Olympiad problems

2008 Germany Team Selection Test, 3

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

2001 Italy TST, 3

Tags: number theory

Find all pairs $ (p, q)$ of prime numbers such that $ p$ divides $ 5^q \plus{} 1$ and $ q$ divides $ 5^p \plus{} 1$.

2024 CMIMC Integration Bee, 11

Tags: calculus , integration

\[\int_1^\infty \frac{\lfloor x^2\rfloor}{x^5}\mathrm dx\] [i]Proposed by Robert Trosten[/i]

2016 AMC 10, 2

Tags: function

If $n\heartsuit m=n^3m^2$, what is $\frac{2\heartsuit 4}{4\heartsuit 2}$? $\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$

2001 JBMO ShortLists, 2

Tags: modular arithmetic , number theory

Let $P_n \ (n=3,4,5,6,7)$ be the set of positive integers $n^k+n^l+n^m$, where $k,l,m$ are positive integers. Find $n$ such that: i) In the set $P_n$ there are infinitely many squares. ii) In the set $P_n$ there are no squares.

1997 Bundeswettbewerb Mathematik, 4

Tags: number theory , perfect square , divisible

Prove that if $n$ is a natural number such that both $3n+1$ and $4n+1$ are squares, then $n$ is divisible by $56$.

1996 Romania National Olympiad, 4

Tags: geometry , locus

In the triangle $ABC$ the incircle $J$ touches the sides $BC$, $CA$, $AB$ in $D$, $E$, $F$, respectively. The segments $(BE)$ and $(CF)$ intersect $J$ in $G,H$. If $B$ and $C$ are fixed points, find the loci of points $A, D, E, F, G, H$ if $GH \parallel BC$ and the loci of the same points if $BCHG$ is an inscriptible quadrilateral.

2006 Bulgaria Team Selection Test, 3

Tags: combinatorics

[b] Problem 6.[/b] Let $m\geq 5$ and $n$ are given natural numbers, and $M$ is regular $2n+1$-gon. Find the number of the convex $m$-gons with vertices among the vertices of $M$, who have at least one acute angle. [i]Alexandar Ivanov[/i]

2015 Regional Competition For Advanced Students, 1

Tags: number theory , gcd

Determine all triples $(a,b,c)$ of positive integers satisfying the conditions $$\gcd(a,20) = b$$ $$\gcd(b,15) = c$$ $$\gcd(a,c) = 5$$ (Richard Henner)

KoMaL A Problems 2023/2024, A. 867

Tags: polynomial , algebra

Let $p(x)$ be a monic integer polynomial of degree $n$ that has $n$ real roots, $\alpha_1,\alpha_2,\ldots, \alpha_n$. Let $q(x)$ be an arbitrary integer polynomial that is relatively prime to polynomial $p(x)$. Prove that \[\sum_{i=1}^n \left|q(\alpha_i)\right|\ge n.\] [i]Submitted by Dávid Matolcsi, Berkeley[/i]

2020 Princeton University Math Competition, B2

Tags: number theory

Last year, the U.S. House of Representatives passed a bill which would make Washington, D.C. into the $51$st state. Naturally, the mathematicians are upset that Congress won’t prioritize mathematical interest of flag design in choosing how many U.S. states there should be. Suppose the U.S. flag must contain, as it does now, stars arranged in rows alternating between $n$ and $n - 1$ stars, starting and ending with rows of n stars, where $n \ge 2$ is some integer and the flag has more than one row. What is the minimum number of states that the U.S. would need to contain so that there are at least three different ways, excluding rotations, to arrange the stars on the flag?

2023 HMNT, 2

Tags: geometry

A regular $n$-gon $P_1P_2...P_n$ satisfies $\angle P_1P_7P_8 = 178^o$. Compute $n$.

2004 USAMTS Problems, 2

Tags: perimeter , geometry

Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle's area is numerically equal to $6$ times its perimeter.

