Olympiad problems

2019 Harvard-MIT Mathematics Tournament, 5

Tags: HMMT , combinatorics

Find all positive integers $n$ such that the unit segments of an $n \times n$ grid of unit squares can be partitioned into groups of three such that the segments of each group share a common vertex.

2016 NIMO Problems, 4

Tags:

Triangle $ABC$ has $AB=13$, $BC=14$, and $CA=15$. Let $\omega_A$, $\omega_B$ and $\omega_C$ be circles such that $\omega_B$ and $\omega_C$ are tangent at $A$, $\omega_C$ and $\omega_A$ are tangent at $B$, and $\omega_A$ and $\omega_B$ are tangent at $C$. Suppose that line $AB$ intersects $\omega_B$ at a point $X \neq A$ and line $AC$ intersects $\omega_C$ at a point $Y \neq A$. If lines $XY$ and $BC$ intersect at $P$, then $\tfrac{BC}{BP} = \tfrac{m}{n}$ for coprime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Michael Ren[/i]

2009 Junior Balkan Team Selection Tests - Moldova, 2

Tags: inequalities , algebra , TST

Real positive numbers $a, b, c$ satisfy $abc=1$. Prove the inequality $$\frac{a^2+b^2}{a^4+b^4}+\frac{b^2+c^2}{b^4+c^4}+\frac{c^2+a^2}{c^4+a^4}\leq a+b+c.$$

2024 Iranian Geometry Olympiad, 1

Tags: geometry

Reflect each of the shapes $A,B$ over some lines $l_A,l_B$ respectively and rotate the shape $C$ such that a $4 \times 4$ square is obtained. Identify the lines $l_A,l_B$ and the center of the rotation, and also draw the transformed versions of $A,B$ and $C$ under these operations. [img]https://s8.uupload.ir/files/photo14908574605_i39w.jpg[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

2004 Switzerland Team Selection Test, 12

Tags: naturals , number theory

Find all natural numbers which can be written in the form $\frac{(a+b+c)^2}{abc}$ , where $a,b,c \in N$.

2014 Ukraine Team Selection Test, 12

Tags: prime , number theory , Divisibility

Prove that for an arbitrary prime $p \ge 3$ the number of positive integers $n$, for which $p | n! +1$ does not exceed $cp^{2/3}$, where c is a constant that does not depend on $p$.

2001 USAMO, 5

Tags: AMC , USAMO , modular arithmetic , number theory

Let $S$ be a set of integers (not necessarily positive) such that (a) there exist $a,b \in S$ with $\gcd(a,b)=\gcd(a-2,b-2)=1$; (b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2-y$ also belongs to $S$. Prove that $S$ is the set of all integers.

2014 Putnam, 2

Tags: Putnam , function , integration , real analysis , absolute value , Putnam calculus , Putnam 2014

Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be?

2007 Iran Team Selection Test, 1

Tags: geometry , geometric transformation , reflection , analytic geometry , geometry proposed

In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position [i]By Sam Nariman[/i]

2019 Miklós Schweitzer, 10

Tags: linear algebra

Let $A$ and $B$ be positive self-adjoint operators on a complex Hilbert space $H$. Prove that \[\limsup_{n \to \infty} \|A^n x\|^{1/n} \le \limsup_{n \to \infty} \|B^n x\|^{1/n}\] holds for every $x \in H$ if and only if $A^n \le B^n$ for each positive integer $n$.

1995 Korea National Olympiad, Day 1

Tags: geometry , perpendicular lines , area of a triangle

Let $O$ and $R$ be the circumcenter and circumradius of a triangle $ABC$, and let $P$ be any point in the plane of the triangle. The perpendiculars $PA_1,PB_1,PC_1$ are drawn from $P$ on $BC,CA,AB$. Express $S_{A_1B_1C_1}/S_{ABC}$ in terms of $R$ and $d = OP$, where $S_{XYZ}$ is the area of $\triangle XYZ$.

2004 Junior Balkan Team Selection Tests - Moldova, 7

Tags: circumcircle , geometry , angle bisector

Let the triangle $ABC$ have area $1$. The interior bisectors of the angles $\angle BAC,\angle ABC, \angle BCA$ intersect the sides $(BC), (AC), (AB) $ and the circumscribed circle of the respective triangle $ABC$ at the points $L$ and $G, N$ and $F, Q$ and $E$. The lines $EF, FG,GE$ intersect the bisectors $(AL), (CQ) ,(BN)$ respectively at points $P, M, R$. Determine the area of the hexagon $LMNPR$.

