Olympiad problems

1997 Belarusian National Olympiad, 1

Different points $A_1,A_2,A_3,A_4,A_5$ lie on a circle so that $A_1A_2 = A_2A_3 = A_3A_4 =A_4A_5$. Let $A_6$ be the diametrically opposite point to $A_2$, and $A_7$ be the intersection of $A_1A_5$ and $A_3A_6$. Prove that the lines $A_1A_6$ and $A_4A_7$ are perpendicular

2017 ASDAN Math Tournament, 6

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The sum $$\sum_{n=0}^{2016\cdot2017^2}2018^n$$ can be represented uniquely in the form $\sum_{i=0}^{\infty}a_i\cdot2017^i$ for nonnegative integers $a_i$ less than $2017$. Compute $a_0+a_1$.

1985 Polish MO Finals, 3

Tags: function , continuous , inequalities

The function $f : R \to R$ satisfies $f(3x) = 3f(x) - 4f(x)^3$ for all real $x$ and is continuous at $x = 0$. Show that $|f(x)| \le 1$ for all $x$.

2012 IMO Shortlist, A7

Tags: function , linear algebra , algebra , polynomial

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integers $m$ and $n$, it can be represented in the form \[f(x_1,\cdots , x_k )=\max_{i=1,\cdots , m} \min_{j=1,\cdots , n}P_{i,j}(x_1,\cdots , x_k),\] where $P_{i,j}$ are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.

1992 Balkan MO, 4

Tags: number theory

For each integer $n\geq 3$, find the least natural number $f(n)$ having the property $\star$ For every $A \subset \{1, 2, \ldots, n\}$ with $f(n)$ elements, there exist elements $x, y, z \in A$ that are pairwise coprime.

2007 Putnam, 2

Tags: geometry , hyperbola , analytic geometry , inequalities , rectangle , conic

Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$)

2016 Chile National Olympiad, 1

Tags: number theory , divisible

The natural number $a_n$ is obtained by writing together and ordered, in decimal notation , all natural numbers between $1$ and $n$. So we have for example that $a_1 = 1$,$a_2 = 12$, $a_3 = 123$, $. . .$ , $a_{11} = 1234567891011$, $...$ . Find all values of $n$ for which $a_n$ is not divisible by $3$.

2001 Hungary-Israel Binational, 4

Tags: polynomial , trigonometry , function , algebra

Let $P (x) = x^{3}-3x+1.$ Find the polynomial $Q$ whose roots are the ﬁfth powers of the roots of $P$.

2024-25 IOQM India, 13

Tags:

Three positive integers $a,b,c$ with $a>c$ satisfy the following equations : $$ac + b+c = bc + a + 66, \; \; \; \; a+b+c=32$$ Find the value of $a$.

2007 Stanford Mathematics Tournament, 18

Tags: geometry

A farmer wants to build a rectangular region, using a river as one side and some fencing as the other three sides. He has 1200 feet of fence which he can arrange to different dimensions. He creates the rectangular region with length $ L$ and width $ W$ to enclose the greatest area. Find $ L\plus{}W$.

2004 Harvard-MIT Mathematics Tournament, 6

Tags: number theory

$a$ and $b$ are positive integers. When written in binary, $a$ has $2004$ $1$'s, and $b$ has $2005$ $1$'s (not necessarily consecutive). What is the smallest number of $1$'s $a + b$ could possibly have?

2001 Turkey MO (2nd round), 2

Tags: trigonometry , circumcircle , perpendicular bisector , geometric transformation , geometry

Two nonperpendicular lines throught the point $A$ and a point $F$ on one of these lines different from $A$ are given. Let $P_{G}$ be the intersection point of tangent lines at $G$ and $F$ to the circle through the point $A$, $F$ and $G$ where $G$ is a point on the given line different from the line $FA$. What is the locus of $P_{G}$ as $G$ varies.

2014 Contests, Problem 1

Tags: function

Let $g:[2013,2014]\to\mathbb{R}$ a function that satisfy the following two conditions: i) $g(2013)=g(2014) = 0,$ ii) for any $a,b \in [2013,2014]$ it hold that $g\left(\frac{a+b}{2}\right) \leq g(a) + g(b).$ Prove that $g$ has zeros in any open subinterval $(c,d) \subset[2013,2014].$

2001 239 Open Mathematical Olympiad, 7

Tags: geometric inequality , circles , radii , tangent circle , geometry

The quadrangle $ ABCD $ contains two circles of radii $ R_1 $ and $ R_2 $ tangent externally. The first circle touches the sides of $ DA $,$ AB $ and $ BC $, moreover, the sides of $ AB $ at the point $ E $. The second circle touches sides $ BC $, $ CD $ and $ DA $, and sides $ CD $ at $ F $. Diagonals of the quadrangle intersect at $ O $. Prove that $ OE + OF \leq 2 (R_1 + R_2) $. (F. Bakharev, S. Berlov)