2023 CCA Math Bonanza, I7

Tags:

Of all positive integral solutions $(x,y,z)$ to the equation \[x^3+y^3+z^3-3xyz=607,\] compute the minimum possible value of $x+2y+3z.$ [i]Individual #7[/i]

2003 District Olympiad, 4

Tags: function , real analysis

Let $\alpha>1$ and $f:\left[\frac{1}{\alpha},\alpha\right]\rightarrow \left[\frac{1}{\alpha},\alpha\right]$, a bijective function. If $f^{-1}(x)=\frac{1}{f(x)},\ \forall x\in \left[\frac{1}{\alpha},\alpha\right]$, prove that: a)$f$ has at least one point of discontinuity; b)if $f$ is continuous in $1$, then $f$ has an infinity points of discontinuity; c)there is a function $f$ which satisfies the conditions from the hypothesis and has a finite number of points of dicontinuity. [i]Radu Mortici [/i]

2005 AMC 12/AHSME, 2

Tags:

The equations $ 2x \plus{} 7 \equal{} 3$ and $ bx\minus{}10 \equal{} \minus{}\!2$ have the same solution for $ x$. What is the value of $ b$? $ \textbf{(A)}\minus{}\!8 \qquad \textbf{(B)}\minus{}\!4 \qquad \textbf{(C)}\minus{}\!2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

2023 Indonesia TST, 3

Tags: function , combinatorics

Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.

XMO (China) 2-15 - geometry, 4.1

Tags: perpendicular , perpendicularity , geometry

As shown in the figure, it is known that $BC= AC$ in $\vartriangle ABC$, $M$ is the midpoint of $AB$, points $D$, $E$ lie on $AB$ such that $\angle DCE= \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F $(different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$, $O_2$ respectively. Prove that $O_1O_2 \perp CF$. [img]https://cdn.artofproblemsolving.com/attachments/8/c/62d4ecbc18458fb4f2bf88258d5024cddbc3b0.jpg[/img]

2002 AMC 10, 3

Tags:

According to the standard convention for exponentiation, \[2^{2^{2^2}} \equal{} 2^{\left(2^{\left(2^2\right)}\right)} \equal{} 2^{16} \equal{} 65,\!536.\] If the order in which the exponentiations are performed is changed, how many [u]other[/u] values are possible? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

1976 Euclid, 10

Tags: function , equation

Source: 1976 Euclid Part A Problem 10 ----- If $f$, $g$, $h$, and $k$ are functions and $a$ and $b$ are numbers such that $f(x)=(x-1)g(x)+3=(x+1)h(x)+1=(x^2-1)k(x)+ax+b$ for all $x$, then $(a,b)$ equals $\textbf{(A) } (-2,1) \qquad \textbf{(B) } (-1,2) \qquad \textbf{(C) } (1,1) \qquad \textbf{(D) } (1,2) \qquad \textbf{(E) } (2,1)$

2005 Tournament of Towns, 1

Tags: analytic geometry , polynomial , algebra

On the graph of a polynomial with integral coefficients are two points with integral coordinates. Prove that if the distance between these two points is integral, then the segment connecting them is parallel to the $x$-axis. [i](4 points)[/i]

2018 Oral Moscow Geometry Olympiad, 4

Tags: arc midpoint , construction , geometry

Given a triangle $ABC$ ($AB> AC$) and a circle circumscribed around it. Construct with a compass and a ruler the midpoint of the arc $BC$ (not containing vertex $A$), with no more than two lines (straight or circles).

2013 Macedonia National Olympiad, 1

Tags: modular arithmetic , number theory

Let $ p,q,r $ be prime numbers. Solve the equation $ p^{2q}+q^{2p}=r $

2006 Poland - Second Round, 1

Tags: pigeonhole principle , number theory

Let $c$ be fixed natural number. Sequence $(a_n)$ is defined by: $a_1=1$, $a_{n+1}=d(a_n)+c$ for $n=1,2,...$. where $d(m)$ is number of divisors of $m$. Prove that there exist $k$ natural such that sequence $a_k,a_{k+1},...$ is periodic.

2006 MOP Homework, 1

Tags: geometry , rectangle , geometric transformation , rotation , combinatorics

Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can not use a needle to peer through the tiling? Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can use a needle to through the tiling? What if it is a $6 \times 6$ board?