Olympiad problems

2005 All-Russian Olympiad Regional Round, 11.4

11.4 Let $AA_1$ and $BB_1$ are altitudes of an acute non-isosceles triangle $ABC$, $A'$ is a midpoint of $BC$ and $B'$ is a midpoint of $AC$. A segement $A_1B_1$ intersects $A'B'$ at point $C'$. Prove that $CC'\perp HO$, where $H$ is a orthocenter and $O$ is a circumcenter of $ABC$. ([i]L. Emel'yanov[/i])

2014 IFYM, Sozopol, 2

Tags: quadratic , number theory

Polly can do the following operations on a quadratic trinomial: 1) Swapping the places of its leading coefficient and constant coefficient (swapping $a_2$ with $a_0$); 2) Substituting (changing) $x$ with $x-m$, where $m$ is an arbitrary real number; Is it possible for Polly to get $25x^2+5x+2014$ from $6x^2+2x+1996$ with finite applications of the upper operations?

2020 CMIMC Algebra & Number Theory, 10

Tags: algebra , number theory

We call a polynomial $P$ [i]square-friendly[/i] if it is monic, has integer coefficients, and there is a polynomial $Q$ for which $P(n^2)=P(n)Q(n)$ for all integers $n$. We say $P$ is [i]minimally square-friendly[/i] if it is square-friendly and cannot be written as the product of nonconstant, square-friendly polynomials. Determine the number of nonconstant, minimally square-friendly polynomials of degree at most $12$.

2010 LMT, 19

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Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$

2021 Junior Balkan Team Selection Tests - Moldova, 8

Tags: combinatorics

In a box there are $n$ balls, each colored in one of the following colors: green, red, blue or yellow. It is known that among any $28$ balls in the box at least one is green. Among any $26$ balls at least one is red. Among any $24$ balls at least one is blue. Among any $23$ balls at least one is yellow. Find the largest possible value of the number $n$.

2001 Baltic Way, 16

Tags: function , induction , number theory

Let $f$ be a real-valued function defined on the positive integers satisfying the following condition: For all $n>1$ there exists a prime divisor $p$ of $n$ such that $f(n)=f\left(\frac{n}{p}\right)-f(p)$. Given that $f(2001)=1$, what is the value of $f(2002)$?

2012 Indonesia TST, 1

Tags: combinatorics

A cycling group that has $4n$ members will have several cycling events, such that: a) Two cycling events are done every week; once on Saturday and once on Sunday. b) Exactly $2n$ members participate in any cycling event. c) No member may participate in both cycling events of a week. d) After all cycling events are completed, the number of events where each pair of members meet is the same for all pairs of members. Prove that after all cycling events are completed, the number of events where each group of three members meet is the same value $t$ for all groups of three members, and that for $n \ge 2$, $t$ is divisible by $n-1$.

Indonesia MO Shortlist - geometry, g9

Tags: right angle , arc midpoint , circles , geometry

Given two circles $\Gamma_1$ and $\Gamma_2$ which intersect at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at points $C$ and $D$, respectively. Let $M$ be the midpoint of arc $BC$ in $\Gamma_1$ ,which does not contains $A$, and $N$ is the midpoint of the arc $BD$ in $\Gamma_2$, which does not contain $A$. If $K$ is the midpoint of $CD$, prove that $\angle MKN = 90^o.$

2012 Regional Competition For Advanced Students, 4

Tags: trigonometry , geometry

In a triangle $ABC$, let $H_a$, $H_b$ and $H_c$ denote the base points of the altitudes on the sides $BC$, $CA$ and $AB$, respectively. Determine for which triangles $ABC$ two of the lengths $H_aH_b$, $H_bH_c$ and $H_aH_c$ are equal.

2000 Saint Petersburg Mathematical Olympiad, 10.4

Tags: number theory , existence

The number $N$ is the product of $200$ distinct positive integers. Prove that it has at least 19901 distinct divisors (including 0 and itself). [I]Proposed by A. Golovanov[/i]

2008 239 Open Mathematical Olympiad, 3

Tags: graph theory , combinatorics

A connected graph has $100$ vertices, the degrees of all the vertices do not exceed $4$ and no two vertices of degree $4$ are adjacent. Prove that it is possible to remove several edges that have no common vertices from this graph such that there would be no triangles in the resulting graph.

1989 Federal Competition For Advanced Students, P2, 1

Tags: induction , polynomial , algebra

Consider the set $ S_n$ of all the $ 2^n$ numbers of the type $ 2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$ where number $ 2$ appears $ n\plus{}1$ times. $ (a)$ Show that all members of $ S_n$ are real. $ (b)$ Find the product $ P_n$ of the elements of $ S_n$.

