Olympiad problems

2023-24 IOQM India, 22

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In an equilateral triangle of side length 6 , pegs are placed at the vertices and also evenly along each side at a distance of 1 from each other. Four distinct pegs are chosen from the 15 interior pegs on the sides (that is, the chosen ones are not vertices of the triangle) and each peg is joined to the respective opposite vertex by a line segment. If $N$ denotes the number of ways we can choose the pegs such that the drawn line segments divide the interior of the triangle into exactly nine regions, find the sum of the squares of the digits of $N$.

2017 Thailand Mathematical Olympiad, 9

Determine all functions $f$ on the set of positive rational numbers such that $f(xf(x) + f(y)) = f(x)^2 + y$ for all positive rational numbers $x, y$.

2020 MBMT, 17

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$\triangle KWU$ is an equilateral triangle with side length $12$. Point $P$ lies on minor arc $\overarc{WU}$ of the circumcircle of $\triangle KWU$. If $\overline{KP} = 13$, find the length of the altitude from $P$ onto $\overline{WU}$. [i]Proposed by Bradley Guo[/i]

2017 Philippine MO, 1

Tags: inequalities , number theory , function

Given $n \in \mathbb{N}$, let $\sigma (n)$ denote the sum of the divisors of $n$ and $\phi (n)$ denote the number of integers $n \geq m$ for which $\gcd(m,n) = 1$. Show that for all $n \in \mathbb{N}$, \[\large \frac{1}{\sigma (n)} + \frac{1}{\phi (n)} \geq \frac{2}{n}\] and determine when equality holds.

1990 Bulgaria National Olympiad, Problem 4

Tags: number theory , set , Sets

Suppose $M$ is an infinite set of natural numbers such that, whenever the sum of two natural numbers is in $M$, one of these two numbers is in $M$ as well. Prove that the elements of any finite set of natural numbers not belonging to $M$ have a common divisor greater than $1$.

III Soros Olympiad 1996 - 97 (Russia), 11.7

Tags: algebra , polynomial

Let us assume that each of the equations $x^7 + x^2 + 1= 0$ and $x^5- x^4 + x^2- x + 1.001 = 0$ has a single root. Which of these roots is larger?

1950 AMC 12/AHSME, 10

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After rationalizing the numerator of $ \frac {\sqrt{3}\minus{}\sqrt{2}}{\sqrt{3}}$, the denominator in simplest form is: $\textbf{(A)}\ \sqrt{3}(\sqrt{3}+\sqrt{2}) \qquad \textbf{(B)}\ \sqrt{3}(\sqrt{3}-\sqrt{2}) \qquad \textbf{(C)}\ 3-\sqrt{3}\sqrt{2} \qquad\\ \textbf{(D)}\ 3+\sqrt6 \qquad \textbf{(E)}\ \text{None of these answers}$

2024 Simon Marais Mathematical Competition, A4

Tags: number theory

Define a sequence by $s_0 = 1$ and for $d \geq 1$, $s_d = s_{d-1} + X_d$, where $X_d$ is chosen uniformly at random from the set $\{1, 2, \dots, d\}$. What is the probability that the sequence $s_0, s_1, s_2, \dots$ contains infinitely many primes?

2020 HK IMO Preliminary Selection Contest, 3

Tags: combinatorics

A child lines up $2020^2$ pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated until the number of bricks remaining drops below $250$. How many bricks remain in the end?

2001 Turkey MO (2nd round), 2

Tags: algebra unsolved , algebra

$(x_{n})_{-\infty<n<\infty}$ is a sequence of real numbers which satisfies $x_{n+1}=\frac{x_{n}^2+10}{7}$ for every $n \in \mathbb{Z}$. If there exist a real upperbound for this sequence, find all the values $x_{0}$ can take.

