Olympiad problems

2021 Princeton University Math Competition, A5 / B7

Tags: number theory

Suppose that $f : Z\times Z \to R$, satisfies the equation $f(x, y) = f(3x+y, 2x+ 2y)$ for all $x, y \in Z$. Determine the maximal number of distinct values of $f(x, y)$ for $1 \le x, y \le 100$.

2023 Math Prize for Girls Problems, 6

Tags:

Solve for $x$: \begin{eqnarray*} v - w + x - y + z & = & 79 \\ v + w + x + y + z & = & -1 \\ v + 2w + 4x + 8y + 16z & = & -2 \\ v + 3w + 9x + 27y + 81z & = & -1 \\ v + 5w + 25x + 125y + 625z & = & 79. \end{eqnarray*}

2004 Federal Math Competition of S&M, 3

Tags: inequalities

Let $M, N, P$ be arbitrary points on the sides $BC, CA, AB$ respectively of an acute-angled triangle $ABC$. Prove that at least one of the following inequalities is satisfied: $NP \geq \frac{1}{2}BC; PM \geq \frac{1}{2}CA; MN \geq \frac{1}{2}AB$

Kvant 2022, M2721

Tags: polynomial , number theory

Let $n{}$ be a natural number and $f{}$ be polynomial with integer coefficients. It is known that for any integer $m{}$ there is an integer $k{}$ such that $f(k)-m$ is divisible by $n{}$. Prove that there exists a polynomial $g{}$ with integer coefficients such that $f(g(m))-m$ is divisible by $n{}$ for any integer $m{}$. [i]From the folklore[/i]

2021 IMO Shortlist, C6

Tags: combinatorics , game

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

2005 MOP Homework, 7

Tags: trigonometry , geometry , inequalities

Let $ABC$ be a triangle. Prove that \[\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab} \ge 4\left(\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}\right).\]

2021 Taiwan TST Round 2, G

Tags: circumcircle , geometry , euler

Let $ABCD$ be a convex quadrilateral with pairwise distinct side lengths such that $AC\perp BD$. Let $O_1,O_2$ be the circumcenters of $\Delta ABD, \Delta CBD$, respectively. Show that $AO_2, CO_1$, the Euler line of $\Delta ABC$ and the Euler line of $\Delta ADC$ are concurrent. (Remark: The [i]Euler line[/i] of a triangle is the line on which its circumcenter, centroid, and orthocenter lie.) [i]Proposed by usjl[/i]

2002 AMC 12/AHSME, 3

Tags:

The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002\text{ in}^3$. Find the minimum possible sum in inches of the three dimensions. $\textbf{(A) }36\qquad\textbf{(B) }38\qquad\textbf{(C) }42\qquad\textbf{(D) }44\qquad\textbf{(E) }92$

2009 AMC 8, 23

Tags: algebra , quadratic , quadratic formula

On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $ 400$ jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class? $ \textbf{(A)}\ 26 \qquad \textbf{(B)}\ 28 \qquad \textbf{(C)}\ 30 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 34$

2011 Tuymaada Olympiad, 4

Tags: polynomial , algebra , quadratic , analytic number theory , integer sequence , integer polynomial

Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$, the number $P(n)$ has a proper divisor $d_n$, i.e., $1<d_n<P(n)$, such that the sequence $d_1,d_2,d_3,\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomials with integer coefficients or all the values of $P(n)$, for positive integers $n$, are divisible by the same integer $m>1$.

2024 Middle European Mathematical Olympiad, 4

Tags: number theory , polynomial , sum of divisors , divisibility , factorization

Determine all polynomials $P(x)$ with integer coefficients such that $P(n)$ is divisible by $\sigma(n)$ for all positive integers $n$. (As usual, $\sigma(n)$ denotes the sum of all positive divisors of $n$.)

Kyiv City MO Juniors 2003+ geometry, 2022.9.3

Tags: geometry

Let $AL$ be the inner bisector of triangle $ABC$. The circle centered at $B$ with radius $BL$ meets the ray $AL$ at points $L$ and $E$, and the circle centered at $C$ with radius $CL$ meets the ray $AL$ at points $L$ and $D$. Show that $AL^2 = AE\times AD$. [i](Proposed by Mykola Moroz)[/i]

Russian TST 2017, P2

Tags: number theory

Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that [list] [*]$m = 1$ and $l = 2k$; or [*]$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$. [/list]

2005 Baltic Way, 11

Tags: circumcircle , geometry , geometric transformation , rotation , rhombus , power of a point , radical axis

Let the points $D$ and $E$ lie on the sides $BC$ and $AC$, respectively, of the triangle $ABC$, satisfying $BD=AE$. The line joining the circumcentres of the triangles $ADC$ and $BEC$ meets the lines $AC$ and $BC$ at $K$ and $L$, respectively. Prove that $KC=LC$.