1985 All Soviet Union Mathematical Olympiad, 395

Tags: geometry , hexagon , area

Two perpendiculars are drawn from the midpoints of each side of the acute-angle triangle to two other sides. Those six segments make hexagon. Prove that the hexagon area is a half of the triangle area.

1977 IMO Longlists, 8

Tags: geometry

A hexahedron $ABCDE$ is made of two regular congruent tetrahedra $ABCD$ and $ABCE.$ Prove that there exists only one isometry $\mathbf Z$ that maps points $A, B, C, D, E$ onto $B, C, A, E, D,$ respectively. Find all points $X$ on the surface of hexahedron whose distance from $\mathbf Z(X)$ is minimal.

2006 MOP Homework, 3

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Let $X=\{A_{1},...,A_{n}\}$ be a set of distinct 3-element subsets of the set $\{1,2,...,36\}$ such that (a) $A_{i},A_{j}$ have nonempty intersections for all $i,j$ (b) The intersection of all elements of $X$ is the empty set. Show that $n\leq 100$. Determine the number of such sets $X$ when $n=100$

2020 LMT Fall, A9

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$\triangle ABC$ has a right angle at $B$, $AB = 12$, and $BC = 16$. Let $M$ be the midpoint of $AC$. Let $\omega_1$ be the incircle of $\triangle ABM$ and $\omega_2$ be the incircle of $\triangle BCM$. The line externally tangent to $\omega_1$ and $\omega_2$ that is not $AC$ intersects $AB$ and $BC$ at $X$ and $Y$, respectively. If the area of $\triangle BXY$ can be expressed as $\frac{m}{n}$, compute is $m+n$. [i]Proposed by Alex Li[/i]

2023 Grosman Mathematical Olympiad, 1

Tags: number theory , divisibility , arithmetic sequence

An arithmetic progression of natural numbers of length $10$ and with difference $11$ is given. Prove that the product of the numbers in this progression is divisible by $10!$.

2013 Moldova Team Selection Test, 4

Tags: inequalities

Prove that for any positive real numbers $a_i,b_i,c_i$ with $i=1,2,3$, $(a_1^3+b_1^3+c_1^3+1)(a_2^3+b_2^3+c_2^3+1)(a_3^3+b_3^3+c_3^3+1)\geq \frac{3}{4} (a_1+b_1+c_1)(a_2+b_2+c_2)(a_3+b_3+c_3)$

2022 HMNT, 1

Tags: clock

Emily’s broken clock runs backwards at five times the speed of a regular clock. Right now, it is displaying the wrong time. How many times will it display the correct time in the next 24 hours? It is an analog clock (i.e. a clock with hands), so it only displays the numerical time, not AM or PM. Emily’s clock also does not tick, but rather updates continuously.

2013 Hanoi Open Mathematics Competitions, 13

Tags: algebra , system of equations

Solve the system of equations $\begin{cases} \frac{1}{x}+\frac{1}{y}=\frac{1}{6} \\ \frac{3}{x}+\frac{2}{y}=\frac{5}{6} \end{cases}$

2010 AMC 12/AHSME, 20

Tags: number theory , greatest common divisor , modular arithmetic

Arithmetic sequences $ (a_n)$ and $ (b_n)$ have integer terms with $ a_1 \equal{} b_1 \equal{} 1 < a_2 \le b_2$ and $ a_nb_n \equal{} 2010$ for some $ n$. What is the largest possible value of $ n$? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 288 \qquad \textbf{(E)}\ 2009$

2012 Mathcenter Contest + Longlist, 6 sl14

Tags: algebra , inequalities , max

For a real number $a,b,c>0$ where $bc-ca-ab=1$ find the maximum value of $$P=\frac{4024}{1+a^2}-\frac{4024}{1+b^2}-\frac{2555}{1+c^2}$$ and find out when that holds . [i](PP-nine)[/i]

1966 Swedish Mathematical Competition, 4

Tags: composition , sequence , algebra

Let $f(x) = 1 + \frac{2}{x}$. Put $f_1(x) = f(x)$, $f_2(x) = f(f_1(x))$, $f_3(x) = f(f_2(x))$, $... $. Find the solutions to $x = f_n(x)$ for $n > 0$.

2013 Junior Balkan Team Selection Tests - Romania, 5

Tags: integer , number theory , set

a) Prove that for every positive integer n, there exist $a, b \in R - Z$ such that the set $A_n = \{a - b, a^2 - b^2, a^3 - b^3,...,a^n - b^n\}$ contains only positive integers. b) Let $a$ and $b$ be two real numbers such that the set $A = \{a^k - b^k | k \in N*\}$ contains only positive integers. Prove that $a$ and $b$ are integers.

2005 National Olympiad First Round, 18

Tags: modular arithmetic , quadratic

How many integers $0\leq x < 121$ are there such that $x^5+5x^2 + x + 1 \equiv 0 \pmod{121}$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $