Olympiad problems

2009 Germany Team Selection Test, 2

Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$

PEN H Problems, 66

Tags: diophantine equation , inequalities

Let $b$ be a positive integer. Determine all $2002$-tuples of non-negative integers $(a_{1}, a_{2}, \cdots, a_{2002})$ satisfying \[\sum^{2002}_{j=1}{a_{j}}^{a_{j}}=2002{b}^{b}.\]

1997 National High School Mathematics League, 9

Tags: complex number

$z$ is a complex number that $\left|2z+\frac{1}{z}\right|=1$, then the range value of $\arg(z)$ is________.

2013 Iran MO (3rd Round), 2

Tags: circumcircle , incenter , geometric transformation , geometry

Let $ABC$ be a triangle with circumcircle $(O)$. Let $M,N$ be the midpoint of arc $AB,AC$ which does not contain $C,B$ and let $M',N'$ be the point of tangency of incircle of $\triangle ABC$ with $AB,AC$. Suppose that $X,Y$ are foot of perpendicular of $A$ to $MM',NN'$. If $I$ is the incenter of $\triangle ABC$ then prove that quadrilateral $AXIY$ is cyclic if and only if $b+c=2a$.

2014-2015 SDML (Middle School), 15

Tags:

How many triangles formed by three vertices of a regular $17$-gon are obtuse? $\text{(A) }156\qquad\text{(B) }204\qquad\text{(C) }357\qquad\text{(D) }476\qquad\text{(E) }524$

2006 Sharygin Geometry Olympiad, 9.3

Tags: equal ratio , similar triangles , collinear , geometry

Triangles $ABC$ and $A_1B_1C_1$ are similar and differently oriented. On the segment $AA_1$, a point $A'$ is taken such that $AA' / A_1A'= BC / B_1C_1$. We similarly construct $B'$ and $C'$. Prove that $A', B',C'$ lie on one straight line.

2022 International Zhautykov Olympiad, 5

Tags: algebra , polynomial

A polynomial $f(x)$ with real coefficients of degree greater than $1$ is given. Prove that there are infinitely many positive integers which cannot be represented in the form \[f(n+1)+f(n+2)+\cdots+f(n+k)\] where $n$ and $k$ are positive integers.

2001 National High School Mathematics League, 14

Tags: analytic geometry , geometry

$C_1:\frac{x^2}{a^2}+y^2=1(a>0),C_2:y^2=2(x+m)$, one intersection of $C_1$ and $C_2$ is $P$, and $P$ is above the $x$-axis. [b](a)[/b] Find the range value of $m$ (express with $a$). [b](b)[/b] $O(0,0),A(-a,0)$. If $0<a<\frac{1}{2}$, find the maximum value of $S_{\triangle OAP}$.

2018 Iran MO (1st Round), 3

Tags: number theory , divisibility tests

How many $8$-digit numbers in base $4$ formed of the digits $1,2, 3$ are divisible by $3$?

2008 ITest, 39

Tags: function , number theory , relatively prime

Let $\phi(n)$ denote $\textit{Euler's phi function}$, the number of integers $1\leq i\leq n$ that are relatively prime to $n$. (For example, $\phi(6)=2$ and $\phi(10)=4$.) Let \[S=\sum_{d|2008}\phi(d),\] in which $d$ ranges through all positive divisors of $2008$, including $1$ and $2008$. Find the remainder when $S$ is divided by $1000$.

1980 All Soviet Union Mathematical Olympiad, 298

Tags: geometry , equilateral , angle

Given equilateral triangle $ABC$. Some line, parallel to $[AC]$ crosses $[AB]$ and $[BC]$ in $M$ and $P$ points respectively. Let $D$ be the centre of $PMB$ triangle, $E$ be the midpoint of the $[AP]$ segment. Find the angles of triangle $DEC$ .

OMMC POTM, 2023 11

Tags: algebra , inequalities , strictly increasing

Consider an infinite strictly increasing sequence of positive integers $a_1$, $a_2$,$...$ where for any real number $C$, there exists an integer $N$ where $a_k >Ck$ for any $k >N$. Do there necessarily exist inifinite many indices $k$ where $2a_k <a_{k-1}+a_{k+1}$ for any $0<i<k$?

2012 Bogdan Stan, 3

Tags: geometric sequence , algebra

Consider $ 2011 $ positive real numbers $ a_1,a_2,\ldots ,a_{2011} . $ If they are in geometric progression, show that there exists a real number $ \lambda $ such that any $ i\in\{ 1,2,\ldots , 1005 \} $ implies $ \lambda =a_i\cdot a_{2012-i} . $ Disprove the converse. [i]Teodor Radu[/i]

2015 Thailand Mathematical Olympiad, 3

Tags: combinatorics , combinatorial geometry

Let $P = \{(x, y) | x, y \in \{0, 1, 2,... , 2015\}\}$ be a set of points on the plane. Straight wires of unit length are placed to connect points in $P$ so that each piece of wire connects exactly two points in $P$, and each point in $P$ is an endpoint of exactly one wire. Prove that no matter how the wires are placed, it is always possible to draw a straight line parallel to either the horizontal or vertical axis passing through midpoints of at least $506$ pieces of wire.

