Olympiad problems

1991 Tournament Of Towns, (296) 3

The numbers $x_1,x_2,x_3, ..., x_n$ satisfy the two conditions $$\sum^n_{i=1}x_i=0 \,\, , \,\,\,\,\sum^n_{i=1}x_i^2=1$$ Prove that there are two numbers among them whose product is no greater than $- 1/n$. (Stolov, Kharkov)

2014 IMS, 5

Tags: superior algebra , group theory , abstract algebra

Let $G_1$ and $G_2$ be two finite groups such that for any finite group $H$, the number of group homomorphisms from $G_1$ to $H$ is equal to the number of group homomorphisms from $G_2$ to $H$. Prove that $G_1$ and $G_2$ are Isomorphic.

2017 District Olympiad, 1

Tags: number theory , easy

[b]a)[/b] Let $ m,n,p\in\mathbb{Z}_{\ge 0} $ such that $ m>n $ and $ \sqrt{m} -\sqrt n=p. $ Prove that $ m $ and $ n $ are perfect squares. [b]b)[/b] Find the numbers of four digits $ \overline{abcd} $ that satisfy the equation: $$ \sqrt {\overline{abcd} } -\sqrt{\overline{acd}} =\overline{bb} . $$

1963 Putnam, B1

Tags: polynomial

For what integers $a$ does $x^2 -x+a$ divide $x^{13}+ x +90$ ?

1993 Denmark MO - Mohr Contest, 2

Tags: rectangle , paper folding , geometry

A rectangular piece of paper has the side lengths $12$ and $15$. A corner is bent about as shown in the figure. Determine the area of the gray triangle. [img]https://1.bp.blogspot.com/-HCfqWF0p_eA/XzcIhnHS1rI/AAAAAAAAMYg/KfY14frGPXUvF-H6ZVpV4RymlhD_kMs-ACLcBGAsYHQ/s0/1993%2BMohr%2Bp2.png[/img]

2022 AMC 8 -, 18

Tags: geometry , rectangle

The midpoints of the four sides of a rectangle are $(-3, 0), (2, 0), (5, 4)$ and $(0, 4)$. What is the area of the rectangle? $\textbf{(A)} ~20\qquad\textbf{(B)} ~25\qquad\textbf{(C)} ~40\qquad\textbf{(D)} ~50\qquad\textbf{(E)} ~80\qquad$

1998 USAMTS Problems, 2

Tags: quadratic , 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.

2011 Morocco National Olympiad, 4

Tags: geometry , inequalities

Let $ABCD$ be a convex quadrilateral with angles $\angle ABC$ and $\angle BCD$ not less than $120^{\circ}$. Prove that \[AC + BD> AB+BC+CD\]

2012 APMO, 1

Tags: geometry , polynomial , inequalities , algebra

Let $ P $ be a point in the interior of a triangle $ ABC $, and let $ D, E, F $ be the point of intersection of the line $ AP $ and the side $ BC $ of the triangle, of the line $ BP $ and the side $ CA $, and of the line $ CP $ and the side $ AB $, respectively. Prove that the area of the triangle $ ABC $ must be $ 6 $ if the area of each of the triangles $ PFA, PDB $ and $ PEC $ is $ 1 $.

PEN A Problems, 58

Tags: divisibility theory

Let $k\ge 14$ be an integer, and let $p_k$ be the largest prime number which is strictly less than $k$. You may assume that $p_k\ge \tfrac{3k}{4}$. Let $n$ be a composite integer. Prove that [list=a] [*] if $n=2p_k$, then $n$ does not divide $(n-k)!$, [*] if $n>2p_k$, then $n$ divides $(n-k)!$. [/list]

1980 Swedish Mathematical Competition, 6

Tags: combinatorial geometry , combinatorics , geometric inequalities

Find the smallest constant $c$ such that for every $4$ points in a unit square there are two a distance $\leq c$ apart.

2016 Philippine MO, 4

Tags: combinatorics , game , nim

Two players, $A$ (first player) and $B$, take alternate turns in playing a game using 2016 chips as follows: [i]the player whose turn it is, must remove $s$ chips from the remaining pile of chips, where $s \in \{ 2,4,5 \}$[/i]. No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?

