Olympiad problems

1997 Iran MO (3rd Round), 1

Find all strictly ascending functions $f$ such that for all $x\in \mathbb R$, \[f(1-x)=1-f(f(x)).\]

2021 Oral Moscow Geometry Olympiad, 2

Tags: geometry , perimeter , congruent

Two quadrangles have equal areas, perimeters and corresponding angles. Are such quadrilaterals necessarily congurent ?

Five players $(A,B,C,D,E)$ take part in a bridge tournament. Every two players must play (as partners) against every other two players. Any two given players can be partners not more than once per a day. What is the least number of days needed for this tournament?

2020 Harvard-MIT Mathematics Tournament, 5

Tags:

A positive integer $N$ is \emph{piquant} if there exists a positive integer $m$ such that if $n_i$ denotes the number of digits in $m^i$ (in base $10$), then $n_1+n_2+\cdots + n_{10}=N$. Let $p_M$ denote the fraction of the first $M$ positive integers that are piquant. Find $\lim\limits_{M\to \infty} p_M$. [i]Proposed by James Lin.[/i]

2016 Auckland Mathematical Olympiad, 4

Tags: algebra , number theory

If $m, n$, and $p$ are three different natural numbers, each between $2$ and $9$, what then are all the possible integer value(s) of the expression $\frac{m+n+p}{m+n}$?

1988 ITAMO, 6

Tags: 3D geometry , geometry , tetrahedron , geometric inequality

The edge lengths of the base of a tetrahedron are $a,b,c$, and the lateral edge lengths are $x,y,z$. If $d$ is the distance from the top vertex to the centroid of the base, prove that $x+y+z \le a+b+c+3d$.

1966 Miklós Schweitzer, 8

Tags: algebra , polynomial , function , Ring Theory , superior algebra , superior algebra unsolved

Prove that in Euclidean ring $ R$ the quotient and remainder are always uniquely determined if and only if $ R$ is a polynomial ring over some field and the value of the norm is a strictly monotone function of the degree of the polynomial. (To be precise, there are two trivial cases: $ R$ can also be a field or the null ring.) [i]E. Fried[/i]

2007 Italy TST, 3

Tags: inequalities , geometry , geometric transformation , reflection , number theory , prime numbers , number theory proposed

Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

2016 Polish MO Finals, 3

Tags: combinatorics , counting

Let $a, \ b \in \mathbb{Z_{+}}$. Denote $f(a, b)$ the number sequences $s_1, \ s_2, \ ..., \ s_a$, $s_i \in \mathbb{Z}$ such that $|s_1|+|s_2|+...+|s_a| \le b$. Show that $f(a, b)=f(b, a)$.

2023 Moldova Team Selection Test, 3

Tags: combinatorics

Let $ n $ be a positive integer. A sequence $(a_1,a_2,\ldots,a_n)$ of length is called $balanced$ if for every $ k $ $(1\leq k\leq n)$ the term $ a_k $ is equal with the number of distinct numbers from the subsequence $(a_1,a_2,\ldots,a_k).$ a) How many balanced sequences $(a_1,a_2,\ldots,a_n)$ of length $ n $ do exist? b) For every positive integer $m$ find how many balanced sequences $(a_1,a_2,\ldots,a_n)$ of length $ n $ exist such that $a_n=m.$

1964 Vietnam National Olympiad, 3

Tags: geometry , 3-Dimensional Geometry , Locus

Let $P$ be a plane and two points $A \in (P),O \notin (P)$. For each line in $(P)$ through $A$, let $H$ be the foot of the perpendicular from $O$ to the line. Find the locus $(c)$ of $H$. Denote by $(C)$ the oblique cone with peak $O$ and base $(c)$. Prove that all planes, either parallel to $(P)$ or perpendicular to $OA$, intersect $(C)$ by circles. Consider the two symmetric faces of $(C)$ that intersect $(C)$ by the angles $\alpha$ and $\beta$ respectively. Find a relation between $\alpha$ and $\beta$.