Ukrainian TYM Qualifying - geometry, 2014.22

Tags: orthocenter , geometry , incenter , symmetric

In $\vartriangle ABC$ on the sides $BC, CA, AB$ mark feet of altitudes $H_1, H_2, H_3$ and the midpoint of sides $M_1, M_3, M_3$. Let $H$ be orthocenter $\vartriangle ABC$. Suppose that $X_2, X_3$ are points symmetric to $H_1$ wrt $BH_2$ and $CH_3$. Lines $M_3X_2$ and $M_2X_3$ intersect at point $X$. Similarly, $Y_3,Y_1$ are points symmetric to $H_2$ wrt $C_3H$ and $AH_1$.Lines $M_1Y_3$ and $M_3Y_1$ intersect at point $Y.$ Finally, $Z_1,Z_2$ are points symmetric to $H_3$ wrt $AH_1$ and $BH_2$. Lines $M_1Z_2$ and $M_2Z_1$ intersect at the point $Z$ Prove that $H$ is the incenter $\vartriangle XYZ$ .

2011 Kosovo National Mathematical Olympiad, 5

Tags: combinatorics

Let $n>1$ be an integer and $S_n$ the set of all permutations $\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \}$ where $\pi$ is bijective function. For every permutation $\pi \in S_n$ we define: \[ F(\pi)= \sum_{k=1}^n |k-\pi(k)| \ \ \text{and} \ \ M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi) \] where $M_n$ is taken with all permutations $\pi \in S_n$. Calculate the sum $M_n$.

1994 IberoAmerican, 2

Tags: function , ceiling function , combinatorics

Let $n$ and $r$ two positive integers. It is wanted to make $r$ subsets $A_1,\ A_2,\dots,A_r$ from the set $\{0,1,\cdots,n-1\}$ such that all those subsets contain exactly $k$ elements and such that, for all integer $x$ with $0\leq{x}\leq{n-1}$ there exist $x_1\in{}A_1,\ x_2\in{}A_2 \dots,x_r\in{}A_r$ (an element of each set) with $x=x_1+x_2+\cdots+x_r$. Find the minimum value of $k$ in terms of $n$ and $r$.

2015 Thailand TSTST, 3

Tags: inequalities

Let $a, b, c$ be positive real numbers. Prove that $$\frac {3(ab + bc + ca)}{2(a^2b^2+b^2c^2+c^2a^2)}\leq \frac1{a^2 + bc} + \frac1{b^2 + ca} + \frac1{c^2 + ab}\leq\frac{a+b+c}{2abc}.$$

2015 239 Open Mathematical Olympiad, 5

Tags: geometry , 3d geometry

The nodes of a three dimensional unit cube lattice with all three coordinates even are coloured red and blue otherwise. A convex polyhedron with all vertices red is given. Assuming the number of red points on its border is $n$. How many blue vertices can be on its border?

2021 Saint Petersburg Mathematical Olympiad, 1

Tags: prime number , number theory

Let $p$ be a prime number. All natural numbers from $1$ to $p$ are written in a row in ascending order. Find all $p$ such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum. [i]A. Khrabov[/i]

2023-24 IOQM India, 5

Tags: geometry , area , area of a triangle , barycentric coordinates , barycentric

In a triangle $A B C$, let $E$ be the midpoint of $A C$ and $F$ be the midpoint of $A B$. The medians $B E$ and $C F$ intersect at $G$. Let $Y$ and $Z$ be the midpoints of $B E$ and $C F$ respectively. If the area of triangle $A B C$ is 480 , find the area of triangle $G Y Z$.

2022 Germany Team Selection Test, 3

Tags: number theory

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2024 Kazakhstan National Olympiad, 1

Tags: algebra , number theory

Positive integers $a,b,c$ satisfy the equations $a^2=b^3+ab$ and $c^3=a+b+c$. Prove that $a=bc$.

2019 Sharygin Geometry Olympiad, 7

Tags: geometry

Let $AH_A$, $BH_B$, $CH_C$ be the altitudes of the acute-angled $\Delta ABC$. Let $X$ be an arbitrary point of segment $CH_C$, and $P$ be the common point of circles with diameters $H_CX$ and BC, distinct from $H_C$. The lines $CP$ and $AH_A$ meet at point $Q$, and the lines $XP$ and $AB$ meet at point $R$. Prove that $A, P, Q, R, H_B$ are concyclic.

1994 Argentina National Olympiad, 5

Tags: combinatorics , combinatorial geometry , analytic geometry , geometry

Let $A$ be an infinite set of points in the plane such that inside each circle there are only a finite number of points of $A$, with the following properties: $\bullet$ $(0, 0)$ belongs to $A$. $\bullet$ If $(a, b)$ and $(c, d)$ belong to $A$, then $(a-c, b-d)$ belongs to $A$. $\bullet$ There is a value of $\alpha$ such that by rotating the set $A$ with center at $(0, 0)$ and angle $\alpha$, the set $A$ is obtained again. Prove that $\alpha$ must be equal to $\pm 60^{\circ}$ or $\pm 90^{\circ}$ or $\pm 120^{\circ}$ or $\pm 180^{\circ}$.