Olympiad problems

2004 Junior Balkan MO, 1

Prove that the inequality \[ \frac{ x+y}{x^2-xy+y^2 } \leq \frac{ 2\sqrt 2 }{\sqrt{ x^2 +y^2 } } \] holds for all real numbers $x$ and $y$, not both equal to 0.

2012 Purple Comet Problems, 9

Tags: logarithm

Find the value of $x$ that satisfies $\log_{3}(\log_9x)=\log_9(\log_3x)$

1986 Czech And Slovak Olympiad IIIA, 6

Tags: combinatorics , subset , inequalities

Assume that $M \subset N$ has the property that every two numbers $m,n$ of $M$ satisfy $|m-n| \ge mn/25$. Prove that the set $M$ contains no more than $9$ elements. Decide whether there exists such set M.

2003 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , consecutive , divide , sum of powers , prime

Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.

1982 IMO Shortlist, 8

Tags: symmetry , geometry , rectangle , convex polygon , circles

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.

2023 Balkan MO Shortlist, A5

Tags: algebra

Are there polynomials $P, Q$ with real coefficients, such that $P(P(x))\cdot Q(Q(x))$ has exactly $2023$ distinct real roots and $P(Q(x)) \cdot Q(P(x))$ has exactly $2024$ distinct real roots?

2016 Mathematical Talent Reward Programme, SAQ: P 4

Tags: geometry

For any given $k$ points in a plane, we define the diameter of the points as the maximum distance between any two points among the given points. Suppose $n$ points are there in a plane with diameter $d$. Show that we can always find a circle with radius $\frac{\sqrt{3}}{2}d$ such that all points lie inside the circle.

2000 USAMO, 3

Tags: induction

A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.

2015 AIME Problems, 4

Tags: geometry , trapezoid , perimeter

In an isosceles trapezoid, the parallel bases have lengths $\log3$ and $\log192$, and the altitude to these bases has length $\log16$. The perimeter of the trapezoid can be written in the form $\log2^p3^q$, where $p$ and $q$ are positive integers. Find $p+q$.

1989 Czech And Slovak Olympiad IIIA, 3

Tags: number theory

For given coprime numbers $p > q > 0$, find all pairs of real numbers $c,d$ such that for the sets $$A = \left\{ \left[n\frac{p}{q}\right] , n \in N \right\} \ \ and \ \ B = \{[cn + d], n \in N\}$$ where $A \cap B = \emptyset$, $A \cup B = N$, where $N = \{1, 2, 3, ...\}$ is the set of all natural numbers.

PEN O Problems, 25

Tags: limit

Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_{1}$, $\cdots$, $b_{n}$ and $c_{1}$, $\cdots$, $c_{n}$ such that [list] [*] for each $i$ the set $b_{i}A+c_{i}=\{b_{i}a+c_{i}\vert a \in A \}$ is a subset of $A$, [*] the sets $b_{i}A+c_{i}$ and $b_{j}A+c_{j}$ are disjoint whenever $i \neq j$.[/list] Prove that \[\frac{1}{b_{1}}+\cdots+\frac{1}{b_{n}}\le 1.\]

2021-IMOC qualification, C1

Tags: combinatorics

There are $3n$ $A$s and $2n$ $B$s in a string, where $n$ is a positive integer, prove that you can find a substring in this string that contains $3$ $A$s and $2$ $B$s.

2004 All-Russian Olympiad Regional Round, 9.6

Tags: algebra , inequalities

Positive numbers $x, y, z$ are such that the absolute value of the difference of any two of them are less than $2$. Prove that $$ \sqrt{xy +1}+\sqrt{yz + 1}+\sqrt{zx+ 1} > x+ y + z.$$

2009 Regional Olympiad of Mexico Center Zone, 2

Tags: number theory , prime number

Let $p \ge 2$ be a prime number and $a \ge 1$ a positive integer with $p \neq a$. Find all pairs $(a,p)$ such that: $a+p \mid a^2+p^2$

2022 Moldova Team Selection Test, 1

Tags: number theory

Show that for every integer $n \geq 2$ the number $$a=n^{5n-1}+n^{5n-2}+n^{5n-3}+n+1$$ is a composite number.

