Olympiad problems

2020 Caucasus Mathematical Olympiad, 8

Peter wrote $100$ distinct integers on a board. Basil needs to fill the cells of a table $100\times{100}$ with integers so that the sum in each rectangle $1\times{3}$ (either vertical, or horizontal) is equal to one of the numbers written on the board. Find the greatest $n$ such that, regardless of numbers written by Peter, Basil can fill the table so that it would contain each of numbers $(1,2,...,n)$ at least once (and possibly some other integers).

2008 Germany Team Selection Test, 3

Tags: geometry , rectangle , combinatorics , dissection

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

2017 Turkey EGMO TST, 5

Tags: combinatorics

In a $12\times 12$ square table some stones are placed in the cells with at most one stone per cell. If the number of stones on each line, column, and diagonal is even, what is the maximum number of the stones? [b]Note[/b]. Each diagonal is parallel to one of two main diagonals of the table and consists of $1,2\ldots,11$ or $12$ cells.

1999 Czech And Slovak Olympiad IIIA, 3

Tags: median , sum , ratio , geometry

Show that there exists a triangle $ABC$ such that $a \ne b$ and $a+t_a = b+t_b$, where $t_a,t_b$ are the medians corresponding to $a,b$, respectively. Also prove that there exists a number $k$ such that every such triangle satisfies $a+t_a = b+t_b = k(a+b)$. Finally, find all possible ratios $a : b$ in such triangles.

2004 German National Olympiad, 4

Tags: integer , square root , algebra , sum

For a positive integer $n,$ let $a_n$ be the integer closest to $\sqrt{n}.$ Compute $$ \frac{1}{a_1 } + \frac{1}{a_2 }+ \cdots + \frac{1}{a_{2004}}.$$

1987 IMO Longlists, 43

Tags: combinatorics

Let $2n + 3$ points be given in the plane in such a way that no three lie on a line and no four lie on a circle. Prove that the number of circles that pass through three of these points and contain exactly $n$ interior points is not less than $\frac 13 \binom{2n+3}{2}.$

2007 AMC 12/AHSME, 9

Tags: ratio

Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides $ 7$ times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium? $ \textbf{(A)}\ \frac {2}{3}\qquad \textbf{(B)}\ \frac {3}{4}\qquad \textbf{(C)}\ \frac {4}{5}\qquad \textbf{(D)}\ \frac {5}{6}\qquad \textbf{(E)}\ \frac {6}{7}$

2009 Czech and Slovak Olympiad III A, 4

Tags: number theory

A positive integer $n$ is called [i]good[/i] if and only if there exist exactly $4$ positive integers $k_1, k_2, k_3, k_4$ such that $n+k_i|n+k_i^2$ ($1 \leq k \leq 4$). Prove that: [list] [*]$58$ is [i]good[/i]; [*]$2p$ is [i]good[/i] if and only if $p$ and $2p+1$ are both primes ($p>2$).[/list]

1974 Bulgaria National Olympiad, Problem 6

Tags: inequalities , geometry , 3d geometry , geometrical inequalities , pyramid , geometric inequalities

In triangle pyramid $MABC$ at least two of the plane angles next to the edge $M$ are not equal to each other. Prove that if the bisectors of these angles form the same angle with the angle bisector of the third plane angle, the following inequality is true $$8a_1b_1c_1\le a^2a_1+b^2b_1+c^2c_1$$ where $a,b,c$ are sides of triangle $ABC$ and $a_1,b_1,c_1$ are edges crossed respectively with $a,b,c$. [i]V. Petkov[/i]

2006 Princeton University Math Competition, 4

Tags: algebra

Suppose that $n>1$ and $P_n(x)$ is a polynomial of degree $n$. For $k =1,2, . . . ,n$ we have $P_n(k)=k(k+1)$. Also $P_n(0) = 1$. For all $n$ there exists an integer $m > n$ such that $P_n(m) = P_{n+2}(m)$. Find the value of $m$ for $n = 10$.

