Olympiad problems

2011 Princeton University Math Competition, A1 / B3

Tags: number theory

The only prime factors of an integer $n$ are 2 and 3. If the sum of the divisors of $n$ (including itself) is $1815$, find $n$.

2017 Math Prize for Girls Problems, 12

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Let $S$ be the set of all real values of $x$ with $0 < x < \pi/2$ such that $\sin x$, $\cos x$, and $\tan x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\tan^2 x$ over all $x$ in $S$.

2016 China Team Selection Test, 6

Tags: function , algebra

Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$.

2008 Indonesia TST, 1

Tags: combinatorics , subset

Let $A$ be the subset of $\{1, 2, ..., 16\}$ that has $6$ elements. Prove that there exist $2$ subsets of $A$ that are disjoint, and the sum of their elements are the same.

2004 Nicolae Coculescu, 2

Tags: equation , monotone , quadratic , algebra , trigonometry , logarithm

Solve in the real numbers the equation: $$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$ [i]Gheorghe Mihai[/i]

2001 Miklós Schweitzer, 10

Tags: differential equation , topology , linear algebra

Show that if a connected, nowhere zero sectional curvature of Riemannian manifold, where symmetric (1,1)-tensor of the Levi-Civita connection covariant derivative vanishes, then the tensor is constant times the unit tensor. (translated by j___d)

2022 Assara - South Russian Girl's MO, 1

Tags: number theory , perfect square

Given three natural numbers $a$, $b$ and $c$. It turned out that they are coprime together. And their least common multiple and their product are perfect squares. Prove that $a$, $b$ and $c$ are perfect squares.

1994 Nordic, 4

Tags: pell equation , number theory , perfect square , integer

Determine all positive integers $n < 200$, such that $n^2 + (n+ 1)^2$ is the square of an integer.

2017 Regional Olympiad of Mexico West, 1

Tags: number theory

The Occidentalia bank issues coins with denominations of $1$ peso, $8$ pesos, $27$ pesos... and any amount that is a perfect cube ($n^3$) of pesos. Determine what is the least amount $k$ of coins needed to give $2017$ pesos. For that amount, find all the possible ways to give $2017$ pesos using exactly $k$ currency.

1983 Polish MO Finals, 6

Tags: tetrahedron , dihedral angle , faces , 3d geometry , geometry

Prove that if all dihedral angles of a tetrahedron are acute, then all its faces are acute-angled triangles.

2022 HMNT, 29

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Consider the set $S$ of all complex numbers $z$ with nonnegative real and imaginary part such that $$|z^2+2| \le |z|.$$ Across all $z \in S,$ compute the minimum possible value of $\tan(\theta),$ where $\theta$ is the angle formed between $z$ and the real axis.

2019 Online Math Open Problems, 26

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There exists a unique prime $p > 5$ for which the decimal expansion of $\tfrac{1}{p}$ repeats with a period of exactly 294. Given that $p > 10^{50}$, compute the remainder when $p$ is divided by $10^9$. [i]Proposed by Ankan Bhattacharya[/i]

2023-24 IOQM India, 19

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For $n \in \mathbb{N}$, let $P(n)$ denote the product of the digits in $n$ and $S(n)$ denote the sum of the digits in $n$. Consider the set $A=\{n \in \mathbb{N}: P(n)$ is non-zero, square free and $S(n)$ is a proper divisor of $P(n)\}$. Find the maximum possible number of digits of the numbers in $A$.

2001 China Team Selection Test, 2

Tags: inequality , sequence , inequalities , analysis , trigonometry

Let ${a_n}$ be a non-increasing sequence of positive numbers. Prove that if for $n \ge 2001$, $na_{n} \le 1$, then for any positive integer $m \ge 2001$ and $x \in \mathbb{R}$, the following inequality holds: $\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$

2005 AMC 10, 11

Tags: modular arithmetic , algebra , recursion , sequence

The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence? $ \textbf{(A)}\ 29\qquad \textbf{(B)}\ 55\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 133\qquad \textbf{(E)}\ 250$

2020 International Zhautykov Olympiad, 1

Tags: number theory , zhautykov , euler s phi function , order , prime number

Given natural number n such that, for any natural $a,b$ number $2^a3^b+1$ is not divisible by $n$.Prove that $2^c+3^d$ is not divisible by $n$ for any natural $c$ and $d$

2000 Nordic, 1

Tags: integer , combinatorics , sum

In how many ways can the number $2000$ be written as a sum of three positive, not necessarily different integers? (Sums like $1 + 2 + 3$ and $3 + 1 + 2$ etc. are the same.)

2006 Korea National Olympiad, 5

Tags: number theory , function

Find all positive integers $n$ such that $\phi(n)$ is the fourth power of some prime.

2015 Princeton University Math Competition, A4/B6

Tags: algebra

Define the sequence $a_i$ as follows: $a_1 = 1, a_2 = 2015$, and $a_n = \frac{na_{n-1}^2}{a_{n-1}+na_{n-2}}$ for $n > 2$. What is the least $k$ such that $a_k < a_{k-1}$?

1998 French Mathematical Olympiad, Problem 2

Tags: sequence , algebra

Let $(u_n)$ be a sequence of real numbers which satisfies $$u_{n+2}=|u_{n+1}|-u_n\qquad\text{for all }n\in\mathbb N.$$Prove that there exists a positive integer $p$ such that $u_n=u_{n+p}$ holds for all $n\in\mathbb N$.

2023 AMC 12/AHSME, 10

Tags: analytic geometry , graphing lines , slope , geometry

In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect? $\textbf{(A)}\ \dfrac{2}{7} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{29}} \qquad\textbf{(D)}\ \dfrac{1}{\sqrt{29}} \qquad\textbf{(E)}\ \dfrac{2}{5}$

2023 Novosibirsk Oral Olympiad in Geometry, 3

Tags: square , geometry , rectangle

The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]

Indonesia Regional MO OSP SMA - geometry, 2010.1

Tags: geometry , ratio , centroid , collinear

Given triangle $ABC$. Suppose $P$ and $P_1$ are points on $BC, Q$ lies on $CA, R$ lies on $AB$, such that $$\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP_1}{P_1B}$$ Let $G$ be the centroid of triangle $ABC$ and $K = AP_1 \cap RQ$. Prove that points $P,G$, and $K$ are collinear.

2018 ITAMO, 1

Tags: geometry , 3d geometry

$1.$A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high. Find the height of the bottle.

2003 Gheorghe Vranceanu, 3

Tags: function , convergence , real analysis , series , floor function

Let be a sequence of functions $ a_n:\mathbb{R}\longrightarrow\mathbb{Z} $ defined as $ a_n(x)=\sum_{i=1}^n (-1)^i\lfloor xi\rfloor . $ [b]a)[/b] Find the real numbers $ y $ such that $ \left( a_n(y) \right)_{n\ge 1} $ converges to $ 1. $ [b]b)[/b] Find the real numbers $ z $ such that $ \left( a_n(z) \right)_{n\ge 1} $ converges.