Olympiad problems

2021 Middle European Mathematical Olympiad, 4

Let $n$ be a positive integer. Prove that in a regular $6n$-gon, we can draw $3n$ diagonals with pairwise distinct ends and partition the drawn diagonals into $n$ triplets so that: [list] [*] the diagonals in each triplet intersect in one interior point of the polygon and [*] all these $n$ intersection points are distinct. [/list]

2011 All-Russian Olympiad Regional Round, 11.2

Tags: number theory

2011 non-zero integers are given. It is known that the sum of any one of them with the product of the remaining 2010 numbers is negative. Prove that if all numbers are split arbitrarily into two groups, the sum of the two products will also be negative. (Authors: N. Agahanov & I. Bogdanov)

1995 India National Olympiad, 2

Tags: quadratic , number theory , relatively prime , diophantine equation , algebra

Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.

2024 Durer Math Competition Finals, 5

Tags: number theory , quadratic residue

Let $p{}$ be a fixed prime number. Determine the number of ordered $k$-tuples $(a_1,\ldots,a_k)$ of non-negative integers smaller than $p{}$ for which $p\mid a_1^2+\cdots+a_k^2$ where a) $k=3$ and b) $k$ is an arbitrary odd positive integer.

2007 Indonesia TST, 2

Tags: parameterization , algebra

Solve the equation \[ x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x}\] where $ a$ is a real parameter.

1979 IMO Longlists, 1

Tags: geometry , polygon , dissection , rhombus

Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).

1952 AMC 12/AHSME, 27

Tags: ratio , geometry , perimeter

The ratio of the perimeter of an equilateral triangle having an altitude equal to the radius of a circle, to the perimeter of an equilateral triangle inscribed in the circle is: $ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 1: 3 \qquad\textbf{(C)}\ 1: \sqrt {3} \qquad\textbf{(D)}\ \sqrt {3}: 2 \qquad\textbf{(E)}\ 2: 3$

2019 Taiwan APMO Preliminary Test, P4

Tags: algebra , floor function , sequence , recursive

We define a sequence ${a_n}$: $$a_1=1,a_{n+1}=\sqrt{a_n+n^2},n=1,2,...$$ (1)Find $\lfloor a_{2019}\rfloor$ (2)Find $\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor$

2001 Grosman Memorial Mathematical Olympiad, 5

Tags: construction , geometric construction , geometry

Triangle $ABC$ in the plane $\Pi$ is called [i]good [/i] if it has the following property: For any point $D$ in space outside the plane $\Pi$, it is possible to construct a triangle with sides of lengths $CD,BD,AD$. Find all good triangles

2018 Sharygin Geometry Olympiad, 11

Tags: geometry

Let $I$ be the incenter of a nonisosceles triangle $ABC$. Prove that there exists a unique pair of points $M$, $N$ lying on the sides $AC$, $BC$ respectively, such that $\angle AIM = \angle BIN$ and $MN|| AB$.

2021 Science ON grade V, 4

Tags: number theory

Find the least positive integer which is a multiple of $13$ and all its digits are the same. [i](Adapted from Gazeta Matematică 1/1982, Florin Nicolăită)[/i]

2006 Petru Moroșan-Trident, 2

Tags: complex number , algebra

Consider $ n\ge 1 $ complex numbers $ z_1,z_2,\ldots ,z_n $ that have the same nonzero modulus, and which verify $$ 0=\Re\left( \sum_{a=1}^n\sum_{b=1}^n\sum_{c=1}^n\sum_{d=1}^n \frac{z_bz_c}{z_az_d} \right) . $$ Prove that $ n\left( -1+\left| z_1 \right|^2 \right) =\sum_{k=1}^n\left| 1-z_k \right| . $ [i]Botea Viorel[/i]

2016 Harvard-MIT Mathematics Tournament, 7

Tags: geometry

For $i=0,1,\dots,5$ let $l_i$ be the ray on the Cartesian plane starting at the origin, an angle $\theta=i\frac{\pi}{3}$ counterclockwise from the positive $x$-axis. For each $i$, point $P_i$ is chosen uniformly at random from the intersection of $l_i$ with the unit disk. Consider the convex hull of the points $P_i$, which will (with probability 1) be a convex polygon with $n$ vertices for some $n$. What is the expected value of $n$?

2007 Greece JBMO TST, 3

Tags: geometry , rectangle , tangent , tangent circle

Let $ABCD$ be a rectangle with $AB=a >CD =b$. Given circles $(K_1,r_1) , (K_2,r_2)$ with $r_1<r_2$ tangent externally at point $K$ and also tangent to the sides of the rectangle, circle $(K_1,r_1)$ tangent to both $AD$ and $AB$, circle $(K_2,r_2)$ tangent to both $AB$ and $BC$. Let also the internal common tangent of those circles pass through point $D$. (i) Express sidelengths $a$ and $b$ in terms of $r_1$ and $r_2$. (ii) Calculate the ratios $\frac{r_1}{r_2}$ and $\frac{a}{b}$ . (iii) Find the length of $DK$ in terms of $r_1$ and $r_2$.

