Olympiad problems

2017 EGMO, 2

Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties: $(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$ $(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$ [i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]

2010 AMC 10, 24

Tags: modular arithmetic , ratio , geometric sequence

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

V Soros Olympiad 1998 - 99 (Russia), 11.8

Tags: geometry , angle

Inside triangle $ABC$, point $P$ is taken so that angles $\angle ARB= \angle BPC = \angle CPA= 120^o$. Lines $BP$ and $CP$ intersect lines $AC$ and $AB$ at points $M$ and $K$. It is known that the quadrilateral $AMPK$ has same areq with the triangle $BCP$. What is the angle $\angle BAC$?

2013 Estonia Team Selection Test, 3

Tags: inequalities , sum , algebra

Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros. (i) Prove that $$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$ (ii) Show that, for each $n > 1$, both inequalities can hold as equalities.

2012 Grand Duchy of Lithuania, 2

Tags: geometry , trapezoid , right angle

The base $AB$ of a trapezium $ABCD$ is longer than the base $CD$, and $\angle ADC$ is a right angle. The diagonals $AC$ and $BD$ are perpendicular. Let $E$ be the foot of the altitude from $D$ to the line $BC$. Prove that $$\frac{AE}{BE} =\frac{ AC \cdot CD}{AC^2 - CD^2}$$ .

2015 IFYM, Sozopol, 6

Tags: number theory

A natural number is called [i]“sozopolian”[/i], if it has exactly two prime divisors. Does there exist 12 consecutive [i]“sozopolian”[/i] numbers?

2018 PUMaC Number Theory B, 1

Tags: number theory

Find the largest prime factor of $8001$.

2023 IMAR Test, P3

Tags: permutation , number theory , prime number

Let $p{}$ be an odd prime number. Determine whether there exists a permutation $a_1,\ldots,a_p$ of $1,\ldots,p$ satisfying \[(i-j)a_k+(j-k)a_i+(k-i)a_j\neq 0,\] for all pairwise distinct $i,j,k.$

2024 JHMT HS, 14

Tags: trigonometry

Compute \[ \frac{1}{2}\sin\frac{3\pi}{7}+\sin\frac{2\pi}{7}\cos\frac{3\pi}{7}. \]

2019 Canadian Mathematical Olympiad Qualification, 6

Tags: pentagon , concurrency , geometry , perpendicular

Pentagon $ABCDE$ is given in the plane. Let the perpendicular from $A$ to line $CD$ be $F$, the perpendicular from $B$ to $DE$ be $G$, from $C$ to $EA$ be $H$, from $D$ to $AB$ be $I$,and from $E$ to $BC$ be $J$. Given that lines $AF,BG,CH$, and $DI$ concur, show that they also concur with line $EJ$.

2012 IMAR Test, 3

Tags: equal angles , geometry , reflection , angle bisector

Given a triangle $ABC$, let $D$ be a point different from $A$ on the external bisectrix $\ell$ of the angle $BAC$, and let $E$ be an interior point of the segment $AD$. Reflect $\ell$ in the internal bisectrices of the angles $BDC$ and $BEC$ to obtain two lines that meet at some point $F$. Show that the angles $ABD$ and $EBF$ are congruent.

2021 IMO Shortlist, A7

Tags: inequality , inequalities , algebra

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]

2016 HMNT, 1

Tags: hmmt

If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$, determine the product of all possible values of $ab$.

2000 Greece JBMO TST, 4

Tags: geometry , geometric inequality , inequalities

Let $a,b,c$ be sidelengths with $a\ge b\ge c$ and $s\ge a+1$ where $s$ be the semiperimeter of the triangle. Prove that $$ \frac{s-c}{\sqrt{a}}+\frac{s-b}{\sqrt{c}}+\frac{s-a}{\sqrt{b}}\ge \frac{s-b}{\sqrt{a}}+\frac{s-c}{\sqrt{b}}+\frac{s-a}{\sqrt{c}}$$

1999 IMC, 6

Tags: fourier series , complex number , pigeonhole principle

Let $A$ be a subset of $\mathbb{Z}/n\mathbb{Z}$ with at most $\frac{\ln(n)}{100}$ elements. Define $f(r)=\sum_{s\in A} e^{\dfrac{2 \pi i r s}{n}}$. Show that for some $r \ne 0$ we have $|f(r)| \geq \frac{|A|}{2}$.

