Olympiad problems

2010 AMC 10, 11

Tags:

A shopper plans to purchase an item that has a listed price greater than $ \$100$ and can use any one of the three coupons. Coupon A gives $ 15\%$ off the listed price, Coupon B gives $ \$30$ the listed price, and Coupon C gives $ 25\%$ off the amount by which the listed price exceeds $ \$100$. Let $ x$ and $ y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $ y\minus{}x$? $ \textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 80\qquad\textbf{(E)}\ 100$

2018 Iran MO (1st Round), 6

Let $n$ be the smallest positive integer such that the remainder of $3n+45$, when divided by $1060$, is $16$. Find the remainder of $18n+17$ upon division by $1920$.

2005 Putnam, B6

Tags: integration , linear algebra , matrix , algebra , polynomial , calculus

Let $S_n$ denote the set of all permutations of the numbers $1,2,\dots,n.$ For $\pi\in S_n,$ let $\sigma(\pi)=1$ if $\pi$ is an even permutation and $\sigma(\pi)=-1$ if $\pi$ is an odd permutation. Also, let $v(\pi)$ denote the number of fixed points of $\pi.$ Show that \[ \sum_{\pi\in S_n}\frac{\sigma(\pi)}{v(\pi)+1}=(-1)^{n+1}\frac{n}{n+1}. \]

Revenge EL(S)MO 2024, 7

Tags: geometry , incenter , feuerbach point , nagel point , constructive geometry

A scalene triangle $ABC$ was drawn, and Elmo marked its incenter $I$, Feuerbach point $X$, and Nagel point $N$. Sadly, after taking the abcdEfghijkLMnOpqrstuvwxyz, Elmo lost the triangle $ABC$. Can Elmo use only a ruler and compass to reconstruct the triangle? Proposed by [i]Karn Chutinan[/i]

2017 India IMO Training Camp, 1

Tags: algebra , polynomial

Let $P_c(x)=x^4+ax^3+bx^2+cx+1$ and $Q_c(x)=x^4+cx^3+bx^2+ax+1$ with $a,b$ real numbers, $c \in \{1,2, \dots, 2017\}$ an integer and $a \ne c$. Define $A_c=\{\alpha | P_c(\alpha)=0\}$ and $B_c=\{\beta | P(\beta)=0\}$. (a) Find the number of unordered pairs of polynomials $P_c(x), Q_c(x)$ with exactly two common roots. (b) For any $1 \le c \le 2017$, find the sum of the elements of $A_c \Delta B_c$.

1966 IMO Shortlist, 20

Tags: geometry , rectangle , 3d geometry , icosahedron

Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles. [b]a.)[/b] What is the volume of this polyhedron ? [b]b.)[/b] Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?

2014 IMS, 8

Tags: trigonometry , real analysis

Is $\sum_{n=1}^{+\infty}\frac{\cos n}{n}(1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}})$ convergent? why?

2006 District Olympiad, 4

Tags: inequalities , integration , function , real analysis

Let $\mathcal F = \{ f: [0,1] \to [0,\infty) \mid f$ continuous $\}$ and $n$ an integer, $n\geq 2$. Find the smallest real constant $c$ such that for any $f\in \mathcal F$ the following inequality takes place \[ \int^1_0 f \left( \sqrt [n] x \right) dx \leq c \int^1_0 f(x) dx. \]

1993 Tournament Of Towns, (378) 7

Tags: combinatorics

In a handbook of plants each plant is characterized by $100$ attributes (each attribute may either be present in a plant or not). Two plants are called [i]dissimilar [/i] if they differ by no less than $51$ attributes. (a) Prove that the handbook cannot describe more than $50$ pair-wise dissimilar plants. (b) Can it describe $50$ pairwise dissimilar plants? (Dima Tereshin)

2004 AMC 10, 15

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Patty has $ 20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $ 70$ cents more. How much are her coins worth? $ \textbf{(A)}\ \$1.15\qquad \textbf{(B)}\ \$1.20\qquad \textbf{(C)}\ \$1.25\qquad \textbf{(D)}\ \$1.30\qquad \textbf{(E)}\ \$1.35$

