Olympiad problems

Kvant 2019, M2580

We are given a convex four-sided pyramid with apex $S$ and base face $ABCD$ such that the pyramid has an inscribed sphere (i.e., it contains a sphere which is tangent to each race). By making cuts along the edges $SA,SB,SC,SD$ and rotating the faces $SAB,SBC,SCD,SDA$ outwards into the plane $ABCD$, we unfold the pyramid into the polygon $AKBLCMDN$ as shown in the figure. Prove that $K,L,M,N$ are concyclic. [i] Tibor Bakos and Géza Kós [/i]

1992 French Mathematical Olympiad, Problem 4

Tags: sequence , recurrence relation , algebra

Given $u_0,u_1$ with $0<u_0,u_1<1$, define the sequence $(u_n)$ recurrently by the formula $$u_{n+2}=\frac12\left(\sqrt{u_{n+1}}+\sqrt{u_n}\right).$$(a) Prove that the sequence $u_n$ is convergent and find its limit. (b) Prove that, starting from some index $n_0$, the sequence $u_n$ is monotonous.

1979 All Soviet Union Mathematical Olympiad, 269

Tags: area , inscribed , geometry , isosceles

What is the least possible ratio of two isosceles triangles areas, if three vertices of the first one belong to three different sides of the second one?

PEN S Problems, 11

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For each positive integer $n$, prove that there are two consecutive positive integers each of which is the product of $n$ positive integers greater than $1$.

1998 IberoAmerican Olympiad For University Students, 3

Tags: number theory

The positive divisors of a positive integer $n$ are written in increasing order starting with 1. \[1=d_1<d_2<d_3<\cdots<n\] Find $n$ if it is known that: [b]i[/b]. $\, n=d_{13}+d_{14}+d_{15}$ [b]ii[/b]. $\,(d_5+1)^3=d_{15}+1$

2000 Federal Competition For Advanced Students, Part 2, 2

Tags: number theory

Find all pairs of integers $(m, n)$ such that \[ \left| (m^2 + 2000m+ 999999)- (3n^3 + 9n^2 + 27n) \right|= 1.\]

Kyiv City MO Juniors 2003+ geometry, 2021.8.4

Tags: angle , equal angles , geometry

Let $BM$ be the median of the triangle $ABC$, in which $AB> BC$. Point $P$ is chosen so that $AB \parallel PC$ and $PM \perp BM$. Prove that $\angle ABM = \angle MBP$. (Mikhail Standenko)

1964 AMC 12/AHSME, 27

Tags: inequalities

If $x$ is a real number and $|x-4|+|x-3|<a$ where $a>0$, then: $ \textbf{(A)}\ 0<a<.01\qquad\textbf{(B)}\ .01<a<1 \qquad\textbf{(C)}\ 0<a<1\qquad$ $\textbf{(D)}\ 0<a \le 1\qquad\textbf{(E)}\ a>1 $

2024 IFYM, Sozopol, 6

Tags: polynomial , number theory

Let $P(x)$ be a polynomial in one variable with integer coefficients. Prove that the number of pairs $(m,n)$ of positive integers such that $2^n + P(n) = m!$, is finite.

2021 USAMO, 1

Tags: geometry

Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that \[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\] Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.

1969 AMC 12/AHSME, 9

Tags: arithmetic sequence

The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning with $2$ is: $\textbf{(A) }27\qquad \textbf{(B) }27\tfrac14\qquad \textbf{(C) }27\tfrac12\qquad \textbf{(D) }28\qquad \textbf{(E) }28\tfrac12$

1998 Tournament Of Towns, 4

Tags: combinatorics

Twelve candidates for mayor participate in a TV talk show. At some point a candidate said: "One lie has been told." Another said: "Now two lies have been told". "Now three lies," said a third. This continued until the twelfth said: "Now twelve lies have been told". At this point the moderator ended the discussion. It turned out that at least one of the candidates correctly stated the number of lies told before he made the claim. How many lies were actually told by the candidates?

1948 Moscow Mathematical Olympiad, 145

Tags: inequalities , logarithm , algebra

Without tables and such, prove that $\frac{1}{\log_2 \pi}+\frac{1}{\log_5 \pi} >2$

1996 All-Russian Olympiad, 7

Tags: inequalities , polynomial , algebra

Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coeficients in $M$, all of whose roots are real and belong to $M$? [i]E. Malinnikova[/i]

1975 Chisinau City MO, 94

Tags: locus , geometry

A straight line $\ell$ and a point $A$ outside of it are given on the plane. Find the locus of the vertices $C$ of the equilateral triangle $ABC$, the vertex $B$ of which lies on the straight line $\ell$.

2008 China Girls Math Olympiad, 2

Tags: polynomial , inequalities , algebra

Let $ \varphi(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$ be a polynomial with real coefficients. Given that $ \varphi(x)$ has three positive real roots and that $ \varphi(0) < 0$, prove that \[ 2b^3 \plus{} 9a^2d \minus{} 7abc \leq 0. \]

2008 IMO Shortlist, 3

Tags: point set , bezout s identity , pigeonhole principle , extremal combinatorics , combinatorics , geometry

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

2016 Switzerland - Final Round, 9

Tags: lcm , combinatorics , number theory

Let $n \ge 2$ be a natural number. For an $n$-element subset $F$ of $\{1, . . . , 2n\}$ we define $m(F)$ as the minimum of all $lcm \,\, (x, y)$ , where $x$ and $y$ are two distinct elements of $F$. Find the maximum value of $m(F)$.

2000 South africa National Olympiad, 2

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Solve for $x$, given $36x^4 + 36x^3 - 7x^2 - 6x + 1 = 0$.

2011 Saudi Arabia Pre-TST, 4

Tags: fixed , geometry , square

Points $A ,B ,C ,D$ lie on a line in this order. Draw parallel lines $a$ and $b$ through $A$ and $B$, respectively, and parallel lines $c$ and $d$ through $C$ and $D$, respectively, such that their points of intersection are vertices of a square. Prove that the side length of this square does not depend on the length of segment $BC$.

1991 IMTS, 1

Tags: prime number , number theory

Use each of the digits 1,2,3,4,5,6,7,8,9 exactly twice to form distinct prime numbers whose sum is as small as possible. What must this minimal sum be? (Note: The five smallest primes are 2,3,5,7, and 11)

2007 Princeton University Math Competition, 2

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Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits.

2006 Iran MO (3rd Round), 4

Tags: polynomial , algebra

$p(x)$ is a real polynomial that for each $x\geq 0$, $p(x)\geq 0$. Prove that there are real polynomials $A(x),B(x)$ that $p(x)=A(x)^{2}+xB(x)^{2}$

2022 IFYM, Sozopol, 2

Tags: algebra

Does there exist a solution in integers for the equation $a^2+b^2+c^2+d^2+e^2=abcde-78$ where $a,b,c,d,e>2022$?

1991 Cono Sur Olympiad, 3

Tags: algebra , number theory , prime number , inequalities

Given a positive integrer number $n$ ($n\ne 0$), let $f(n)$ be the average of all the positive divisors of $n$. For example, $f(3)=\frac{1+3}{2}=2$, and $f(12)=\frac{1+2+3+4+6+12}{6}=\frac{14}{3}$. [b]a[/b] Prove that $\frac{n+1}{2} \ge f(n)\ge \sqrt{n}$. [b]b[/b] Find all $n$ such that $f(n)=\frac{91}{9}$.