Olympiad problems

1997 Balkan MO, 2

Let $S = \{A_1,A_2,\ldots ,A_k\}$ be a collection of subsets of an $n$-element set $A$. If for any two elements $x, y \in A$ there is a subset $A_i \in S$ containing exactly one of the two elements $x$, $y$, prove that $2^k\geq n$. [i]Yugoslavia[/i]

2002 Junior Balkan Team Selection Tests - Moldova, 9

Tags: inequalities , algebra

The real numbers $a$ and $b$ satisfy the relation $a + b \ge 1$. Show that $8 (a^4 + b^4) \ge 1$.

2007 Hong kong National Olympiad, 4

Tags: floor function , number theory proposed , number theory

find all positive integer pairs $(m,n)$,satisfies: (1)$gcd(m,n)=1$,and $m\le\ 2007$ (2)for any $k=1,2,...2007$,we have $[\frac{nk}{m}]=[\sqrt{2}k]$

2017 IMO Shortlist, N5

Tags: number theory , IMO Shortlist

Find all pairs $(p,q)$ of prime numbers which $p>q$ and $$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$ is an integer.

2000 Saint Petersburg Mathematical Olympiad, 10.7

Tags: number theory , prime numbers , Divisibility

We'll call a positive integer "almost prime", if it is not divisible by any prime from the interval $[3,19]$. We'll call a number "very non-prime", if it has at least 2 primes from interval $[3,19]$ dividing it. What is the greatest amount of almost prime numbers can be selected, such that the sum of any two of them is a very non-prime number? [I]Proposed by S. Berlov, S. Ivanov[/i]

2023 Ukraine National Mathematical Olympiad, 10.4

Tags: algebra , Sequences

Let $(x_n)$ be an infinite sequence of real numbers from interval $(0, 1)$. An infinite sequence $(a_n)$ of positive integers is defined as follows: $a_1 = 1$, and for $i \ge 1$, $a_{i+1}$ is equal to the smallest positive integer $m$, for which $[x_1 + x_2 + \ldots + x_m] = a_i$. Show that for any indexes $i, j$ holds $a_{i+j} \ge a_i + a_j$. [i]Proposed by Nazar Serdyuk[/i]

2005 Putnam, A1

Tags: Putnam , college contests

Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another. (For example, $23=9+8+6.)$

2003 National High School Mathematics League, 8

Tags: conics , ellipse , geometry

$F_1,F_2$ are two focal points of ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, $P$ is a point on the ellipse, and $|PF_1|:|PF_2|=2:1$, then the area of $\triangle PF_1F_2$ is________.

1992 India Regional Mathematical Olympiad, 8

Tags: geometry , circumcircle , symmetry , trigonometry , trig identities , Law of Cosines

The cyclic octagon $ABCDEFGH$ has sides $a,a,a,a,b,b,b,b$ respectively. Find the radius of the circle that circumscribes $ABCDEFGH.$

2010 India IMO Training Camp, 6

Tags: floor function , induction , combinatorics unsolved , combinatorics

Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to \[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]

1976 Yugoslav Team Selection Test, Problem 3

Tags: inequalities

Find the minimum and maximum values of the function $$f(x,y,z,t)=\frac{ax^2+by^2}{ax+by}+\frac{az^2+bt^2}{az+bt},~(a>0,b>0),$$given that $x+z=y+t=1$, and $x,y,z,t\ge0$.

2017 Serbia JBMO TST, 1

Tags: combinatorics

15 of the cells of a chessboard 8x8 are chosen. We draw the segments which unite the centers of every two of the chosen squares. Prove that among these segments there are four segments which have the same length.

2000 French Mathematical Olympiad, Exercise 2

Tags: 3D geometry , geometry , sphere , parameterization

Let $A,B,C$ be three distinct points in space, $(A)$ the sphere with center $A$ and radius $r$. Let $E$ be the set of numbers $R>0$ for which there is a sphere $(H)$ with center $H$ and radius $R$ such that $B$ and $C$ are outside the sphere, and the points of the sphere $(A)$ are strictly inside it. (a) Suppose that $B$ and $C$ are on a line with $A$ and strictly outside $(A)$. Show that $E$ is nonempty and bounded, and determine its supremum in terms of the given data. (b) Find a necessary and sufficient condition for $E$ to be nonempty and bounded (c) Given $r$, compute the smallest possible supremum of $E$, if it exists.