1987 Romania Team Selection Test, 2

Tags: modular arithmetic , ratio , number theory

Find all positive integers $A$ which can be represented in the form: \[ A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right) \] where $m\geq n\geq p \geq 1$ are integer numbers. [i]Ioan Bogdan[/i]

2016 Estonia Team Selection Test, 4

Tags: algebra , inequalities

Prove that for any positive integer $n\ge $, $2 \cdot \sqrt3 \cdot \sqrt[3]{4} ...\sqrt[n-1]{n} > n$

2005 Pan African, 3

Tags: function , induction

Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function such that: For all $a$ and $b$ in $\mathbb{Z} - \{0\}$, $f(ab) \geq f(a) + f(b)$. Show that for all $a \in \mathbb{Z} - \{0\}$ we have $f(a^n) = nf(a)$ for all $n \in \mathbb{N}$ if and only if $f(a^2) = 2f(a)$

1972 Dutch Mathematical Olympiad, 5

Tags: geometry , ratio

Given is an acute-angled triangle $ABC$ with angles $\alpha$, $\beta$ and $\gamma$. On side $AB$ lies a point $P$ such that the line connecting the feet of the perpendiculars from $P$ on $AC$ and $BC$ is parallel to $AB$. Express the ratio $\frac{AP}{BP}$ in terms of $\alpha$ and $\beta$.

1997 Croatia National Olympiad, Problem 1

Tags: geometry , hexagon

In a regular hexagon $ABCDEF$ with center $O$, points $M$ and $N$ are the midpoints of the sides $CD$ and $DE$, and $L$ the intersection point of $AM$ and $BN$. Prove that: (a) $ABL$ and $DMLN$ have equal areas; (b) $\angle ALD=\angle OLN=60^\circ$; (c) $\angle OLD=90^\circ$.

2008 Vietnam Team Selection Test, 3

Tags: combinatorics

Let an integer $ n > 3$. Denote the set $ T\equal{}\{1,2, \ldots,n\}.$ A subset S of T is called [i]wanting set[/i] if S has the property: There exists a positive integer $ c$ which is not greater than $ \frac {n}{2}$ such that $ |s_1 \minus{} s_2|\ne c$ for every pairs of arbitrary elements $ s_1,s_2\in S$. How many does a [i]wanting set[/i] have at most are there ?

2019 Finnish National High School Mathematics Comp, 2

Tags: floor function , prime , number theory , algebra

Prove that the number $\lfloor (2+\sqrt5)^{2019} \rfloor$ is not prime.

2018 Thailand TSTST, 1

Tags: polynomial , algebra

Find all polynomials $P(x)$ with real coefficients satisfying: $P(2017) = 2016$ and $$(P(x)+1)^2=P(x^2+1).$$

1954 AMC 12/AHSME, 13

Tags:

A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, their sum will be: $ \textbf{(A)}\ 180^\circ \qquad \textbf{(B)}\ 540^\circ \qquad \textbf{(C)}\ 360^\circ \qquad \textbf{(D)}\ 450^\circ \qquad \textbf{(E)}\ 1080^\circ$

2007 Polish MO Finals, 5

Tags: 3d geometry , tetrahedron , sphere , parallelogram , circumcircle , geometry

5. In tetrahedron $ABCD$ following equalities hold: $\angle BAC+\angle BDC=\angle ABD+\angle ACD$ $\angle BAD+\angle BCD=\angle ABC+\angle ADC$ Prove that center of sphere circumscribed about ABCD lies on a line through midpoints of $AB$ and $CD$.

2025 Romania National Olympiad, 4

Tags: finite field , number theory , prime number , algebra , polynomial

Let $p$ be an odd prime number, and $k$ be an odd number not divisible by $p$. Consider a field $K$ be a field with $kp+1$ elements, and $A = \{x_1,x_2, \dots, x_t\}$ be the set of elements of $K^*$, whose order is not $k$ in the multiplicative group $(K^*,\cdot)$. Prove that the polynomial $P(X)=(X+x_1)(X+x_2)\dots(X+x_t)$ has at least $p$ coefficients equal to $1$.

1974 AMC 12/AHSME, 12

Tags: function

If $ g(x)\equal{}1\minus{}x^2$ and $ f(g(x)) \equal{} \frac{1\minus{}x^2}{x^2}$ when $ x\neq0$, then $ f(1/2)$ equals $ \textbf{(A)}\ 3/4 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \sqrt2/2 \qquad \textbf{(E)}\ \sqrt2$

2004 Poland - First Round, 1

Tags: algebra

1. Solve in real numbers x,y,z : $\{\begin{array}{ccc} x^2=yz+1 \\ y^2=zx+2 \\ z^2=xy+4 \\ \end{array}$

Croatia MO (HMO) - geometry, 2016.7

Tags: circumcircle , incircle , geometry

Let $P$ be a point inside a triangle $ABC$ such that $$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$ Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.

2013 Princeton University Math Competition, 16

Tags: trigonometry , algebra , polynomial , induction , inequalities

Is $\cos 1^\circ$ rational? Prove.