Olympiad problems

2008 IMS, 9

Let $ \gamma: [0,1]\rightarrow [0,1]\times [0,1]$ be a mapping such that for each $ s,t\in [0,1]$ \[ |\gamma(s) \minus{} \gamma(t)|\leq M|s \minus{} t|^\alpha \] in which $ \alpha,M$ are fixed numbers. Prove that if $ \gamma$ is surjective, then $ \alpha\leq\frac12$

2002 Federal Math Competition of S&M, Problem 2

Tags: geometry , geometric inequality , area , inequalities

Points $A_0,A_1,\ldots,A_{2k}$, in this order, divide a circumference into $2k+1$ equal arcs. Point $A_0$ is connected by chords to all the other points. These $2k$ chords divide the interior of the circle into $2k+1$ parts. These parts are alternately painted red and blue so that there are $k+1$ red and $k$ blue parts. Show that the blue area is larger than the red area.

1998 AMC 12/AHSME, 4

Tags:

Define $[a,b,c]$ to mean $\frac{a+b}{c},$ where $c \neq 0$. What is the value of \[[[60,30,90],[2,1,3],[10,5,15]]?\] $\text{(A)} \ 0 \qquad \text{(B)} \ 0.5 \qquad \text{(C)} \ 1 \qquad \text{(D)} \ 1.5 \qquad \text{(E)} \ 2$

2014 Regional Competition For Advanced Students, 1

Tags: algebra , equation

Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .

2018 Junior Balkan Team Selection Tests - Romania, 3

Tags: algebra , set

Let $A =\left\{a = q + \frac{1}{q }/ q \in Q^*,q > 0 \right\}$, $A + A = \{a + b |a,b \in A\}$,$A \cdot A =\{a \cdot b | a, b \in A\}$. Prove that: i) $A + A \ne A \cdot A$ ii) $(A + A) \cap N = (A \cdot A) \cap N$. Vasile Pop

2023 Indonesia TST, C

Tags:

There are $2023$ distinct points on a plane, which are coloured in white or red, such that for each white point, there are exactly two red points whose distance is $2023$ to that white point. Find the minimum number of red points.

2017 Hanoi Open Mathematics Competitions, 11

Tags: geometry , inequalities , geometric inequality , equilateral , angle

Let $ABC$ be an equilateral triangle, and let $P$ stand for an arbitrary point inside the triangle. Is it true that $| \angle PAB - \angle PAC| \ge | \angle PBC - \angle PCB|$ ?

2005 USAMTS Problems, 5

Tags: geometry , 3d geometry , sphere

Sphere $S$ is inscribed in cone $C$. The height of $C$ equals its radius, and both equal $12+12\sqrt2$. Let the vertex of the cone be $A$ and the center of the sphere be $B$. Plane $P$ is tangent to $S$ and intersects $\overline{AB}$. $X$ is the point on the intersection of $P$ and $C$ closest to $A$. Given that $AX=6$, find the area of the region of $P$ enclosed by the intersection of $C$ and $P$.

2022 IFYM, Sozopol, 6

Tags: polynomial , algebra

Let $n$ be a natural number and $P_1, P_2, ... , P_n$ are polynomials with integer coefficients, each of degree at least $2$. Let $S$ be the set of all natural numbers $N$ for which there exists a natural number $a$ and an index $1 \le i \le n$ such that $P_i(a) = N$. Prove, that there are infinitely many primes that do not belong to $S$.

2014 Contests, 3

Tags: number theory

Find all positive integers $n$ so that $$17^n +9^{n^2} = 23^n +3^{n^2} .$$

2016 Brazil Team Selection Test, 4

Tags: combinatorics , additive combinatorics

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

2008 Stanford Mathematics Tournament, 6

Tags: geometry , 3d geometry

A round pencil has length $ 8$ when unsharpened, and diameter $ \frac {1}{4}$. It is sharpened perfectly so that it remains $ 8$ inches long, with a $ 7$ inch section still cylindrical and the remaining $ 1$ inch giving a conical tip. What is its volume?