2017 Tuymaada Olympiad, 6

Tags: number theory

Let $\sigma(n) $ denote the sum of positive divisors of a number $n $. A positive integer $N=2^rb $ is given,where $r $ and $b $ are positive integers and $b $ is odd. It is known that $\sigma(N)=2N-1$. Prove that $b$ and $\sigma (b) $ are coprime. Tuymaada Q6 Juniors

1997 USAMO, 6

Tags: floor function , induction , inequalities , algorithm , algebra proposed , algebra

Suppose the sequence of nonnegative integers $a_1, a_2, \ldots, a_{1997}$ satisfies \[ a_i + a_j \leq a_{i+j} \leq a_i + a_j + 1 \] for all $i,j \geq 1$ with $i + j \leq 1997$. Show that there exists a real number $x$ such that $a_n = \lfloor nx \rfloor$ (the greatest integer $\leq nx$) for all $1 \leq n \leq 1997$.

2012 May Olympiad, 2

Tags: number theory , sum of digits

We call S $(n)$ the sum of the digits of the integer $n$. For example, $S (327)=3+2+7=12$. Find the value of $$A=S(1)-S(2)+S(3)-S(4)+...+S(2011)-S(2012).$$ ($A$ has $2012$ terms).

2006 Kyiv Mathematical Festival, 1

Tags: geometry , conics , hyperbola , Asymptote , geometric transformation , reflection , geometry proposed

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Triangle $ABC$ and straight line $l$ are given at the plane. Construct using a compass and a ruler the straightline which is parallel to $l$ and bisects the area of triangle $ABC.$

1964 AMC 12/AHSME, 32

Tags: AMC

If $\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}$, then: $\textbf{(A)}\ a\text{ must equal }c \qquad \textbf{(B)}\ a+b+c+d\text{ must equal zero }\qquad$ $ \textbf{(C)}\ \text{either }a=c\text{ or }a+b+c+d=0,\text{ or both} \qquad$ $ \textbf{(D)}\ a+b+c+d\neq 0\text{ if }a=c \qquad \textbf{(E)}\ a(b+c+d)=c(a+b+d)$

2024-IMOC, N7

Tags: number theory , functional equation

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$|xf(y)-yf(x)|$$ is a perfect square for every $x,y \in \mathbb{N}$

1997 Belarusian National Olympiad, 3

Tags: algebra

$$Problem3;$$If distinct real numbers x,y satisfy $\{x\} = \{y\}$ and $\{x^3\}=\{y^3\}$ prove that $x$ is a root of a quadratic equation with integer coefficients.

1988 AMC 12/AHSME, 25

Tags: ratio , AMC

$X$, $Y$ and $Z$ are pairwise disjoint sets of people. The average ages of people in the sets $X$, $Y$, $Z$, $X \cup Y$, $X \cup Z$ and $Y \cup Z$ are given in the table below. \begin{tabular}{|c|c|c|c|c|c|c|} \hline \rule{0pt}{1.1em} Set & $X$ & $Y$ & $Z$ & $X\cup Y$ & $X\cup Z$ & $Y\cup Z$\\[0.5ex] \hline \rule{0pt}{2.2em} \shortstack{Average age of \\ people in the set} & 37 & 23 & 41 & 29 & 39.5 & 33\\[1ex]\hline\end{tabular} Find the average age of the people in set $X \cup Y \cup Z$. $ \textbf{(A)}\ 33\qquad\textbf{(B)}\ 33.5\qquad\textbf{(C)}\ 33.6\overline{6}\qquad\textbf{(D)}\ 33.83\overline{3}\qquad\textbf{(E)}\ 34 $

2019 China Team Selection Test, 2

Tags: graph theory , combinatorics

A graph $G(V,E)$ is triangle-free, but adding any edges to the graph will form a triangle. It's given that $|V|=2019$, $|E|>2018$, find the minimum of $|E|$ .

2016 Kosovo National Mathematical Olympiad, 2

Tags: polynomial

Sum of all coefficients of polynomial $P(x)$ is equal with $2$ . Also the sum of coefficients which are at odd exponential in $x^k$ are equal to sum of coefficients which are at even exponential in $x^k$ . Find the residue of polynomial $P(x)$ when it is divide by $x^2-1$ .

2019-2020 Winter SDPC, 7

Tags: algebra

Let $a,b$ be positive integers. Find, with proof, the maximum possible value of $a\lceil b\lambda \rceil - b \lfloor a \lambda \rfloor$ for irrational $\lambda$.

2022 Sharygin Geometry Olympiad, 8.8

Tags: inequalities , geometry , geometric inequality , isosceles , trapezoid

An isosceles trapezoid $ABCD$ ($AB = CD$) is given. A point $P$ on its circumcircle is such that segments $CP$ and $AD$ meet at point $Q$. Let $L$ be tha midpoint of$ QD$. Prove that the diagonal of the trapezoid is not greater than the sum of distances from the midpoints of the lateral sides to ana arbitrary point of line $PL$.

2009 AMC 12/AHSME, 17

Tags: geometry , 3D geometry , probability , AMC

Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? $ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {3}{16}\qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {3}{8}\qquad \textbf{(E)}\ \frac {1}{2}$