1995 Cono Sur Olympiad, 3

Tags: floor function , function , modular arithmetic , inequalities , number theory , relatively prime

Let $n$ be a natural number and $f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor$($\lfloor$ $\rfloor$ denotes the floor function). 1. Show that if for some integer $r$: $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times), then $n$ is multiple of $1995$. 2. Show that if $n$ is multiple of 1995, then there exists r such that:$f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times). Determine $r$ if $n=1995.500=997500$

2005 National High School Mathematics League, 3

Tags: calculus , integration

For positive integer $n$, define $f(n)=\begin{cases} 0, \text{if }n\text{ is a perfect square}\\ \displaystyle \left[\frac{1}{\{\sqrt{n}\}}\right], \text{if }n\text{ is not a perfect square}\\ \end{cases}$. Find the value of $\sum_{k=1}^{240} f(k)$. Note: $[x]$ is the integral part of real number $x$, and $\{x\}=x-[x]$.

2022 JBMO Shortlist, N6

Tags: number theory , Sum of Squares , Digits , Junior , Balkan , shortlist

Find all positive integers $n$ for which there exists an integer multiple of $2022$ such that the sum of the squares of its digits is equal to $n$.

2006 AMC 12/AHSME, 25

Tags: algorithm , number theory , greatest common divisor , Euler , function , Euclidean algorithm , relatively prime

A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n \plus{} 2} \equal{} |a_{n \plus{} 1} \minus{} a_n|$ for $ n\ge 1$. If $ a_1 \equal{} 999, a_2 < 999,$ and $ a_{2006} \equal{} 1$, how many different values of $ a_2$ are possible? $ \textbf{(A) } 165 \qquad \textbf{(B) } 324 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 499 \qquad \textbf{(E) } 660$

2018 Kyiv Mathematical Festival, 4

Tags: Kyiv mathematical festival , inequalities

For every $x,y\ge0$ prove that $(x+1)^2+(y-1)^2\ge\frac{8y\sqrt{xy}}{3\sqrt{3}}.$

2005 AMC 10, 17

Tags: logarithms

Suppose that $ 4^a \equal{} 5$, $ 5^b \equal{} 6$, $ 6^c \equal{} 7$, and $ 7^d \equal{} 8$. What is $ a\cdot b\cdot c\cdot d$? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ \frac{3}{2}\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ \frac{5}{2}\qquad \textbf{(E)}\ 3$

2024 Harvard-MIT Mathematics Tournament, 19

Tags: guts

let $A_1A_2\ldots A_{19}$ be a regular nonadecagon. Lines $A_1A_5$ and $A_3A_4$ meet at $X.$ Compute $\angle A_7 X A_5.$

1997 Turkey Team Selection Test, 1

Tags: geometry , inradius , incenter , area of a triangle , geometry proposed

In a triangle $ABC$ with a right angle at $A$, $H$ is the foot of the altitude from $A$. Prove that the sum of the inradii of the triangles $ABC$, $ABH$, and $AHC$ is equal to $AH$.

1895 Eotvos Mathematical Competition, 1

Tags: Proof , cards , combinatorics

Prove that there are exactly $2(2^{n-1}-1)$ ways of dealing $n$ cards to two persons. (The persons may receive unequal numbers of cards.)

2021 Indonesia TST, A

Tags: Inequality , inequalities , algebra

A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying $$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$ then the following inequality holds: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$ (a) Prove that $M=20-\frac{1}{20}$ is not $strong$. (b) Prove that $M=20-\frac{1}{21}$ is $strong$.

2016 AMC 10, 8

Tags: AMC , AMC 10 , AMC 10 A

Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling, Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning? $\textbf{(A)}\ 20 \qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 45$

2025 Austrian MO National Competition, 2

Tags: geometry , Inversion , homothety

Let $\triangle{ABC}$ be an acute triangle with $BC > AC$. Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$. The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$. Let $M$ be the point where $CS$ intersects $AB$. The line $SF$ intersects the circle $\gamma$ at a point $Q$, such that $F$ lies between $S$ and $Q$. Prove that the points $M,P,Q$ and $F$ lie on a circle. [i](Karl Czakler)[/i]

1995 Romania Team Selection Test, 4

Tags: algebra

Let $m,n$ be positive integers, greater than 2.Find the number of polynomials of degree $2n-1$ with distinct coefficients from the set $\left\{ 1,2,\ldots,m\right\}$ which are divisible by $x^{n-1}+x^{n-2}+\ldots+1.$

2005 Sharygin Geometry Olympiad, 12

Tags: geometry , construction , sidelengths

Construct a quadrangle along the given sides $a, b, c$, and $d$ and the distance $I$ between the midpoints of its diagonals.