2001 Greece JBMO TST, 2

Tags: geometry , area

Let $ABCD$ be a quadrilateral with $\angle DAB=60^o$, $\angle ABC=60^o$ and $\angle BCD=120^o$. Diagonals $AC$, $BD$ intersect at point $M$ and $BM=a, MD=2a$. Let $O$ be the midpoint of side $AC$ and draw $OH \perp BD, H \in BD$ and $MN\perp OB, N \in OB$. Prove that i) $HM=MN=\frac{a}{2}$ ii) $AD=DC$ iii) $S_{ABCD}=\frac{9a^2}{2}$

1980 Miklós Schweitzer, 5

Tags: superior algebra , group theory , abstract algebra

Let $ G$ be a transitive subgroup of the symmetric group $ S_{25}$ different from $ S_{25}$ and $ A_{25}$. Prove that the order of $ G$ is not divisible by $ 23$. [i]J. Pelikan[/i]

2020 IMO Shortlist, G6

Tags: excenter , geometry , incenter , computer problems , tangent circle

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other

Russian TST 2019, P3

Tags: number theory , diophantine

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

2023 Bangladesh Mathematical Olympiad, P3

Tags: number theory , factorization , diophantine equation

Solve the equation for the positive integers: $$(x+2y)^2+2x+5y+9=(y+z)^2$$

2017 IMC, 1

Tags: linear algebra , matrix

Determine all complex numbers $\lambda$ for which there exists a positive integer $n$ and a real $n\times n$ matrix $A$ such that $A^2=A^T$ and $\lambda$ is an eigenvalue of $A$.

2019 Saudi Arabia JBMO TST, 2

Tags: inequalities , algebra

Let $a, b, c$ be non-negative real numbers. Prove that $$a\sqrt{3a^2+6b^2}+b\sqrt{3b^2+6c^2}+c\sqrt{3c^2+6a^2}=>(a+b+c)^2$$

1975 Bulgaria National Olympiad, Problem 4

Tags: inequalities , circles , geometry , geometrical inequalities , geometric inequalities

In the plane are given a circle $k$ with radii $R$ and the points $A_1,A_2,\ldots,A_n$, lying on $k$ or outside $k$. Prove that there exist infinitely many points $X$ from the given circumference for which $$\sum_{i=1}^n A_iX^2\ge2nR^2.$$ Does there exist a pair of points on different sides of some diameter, $X$ and $Y$ from $k$, such that $$\sum_{i=1}^n A_iX^2\ge2nR^2\text{ and }\sum_{i=1}^n A_iY^2\ge2nR^2?$$ [i]H. Lesov[/i]

2021 German National Olympiad, 6

Tags: arithmetic progression , perfect square , number theory

Determine whether there are infinitely many triples $(u,v,w)$ of positive integers such that $u,v,w$ form an arithmetic progression and the numbers $uv+1, vw+1$ and $wu+1$ are all perfect squares.

2024 May Olympiad, 2

Tags: number theory

A number is [i]special[/i] if its tens digit is $9$. For example, $499$ and $1092$ are special, but $509$ is not. Diego has several cards. On each of them, he wrote a special number (he may write the same number on more than one card). When he adds up the numbers on the cards, the total is $2024$. What is the smallest number of cards Diego can have?

2008 Princeton University Math Competition, 3

Tags: number theory

What is the smallest positive integer value of $x$ for which $x \equiv 4$ (mod $9$) and $x \equiv 7$ (mod $8$)?

2015 Argentina National Olympiad Level 2, 5

Tags: number theory

Find all positive integers $n$ that can be represented in the form $$n=\mathrm{lcm}(a,b)+\mathrm{lcm}(b,c)+\mathrm{lcm}(c,a)$$ where $a,b,c$ are positive integers.

III Soros Olympiad 1996 - 97 (Russia), 9.7

Tags: system of equations , algebra

Solve the system of equations: $$\begin{cases} xy+zu=14 \\ xz+yu=11 \\ xu+yz=10 \\ x+y+z+u=10 \end{cases}$$

1994 Baltic Way, 16

Tags: geometry , circumcircle , inradius , search , combinatorics

The Wonder Island is inhabited by Hedgehogs. Each Hedgehog consists of three segments of unit length having a common endpoint, with all three angles between them $120^{\circ}$. Given that all Hedgehogs are lying flat on the island and no two of them touch each other, prove that there is a finite number of Hedgehogs on Wonder Island.

2014 Singapore Senior Math Olympiad, 16

Tags: factorial

Evaluate the sum $\frac{3!+4!}{2(1!+2!)}+\frac{4!+5!}{3(2!+3!)}+\cdots+\frac{12!+13!}{11(10!+11!)}$