Olympiad problems

1967 German National Olympiad, 5

For each natural number $n$, determine the number $A(n)$ of all integer nonnegative solutions the equation $$5x + 2y + z = 10n.$$

2009 Postal Coaching, 3

Tags: polynomial , algebra

Let $n \ge 3$ be a positive integer. Find all nonconstant real polynomials $f_1(x), f_2(x), ..., f_n(x)$ such that $f_k(x)f_{k+1}(x) = f_{k+1}(f_{k+2}(x))$, $1 \le k \le n$ for all real x. [All suffixes are taken modulo $n$.]

1983 Swedish Mathematical Competition, 5

Tags: geometry , combinatorial geometry , combinatorics , disks , discs , Geometric Inequalities , min

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$. What is the smallest possible radius?

2008 Alexandru Myller, 3

Tags: geometry , area

For a convex pentagon, prove that $ \frac{\text{area} (ABC)}{\text{area} (ABCD)} +\frac{\text{area} (CDE)}{\text{area} (BCDE)} <1. $ [i]Dan Ismailescu[/i]

2001 Singapore Senior Math Olympiad, 3

Tags: combinatorics

Each of the squares in a $50 \times 50$ square board is filled with a number from $1$ to $50$ so that each of the numbers $1,2, ..., 50$ appears exactly $50$ times. Prove that there is a row or a column containing at least $8$ distinct numbers.

2020 Czech-Austrian-Polish-Slovak Match, 1

Tags: geometry , parallelogram , circumcircle , Computer problems

Let $ABCD$ be a parallelogram whose diagonals meet at $P$. Denote by $M$ the midpoint of $AB$. Let $Q$ be a point such that $QA$ is tangent to the circumcircle of $MAD$ and $QB$ is tangent to the circumcircle of $MBC$. Prove that points $Q,M,P$ are collinear. (Patrik Bak, Slovakia)

2008 Mexico National Olympiad, 3

Tags: geometry , circumcircle , power of a point , radical axis , geometry unsolved

The internal angle bisectors of $A$, $B$, and $C$ in $\triangle ABC$ concur at $I$ and intersect the circumcircle of $\triangle ABC$ at $L$, $M$, and $N$, respectively. The circle with diameter $IL$ intersects $BC$ at $D$ and $E$; the circle with diameter $IM$ intersects $CA$ at $F$ and $G$; the circle with diameter $IN$ intersects $AB$ at $H$ and $J$. Show that $D$, $E$, $F$, $G$, $H$, and $J$ are concyclic.

2000 Romania National Olympiad, 3

Tags: geometry , 3D geometry , pyramid

Let $SABC$ be the pyramid where$ m(\angle ASB) = m(\angle BSC) = m(\angle CSA) = 90^o$. Show that, whatever the point $M \in AS$ is and whatever the point $N \in BC$ is, holds the relation $$\frac{1}{MN^2} \le \frac{1}{SB^2} + \frac{1}{SC^2}.$$

2005 Austrian-Polish Competition, 9

Tags: number theory , Diophantine equation , sum of cubes , infinitely many solutions

Consider the equation $x^3 + y^3 + z^3 = 2$. a) Prove that it has infinitely many integer solutions $x,y,z$. b) Determine all integer solutions $x, y, z$ with $|x|, |y|, |z| \leq 28$.

2006 Croatia Team Selection Test, 4

Tags: inequalities , number theory proposed , number theory

Find all natural solutions of $3^{x}= 2^{x}y+1.$

2015 IFYM, Sozopol, 4

Tags: number theory , injective function , function

Let $k$ be a natural number. For each natural number $n$ we define $f_k (n)$ to be the least number, greater than $kn$, for which $nf_k (n)$ is a perfect square. Prove that $f_k (n)$ is injective.

2012 AMC 12/AHSME, 22

Tags: geometry , 3D geometry , tetrahedron , prism , AMC

Distinct planes $p_1,p_2,....,p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $ P =\bigcup_{j=1}^{k}p_{j} $. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$? $ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 24 $

1998 IMO, 4

Tags: number theory , Divisibility , polynomial , IMO , IMO 1998 , david monk , Hi

Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.