Durer Math Competition CD Finals - geometry, 2011.D2

Tags: geometry , right triangle , isosceles

In an right isosceles triangle $ABC$, there are two points on the hypotenuse $AB, K$ and $M$, respectively, such that $KCM$ angle is $45^o$ (point $K$ lies between $A$ and $M$). Prove that $AK^2 + MB^2 = KM^2$ [img]https://cdn.artofproblemsolving.com/attachments/2/c/e7c57e0651e5a4c492cc4ae4b115bf68a7a833.png[/img]

2013 ELMO Shortlist, 11

Tags: circumcircle , geometry

Let $\triangle ABC$ be a nondegenerate isosceles triangle with $AB=AC$, and let $D, E, F$ be the midpoints of $BC, CA, AB$ respectively. $BE$ intersects the circumcircle of $\triangle ABC$ again at $G$, and $H$ is the midpoint of minor arc $BC$. $CF\cap DG=I, BI\cap AC=J$. Prove that $\angle BJH=\angle ADG$ if and only if $\angle BID=\angle GBC$. [i]Proposed by David Stoner[/i]

2013 Dutch IMO TST, 5

Tags: algebra , inequalities

Let $a, b$, and $c$ be positive real numbers satisfying $abc = 1$. Show that $a + b + c \ge \sqrt{\frac13 (a + 2)(b + 2)(c + 2)}$

1986 China National Olympiad, 5

Tags: combinatorics

Given a sequence $1,1,2,2,3,3,\ldots,1986,1986$, determine, with proof, if we can rearrange the sequence so that for any integer $1\le k \le 1986$ there are exactly $k$ numbers between the two “$k$”s.

2016 Saint Petersburg Mathematical Olympiad, 5

Tags: geometry , circumcircle , power of point , coaxial circles

Points $A$ and $P$ are marked in the plane not lying on the line $\ell$. For all right triangles $ABC$ with hypotenuse on $\ell$, show that the circumcircle of triangle $BPC$ passes through a fixed point other than $P$.

2004 Greece National Olympiad, 4

Tags: combinatorics

Let $M\subset \Bbb{N}^*$ such that $|M|=2004.$ If no element of $M$ is equal to the sum of any two elements of $M,$ find the least value that the greatest element of $M$ can take.

2020 Germany Team Selection Test, 2

Tags: geometry

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

2024 Dutch IMO TST, 3

Tags: algebra , inequalities

Let $a,b,c$ be real numbers such that $0 \le a \le b \le c$ and $a+b+c=1$. Show that \[ab\sqrt{b-a}+bc\sqrt{c-b}+ac\sqrt{c-a}<\frac{1}{4}.\]

2023 CUBRMC, 1

Tags: number theory

Ben starts with an integer greater than $9$ and subtracts the sum of its digits from it to get a new integer. He repeats this process with each new integer he gets until he gets a positive $1$-digit integer. Find all possible $1$-digit integers Ben can end with from this process.

2003 AMC 12-AHSME, 8

Tags: probability

What is the probability that a randomly drawn positive factor of $ 60$ is less than $ 7$? $ \textbf{(A)}\ \frac{1}{10} \qquad \textbf{(B)}\ \frac{1}{6} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{1}{3} \qquad \textbf{(E)}\ \frac{1}{2}$

2016 Serbia National Math Olympiad, 3

Tags: geometry , circumcircle

Let $ABC$ be a triangle and $O$ its circumcentre. A line tangent to the circumcircle of the triangle $BOC$ intersects sides $AB$ at $D$ and $AC$ at $E$. Let $A'$ be the image of $A$ under $DE$. Prove that the circumcircle of the triangle $A'DE$ is tangent to the circumcircle of triangle $ABC$.

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 $M_n$.

2005 Federal Math Competition of S&M, Problem 4

Tags: area , circles , geometry

Inside a circle $k$ of radius $R$ some round spots are made. The area of each spot is $1$. Every radius of circle $k$, as well as every circle concentric with $k$, meets in no more than one spot. Prove that the total area of all the spots is less than $$\pi\sqrt R+\frac12R\sqrt R.$$