Olympiad problems

2021 IMO Shortlist, C2

Tags: pigeonhole principle , c2

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

1972 Swedish Mathematical Competition, 2

Tags: combinatorial geometry , combinatorics

A rectangular grid of streets has $m$ north-south streets and $n$ east-west streets. For which $m, n > 1$ is it possible to start at an intersection and drive through each of the other intersections just once before returning to the start?

2017 QEDMO 15th, 4

Tags: functional equation , functional , algebra

Find all functions $f: R \to R$ for which the image $f ([a, b])$ for all real $a \le b$ is (not necessarily closed!) interval of length $b - a$.

2023 New Zealand MO, 6

Tags: geometry

Let triangle $ABC$ be right-angled at $A$. Let $D$ be the point on $AC$ such that $BD$ bisects angle $\angle ABC$. Prove that $BC - BD = 2AB$ if and only if $\frac{1}{BD} - \frac{1}{BC} =\frac{1}{2AB}$.

2024 Iberoamerican, 2

Tags: geometry , circumcircle

Let $\triangle ABC$ be an acute triangle and let $M, N$ be the midpoints of $AB, AC$ respectively. Given a point $D$ in the interior of segment $BC$ with $DB<DC$, let $P, Q$ the intersections of $DM, DN$ with $AC, AB$ respectively. Let $R \ne A$ be the intersection of circumcircles of triangles $\triangle PAQ$ and $\triangle AMN$. If $K$ is midpoint of $AR$, prove that $\angle MKN=2\angle BAC$

1980 AMC 12/AHSME, 2

Tags: algebra , polynomial

The degree of $(x^2+1)^4 (x^3+1)^3$ as a polynomial in $x$ is $\text{(A)} \ 5 \qquad \text{(B)} \ 7 \qquad \text{(C)} \ 12 \qquad \text{(D)} \ 17 \qquad \text{(E)} \ 72$

1991 Kurschak Competition, 3

Tags: combinatorics

Consider $998$ red points on the plane with no three collinear. We select $k$ blue points in such a way that inside each triangle whose vertices are red points, there is a blue point as well. Find the smallest $k$ for which the described selection of blue points is possible for any configuration of $998$ red points.

2018 ISI Entrance Examination, 8

Tags: linear algebra

Let $n\geqslant 3$. Let $A=((a_{ij}))_{1\leqslant i,j\leqslant n}$ be an $n\times n$ matrix such that $a_{ij}\in\{-1,1\}$ for all $1\leqslant i,j\leqslant n$. Suppose that $$a_{k1}=1~~\text{for all}~1\leqslant k\leqslant n$$ and $~~\sum_{k=1}^n a_{ki}a_{kj}=0~~\text{for all}~i\neq j$. Show that $n$ is a multiple of $4$.

2015 South East Mathematical Olympiad, 4

Tags: number theory

For any positive integer $n$, we have the set $P_n = \{ n^k \mid k=0,1,2, \ldots \}$. For positive integers $a,b,c$, we define the group of $(a,b,c)$ as lucky if there is a positive integer $m$ such that $a-1$, $ab-12$, $abc-2015$ (the three numbers need not be different from each other) belong to the set $P_m$. Find the number of lucky groups.

2000 Singapore Team Selection Test, 3

Tags: number theory , sum

Let $n$ be any integer $\ge 2$. Prove that $\sum 1/pq = 1/2$, where the summation is over all integers$ p, q$ which satisfy $0 < p < q \le n$,$ p + q > n$, $(p, q) = 1$.

2010 VJIMC, Problem 1

Tags: number theory

Let $a$ and $b$ be given positive coprime integers. Then for every integer $n$ there exist integers $x,y$ such that $$n=ax+by.$$Prove that $n=ab$ is the greatest integer for which $xy\le0$ in all such representations of $n$.

2025 Bulgarian Spring Mathematical Competition, 9.2

Tags: geometry , angle chasing , similarity , circumcircle

Let $ABC$ be an acute scalene triangle inscribed in a circle $ \Gamma $. The angle bisector of $ \angle BAC $ intersects $ BC $ at $ L $ and $ \Gamma $ at $ S $. The point $ M $ is the midpoint of $ AL $. Let $ AD $ be the altitude in $ \triangle ABC $, and the circumcircle of $ \triangle DSL $ intersects $ \Gamma $ again at $ P $. Let $ N $ be the midpoint of $ BC $, and let $ K $ be the reflection of $ D $ with respect to $ N $. Prove that the triangles $ \triangle MPS $ and $ \triangle ADK $ are similar.