2016 Croatia Team Selection Test, Problem 4

Tags: number theory , prime numbers , prime , primitive root , decimal representation

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

2025 Belarusian National Olympiad, 11.4

Tags: number theory

A finite set $S$ consists of primes, and $3$ is not in $S$. Prove that there exists a positive integer $M$ such that for every $p \in S$ one can shuffle the digits of $M$ to get a number divisible by $p$ and not divisible by all other numbers in $S$. (Note: the first digit of a positive integer can not be zero). [i]A. Voidelevich[/i]

1977 AMC 12/AHSME, 29

Tags: AMC

Find the smallest integer $n$ such that \[(x^2+y^2+z^2)^2\le n(x^4+y^4+z^4)\] for all real numbers $x,y,$ and $z$. $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad \textbf{(E) }\text{There is no such integer }n.$

2009 Sharygin Geometry Olympiad, 2

Tags: geometry , cyclic quadrilateral , circumradius

A cyclic quadrilateral is divided into four quadrilaterals by two lines passing through its inner point. Three of these quadrilaterals are cyclic with equal circumradii. Prove that the fourth part also is cyclic quadrilateral and its circumradius is the same. (A.Blinkov)

2010 Malaysia National Olympiad, 7

Tags: geometry , perpendicular bisector , geometry unsolved

A line segment of length 1 is given on the plane. Show that a line segment of length $\sqrt{2010}$ can be constructed using only a straightedge and a compass.

1966 German National Olympiad, 6

Tags: sphere , geometry , 3D geometry

Prove the following theorem: If the intersection of any plane that has more than one point in common with the surface $F$ is a circle, then $F$ is a sphere (surface).

2001 Korea Junior Math Olympiad, 7

Tags: algebra , combinatorics , set

Finite set $\{a_1, a_2, ..., a_n, b_1, b_2, ..., b_n\}=\{1, 2, …, 2n\}$ is given. If $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$, show that $$\sum_{i=1}^n |a_i-b_i|=n^2$$

1968 Spain Mathematical Olympiad, 5

Tags: geometry , rectangle , Locus , inscribed

Find the locus of the center of a rectangle, whose four vertices lies on the sides of a given triangle.

VI Soros Olympiad 1999 - 2000 (Russia), 11.2

Tags: algebra , derivative , calculus

Let $$f(x) = (...((x - 2)^2 - 2)^2 - 2)^2... - 2)^2$$ (here there are $n$ brackets $( )$). Find $f''(0)$

2003 SNSB Admission, 3

Tags: complex analysis , function

Let be the set $ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . $ Show that: $ \text{(1)}\sin\in\Lambda $ $ \text{(2)}\sum_{p\in\mathbb{Z}}\frac{1}{(1+2p)^2} =\frac{\pi^2}{4} $ $ \text{(3)} f\in\Lambda\implies \left| f'(0) \right|\le 1 $

II Soros Olympiad 1995 - 96 (Russia), 10.1

Tags: algebra

Find the smallest $a$ for which the equation $x^2-ax +21 = 0$ has a root that is a natural number.

2010 Federal Competition For Advanced Students, P2, 5

Tags: geometry , rectangle , combinatorics , combinatorial geometry

Two decompositions of a square into three rectangles are called substantially different, if reordering the rectangles does not change one into the other. How many substantially different decompositions of a $2010 \times 2010$ square in three rectangles with integer side lengths are there such that the area of one rectangle is equal to the arithmetic mean of the areas of the other rectangles?

2013 Online Math Open Problems, 23

Tags: Online Math Open

Let $ABCDE$ be a regular pentagon, and let $F$ be a point on $\overline{AB}$ with $\angle CDF=55^\circ$. Suppose $\overline{FC}$ and $\overline{BE}$ meet at $G$, and select $H$ on the extension of $\overline{CE}$ past $E$ such that $\angle DHE=\angle FDG$. Find the measure of $\angle GHD$, in degrees. [i]Proposed by David Stoner[/i]

1953 Putnam, B3

Tags: Putnam , differential equation

Solve the equations $$ \frac{dy}{dx}=z(y+z)^n, \;\; \; \frac{dz}{dx} = y(y+z)^n,$$ given the initial conditions $y=1$ and $z=0$ when $x=0.$

1978 USAMO, 4

Tags: AMC , USA(J)MO , USAMO , geometry , 3D geometry , tetrahedron , sphere

(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular. (b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?