2012 Romania National Olympiad, 1

Tags: function , limit , real analysis

[color=darkred]Let $f,g\colon [0,1]\to [0,1]$ be two functions such that $g$ is monotonic, surjective and $|f(x)-f(y)|\le |g(x)-g(y)|$ , for any $x,y\in [0,1]$ . [list] [b]a)[/b] Prove that $f$ is continuous and that there exists some $x_0\in [0,1]$ with $f(x_0)=g(x_0)$ . [b]b)[/b] Prove that the set $\{x\in [0,1]\, |\, f(x)=g(x)\}$ is a closed interval. [/list][/color]

2020 Malaysia IMONST 1, 20

Tags: combinatorics

Geetha wants to cut a cube of size $4 \times 4\times 4$ into $64$ unit cubes (of size $1\times 1\times 1$). Every cut must be straight, and parallel to a face of the big cube. What is the minimum number of cuts that Geetha needs? Note: After every cut, she can rearrange the pieces before cutting again. At every cut, she can cut more than one pieces as long as the pieces are on a straight line.

Brazil L2 Finals (OBM) - geometry, 2005.2

Tags: geometry , perpendicular , right triangle , area

In the right triangle $ABC$, the perpendicular sides $AB$ and $BC$ have lengths $3$ cm and $4$ cm, respectively. Let $M$ be the midpoint of the side $AC$ and let $D$ be a point, distinct from $A$, such that $BM = MD$ and $AB = BD$. a) Prove that $BM$ is perpendicular to $AD$. b) Calculate the area of the quadrilateral $ABDC$.

2021 Science ON all problems, 4

Tags: multiplicative function , functional equation , number theory , algebra

Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$ where both sums are taken over the positive divisors of $n$. [i] (Vlad Robu) [/i]

2024 Harvard-MIT Mathematics Tournament, 9

Tags: hmmt , combinatorics

Compute the number of triples $(f,g,h)$ of permutations on $\{1,2,3,4,5\}$ such that \begin{align*} & f(g(h(x))) = h(g(f(x))) = g(x) \\ & g(h(f(x))) = f(h(g(x))) = h(x), \text{ and } \\ & h(f(g(x))) = g(f(h(x))) = f(x), \\ \end{align*} for all $x\in \{1,2,3,4,5\}$.

2019 CMIMC, 6

Tags: algebra , polynomial , vieta

Let $a, b$ and $c$ be the distinct solutions to the equation $x^3-2x^2+3x-4=0$. Find the value of $$\frac{1}{a(b^2+c^2-a^2)}+\frac{1}{b(c^2+a^2-b^2)}+\frac{1}{c(a^2+b^2-c^2)}.$$

2020 Iranian Geometry Olympiad, 2

Tags: geometry , parallelogram , angle bisector

A parallelogram $ABCD$ is given ($AB \neq BC$). Points $E$ and $G$ are chosen on the line $\overline{CD}$ such that $\overline{AC}$ is the angle bisector of both angles $\angle EAD$ and $\angle BAG$. The line $\overline{BC}$ intersects $\overline{AE}$ and $\overline{AG}$ at $F$ and $H$, respectively. Prove that the line $\overline{FG}$ passes through the midpoint of $HE$. [i]Proposed by Mahdi Etesamifard[/i]

Kvant 2019, M2548

Tags: number theory

A non-negative integer $n$ is called [I]redundant[/I] if the sum of all his proper divisors is bigger than $n$. Prove that for each non-negative integer $N$ there are $N$ consecutive redundant non-negative integers. [I]Proposed by V. Bragin[/I]

2009 Baltic Way, 14

Tags: similar triangles , geometry

For which $n\ge 2$ is it possible to find $n$ pairwise non-similar triangles $A_1, A_2,\ldots , A_n$ such that each of them can be divided into $n$ pairwise non-similar triangles, each of them similar to one of $A_1,A_2 ,\ldots ,A_n$?

2007 Germany Team Selection Test, 1

Tags: combinatorics

Let $ n > 1, n \in \mathbb{Z}$ and $ B \equal{}\{1,2,\ldots, 2^n\}.$ A subset $ A$ of $ B$ is called weird if it contains exactly one of the distinct elements $ x,y \in B$ such that the sum of $ x$ and $ y$ is a power of two. How many weird subsets does $ B$ have?

2010 Indonesia TST, 4

Tags: inequalities , algebra

Given a positive integer $n$ and $I = \{1, 2,..., k\}$ with $k$ is a positive integer. Given positive integers $a_1, a_2, ..., a_k$ such that for all $i \in I$: $1 \le a_i \le n$ and $$\sum_{i=1}^k a_i \ge 2(n!).$$ Show that there exists $J \subseteq I$ such that $$n! + 1 \ge \sum_{j \in J}a_j >\sqrt {n! + (n - 1)n}$$