MathNet 2026 — Part 09

Determine all integers $n$ for which $n^{2}+15$ is the square of an integer.

Find the exact value of $1+\frac{1}{1+\frac{2}{1+\frac{1}{1+\frac{2}{1+\ldots}}}}$.

Several weights are given, each of which is not heavier than $1 \mathrm{lb}$. It is known that they cannot be divided into two groups such that the weight of each group is greater than $1 \mathrm{lb}$. Find the maximum possible total weight of these weights.

The country of Squareland is shaped like a square and is divided into $64$ congruent square cities. We want to divide Squareland into states and assign to each state a capital city so that the following rules are satisfied:


a. Every city lies entirely within one state.


b. Given any two states, the numbers of cities in them differ by at most $1$.


c. Any city in a state shares at least one corner with the state's capital.

What is the smallest possible number of states?

An integer is called formidable if it can be written as a sum of distinct powers of $4$, and successful if it can be written as a sum of distinct powers of $6$. Can $2005$ be written as a sum of a formidable number and a successful number? Prove your answer.

Determine whether there exist two distinct finite subsets $A$ and $B$ of the reals such that for every polynomial $f$ of degree $2012$ with real coefficients,
$$
\sum_{x \in A} f(x) = \sum_{x \in B} f(x)
$$

Given a quadrilateral $ABCD$, show that the midpoints of its four edges form the vertices of a parallelogram.

Let $n \geq 1$ be an integer. How many ways can the rectangle having vertices $(0,0)$, $(n, 0)$, $(n, 1)$, $(0,1)$ be dissected into $2n$ triangles, all vertices of which have integer coordinates?

The triangles are considered as positioned on the coordinate plane; in particular, tilings related by rotation and reflection are considered distinct.

Find all integer solutions to $x y = 2003(x + y)$

If $k$ is an integer, prove that the number $k^{2}+k+1$ is not divisible by $2006$.

Find all primes $p$ such that there exist integers $a, b, c$, and $k$ satisfying the equations
$$
\begin{aligned}
& a^{2}+b^{2}+c^{2}=p \\
& a^{4}+b^{4}+c^{4}=k p .
\end{aligned}
$$

Alice, Bob, and Charlie each pick a 2-digit number at random. What is the probability that all of their numbers' tens' digits are different from each others' tens' digits and all of their numbers' ones digits are different from each others' ones' digits?

Suppose you have an unlimited number pennies, nickels, dimes, and quarters. Determine the number of ways to make 30 cents using these coins.

Bart has 17 and 19 dollar bills only.


a. Prove that these bills are fake.


b. Prove that there exists $m\gt 0$ such that for each $n \geq m$ Bart can give to Lisa exactly $n$ dollars using his bills.

Let $ABC$ be a triangle such that $\angle A = 90^{\circ}$ and $\angle B \lt \angle C$. The tangent at $A$ to its circumcircle $\omega$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ across $BC$, $X$ the foot of the perpendicular from $A$ to $BE$, and $Y$ the midpoint of $AX$. Let the line $BY$ meet $\omega$ again at $Z$. Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$.

If $\frac{1}{9}$ of $60$ is $5$, what is $\frac{1}{20}$ of $80$?

Find the number of real zeros of $x^{3}-x^{2}-x+2$.

Let $A O B$ be a $60$-degree angle. For any point $P$ in the interior of $\angle A O B$, let $A'$ and $B'$ be the feet of the perpendiculars from $P$ to $A O$ and $B O$ respectively. Denote by $r$ and $s$ the distances $O P$ and $A' B'$. Find all possible pairs of real numbers $(r, s)$.

Prove that
$$
tan \left(\frac{3 \pi}{11}\right)+4 \sin \left(\frac{2 \pi}{11}\right)=\sqrt{11}
$$

After walking so much that his feet get really tired, the beaver staggers so that, at each step, his coordinates change by either $(+1,+1)$ or $(+1,-1)$. Now he walks from $(0,0)$ to $(8,0)$ without ever going below the $x$-axis. How many such paths are there?