Olympiad problems

1995 All-Russian Olympiad Regional Round, 11.6

Tags: induction , algebra

The sequence $ a_n$ satisfies $ a_{m\plus{}n}\plus{} a_{m\minus{}n}\equal{}\frac12(a_{2m}\plus{}a_{2n})$ for all $ m\geq n\geq 0$. If $ a_1\equal{}1$, find $ a_{1995}$.

1969 Poland - Second Round, 5

Tags: geometry , projection

Prove that if, in parallel projection of one plane onto another plane, the image of a certain square is a square, then the image of every figure is the figure congruent to it.

MathLinks Contest 5th, 3.3

Tags: inequalities , algebra

Let $x_1, x_2,... x_n$ be positive numbers such that $S = x_1+x_2+...+x_n =\frac{1}{x_1}+...+\frac{1}{x_n}$ Prove that $$\sum_{i=1}^{n} \frac{1}{n - 1 + x_i} \ge \sum_{i=1}^{n} \frac{1}{1+S - x_i}$$

2021 Harvard-MIT Mathematics Tournament., 2

Tags: combi

Ava and Tiffany participate in a knockout tournament consisting of a total of $32$ players. In each of $5$ rounds, the remaining players are paired uniformly at random. In each pair, both players are equally likely to win, and the loser is knocked out of the tournament. The probability that Ava and Tiffany play each other during the tournament is $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Compute $100a + b.$

2022 JHMT HS, 1

Tags: geometry

The side lengths of an equiangular octagon alternate between $20$ and $22$. Find its area.

1992 IMO Longlists, 48

Tags: function , algebra , functional equation

Find all the functions $f : \mathbb R^+ \to \mathbb R$ satisfying the identity \[f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \qquad \forall x,y \in \mathbb R^+\] Where $\alpha,\beta$ are given real numbers.

2013 Portugal MO, 3

Tags: combinatorics

In the Republic of Unistan there are $n$ national roads, each of them links two cities exactly. You can travel from one city to another of your choice using a sequence of roads. The President of Unistan ordered to label the national roads with the integers from $1$ to $n$ by an old law: if a city is adjacent to more than one road, the greatest common divisor of the numbers of that roads must be one. Show that you can label the national roads without breaking the law.

2008 Hanoi Open Mathematics Competitions, 8

Tags: geometry , rhombus , diagonal

The sides of a rhombus have length $a$ and the area is $S$. What is the length of the shorter diagonal?

2011 China Team Selection Test, 2

Tags: ceiling function , floor function , number theory

Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.

2012 Czech And Slovak Olympiad IIIA, 6

Tags: system of equations , algebra

In the set of real numbers solve the system of equations $x^4+y^2+4=5yz$ $y^4+z^2+4=5zx$ $z^4+x^2+4=5xy$

1964 AMC 12/AHSME, 9

Tags:

A jobber buys an article at $\$24$ less $12\frac{1}{2}\%$. He then wishes to sell the article at a gain of $33\frac{1}{3}\%$ of his cost after allowing a $20\%$ discount on his marked price. At what price, in dollars, should the article be marked? ${{ \textbf{(A)}\ 25.20 \qquad\textbf{(B)}\ 30.00 \qquad\textbf{(C)}\ 33.60 \qquad\textbf{(D)}\ 40.00 }\qquad\textbf{(E)}\ \text{none of these} } $

2015 Regional Olympiad of Mexico Southeast, 5

Tags: geometry , angle bisector , ratio

In the triangle $ABC$, let $AM$ and $CN$ internal bisectors, with $M$ in $BC$ and $N$ in $AB$. Prove that if $$\frac{\angle BNM}{\angle MNC}=\frac{\angle BMN}{\angle NMA}$$ then $ABC$ is isosceles.

2010 May Olympiad, 1

Tags: number theory , multiple

Determine the smallest positive integer that has all its digits equal to $4$, and is a multiple of $169$.

2011 NZMOC Camp Selection Problems, 4

Tags: diophantine , diophantine equation , number theory

Find all pairs of positive integers $m$ and $n$ such that $$(m + 1)! + (n + 1)! = m^2n.$$

1971 IMO Longlists, 50

Tags: geometry , polyhedron , 3d geometry , combinatorial geometry , volume

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

2013 Indonesia MO, 7

Tags: geometry

Let $ABCD$ be a parallelogram. Construct squares $ABC_1D_1, BCD_2A_2, CDA_3B_3, DAB_4C_4$ on the outer side of the parallelogram. Construct a square having $B_4D_1$ as one of its sides and it is on the outer side of $AB_4D_1$ and call its center $O_A$. Similarly do it for $C_1A_2, D_2B_3, A_3C_4$ to obtain $O_B, O_C, O_D$. Prove that $AO_A = BO_B = CO_C = DO_D$.

