Olympiad problems

2007 QEDMO 5th, 7

Tags: combinatorics

In a group of $20$ people, each person sends a letter to $10$ of the others. Prove that there are two persons who send a letter to each other.

2007 Harvard-MIT Mathematics Tournament, 4

Tags:

A sequence consists of the digits $122333444455555\ldots$ such that the each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the $4501$st and $4052$nd digits of this sequence.

2012 Czech And Slovak Olympiad IIIA, 2

Tags: geometry , area of a triangle , maximum , median

Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$.

2023 Israel TST, P2

Tags: combinatorics , modular arithmetic , abstract algebra

Let $n>3$ be an integer. Integers $a_1, \dots, a_n$ are given so that $a_k\in \{k, -k\}$ for all $1\leq k\leq n$. Prove that there is a sequence of indices $1\leq k_1, k_2, \dots, k_n\leq n$, not necessarily distinct, for which the sums \[a_{k_1}\] \[a_{k_1}+a_{k_2}\] \[a_{k_1}+a_{k_2}+a_{k_3}\] \[\vdots\] \[a_{k_1}+a_{k_2}+\cdots+a_{k_n}\] have distinct residues modulo $2n+1$, and so that the last one is divisible by $2n+1$.

1980 IMO, 20

Tags: inequalities , inradius , circumcircle , geometry

The radii of the circumscribed circle and the inscribed circle of a regular $n$-gon, $n\ge 3$ are denoted by $R_n$ and $r_n$, respectively. Prove that \[\frac{r_n}{R_n}\ge\left(\frac{r_{n+1}}{R_{n+1}}\right)^2.\]

2021 MOAA, 3

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What is the last digit of $2021^{2021}$? [i]Proposed by Yifan Kang[/i]

2022 Olympic Revenge, Problem 4

Tags: number theory

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers such that $a_1=1$. For each $n \geq 1$, $a_{n+1}$ is the smallest positive integer, distinct from $a_1,a_2,...,a_n$, such that $\gcd(a_{n+1}a_n+1,a_i)=1$ for each $i=1,2,...,n$. Prove that every positive integer appears in $\{a_n\}_{n=1}^{\infty}$.

1992 Austrian-Polish Competition, 1

Tags: sum , positive , number theory , product , even , divisor

For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

1991 Arnold's Trivium, 86

Tags: geometry , 3d geometry , tetrahedron , icosahedron

Through the centre of a cube (tetrahedron, icosahedron) draw a straight line in such a way that the sum of the squares of its distances from the vertices is a) minimal, b) maximal.

2019 IMO Shortlist, A2

Tags: inequalities , algebra

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

2014 USA Team Selection Test, 2

Tags: quadratic residue , quadratic , induction , modular arithmetic , number theory

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

2014 Miklós Schweitzer, 10

Tags: 3d geometry , sphere , geometry

To each vertex of a given triangulation of the two-dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two-dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

2010 Today's Calculation Of Integral, 614

Tags: calculus , integration , ratio , trigonometry , calculus computations

Evaluate $\int_0^1 \{x(1-x)\}^{\frac 32}dx.$ [i]2010 Hirosaki University School of Medicine entrance exam[/i]

2023 Romania Team Selection Test, P4

Tags: geometry , combinatorics , combinatorial geometry

Fix a positive integer $n.{}$ Consider an $n{}$-point set $S{}$ in the plane. An [i]eligible[/i] set is a non-empty set of the form $S\cap D,{}$ where $D$ is a closed disk in the plane. In terms of $n,$ determine the smallest possible number of eligible subsets $S{}$ may contain. [i]Proposed by Cristi Săvescu[/i]

2006 Iran MO (3rd Round), 1

Tags: incenter , geometric transformation , homothety , ratio , radical axis , geometry , angle bisector

Prove that in triangle $ABC$, radical center of its excircles lies on line $GI$, which $G$ is Centroid of triangle $ABC$, and $I$ is the incenter.

2016 Korea Junior Math Olympiad, 5

Tags: polynomial , number theory , algebra

$n \in \mathbb {N^+}$ Prove that the following equation can be expressed as a polynomial about $n$. $$\left[2\sqrt {1}\right]+\left[2\sqrt {2}\right]+\left[2\sqrt {3}\right]+ . . . +\left[2\sqrt {n^2}\right]$$

2023 Middle European Mathematical Olympiad, 3

Tags: combinatorics

Find the smallest integer $b$ with the following property: For each way of colouring exactly $b$ squares of an $8 \times 8$ chessboard green, one can place $7$ bishops on $7$ green squares so that no two bishops attack each other.

1999 All-Russian Olympiad, 1

Tags: number theory

The decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digits of $9A$.

2012 Swedish Mathematical Competition, 6

Tags: geometry , trapezoid , diameter , tangential

A circle is inscribed in an trapezoid. Show that the diagonals of the trapezoid intersect at a point on the diameter of the circle perpendicular to the two parallel sides.

2000 Croatia National Olympiad, Problem 3

Tags: number theory , inequalities

Let $n\ge3$ positive integers $a_1,\ldots,a_n$ be written on a circle so that each of them divides the sum of its two neighbors. Let us denote $$S_n=\frac{a_n+a_2}{a_1}+\frac{a_1+a_3}{a_2}+\ldots+\frac{a_{n-2}+a_n}{a_{n-1}}+\ldots+\frac{a_{n-1}+a_1}{a_n}.$$Determine the minimum and maximum values of $S_n$.

VMEO IV 2015, 12.3

Tags: number theory , arithmetic sequence , prime number

Find all integes $a,b,c,d$ that form an arithmetic progression satisfying $d-c+1$ is prime number and $a+b^2+c^3=d^2b$

2023 Romania Team Selection Test, P3

Tags:

Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that $$f(x + f(y)) = f(x) + f(y)$$ for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.

1954 Miklós Schweitzer, 10

Tags: geometry

[b]10.[/b] Given a triangle $ABC$, construct outwards over the sides $AB, BC, CA$ similiar isosceles triangles $ABC_{1}, BCA_{1}$ and $CAB_{1}$. Prove that the straight lines $AA_{1}. BB_{1}$ and $CC_{1}$ are concurrent. Is this statemente true in elliptic and hyperbolic geometry, too? [b](G. 19)[/b]

Swiss NMO - geometry, 2021.2

Tags: geometry

Let $\triangle ABC$ be an acute triangle with $AB =AC$ and let $D$ be a point on the side $BC$. The circle with centre $D$ passing through $C$ intersects $\odot(ABD)$ at points $P$ and $Q$, where $Q$ is the point closer to $B$. The line $BQ$ intersects $AD$ and $AC$ at points $X$ and $Y$ respectively. Prove that quadrilateral $PDXY$ is cyclic.

1999 All-Russian Olympiad Regional Round, 8.2

Tags: algebra , number theory

The natural number $A$ has three digits added to its right. The resulting number turned out to be equal to the sum of all natural numbers from $1$ to $A$. Find $A$.