Olympiad problems

Russian TST 2022, P3

Tags: combinatorics

Write the natural numbers from left to right in ascending order. Every minute, we perform an operation. After $m$ minutes, we divide the entire available series into consecutive blocks of $m$ numbers. We leave the first block unchanged and in each of the other blocks we move all the numbers except the first one one place to the left, and move the first one to the end of the block. Prove that throughout the process, each natural number will only move a finite number of times.

MathLinks Contest 7th, 7.2

Tags: analytic geometry , function , number theory , relatively prime

Prove that the set of all the points with both coordinates begin rational numbers can be written as a reunion of two disjoint sets $ A$ and $ B$ such that any line that that is parallel with $ Ox$, and respectively $ Oy$ intersects $ A$, and respectively $ B$ in a finite number of points.

2002 Moldova Team Selection Test, 4

Tags: algebra

The sequence Pn (x), n ∈ N of polynomials is defined as follows: P0 (x) = x, P1 (x) = 4x³ + 3x Pn+1 (x) = (4x² + 2)Pn (x) − Pn−1 (x), for all n ≥ 1 For every positive integer m, we consider the set A(m) = { Pn (m) | n ∈ N }. Show that the sets A(m) and A(m+4) have no common elements.

1961 Putnam, A1

Tags: Putnam , analytic geometry , Intersection of set

The graph of the equation $x^y =y^x$ in the first quadrant consists of a straight line and a curve. Find the coordinates of the intersection of the line and the curve.

1987 IMO Longlists, 18

Tags: geometry , geometric inequality , 3D geometry , polyhedron , IMO Shortlist

Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality \[AF + AH + AC \leq AB + AD + AE + AG.\] In what cases does equality hold? [i]Proposed by France.[/i]

2016 Kosovo National Mathematical Olympiad, 2

Tags: equation

Find all real numbers $x$ which satisfied $|2x+1|+|x-1|=2-x$ .

1997 AIME Problems, 8

Tags:

How many different $4\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entires in each column is 0?

1969 Bulgaria National Olympiad, Problem 2

Tags: Sequences , inequalities

Prove that $$S_n=\frac1{1^2}+\frac1{2^2}+\ldots+\frac1{n^2}<2$$for every $n\in\mathbb N$.

Estonia Open Senior - geometry, 1997.2.3

Tags: geometry , equal circles , circles , square , tangent circles

The figure shows a square and three circles of equal radius tangent to each other and square passes. Find the radius of the circles if the square length is $1$. [img]http://3.bp.blogspot.com/-iIjwupkz7DQ/XnrIRhKIJnI/AAAAAAAALhA/clERrIDqEtcujzvZk_qu975wsTjKaxCLQCK4BGAYYCw/s400/97%2Bestonia%2Bopen%2Bs2.3.png[/img]

2024 Princeton University Math Competition, 13

Tags: Team Round

Consider the square with vertices $(0, 0),(1, 0),(1, 1),(0, 1).$ The line segments from $(t, 0)$ to $(0, 1 - t)$ are drawn for $0 \le t \le 1.$ The set of points inside the square but not on one of these line segments has area $\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ Find $m + n.$

1955 Putnam, A5

Tags: Putnam

If a parabola is given in the plane, find a geometric construction (ruler and compass) for the focus.

2010 AMC 12/AHSME, 8

Tags: trigonometry , geometry , ratio , trig identities , Law of Sines , exterior angle , AMC 10 2010 number 14

Triangle $ ABC$ has $ AB \equal{} 2 \cdot AC$. Let $ D$ and $ E$ be on $ \overline{AB}$ and $ \overline{BC}$, respectively, such that $ \angle{BAE} \equal{} \angle{ACD}.$ Let $ F$ be the intersection of segments $ AE$ and $ CD$, and suppose that $ \triangle{CFE}$ is equilateral. What is $ \angle{ACB}$? $ \textbf{(A)}\ 60^{\circ}\qquad \textbf{(B)}\ 75^{\circ}\qquad \textbf{(C)}\ 90^{\circ}\qquad \textbf{(D)}\ 105^{\circ}\qquad \textbf{(E)}\ 120^{\circ}$

TNO 2008 Senior, 9

Tags: Chile , number theory , algebra , induction

Let $f: \mathbb{N} \to \mathbb{N}$ be a function that satisfies: \[ f(1) = 2008, \] \[ f(4n^2) = 4f(n^2), \] \[ f(4n^2 + 2) = 4f(n^2) + 3, \] \[ f(4n(n+1)) = 4f(n(n+1)) + 1, \] \[ f(4n(n+1) + 3) = 4f(n(n+1)) + 4. \] Determine whether there exists a natural number $m$ such that: \[ 1^2 + 2^2 + \dots + m^2 + f(1^2) + \dots + f(m^2) = 2008m + 251. \]

I Soros Olympiad 1994-95 (Rus + Ukr), 10.8

Tags: algebra , inequalities , trigonometry

Find all $x$ for which the inequality holds $$\sqrt{7+8x-16x^2} \ge 2^{\cos^2 \pi x}+2^{\sin ^2 \pi x}$$

2014 PUMaC Team, 7

Tags: function

Let us consider a function $f:\mathbb{N}\to\mathbb{N}$ for which $f(1)=1$, $f(2n)=f(n)$ and $f(2n+1)=f(2n)+1$. Find the number of values at which the maximum value of $f(n)$ is attained for integer $n$ satisfying $0<n<2014$.

1988 IMO Longlists, 64

Tags: inequalities , number theory unsolved , number theory

Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$

2012 Stanford Mathematics Tournament, 4

Tags: probability

Two different squares are randomly chosen from an $8\times8$ chessboard. What is the probability that two queens placed on the two squares can attack each other? Recall that queens in chess can attack any square in a straight line vertically, horizontally, or diagonally from their current position.

KoMaL A Problems 2020/2021, A. 798

Tags: combinatorics , probability , komal , expected value

Let $0<p<1$ be given. Initially, we have $n$ coins, all of which have probability $p$ of landing on heads, and probability $1-p$ of landing on tails (the results of the tosses are independent of each other). In each round, we toss our coins and remove those that result in heads. We keep repeating this until all our coins are removed. Let $k_n$ denote the expected number of rounds that are needed to get rid of all the coins. Prove that there exists $c>0$ for which the following inequality holds for all $n>0$ \[c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg)<k_n<1+c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg).\]

1988 Romania Team Selection Test, 1

Tags: geometry , 3D geometry , sphere , geometric transformation , rotation , geometry unsolved

Consider a sphere and a plane $\pi$. For a variable point $M \in \pi$, exterior to the sphere, one considers the circular cone with vertex in $M$ and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone. [i]Octavian Stanasila[/i]

PEN E Problems, 24

Tags: inequalities , floor function , logarithms , limit , number theory , prime numbers

Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.

1952 Miklós Schweitzer, 9

Tags: function , real analysis , limit , integration , real analysis unsolved

Let $ C$ denote the set of functions $ f(x)$, integrable (according to either Riemann or Lebesgue) on $ (a,b)$, with $ 0\le f(x)\le1$. An element $ \phi(x)\in C$ is said to be an "extreme point" of $ C$ if it can not be represented as the arithmetical mean of two different elements of $ C$. Find the extreme points of $ C$ and the functions $ f(x)\in C$ which can be obtained as "weak limits" of extreme points $ \phi_n(x)$ of $ C$. (The latter means that $ \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx$ holds for every integrable function $ h(x)$.)