2013 BMT Spring, 3

Tags: combinatorics

Suppose we have $2013$ piles of coins, with the $i$th pile containing exactly $i$ coins. We wish to remove the coins in a series of steps. In each step, we are allowed to take away coins from as many piles as we wish, but we have to take the same number of coins from each pile. We cannot take away more coins than a pile actually has. What is the minimum number of steps we have to take?

1950 Putnam, B3

Tags: Putnam

In the Gregorian calendar: (i) years not divisible by $4$ are common years; (ii) years divisible by $4$ but not by $100$ are leap years; (iii) years divisible by $100$ but not by $400$ are common years; (iv) years divisible by $400$ are leap years; (v) a leap year contains $366$ days; a common year $365$ days. Prove that the probability that Christmas falls on a Wednesday is not $1/7.$

2017 Iran Team Selection Test, 3

Tags: geometry , Iran , Iranian TST , collinear

In triangle $ABC$ let $I_a$ be the $A$-excenter. Let $\omega$ be an arbitrary circle that passes through $A,I_a$ and intersects the extensions of sides $AB,AC$ (extended from $B,C$) at $X,Y$ respectively. Let $S,T$ be points on segments $I_aB,I_aC$ respectively such that $\angle AXI_a=\angle BTI_a$ and $\angle AYI_a=\angle CSI_a$.Lines $BT,CS$ intersect at $K$. Lines $KI_a,TS$ intersect at $Z$. Prove that $X,Y,Z$ are collinear. [i]Proposed by Hooman Fattahi[/i]

2003 Moldova National Olympiad, 12.8

Tags: limit , function , calculus , calculus computations

Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by $F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$. Find the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]

2019 District Olympiad, 1

Tags: algebra , number theory

Determine the numbers $x,y$, with $x$ integer and $y$ rational, for which equality holds: $$5(x^2+xy+y^2) = 7(x+2y)$$

1992 National High School Mathematics League, 14

Tags: geometry , 3D geometry , geometric transformation , reflection

$l,m$ are skew lines. Three points $A,B,C$ on line $l$ satisfy that $AB=BC$. Projection of $A,B,C$ on $m$ are $D,E,F$. If $|AD|=\sqrt{15},|BE|=\frac{7}{2}|CF|=\sqrt{10}$, find the distance between $l$ and $m$.

2003 Baltic Way, 10

Tags: calculus , integration , analytic geometry , combinatorics unsolved , combinatorics

A [i]lattice point[/i] in the plane is a point with integral coordinates. The[i] centroid[/i] of four points $(x_i,y_i )$, $i = 1, 2, 3, 4$, is the point $\left(\frac{x_1 +x_2 +x_3 +x_4}{4},\frac{y_1 +y_2 +y_3 +y_4 }{4}\right)$. Let $n$ be the largest natural number for which there are $n$ distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Prove that $n = 12$.

OIFMAT III 2013, 3

Tags: combinatorics

Legend has it that in a police station in the old west a group of six bandits tried to bribe the Sheriff in charge of the place with six gold coins to free them, the Sheriff was a very honest person so to prevent them from continuing to insist With the idea of bribery, he sat the $6$ bandits around a table and proposed the following: - "Initially the leader will have the six gold coins, in each turn one of you can pass coins to the adjacent companions, but each time you do so you must pass the same amount of coins to each of your neighbors. If at any time they all manage to have the same amount of coins so I will let them go free. " The bandits accepted and began to play. Show that regardless of what moves the bandits make, they cannot win.

1998 USAMTS Problems, 2

Tags: USAMTS , quadratics , arithmetic series , algebra , quadratic formula

There are inﬁnitely many ordered pairs $(m,n)$ of positive integers for which the sum \[ m + ( m + 1) + ( m + 2) +... + ( n - 1 )+n\] is equal to the product $mn$. The four pairs with the smallest values of $m$ are $(1, 1), (3, 6), (15, 35),$ and $(85, 204)$. Find three more $(m, n)$ pairs.

2017 Philippine MO, 2

Tags: inequalities , function

Find all positive real numbers $(a,b,c) \leq 1$ which satisfy \[ \huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\]

2023 Brazil Team Selection Test, 4

Tags: conics , hyperbola , geometry , projective geometry

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

1986 Tournament Of Towns, (108) 2

Tags: number theory , maximum , Digits , Divide

A natural number $N$ is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number $N$ (the digit $0$ is not encountered). What is the maximum number of different digits which there can be in such a representation of $N$? (S . Fomin, Leningrad)