Olympiad problems

2006 National Olympiad First Round, 36

Tags:

In an exam with $n$ problems where $n$ is a positive integer, each problem was answered by at least one student. Each student answered an even number of problems. Any two students answered an even number of problems in common. What is the number of values that $n$ cannot take? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of above} $

2021 China Team Selection Test, 5

Tags: geometry , circumcircle

Given a triangle $ABC$, a circle $\Omega$ is tangent to $AB,AC$ at $B,C,$ respectively. Point $D$ is the midpoint of $AC$, $O$ is the circumcenter of triangle $ABC$. A circle $\Gamma$ passing through $A,C$ intersects the minor arc $BC$ on $\Omega$ at $P$, and intersects $AB$ at $Q$. It is known that the midpoint $R$ of minor arc $PQ$ satisfies that $CR \perp AB$. Ray $PQ$ intersects line $AC$ at $L$, $M$ is the midpoint of $AL$, $N$ is the midpoint of $DR$, and $X$ is the projection of $M$ onto $ON$. Prove that the circumcircle of triangle $DNX$ passes through the center of $\Gamma$.

2007 Nicolae Păun, 1

Tags: function , order , group theory

Consider a finite group $ G $ and the sequence of functions $ \left( A_n \right)_{n\ge 1} :G\longrightarrow \mathcal{P} (G) $ defined as $ A_n(g) = \left\{ x\in G|x^n=g \right\} , $ where $ \mathcal{P} (G) $ is the power of $ G. $ [b]a)[/b] Prove that if $ G $ is commutative, then for any natural numbers $ n, $ either $ A_n(g) =\emptyset , $ or $ \left| A_n(g) \right| =\left| A_n(1) \right| . $ [b]b)[/b] Provide an example of what $ G $ could be in the case that there exists an element $ g_0 $ of $ G $ and a natural number $ n_0 $ such that $ \left| A_{n_0}\left( g_0 \right) \right| >\left| A_{n_0}(1) \right| . $ [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

1991 Arnold's Trivium, 19

Tags: trigonometry

Investigate the path of a light ray in a plane medium with refractive index $n(y)=y^4-y^2+1$ using Snell's law $n(y)\sin\alpha = \text{const}$, where $\alpha$ is the angle made by the ray with the $y$-axis.

2022 IFYM, Sozopol, 7

Tags: algebra

Find the least possible value of the following expression $\lfloor \frac{a+b}{c+d}\rfloor +\lfloor \frac{a+c}{b+d}\rfloor +\lfloor \frac{a+d}{b+c}\rfloor + \lfloor \frac{c+d}{a+b}\rfloor +\lfloor \frac{b+d}{a+c}\rfloor +\lfloor \frac{b+c}{a+d}\rfloor$ where $a$, $b$, $c$ and $d$ are positive real numbers.

2011 Miklós Schweitzer, 2

Tags: graph theory , coloring

Suppose that the minimum degree δ(G) of a simple graph G with n vertices is at least 3n / 4. Prove that in any 2-coloring of the edges of G , there is a connected subgraph with at least δ(G) +1 points, all edges of which are of the same color.

2017 AMC 12/AHSME, 14

Tags: combinatorics

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? $\textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$

2013 BAMO, 4

Tags: algebra

For a positive integer $n>2$, consider the $n-1$ fractions $$\dfrac21, \dfrac32, \cdots, \dfrac{n}{n-1}$$ The product of these fractions equals $n$, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of $n$ for which this is possible and prove that you have found them all.

2017 Thailand TSTST, 5

Tags: inequalities

Let $a, b, c \in \mathbb{R}^+$ such that $a + b + c = 3$. Prove that $$\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)\geq\frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4.$$

2011 Today's Calculation Of Integral, 726

Tags: calculus , integration , calculus computations , logarithm , geometry

Let $P(x,\ y)\ (x>0,\ y>0)$ be a point on the curve $C: x^2-y^2=1$. If $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$, then find the area bounded by the line $OP$, the $x$ axis and the curve $C$ in terms of $u$.