2019 District Olympiad, 4

Tags: number theory

Find all positive integers $p$ for which there exists a positive integer $n$ such that $p^n+3^n~|~p^{n+1}+3^{n+1}.$

2008 ITest, 90

Tags: floor function , inequalities

For $a,b,c$ positive reals, let \[N=\dfrac{a^2+b^2}{c^2+ab}+\dfrac{b^2+c^2}{a^2+bc}+\dfrac{c^2+a^2}{b^2+ca}.\] Find the minimum value of $\lfloor 2008N\rfloor$.

2018 Canada National Olympiad, 4

Tags: algebra , number theory , polynomial

Find all polynomials $p(x)$ with real coefficients that have the following property: there exists a polynomial $q(x)$ with real coefficients such that $$p(1) + p(2) + p(3) +\dots + p(n) = p(n)q(n)$$ for all positive integers $n$.

2008 Purple Comet Problems, 13

Tags:

Let $A_1A_2A_3...A_{12}$ be a regular dodecagon. Find the number of right triangles whose vertices are in the set ${A_1A_2A_3...A_{12}}$

2025 Korea - Final Round, P6

Tags: diophantine equation , number theory

Positive integers $a, b$ satisfy both of the following conditions. For a positive integer $m$, if $m^2 \mid ab$, then $m = 1$. There exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 = z^2 + w^2$ and $z^2 + w^2 > 0$. Prove that there exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 + n = z^2 + w^2$, for each integer $n$.

PEN D Problems, 16

Tags: modular arithmetic , number theory , relatively prime , congruence

Determine all positive integers $n \ge 2$ that satisfy the following condition; For all integers $a, b$ relatively prime to $n$, \[a \equiv b \; \pmod{n}\Longleftrightarrow ab \equiv 1 \; \pmod{n}.\]

2014 Online Math Open Problems, 18

Tags: quadratic

Find the number of pairs $(m,n)$ of integers with $-2014\le m,n\le 2014$ such that $x^3+y^3 = m + 3nxy$ has infinitely many integer solutions $(x,y)$. [i]Proposed by Victor Wang[/i]

Kvant 2024, M2816

Tags: number theory

Find out for which natural numbers $m$ it is possible to find a natural $\ell$ such that the sum of $n+n^2+n^3+\ldots+n^\ell$ will be divisible by $m$ for any natural $n$. [i]A. Skabelin[/i]

2006 AMC 12/AHSME, 24

Tags: trigonometry , geometry , analytic geometry , 3d geometry , probability , quadratic , inequalities

Let $ S$ be the set of all points $ (x,y)$ in the coordinate plane such that $ 0\le x\le \frac \pi2$ and $ 0\le y\le \frac \pi2$. What is the area of the subset of $ S$ for which \[ \sin^2 x \minus{} \sin x\sin y \plus{} \sin^2 y\le \frac 34? \]$ \textbf{(A) } \frac {\pi^2}9 \qquad \textbf{(B) } \frac {\pi^2}8 \qquad \textbf{(C) } \frac {\pi^2}6\qquad \textbf{(D) } \frac {3\pi^2}{16} \qquad \textbf{(E) } \frac {2\pi^2}9$

2007 Estonia Math Open Senior Contests, 4

Tags: algebra , polynomial , number theory

The Fibonacci sequence is determined by conditions $ F_0 \equal{} 0, F1 \equal{} 1$, and $ F_k\equal{}F_{k\minus{}1}\plus{}F_{k\minus{}2}$ for all $ k \ge 2$. Let $ n$ be a positive integer and let $ P(x) \equal{} a_mx^m \plus{}. . .\plus{} a_1x\plus{} a_0$ be a polynomial that satisfies the following two conditions: (1) $ P(F_n) \equal{} F_{n}^{2}$ ; (2) $ P(F_k) \equal{} P(F_{k\minus{}1}) \plus{} P(F_{k\minus{}2}$ for all $ k \ge 2$. Find the sum of the coefficients of P.

1963 Miklós Schweitzer, 3

Tags: superior algebra

Let $ R\equal{}R_1\oplus R_2$ be the direct sum of the rings $ R_1$ and $ R_2$, and let $ N_2$ be the annihilator ideal of $ R_2$ (in $ R_2$). Prove that $ R_1$ will be an ideal in every ring $ \widetilde{R}$ containing $ R$ as an ideal if and only if the only homomorphism from $ R_1$ to $ N_2$ is the zero homomorphism. [Gy. Hajos]