Olympiad problems

1982 All Soviet Union Mathematical Olympiad, 341

Prove that the following inequality is valid for the positive $x$: $$2^{x^{1/12}}+ 2^{x^{1/4}} \ge 2^{1 + x^{1/6} }$$

1998 Taiwan National Olympiad, 6

Tags: combinatorics

In a group of $n\geq 4$ persons, every three who know each other have a common signal. Assume that these signals are not repeatad and that there are $m\geq 1$ signals in total. For any set of four persons in which there are three having a common signal, the fourth person has a common signal with at most one of them. Show that there three persons who have a common signal, such that the number of persons having no signal with anyone of them does not exceed $[n+3-\frac{18m}{n}]$.

1985 Austrian-Polish Competition, 3

Tags: quadrilateral , geometric inequalities , geometry , inequalities , sum , area

In a convex quadrilateral of area $1$, the sum of the lengths of all sides and diagonals is not less than $4+\sqrt 8$. Prove this.

Ukrainian From Tasks to Tasks - geometry, 2016.3

Tags: geometry , angle bisector , concurrency

In fig. the bisectors of the angles $\angle DAC$, $ \angle EBD$, $\angle ACE$, $\angle BDA$ and $\angle CEB$ intersect at one point. Prove that the bisectors of the angles $\angle TPQ$, $\angle PQR$, $\angle QRS$, $\angle RST$ and $\angle STP$ also intersect at one point. [img]https://cdn.artofproblemsolving.com/attachments/6/e/870e4f20bc7fdcb37534f04541c45b1cd5034a.png[/img]

2008 Saint Petersburg Mathematical Olympiad, 1

Tags: incenter , parabola , circumcircle , geometry , conic

The graph $y=x^2+ax+b$ intersects any of the two axes at points $A$, $B$, and $C$. The incenter of triangle $ABC$ lies on the line $y=x$. Prove that $a+b+1=0$.

2024 Israel TST, P2

Tags: function , combinatorics , tree , average

A positive integer $N$ is given. Panda builds a tree on $N$ vertices, and writes a real number on each vertex, so that $1$ plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be $M$ and the minimal number written $m$. Mink then gives Panda $M-m$ kilograms of bamboo. What is the maximum amount of bamboo Panda can get?

IV Soros Olympiad 1997 - 98 (Russia), 11.6

Tags: geometry , combinatorics , combinatorial geometry

There are $6$ points marked on the plane. Find the greatest possible number of acute triangles with vertices at the marked points.

1992 Czech And Slovak Olympiad IIIA, 3

Tags: sum of digits , sum , digit , number theory

Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$

2002 Romania National Olympiad, 1

Tags: inequalities

Let $ab+bc+ca=1$. Show that \[\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\sqrt{3}+\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\]

2020 Poland - Second Round, 1.

Tags: algebra

Assume that for pairwise distinct real numbers $a,b,c,d$ holds: $$ (a^2+b^2-1)(a+b)=(b^2+c^2-1)(b+c)=(c^2+d^2-1)(c+d).$$ Prove that $ a+b+c+d=0.$

2021 Taiwan TST Round 1, C

Tags: combinatorics , recursion

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

2001 National Olympiad First Round, 3

Tags: modular arithmetic , quadratic

How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to square of an integer? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of the preceding} $

2017 Harvard-MIT Mathematics Tournament, 3

Tags: number theory

Find the number of integers $n$ with $1 \le n \le 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer multiple of $1001$.

1961 Polish MO Finals, 6

Tags: combinatorics

Someone wrote six letters to six people and addressed six envelopes to them. How many ways can the letters be put into the envelopes so that none of the letters end up in the correct envelope?

1969 AMC 12/AHSME, 17

Tags: logarithm

The equation $2^{2x}-8\cdot 2^x+12=0$ is satisfied by: $\textbf{(A) }\log3\qquad \textbf{(B) }\tfrac12\log6\qquad \textbf{(C) }1+\log\tfrac34\qquad$ $\textbf{(D) }1+\tfrac{\log3}{\log2}\qquad \textbf{(E) }\text{none of these}$

2021 CMIMC, 4

Tags: combinatorics

How many four-digit positive integers $\overline{a_1a_2a_3a_4}$ have only nonzero digits and have the property that $|a_i-a_j| \neq 1$ for all $1 \leq i<j \leq 4?$ [i]Proposed by Kyle Lee[/i]

2024 Myanmar IMO Training, 6

Tags: induction , number theory

Prove that for all integers $n \geq 3$, there exist odd positive integers $x$, $y$ such that $7x^2 + y^2 = 2^n$.

2014 China Second Round Olympiad, 2

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle such that $\angle BAC \neq 60^\circ$. Let $D,E$ be points such that $BD,CE$ are tangent to the circumcircle of $ABC$ and $BD=CE=BC$ ($A$ is on one side of line $BC$ and $D,E$ are on the other side). Let $F,G$ be intersections of line $DE$ and lines $AB,AC$. Let $M$ be intersection of $CF$ and $BD$, and $N$ be intersection of $CE$ and $BG$. Prove that $AM=AN$.

1986 Traian Lălescu, 1.4

Tags: real analysis , calculus , integration , set theory , function

Let be a parametric set: $$ \mathcal{F}_{\lambda } =\left\{ f:[1,\infty)\longrightarrow\mathbb{R}\bigg| x\in(1,\infty )\implies \int_{x}^{x^2+\lambda^2 x} f\left( \xi\right) d\xi =1\right\} . $$ [b]a)[/b] Show that $ \mathcal{F}_0 =\emptyset . $ [b]b)[/b] Prove that $ \lambda\neq 0 $ implies $ \mathcal{F}_{\lambda }\neq\emptyset . $

1981 Spain Mathematical Olympiad, 8

Tags: number theory , sum of squares , divisible

If $a$ is an odd number, show that $$a^4 + 4a^3 + 11a^2 + 6a+ 2$$ is a sum of three squares and is divisible by $4$.

2003 China Team Selection Test, 2

Tags: function , induction , algebra

Let $S$ be a finite set. $f$ is a function defined on the subset-group $2^S$ of set $S$. $f$ is called $\textsl{monotonic decreasing}$ if when $X \subseteq Y\subseteq S$, then $f(X) \geq f(Y)$ holds. Prove that: $f(X \cup Y)+f(X \cap Y ) \leq f(X)+ f(Y)$ for $X, Y \subseteq S$ if and only if $g(X)=f(X \cup \{ a \}) - f(X)$ is a $\textsl{monotonic decreasing}$ funnction on the subset-group $2^{S \setminus \{a\}}$ of set $S \setminus \{a\}$ for any $a \in S$.

1984 AMC 12/AHSME, 5

Tags:

The largest integer $n$ for which $n^{200} < 5^{300}$ is $\textbf{(A) }8\qquad \textbf{(B) }9\qquad \textbf{(C) }10\qquad \textbf{(D) }11\qquad \textbf{(E) }12$

2014 Greece Team Selection Test, 3

Tags: circumcircle , trapezoid , geometry

Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.

2023 Iran MO (3rd Round), 2

Tags: algebra

Does there exist bijections $f,g$ from positive integers to themselves st: $$g(n)=\frac{f(1)+f(2)+ \cdot \cdot \cdot +f(n)}{n}$$ holds for any $n$?

2006 Italy TST, 2

Tags: modular arithmetic , euler , number theory

Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$. a) Find all $n$ such that $A_{n}\neq \emptyset$ b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero. c) Is there $n$ such that $|{A_{n}}| = 130$?