Olympiad problems

2021 Tuymaada Olympiad, 4

An $n\times n$ square ($n$ is a positive integer) consists of $n^2$ unit squares.A $\emph{monotonous path}$ in this square is a path of length $2n$ beginning in the left lower corner of the square,ending in its right upper corner and going along the sides of unit squares. For each $k$, $0\leq k\leq 2n-1$, let $S_k$ be the set of all the monotonous paths such that the number of unit squares lying below the path leaves remainder $k$ upon division by $2n-1$.Prove that all $S_k$ contain equal number of elements.

2004 National Chemistry Olympiad, 60

Tags:

Most enzymes are a type of $ \textbf{(A) } \text{Carbohydrate} \qquad\textbf{(B) } \text{Lipid} \qquad\textbf{(C) } \text{Nucleic Acid} \qquad\textbf{(D) } \text{Protein} \qquad $

1974 Canada National Olympiad, 1

Tags: induction , logarithm

i) If $x = \left(1+\frac{1}{n}\right)^{n}$ and $y=\left(1+\frac{1}{n}\right)^{n+1}$, show that $y^{x}= x^{y}$. ii) Show that, for all positive integers $n$, \[1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).\]

2007 Pre-Preparation Course Examination, 20

Tags: number theory

Let $m,n$ be two positive integers and $m \geq 2$. We know that for every positive integer $a$ such that $\gcd(a,n)=1$ we have $n|a^m-1$. Prove that $n \leq 4m(2^m-1)$.

2000 Italy TST, 1

Tags: number theory

Determine all triples $(x,y,z)$ of positive integers such that \[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997} \]

1982 Vietnam National Olympiad, 2

Tags: parameterization , algebra

For a given parameter $m$, solve the equation \[x(x + 1)(x + 2)(x + 3) + 1 - m = 0.\]

1990 IMO Longlists, 71

Tags: geometry

Given a point $P = (p_1, p_2, \ldots, p_n)$ in $n$-dimensional space . Find point $X = (x_1, x_2, \ldots, x_n)$, such that $x_1 \leq x_2 \leq\cdots \leq x_n$ and $\sqrt{(x_1-p_1)^2 + (x_2-p_2)^2+\cdots+(x_n-p_n)^2}$ is minimal.

2003 District Olympiad, 2

Tags: algebra

Let $M \subset R$ be a finite set containing at least two elements. We say that the function $f$ has property $P$ if $f : M \to M$ and there are $a \in R^*$ and $b \in R$ such that $f(x) = ax + b$. (a) Show that there is at least a function having property $P$. (b) Show that there are at most two functions having property $P$. (c) If $M$ has $2003$ elements with sum $0$ and if there are two functions with property $P$, prove that $0 \in M$.

2014 Cezar Ivănescu, 2

Tags: function , real analysis

Let be a function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R}_{>0} $ that satisfies the relation $$ \sqrt{x^2-x+1}\le f(x) e^{f(x)}\le \sqrt{x^2+x+1} , $$ for any positive real number $ x. $ Prove that [b]a)[/b] $ \lim_{x\to\infty } f(x)=\infty . $ [b]b)[/b] $ \lim_{x\to\infty } (1/x)^{1/f(x)} =1/e. $

1965 AMC 12/AHSME, 18

Tags: ratio

If $ 1 \minus{} y$ is used as an approximation to the value of $ \frac {1}{1 \plus{} y}$, $ |y| < 1$, the ratio of the error made to the correct value is: $ \textbf{(A)}\ y \qquad \textbf{(B)}\ y^2 \qquad \textbf{(C)}\ \frac {1}{1 \plus{} y} \qquad \textbf{(D)}\ \frac {y}{1 \plus{} y} \qquad \textbf{(E)}\ \frac {y^2}{1 \plus{} y}\qquad$

2008 Sharygin Geometry Olympiad, 10

Tags: angle bisector , geometry

(A.Zaslavsky, 9--10) Quadrilateral $ ABCD$ is circumscribed arounda circle with center $ I$. Prove that the projections of points $ B$ and $ D$ to the lines $ IA$ and $ IC$ lie on a single circle.

2019 Iran Team Selection Test, 5

Tags: combinatorics

A sub-graph of a complete graph with $n$ vertices is chosen such that the number of its edges is a multiple of $3$ and degree of each vertex is an even number. Prove that we can assign a weight to each triangle of the graph such that for each edge of the chosen sub-graph, the sum of the weight of the triangles that contain that edge equals one, and for each edge that is not in the sub-graph, this sum equals zero. [i]Proposed by Morteza Saghafian[/i]

1961 AMC 12/AHSME, 19

Tags: function , algebra , domain , logarithm

Consider the graphs of $y=2\log{x}$ and $y=\log{2x}$. We may say that: $ \textbf{(A)}\ \text{They do not intersect}$ $ \qquad\textbf{(B)}\ \text{They intersect at 1 point only}$ $\qquad\textbf{(C)}\ \text{They intersect at 2 points only}$ $\qquad\textbf{(D)}\ \text{They intersect at a finite number of points but greater than 2 }$ ${\qquad\textbf{(E)}\ \text{They coincide} } $