2010 Romania National Olympiad, 2

Tags: number theory

How many four digit numbers $\overline{abcd}$ simultaneously satisfy the equalities $a+b=c+d$ and $a^2+b^2=c^2+d^2$?

2024 LMT Fall, 12

Tags: guts

Snorlax's weight is modeled by the function $w(t)=t2^t$ where $w(t)$ is Snorlax's weight at time $t$ minutes. Find the smallest integer time $t$ such that Snorlax's weight is greater than $10000.$

2004 Peru MO (ONEM), 4

Tags: geometry , integer , sides of a triangle , area of a triangle , minimum

Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.

1989 IMO Longlists, 23

Tags: geometry

Let $ ABC$ be a triangle. Prove that there is a unique point $ U$ in the plane of $ ABC$ such that there exist real numbers $ \alpha, \beta, \gamma, \delta$ not all zero, such that \[ \alpha PL^2 \plus{} \beta PM^2 \plus{} \gamma PN^2 \plus{} \delta UP^2\] is constant for all points $ P$ of the plane, where $ L,M,N$ are the feet of the perpendiculars from $ P$ to $ BC,CA,AB$ respectively. Identify $ U.$

2001 District Olympiad, 1

Tags: matrix , algebra , polynomial , linear algebra

Let $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A)=d\neq 0$ and $\det(A+dA^*)=0$. Prove that $\det(A-dA^*)=4$. [i]Daniel Jinga[/i]

2001 Polish MO Finals, 3

Tags: algebra

A sequence $x_0=A$ and $x_1=B$ and $x_{n+2}=x_{n+1} +x_n$ is called a Fibonacci type sequence. Call a number $C$ a repeated value if $x_t=x_s=c$ for $t$ different from $s$. Prove one can choose $A$ and $B$ to have as many repeated value as one likes but never infinite.

2009 Postal Coaching, 1

Tags: subset , combinatorics , number theory

Let $n \ge 1$ be an integer. Prove that there exists a set $S$ of $n$ positive integers with the following property: if $A$ and $B$ are any two distinct non-empty subsets of $S$, then the averages $\frac{P_{x\in A} x}{|A|}$ and $\frac{P_{x\in B} x}{|B|}$ are two relatively prime composite integers.

Durer Math Competition CD Finals - geometry, 2020.C4

Tags: geometry , hexagon

Albrecht likes to draw hexagons with all sides having equal length. He calls an angle of such a hexagon [i]nice [/i] if it is exactly $120^o$. He writes the number of its nice angles inside each hexagon. How many different numbers could Albrecht write inside the hexagons? Show examples for as many values as possible and give a reasoning why others cannot appear. [i]Albrecht can also draw concave hexagons[/i]

1995 AIME Problems, 11

Tags: geometry , 3d geometry , prism , calculus , integration , ratio , rectangle

A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?

2006 AMC 8, 6

Tags: geometry , rectangle , perimeter

The letter T is formed by placing two $ 2\times 4$ inch rectangles next to each other, as shown. What is the perimeter of the T, in inches? [asy]size(150); draw((0,6)--(4,6)--(4,4)--(3,4)--(3,0)--(1,0)--(1,4)--(0,4)--cycle, linewidth(1));[/asy] $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 24$

2007 Tournament Of Towns, 5

Tags: combinatorics

A square of side length $1$ centimeter is cut into three convex polygons. Is it possible that the diameter of each of them does not exceed [list][b]a)[/b] $1$ centimeter; [b]b)[/b] $1.01$ centimeters; [b]c)[/b] $1.001$ centimeters?[/list]

2015 China Team Selection Test, 1

Tags: inequalities

Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that \[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]

2024 ELMO Shortlist, A3

Tags: algebra , functional equation

Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$ [i]Andrew Carratu[/i]