Olympiad problems

2006 China Second Round Olympiad, 13

Tags: parabola , conic

Given an integer $n\ge 2$, define $M_0 (x_0, y_0)$ to be an intersection point of the parabola $y^2=nx-1$ and the line $y=x$. Prove that for any positive integer $m$, there exists an integer $k\ge 2$ such that $(x^m_0, y^m_0)$ is an intersection point of $y^2=mx-1$ and the line $y=x$.

2021 LMT Spring, A9

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Find the sum of all positive integers $n$ such that $7<n < 100$ and $1573_{n}$ has $6$ factors when written in base $10$. [i]Proposed by Aidan Duncan[/i]

2015 South East Mathematical Olympiad, 7

Tags: geometry

In $\triangle ABC$, we have $AB>AC>BC$. $D,E,F$ are the tangent points of the inscribed circle of $\triangle ABC$ with the line segments $AB,BC,AC$ respectively. The points $L,M,N$ are the midpoints of the line segments $DE,EF,FD$. The straight line $NL$ intersects with ray $AB$ at $P$, straight line $LM$ intersects ray $BC$ at $Q$ and the straight line $NM$ intersects ray $AC$ at $R$. Prove that $PA \cdot QB \cdot RC = PD \cdot QE \cdot RF$.

1963 Putnam, B4

Tags: geometry , perimeter

Let $C$ be a closed plane curve that has a continuously turning tangent and bounds a convex region. If $T$ is a triangle inscribed in $C$ with maximum perimeter, show that the normal to $C$ at each vertex of $T$ bisects the angle of $T$ at that vertex. If a triangle $T$ has the property just described, does it necessarily have maximum perimeter? What is the situation if $C$ is a circle?

2000 Tournament Of Towns, 1

Tags: inequalities

Can the product of $2$ consecutive natural numbers equal the product of $2$ consecutive even natural numbers? (natural means positive integers)

2013 NZMOC Camp Selection Problems, 6

Tags: geometry , cyclic , tangential , square

$ABCD$ is a quadrilateral having both an inscribed circle (one tangent to all four sides) with center $I,$ and a circumscribed circle with center $O$. Let $S$ be the point of intersection of the diagonals of $ABCD$. Show that if any two of $S, I$ and $O$ coincide, then $ABCD$ is a square (and hence all three coincide).

2016-2017 SDML (Middle School), 6

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There are $4$ pairs of men and women, and all $8$ people are arranged in a row so that in each pair the woman is somewhere to the left of the man. How many such arrangements are there?

2007 Cuba MO, 3

Tags: combinatorics

A tennis competition takes place over four days, the number of participants is $2n$ with $n \ge 5$. Each participant plays exactly once a day (a couple of participants may be more times). Prove that such competition can end with exactly one winner and exactly three players in second place and such that there are no players with four lost games,

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

Tags:

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.)