2004 Putnam, A1

Tags: inequalities , induction , percent

Basketball star Shanille O'Keal's team statistician keeps track of the number, $S(N),$ of successful free throws she has made in her first $N$ attempts of the season. Early in the season, $S(N)$ was less than 80% of $N,$ but by the end of the season, $S(N)$ was more than 80% of $N.$ Was there necessarily a moment in between when $S(N)$ was exactly 80% of $N$?

2012 India IMO Training Camp, 1

Tags: algebra , equation , sequence

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

2025 Belarusian National Olympiad, 8.2

Tags: algebra , physics

The distance between the city and the house of Borya is 2km. Once Borya went from the city to his house with speed 4km/h. Simultaneously with that a dog Sharik started running out of house in the direction to city, and whenever Sharik meets Borya or the house, it starts running back (so the dog runs between Borya and the house), and when the dog runs to the house, its speed is 8km/h, and when it runs from the house, its speed is 12km/h. What distance will Sharik run until Borya comes to the house? [i]Yauheni Barabanau[/i]

2023 Brazil Undergrad MO, 1

Tags: algebra , inequalities , function , real analysis

Let $p$ be the [i]potentioral[/i] function, from positive integers to positive integers, defined by $p(1) = 1$ and $p(n + 1) = p(n)$, if $n + 1$ is not a perfect power and $p(n + 1) = (n + 1) \cdot p(n)$, otherwise. Is there a positive integer $N$ such that, for all $n > N,$ $p(n) > 2^n$?

1951 AMC 12/AHSME, 19

Tags:

A six place number is formed by repeating a three place number; for example, $ 256256$ or $ 678678$, etc. Any number of this form is always exactly divisible by: $ \textbf{(A)}\ 7 \text{ only} \qquad\textbf{(B)}\ 11 \text{ only} \qquad\textbf{(C)}\ 13 \text{ only} \qquad\textbf{(D)}\ 101 \qquad\textbf{(E)}\ 1001$

1998 IberoAmerican, 2

Tags: geometry

The circumference inscribed on the triangle $ABC$ is tangent to the sides $BC$, $CA$ and $AB$ on the points $D$, $E$ and $F$, respectively. $AD$ intersect the circumference on the point $Q$. Show that the line $EQ$ meet the segment $AF$ at its midpoint if and only if $AC=BC$.

2009 China National Olympiad, 3

Tags: induction , number theory , combinatorics , greatest common divisor

Given an integer $ n > 3.$ Prove that there exists a set $ S$ consisting of $ n$ pairwisely distinct positive integers such that for any two different non-empty subset of $ S$:$ A,B, \frac {\sum_{x\in A}x}{|A|}$ and $ \frac {\sum_{x\in B}x}{|B|}$ are two composites which share no common divisors.

PEN I Problems, 17

Tags: floor function

Determine all real numbers $a$ such that \[4\lfloor an\rfloor =n+\lfloor a\lfloor an\rfloor \rfloor \; \text{for all}\; n \in \mathbb{N}.\]

2023 OMpD, 4

Tags: circumcircle , geometry , tangent circle , tangent

Let $ABC$ be a scalene acute triangle with circumcenter $O$. Let $K$ be a point on the side $\overline{BC}$. Define $M$ as the second intersection of $\overleftrightarrow{OK}$ with the circumcircle of $BOC$. Let $L$ be the reflection of $K$ by $\overleftrightarrow{AC}$. Show that the circumcircles of the triangles $LCM$ and $ABC$ are tangent if, and only if, $\overline{AK} \perp \overline{BC}$.

1972 Miklós Schweitzer, 3

Tags: integration , advanced fields , limit , calculus

Let $ \lambda_i \;(i=1,2,...)$ be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form \[ \mu= \sum_{i=1}^{\infty}n_i\lambda_i ,\] where $ n_i \geq 0$ are integers and all but finitely many $ n_i$ are $ 0$. Let \[ L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ \;M(x)= \sum _{\mu \leq x} 1 \ .\] (In the latter sum, each $ \mu$ occurs as many times as its number of representations in the above form.) Prove that if \[ \lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1,\] then \[ \lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1.\] [i]G. Halasz[/i]

1982 Kurschak Competition, 3

Tags: combinatorics , coloring

The set of integers is coloured in $100$ colours in such a way that all the colours are used and the following is true. For any choice of intervals $[a, b]$ and $[c,d]$ of equal length and with integral endpoints, if a and c as well as $b$ and $d$, respectively, have the same colour, then the whole intervals $[a, b]$ and $[c,d]$ are identically coloured in that, for any integer $x$, $0 \le x \le b - a$, the numbers $a + x$ and $c + x$ are of the same colour. Prove that $-1982$ and $1982$ are of different colours