1998 Bosnia and Herzegovina Team Selection Test, 2

Tags: algebra , inequality , inequalities

For positive real numbers $x$, $y$ and $z$ holds $x^2+y^2+z^2=1$. Prove that $$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} \leq \frac{3\sqrt{3}}{4}$$

2018 Middle European Mathematical Olympiad, 7

Tags: number theory , legendre symbol

Let $a_1,a_2,a_3,\cdots$ be the sequence of positive integers such that $$a_1=1 , a_{k+1}=a^3_k+1, $$ for all positive integers $k.$ Prove that for every prime number $p$ of the form $3l +2,$ where $l$ is a non-negative integer ,there exists a positive integer $n$ such that $a_n$ is divisible by $p.$

2013 Sharygin Geometry Olympiad, 2

Tags: inequalities , geometry

Let $ABCD$ is a tangential quadrilateral such that $AB=CD>BC$. $AC$ meets $BD$ at $L$. Prove that $\widehat{ALB}$ is acute. [hide]According to the jury, they want to propose a more generalized problem is to prove $(AB-CD)^2 < (AD-BC)^2$, but this problem has appeared some time ago[/hide]

2024 Durer Math Competition Finals, 1

Tags: algebra , fraction

Describe all ordered sets of four real numbers $(a, b, c, d)$ for which the values $a + b, b + c, c + d, d + a$ are all non-zero and \[\frac{a+2b+3c}{c+d}=\frac{b+2c+3d}{d+a}=\frac{c+2d+3a}{a+b}=\frac{d+2a+3b}{b+c}.\]

VMEO IV 2015, 11.3

Tags: number theory , divisible , floor function

How many natural number $n$ less than $2015$ that is divisible by $\lfloor\sqrt[3]{n}\rfloor$ ?

2005 Uzbekistan National Olympiad, 4

Tags: geometry , parallelogram , circumcircle , euler , geometric transformation

Let $ABCD$ is a cyclic. $K,L,M,N$ are midpoints of segments $AB$, $BC$ $CD$ and $DA$. $H_{1},H_{2},H_{3},H_{4}$ are orthocenters of $AKN$ $KBL$ $LCM$ and $MND$. Prove that $H_{1}H_{2}H_{3}H_{4}$ is a paralelogram.

2022 German National Olympiad, 4

Tags: diophantine equation , cubic equation , number theory

Determine all $6$-tuples $(x,y,z,u,v,w)$ of integers satisfying the equation \[x^3+7y^3+49z^3=2u^3+14v^3+98w^3.\]

2018 Harvard-MIT Mathematics Tournament, 5

Tags: probability

Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is $75\%$. If Lil Wayne doesn't make it rain on one day, the probability that he makes it rain the next day is $25\%$. He decides not to make it rain today. Find the smallest positive integer $n$ such that the probability that Lil Wayne [i]makes it rain[/i] $n$ days from today is greater than $49.9\%$.

2012 Sharygin Geometry Olympiad, 3

Tags: incenter , perpendicular bisector , angle bisector , geometry

A circle with center $I$ touches sides $AB,BC,CA$ of triangle $ABC$ in points $C_{1},A_{1},B_{1}$. Lines $AI, CI, B_{1}I$ meet $A_{1}C_{1}$ in points $X, Y, Z$ respectively. Prove that $\angle Y B_{1}Z = \angle XB_{1}Z$.

2019 ELMO Shortlist, A4

Tags: algebra , functional equation

Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$ [i]Proposed by Carl Schildkraut[/i]

2002 Iran MO (3rd Round), 1

Tags: inequalities

Let $a,b,c\in\mathbb R^{n}, a+b+c=0$ and $\lambda>0$. Prove that \[\prod_{cycle}\frac{|a|+|b|+(2\lambda+1)|c|}{|a|+|b|+|c|}\geq(2\lambda+3)^{3}\]

2008 May Olympiad, 5

Tags: combinatorics

Matthias covered a $7 \times 7$ square board, divided into $1 \times 1$ squares, with pieces of the following three types without gaps or overlaps, and without going off the board. [img]https://cdn.artofproblemsolving.com/attachments/9/9/8a2e63f723cbdf188f22344054f364f1924d47.gif[/img] Each type $1$ piece covers exactly $3$ squares and each type $2$ or type $3$ piece covers exactly $4$ squares. Determine the number of pieces of type $1$ that Matías could have used. (Pieces can be rotated and flipped.)

1972 AMC 12/AHSME, 16

Tags: geometric sequence

There are two positive numbers that may be inserted between $3$ and $9$ such that the first three are in geometric progression while the last three are in arithmetic progression. The sum of those two positive numbers is $\textbf{(A) }13\textstyle\frac{1}{2}\qquad\textbf{(B) }11\frac{1}{4}\qquad\textbf{(C) }10\frac{1}{2}\qquad\textbf{(D) }10\qquad \textbf{(E) }9\frac{1}{2}$

2023 Thailand TST, 1

Tags: integer sequence , combinatorics

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

2024 Bulgarian Winter Tournament, 9.3

Tags: geometry

Let $ABC$ be a triangle, satisfying $2AC=AB+BC$. If $O$ and $I$ are its circumcenter and incenter, show that $\angle OIB=90^{\circ}$.