2012 India Regional Mathematical Olympiad, 5

Tags: ratio , geometry , trigonometry , area of a triangle

Let $ABC$ be a triangle. Let $D,E$ be points on the segment $BC$ such that $BD=DE=EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$.

2011 AMC 12/AHSME, 21

Tags: quadratic , algebra , difference of squares , special factorizations

The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x-y|$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70 $

1997 Turkey Junior National Olympiad, 2

Tags: symmetry , pythagorean theorem , power of a point , geometry

Let $ABC$ be a triangle with $|AB|=|AC|=26$, $|BC|=20$. The altitudes of $\triangle ABC$ from $A$ and $B$ cut the opposite sides at $D$ and $E$, respectively. Calculate the radius of the circle passing through $D$ and tangent to $AC$ at $E$.

2020 Belarusian National Olympiad, 11.7

Tags: geometry

Line $AL$ is an angle bisector in the triangle $ABC$ ($L \in BC$), and $\omega$ is its circumcircle. Chords $X_1X_2$ and $Y_1Y_2$ pass through $L$ such that points $X_1,Y_1$ and $A$ lie in the same half-plane with respect to $BC$. Lines $X_1Y_2$ and $Y_1X_2$ intersect side $BC$ in points $Z_1$ and $Z_2$ respectively. Prove that $\angle BAZ_1=\angle CAZ_2$.

2018 Malaysia National Olympiad, A1

Tags: geometry , area

Quadrilateral $ABCD$ is neither a kite nor a rectangle. It is known that its sidelengths are integers, $AB = 6$, $BC = 7$, and $\angle B = \angle D = 90^o$. Find the area of$ ABCD$.

2010 Nordic, 1

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

A function $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, where $\mathbb{Z}_+$ is the set of positive integers, is non-decreasing and satisfies $f(mn) = f(m)f(n)$ for all relatively prime positive integers $m$ and $n$. Prove that $f(8)f(13) \ge (f(10))^2$.

1991 National High School Mathematics League, 9

Tags:

Devide all odd numbers from small to large into groups: there are $2n-1$ numbers in the $n$th group. For example: the first group is $\{1\}$, the second group is $\{3,5,7\}$, the third group is $\{9,11,13,15,17\}$. Then, $1991$ is in group________.

2002 Greece Junior Math Olympiad, 2

Tags: ratio

In the Mathematical Competition of HMS (Hellenic Mathematical Society) take part boys and girls who are divided into two groups : [i]Juniors[/i] and [i]seniors.[/i]The number of the boys taking part of this year competition is 55% of the number of all participants. The ratio of the number of juniors boys to the number of senior boys is equal to the ratio of the number of juniors to the number of seniors. Find the ratio of the number of junior boys to the number of junior girls.

2014 Hanoi Open Mathematics Competitions, 4

Tags: number theory , diophantine equation , diophantine , prime

If $p$ is a prime number such that there exist positive integers $a$ and $b$ such that $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$ then $p$ is (A): $3$, (B): $5$, (C): $11$, (D): $7$, (E) None of the above.

2007 Singapore Senior Math Olympiad, 5

Tags: inequalities , min , max , algebra

Find the maximum and minimum of $x + y$ such that $x + y = \sqrt{2x-1}+\sqrt{4y+3}$

2018 Harvard-MIT Mathematics Tournament, 6

Tags: combinatorics

Farmer James invents a new currency, such that for every positive integer $n\le 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be [i]nice[/i] if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?

2020 Taiwan TST Round 3, 6

Tags: combinatorics

Alice has a map of Wonderland, a country consisting of $n \geq 2$ towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns. Alice wants to know whether there is at least one town in Wonderland with at most one outgoing road. Prove that she can always find out by asking at most $4n$ questions.

1969 Spain Mathematical Olympiad, 1

Tags: geometry , locus , inversion

Find the locus of the centers of the inversions that transform two points $A, B$ of a given circle $\gamma$ , at diametrically opposite points of the inverse circles of $\gamma$ .