2024 Stars of Mathematics, P1

Tags: combinatorics

Prove that any polygon $A_1A_2\dots A_n$ has three vertices $A_i,A_j,A_k$ such that $[A_iA_jA_k]>\frac{1}{4}[A_1A_2\dots A_n]$. [i]Folklore[/i]

2022 Vietnam National Olympiad, 2

Tags: combinatorics , probability

We are given 4 similar dices. Denote $x_i (1\le x_i \le 6)$ be the number of dots on a face appearing on the $i$-th dice $1\le i \le 4$ a) Find the numbers of $(x_1,x_2,x_3,x_4)$ b) Find the probability that there is a number $x_j$ such that $x_j$ is equal to the sum of the other 3 numbers c) Find the probability that we can divide $x_1,x_2,x_3,x_4$ into 2 groups has the same sum

2004 China Team Selection Test, 1

Tags: function , quadratic , algebra

Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.

2019 Taiwan TST Round 3, 3

Tags: combinatorics

Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. [list=i] [*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. [*] If no such pair exists, we write two times the number $0$. [/list] Prove that, no matter how we make the choices in $(i)$, operation $(ii)$ will be performed only finitely many times. Proposed by [I]Serbia[/I].

1997 Slovenia National Olympiad, Problem 3

Tags: ratio , geometry

In a convex quadrilateral $ABCD$ we have $\angle ADB=\angle ACD$ and $AC=CD=DB$. If the diagonals $AC$ and $BD$ intersect at $X$, prove that $\frac{CX}{BX}-\frac{AX}{DX}=1$.

2020 BMT Fall, 18

Tags: number theory

Let $x$ and $y$ be integers between $0$ and $5$, inclusive. For the system of modular congruences $$ \begin{cases} x + 3y \equiv 1 \,\,(mod \, 2) \\ 4x + 5y \equiv 2 \,\,(mod \, 3) \end{cases}$$, find the sum of all distinct possible values of $x + y$

2018 OMMock - Mexico National Olympiad Mock Exam, 6

Tags: combinatorics , elements of set

Let $A$ be a finite set of positive integers, and for each positive integer $n$ we define \[S_n = \{x_1 + x_2 + \cdots + x_n \;\vert\; x_i \in A \text{ for } i = 1, 2, \dots, n\}\] That is, $S_n$ is the set of all positive integers which can be expressed as sum of exactly $n$ elements of $A$, not necessarily different. Prove that there exist positive integers $N$ and $k$ such that $$\left\vert S_{n + 1} \right\vert = \left\vert S_n \right\vert + k \text{ for all } n\geq N.$$ [i]Proposed by Ariel García[/i]

2010 China Team Selection Test, 1

Tags: geometry , symmetry , geometric transformation , angle bisector

Let $ABCD$ be a convex quadrilateral with $A,B,C,D$ concyclic. Assume $\angle ADC$ is acute and $\frac{AB}{BC}=\frac{DA}{CD}$. Let $\Gamma$ be a circle through $A$ and $D$, tangent to $AB$, and let $E$ be a point on $\Gamma$ and inside $ABCD$. Prove that $AE\perp EC$ if and only if $\frac{AE}{AB}-\frac{ED}{AD}=1$.

2021 Spain Mathematical Olympiad, 6

Tags: geometry , incircle , excircle , incenter

Let $ABC$ be a triangle with $AB \neq AC$, let $I$ be its incenter, $\gamma$ its inscribed circle and $D$ the midpoint of $BC$. The tangent to $\gamma$ from $D$ different to $BC$ touches $\gamma$ in $E$. Prove that $AE$ and $DI$ are parallel.

2007 Tournament Of Towns, 4

Tags: polynomial , quadratic , algebra

Three nonzero real numbers are given. If they are written in any order as coefficients of a quadratic trinomial, then each of these trinomials has a real root. Does it follow that each of these trinomials has a positive root?

LMT Guts Rounds, 2020 F17

Tags:

In a regular square room of side length $2\sqrt{2}$ ft, two cats that can see $2$ feet ahead of them are randomly placed into the four corners such that they do not share the same corner. If the probability that they don't see the mouse, also placed randomly into the room can be expressed as $\frac{a-b\pi}{c},$ where $a,b,c$ are positive integers with a greatest common factor of $1,$ then find $a+b+c.$ [i]Proposed by Ada Tsui[/i]

2023 Brazil Cono Sur TST, 3

Tags: p-adic valuation

Let $a,b,c$ be positive integers satisfying $\gcd(a,b,c)=1$ and $$\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ca}{b}$$ is an integer. Prove that $abc$ is a perfect square.