1996 IMO Shortlist, 7

Tags: function , symmetry , combinatorics , partition

let $ V$ be a finitive set and $ g$ and $ f$ be two injective surjective functions from $ V$to$ V$.let $ T$ and $ S$ be two sets such that they are defined as following" $ S \equal{} \{w \in V: f(f(w)) \equal{} g(g(w))\}$ $ T \equal{} \{w \in V: f(g(w)) \equal{} g(f(w))\}$ we know that $ S \cup T \equal{} V$, prove: for each $ w \in V : f(w) \in S$ if and only if $ g(w) \in S$

2020 LMT Fall, A28 B30

Tags:

Arthur has a regular 11-gon. He labels the vertices with the letters in $CORONAVIRUS$ in consecutive order. Every non-ordered set of 3 letters that forms an isosceles triangle is a member of a set $S$, i.e. $\{C, O, R\}$ is in $S$. How many elements are in $S$? [i]Proposed by Sammy Chareny[/i]

2009 Switzerland - Final Round, 9

Tags: inequalities , functional inequality , functional , algebra

Find all injective functions $f : N\to N$ such that holds for all natural numbers $n$: $$f(f(n)) \le \frac{f(n) + n}{2}$$

2023 Mexican Girls' Contest, 1

Tags: isosceles , geometry

Let $\triangle ABC$ such that $AB=AC$, $D$ and $E$ points on $AB$ and $BC$, respectively, with $DE\parallel AC$. Let $F$ on line $DE$ such that $CADF$ it´s a parallelogram. If $O$ is the circumcenter of $\triangle BDE$, prove that $O,F,A$ and $D$ lie on a circle.

1967 Kurschak Competition, 2

Tags: combinatorics , combinatorial geometry

A convex $n$-gon is divided into triangles by diagonals which do not intersect except at vertices of the n-gon. Each vertex belongs to an odd number of triangles. Show that $n$ must be a multiple of $3$.

Kyiv City MO Juniors Round2 2010+ geometry, 2018.8.31

Tags: geometry , isosceles , equal angles , equal segments

On the sides $AB$, $BC$ and $CA$ of the isosceles triangle $ABC$ with the vertex at the point $B$ marked the points $M$, $D$ and $K$ respectively so that $AM = 2DC$ and $\angle AMD = \angle KDC$. Prove that $MD = KD$.

2001 Romania Team Selection Test, 2

Tags: function , pigeonhole principle , number theory , prime number , combinatorics , algebra

a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function. b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.

2000 National Olympiad First Round, 4

Tags:

What is the sum of real roots of $(x\sqrt{x})^x = x^{x\sqrt{x}}$? $ \textbf{(A)}\ \frac{18}{7} \qquad\textbf{(B)}\ \frac{71}{4} \qquad\textbf{(C)}\ \frac{9}{4} \qquad\textbf{(D)}\ \frac{24}{19} \qquad\textbf{(E)}\ \frac{13}{4} $

1956 Moscow Mathematical Olympiad, 321

Tags: number theory , 2-digit , sum of digits

Find all two-digit numbers $x$ the sum of whose digits is the same as that of $2x$, $3x$, ... , $9x$.

2004 Vietnam Team Selection Test, 3

Tags: geometric transformation , rotation , homothety , function , ratio , geometry

In the plane, there are two circles $\Gamma_1, \Gamma_2$ intersecting each other at two points $A$ and $B$. Tangents of $\Gamma_1$ at $A$ and $B$ meet each other at $K$. Let us consider an arbitrary point $M$ (which is different of $A$ and $B$) on $\Gamma_1$. The line $MA$ meets $\Gamma_2$ again at $P$. The line $MK$ meets $\Gamma_1$ again at $C$. The line $CA$ meets $\Gamma_2 $ again at $Q$. Show that the midpoint of $PQ$ lies on the line $MC$ and the line $PQ$ passes through a fixed point when $M$ moves on $\Gamma_1$. [color=red][Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .][/color]