given the integer $n\ge 3$, let the set $$U_n = \{\cos\frac{2k \pi}{n}+ i\sin\frac{2k \pi}{n}|k=0,1,...,n-1\}.$$Let $a_n$ be the number of non-empty subsets in which the [i]Tao root [/i] of $U_n$ is $0$ , $b_n$ is the number of non-empty subsets of $U_n$ whose [i]Tao root[/i] is $1$. Compare the sizes of $na_n$ and $2b_n$.

Find the sum of all positive integers $n$ with the following properties: [list] [*] $n$ is not divisible by any primes larger than $10$. [*] For some positive integer $k$, the positive divisors of $n$ are \[1=d_1<d_2<d_3\cdots<d_{2k}=n.\] [*] The divisors of $n$ have the property that \[d_1+d_2+\cdots+d_k=3k.\] [/list] [i]2021 CCA Math Bonanza Team Round #7[/i]

On the board was drawn a circle with a marked center, a quadrangle inscribed in it, and a circle inscribed in it, also with a marked center. Then they erased the quadrilateral (keeping one vertex) and the inscribed circle (keeping its center). Restore any of the erased vertices of the quadrilateral using only a ruler and no more than six lines.

Let $\Delta ABC$ be an acute angled triangle. $D$ is a point on side $BC$ such that $AD$ bisects angle $\angle BAC$. A line $l$ is tangent to the circumcircles of triangles $ADB$ and $ADC$ at point $K$ and $L$, respectively. Let $M$, $N$ and $P$ be its midpoints of $BD$, $DC$ and $KL$, respectively. Prove that $l$ is tangent to the circumcircle of $\Delta MNP$.

[asy] size(150); pair A=(0,0),B=(1,0),C=(0,1),D=(-1,0),E=(0,.5),F=(sqrt(2)/2,.25); draw(circle(A,1)^^D--B); draw(circle(E,.5)^^circle( F ,.25)); label("$A$", D, W); label("$K$", A, S); label("$B$", B, dir(0)); label("$L$", E, N); label("$M$",shift(-.05,.05)*F); //Credit to Klaus-Anton for the diagram[/asy] In the adjoining figure, circle $\mathit{K}$ has diameter $\mathit{AB}$; cirlce $\mathit{L}$ is tangent to circle $\mathit{K}$ and to $\mathit{AB}$ at the center of circle $\mathit{K}$; and circle $\mathit{M}$ tangent to circle $\mathit{K}$, to circle $\mathit{L}$ and $\mathit{AB}$. The ratio of the area of circle $\mathit{K}$ to the area of circle $\mathit{M}$ is $\textbf{(A) }12\qquad\textbf{(B) }14\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad \textbf{(E) }\text{not an integer}$

Compute the number of tuples $(a_0, a_1, a_2, a_3, a_4, a_5)$ of (not necessarily positive) integers such that $a_i \le i$ for all $0 \le i \le 5$ and $$a_0+a_1+a_2+a_3+a_4+a_5=6.$$