2010 Stanford Mathematics Tournament, 4

Tags: ratio , quadratics , algebra , quadratic formula

Compute $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}}...}$

2021/2022 Tournament of Towns, P1

Tags: number theory

For each of the $9$ positive integers $n,2n,3n,\dots , 9n$ Alice take the first decimal digit (from the left) and writes it onto a blackboard. She selected $n$ so that among the nine digits on the blackboard there is the least possible number of different digits. What is this number of different digits equals to?

2014 Belarus Team Selection Test, 1

Tags: geometry , analytic geometry , hyperbola , parallel , conics , parabola

All vertices of triangles $ABC$ and $A_1B_1C_1$ lie on the hyperbola $y=1/x$. It is known that $AB \parallel A_1B_1$ and $BC \parallel B_1C_1$. Prove that $AC_1 \parallel A_1C$. (I. Gorodnin)

2004 Purple Comet Problems, 1

Tags:

This year February $29$ fell on a Sunday. In what year will February $29$ next fall on a Sunday?

1998 Canada National Olympiad, 1

Tags: number theory unsolved , number theory

Determine the number of real solutions $a$ to the equation: \[ \left[\,\frac{1}{2}\;a\,\right]+\left[\,\frac{1}{3}\;a\,\right]+\left[\,\frac{1}{5}\;a\,\right] = a. \] Here, if $x$ is a real number, then $[\,x\,]$ denotes the greatest integer that is less than or equal to $x$.

Kyiv City MO Juniors 2003+ geometry, 2012.8.3

Tags: geometry

On the circle $\gamma$ the points $A$ and $B$ are selected. The circle $\omega$ touches the segment $AB$ at the point $K$ and intersects the circle $\gamma$ at the points $M$ and $N$. The points lie on the circle $\gamma$ in the following order: $A, \, \, M, \, \, N, \, \, B$. Prove that $\angle AMK = \angle KNB$. (Yuri Biletsky)

2021 AMC 12/AHSME Fall, 4

Tags: AMC , AMC 10 , AMC 10 B , AMC 12 , AMC 12 B

Let $n = 8^{2022}$. Which of the following is equal to $\frac{n}{4}$? $\textbf{(A) }4^{1010}\qquad\textbf{(B) }2^{2022}\qquad\textbf{(C) }8^{2018}\qquad\textbf{(D) }4^{3031}\qquad\textbf{(E) }4^{3032}$

2001 Moldova National Olympiad, Problem 2

Tags: number theory

Let $S(n)$ denote the sum of digits of a natural number $n$. Find all $n$ for which $n+S(n)=2004$.

2023 Malaysian IMO Training Camp, 4

Tags: number theory

Find the largest constant $c>0$ such that for every positive integer $n\ge 2$, there always exist a positive divisor $d$ of $n$ such that $$d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)}$$ where $\tau(n)$ is the number of divisors of $n$. [i]Proposed by Mohd. Suhaimi Ramly[/i]

2001 Moldova National Olympiad, Problem 2

Tags: Inequality , optimization , inequalities

If $n\in\mathbb N$ and $a_1,a_2,\ldots,a_n$ are arbitrary numbers in the interval $[0,1]$, find the maximum possible value of the smallest among the numbers $a_1-a_1a_2,a_2-a_2a_3,\ldots,a_n-a_na_1$.

1972 Vietnam National Olympiad, 3

Tags: geometry , Parallel Lines , parallel , collinear

$ABC$ is a triangle. $U$ is a point on the line $BC$. $I$ is the midpoint of $BC$. The line through $C$ parallel to $AI$ meets the line $AU$ at $E$. The line through $E$ parallel to $BC$ meets the line $AB$ at $F$. The line through $E$ parallel to $AB$ meets the line $BC$ at $H$. The line through $H$ parallel to $AU$ meets the line $AB$ at $K$. The lines $HK$ and $FG$ meet at $T. V$ is the point on the line $AU$ such that $A$ is the midpoint of $UV$. Show that $V, T$ and $I$ are collinear.

2024 Indonesia TST, 2

Tags: geometry

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.