2004 Junior Balkan Team Selection Tests - Moldova, 3

Tags: geometry , angle bisector , cyclic , parallelogram

Let $ABCD$ be a parallelogram and point $M$ be the midpoint of $[AB]$ so that the quadrilateral $MBCD$ is cyclic. If $N$ is the point of intersection of the lines $DM$ and $BC$, and $P \in BC$, then prove that the ray $(DP$ is the angle bisector of $\angle ADM$ if and only if $PC = 4BC$.

2011 BAMO, 5

Tags: pascal triangle , combinatorics , sum

Does there exist a row of Pascal’s Triangle containing four distinct values $a,b,c$ and $d$ such that $b = 2a$ and $d = 2c$? Recall that Pascal’s triangle is the pattern of numbers that begins as follows [img]https://cdn.artofproblemsolving.com/attachments/2/1/050e56f0f1f1b2a9c78481f03acd65de50c45b.png[/img] where the elements of each row are the sums of pairs of adjacent elements of the prior row. For example, $10 =4+6$. Also note that the last row displayed above contains the four elements $a = 5,b = 10,d = 10,c = 5$, satisfying $b = 2a$ and $d = 2c$, but these four values are NOT distinct.

2021 Latvia Baltic Way TST, P11

Tags: geometry , incenter , perpendicular bisector

Incircle of $\triangle ABC$ has centre $I$ and touches sides $AC, AB$ at $E,F$, respectively. The perpendicular bisector of segment $AI$ intersects side $AC$ at $P$. On side $AB$ a point $Q$ is chosen so that $QI \perp FP$. Prove that $EQ \perp AB$.

2001 Singapore Team Selection Test, 1

Tags: number theory , perfect square

Let $a, b, c, d$ be four positive integers such that each of them is a difference of two squares of positive integers. Prove that $abcd$ is also a difference of two squares of positive integers.

2024 Nigerian MO Round 3, Problem 2

Tags: number theory

Prove that there exist infinitely many distinct positive integers, $x$ and $y$, such that $$x^3+y^2|x^2+y^3$$

2024 Yasinsky Geometry Olympiad, 4

Tags: geometry , construction , incenter , centroid , parallel

Let $ I $ and $ M $ be the incenter and the centroid of a scalene triangle $ ABC $, respectively. A line passing through point $ I $ parallel to $ BC $ intersects $ AC $ and $ AB $ at points $ E $ and $ F $, respectively. Reconstruct triangle $ ABC $ given only the marked points $ E, F, I, $ and $ M $. [i]Proposed by Hryhorii Filippovskyi[/i]

2013 BMT Spring, 10

Tags: combinatorics

Let $\sigma_n$ be a permutation of $\{1,\ldots,n\}$; that is, $\sigma_n(i)$ is a bijective function from $\{1,\ldots,n\}$ to itself. Define $f(\sigma)$ to be the number of times we need to apply $\sigma$ to the identity in order to get the identity back. For example, $f$ of the identity is just $1$, and all other permutations have $f(\sigma)>1$. What is the smallest $n$ such that there exists a $\sigma_n$ with $f(\sigma_n)=k$?

2010 CHKMO, 3

Tags: circumcircle , function , geometry

Let $ \triangle ABC$ be a right-angled triangle with $ \angle C\equal{}90^\circ$. $ CD$ is the altitude from $ C$ to $ AB$, with $ D$ on $ AB$. $ \omega$ is the circumcircle of $ \triangle BCD$. $ \omega_1$ is a circle situated in $ \triangle ACD$, it is tangent to the segments $ AD$ and $ AC$ at $ M$ and $ N$ respectively, and is also tangent to circle $ \omega$. (i) Show that $ BD\cdot CN\plus{}BC\cdot DM\equal{}CD\cdot BM$. (ii) Show that $ BM\equal{}BC$.

2018 Dutch BxMO TST, 2

Tags: geometry , integer , sides of a triangle , coprime

Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.