2010 Malaysia National Olympiad, 5

Tags: ratio , geometry , area

A circle and a square overlap such that the overlapping area is $50\%$ of the area of the circle, and is $25\%$ of the area of the square, as shown in the figure. Find the ratio of the area of the square outside the circle to the area of the whole figure. [img]https://cdn.artofproblemsolving.com/attachments/e/2/c209a95f457dbf3c46f66f82c0a45cc4b5c1c8.png[/img]

2024 Oral Moscow Geometry Olympiad, 6

Tags: geometry

Given an acute-angled triangle $ABC$ and a point $P$ inside it such that $\angle PBA=\angle PCA$. The lines $PB$ and $PC$ intersect the circumcircles of triangles $PCA$ and $PAB$ secondly at points $M$ and $N$, respectively. Let the rays $MC$ and $NB$ intersect at a point $S$, $K$ is the center of the circumscribed circle of the triangle $SMN$. Prove that the lines $AK$ and $BC$ are perpendicular.

2018 Polish Junior MO First Round, 3

Tags: number theory , prime number , divisibility

Prime numbers $a, b, c$ are bigger that $3$. Show that $(a - b)(b - c)(c - a)$ is divisible by $48$.

1998 Brazil Team Selection Test, Problem 5

Tags: inequalities , number theory , sequence

Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that $$k\le2\lfloor\sqrt n\rfloor.$$

2002 National High School Mathematics League, 12

Tags: trigonometry

For all $x\in\mathbb{R}$, $\sin^2 x+a\cos x+a^2\geq 1+\cos x$, then the range value of negative number $a$ is________

2021 Serbia JBMO TSTs, 2

Tags: number theory

Solve the following equation in natural numbers: \begin{align*} x^2=2^y+2021^z \end{align*}

2014 Denmark MO - Mohr Contest, 1

Tags: number theory , algebra

Georg chooses three distinct digits among $1, 2, . . . , 9$ and writes them down on three cards. When the cards are laid down next to each other, a three-digit number is formed. Georg tells his mother that the sum of the largest and the second-largest number that can be formed in this manner is $1732$. Can she figure out which three digits Georg has chosen?

1977 Spain Mathematical Olympiad, 5

Tags: algebra

Using an escalator to go down to the Metro station and walking with a regular pace, I find that I need $50$ steps to go down. if i come back later to run up it, at a speed $5$ times my previous normal pace, I check that I need $125$ steps to reach the top. How many visible steps does the mechanical staircase have when it is stopped?

1988 IMO Longlists, 37

Tags: geometry , 3d geometry , tetrahedron , trigonometry , inradius , incenter , sphere

[b]i.)[/b] Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tedrahedron, each of whose edges has length $ s,$ is circumscribed around the balls. Find the value of $ s.$ [b]ii.)[/b] Suppose that $ ABCD$ and $ EFGH$ are opposite faces of a retangular solid, with $ \angle DHC \equal{} 45^{\circ}$ and $ \angle FHB \equal{} 60^{\circ}.$ Find the cosine of $ \angle BHD.$

2013 Kazakhstan National Olympiad, 3

Tags: circumcircle , similarity , geometry

Let $ABCD$ be cyclic quadrilateral. Let $AC$ and $BD$ intersect at $R$, and let $AB$ and $CD$ intersect at $K$. Let $M$ and $N$ are points on $AB$ and $CD$ such that $\frac{AM}{MB}=\frac{CN}{ND}$. Let $P$ and $Q$ be the intersections of $MN$ with the diagonals of $ABCD$. Prove that circumcircles of triangles $KMN$ and $PQR$ are tangent at a fixed point.