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An Order-Finding Problem

Let $n$ be the least positive integer with

3^3 \cdot 5^5 \cdot 7^7 \;\Big|\; 149^n - 2^n.

How many positive divisors does $n$ have?

number theoryModular Arithmetic

A General Term from a Ratio Recurrence

A sequence $\{a_n\}$ satisfies

\frac{a_{n+1}+a_n-1}{a_{n+1}-a_n+1}=n\qquad(n\in\mathbb N^{*}),

and $a_2=6$ . Find the general term of $\{a_n\}$ .

RecursionsequencesTelescoping

A Centroid Condition

In triangle $ABC$ , the sides opposite angles $A, B, C$ are $a, b, c$ . Given $c = 2\sqrt{5}$ and

2a\sin C\cos B = a\sin A - b\sin B + \frac{\sqrt{5}}{2}b\sin C,

let $O$ be the…

Law of SinestrigonometryLaw of Cosinesvectors

An Exponential-Log Gap

Prove that for all $x > 0$ ,

e^x - \ln x > 2.3.

calculusConvexityEstimationinequality

Sums of Three Powers of Two

Let

A = \left\{2^a + 2^b + 2^c \;:\; a, b, c \in \mathbb{N}\right\}

(the exponents need not be distinct). A subset $B \subseteq A$ consists of $n$ consecutive natural num…

combinatoricsnumber theoryModular ArithmeticCasework

Approximating Functions to Order $k$

Let $f(x)$ be defined on a domain $D$ , and let $g(x)$ be a function different from $f(x)$ . If at some point $x = \xi$ we have

\begin{aligned} f(\xi)&=g(\xi),\\ f'(\xi)&=g'(\xi),…

functionscalculusLogarithms

Counting Lock-Screen Patterns

An Android-style unlock pattern connects some of $9$ points arranged in a $3\times 3$ grid, subject to these rules: choose a starting point, then move the finger through a sequence…

combinatoricsCountingCaseworkSymmetry

A Non-Decreasing Function

A function $f(x)$ on an interval $D$ is non-decreasing if $f(x_1) \leqslant f(x_2)$ whenever $x_1 < x_2$ in $D$ . Suppose $f(x)$ is non-decreasing on $[0, 2]$ with

f(2) = 2, \q…

functionsMonotonicityFunctional EquationsSymmetry

Two Related Zeros

Let

f(x) = -\frac{t}{x} + \ln(ex), \qquad g(x) = \frac{tx}{e} + \ln x,

where $e = 2.71828\ldots$ is the base of natural logarithms.

1. If $t = 0$ , solve the inequality $f(x)…

functionsExtremacalculusMonotonicity+2

A Rotating Recurrence Pair

Real sequences $\{a_n\}$ and $\{b_n\}$ (all terms nonzero) satisfy

\begin{cases}a_n=a_{n-1}\cos\theta-b_{n-1}\sin\theta,\\ b_n=a_{n-1}\sin\theta+b_{n-1}\cos\theta,\end{cases}

w…

complex numberstrigonometryGeometric Progressionsequences+1

Three Radicals on an Interval

Find the minimum and maximum of

y = \sqrt{x + 27} + \sqrt{13 - x} + \sqrt{x}.

functionsExtremaalgebraCauchy-Schwarz+1

Half-Angle Tangents and the Circumcenter

In $\triangle ABC$ ,

\tan\frac{A}{2}\tan\frac{B}{2} = \frac13,

with the usual side notation.

1. Prove that $a, b, c$ can be rearranged into an arithmetic progression. 2. Let …

plane geometrytrigonometryArithmetic ProgressionTriangle Geometry+1

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