Olympiad problems

2019 CCA Math Bonanza, L1.4

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What is the smallest prime number $p$ such that $1+p+p^2+\ldots+p^{p-1}$ is [i]not[/i] prime? [i]2019 CCA Math Bonanza Lightning Round #1.4[/i]

1953 AMC 12/AHSME, 37

Tags: geometry , circumcircle , trigonometry , perpendicular bisector , trig identities , law of sines

The base of an isosceles triangle is $ 6$ inches and one of the equal sides is $ 12$ inches. The radius of the circle through the vertices of the triangle is: $ \textbf{(A)}\ \frac{7\sqrt{15}}{5} \qquad\textbf{(B)}\ 4\sqrt{3} \qquad\textbf{(C)}\ 3\sqrt{5} \qquad\textbf{(D)}\ 6\sqrt{3} \qquad\textbf{(E)}\ \text{none of these}$

1979 IMO Longlists, 41

Tags: 3d geometry , pyramid , geometry

Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.

2016 Romania Team Selection Tests, 1

Tags: geometry

Two circles, $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meet $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1,O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.

2017 Korea Junior Math Olympiad, 6

Tags: geometry , circumcircle

Let triangle $ABC$ be an acute scalene triangle, and denote $D,E,F$ by the midpoints of $BC,CA,AB$, respectively. Let the circumcircle of $DEF$ be $O_1$, and its center be $N$. Let the circumcircle of $BCN$ be $O_2$. $O_1$ and $O_2$ meet at two points $P, Q$. $O_2$ meets $AB$ at point $K(\neq B)$ and meets $AC$ at point $L(\neq C)$. Show that the three lines $EF,PQ,KL$ are concurrent.

2017 BMT Spring, 5

Tags: combinatorics

You enter an elevator on floor $0$ of a building with some other people, and request to go to floor $10$. In order to be efficient, it doesn’t stop at adjacent floors (so, if it’s at floor $0$, its next stop cannot be floor $ 1$). Given that the elevator will stop at floor $10$, no matter what other floors it stops at, how many combinations of stops are there for the elevator?

2017-2018 SDPC, 5

Tags: algebra

Given positive real numbers $a,b,c$ such that $abc=1$, find the maximum possible value of $$\frac{1}{(4a+4b+c)^3}+\frac{1}{(4b+4c+a)^3}+\frac{1}{(4c+4a+b)^3}.$$

1996 All-Russian Olympiad Regional Round, 10.3

Tags: geometry , trapezoid

Given an angle with vertex $B$. Construct point $M$ as follows. Let us take an arbitrary isosceles trapezoid whose sides lie on the sides of a given angle. Through two opposite ones draw tangents to the vertices of the circle circumscribed around it. Let $M$ denote the point of intersection of these tangents. What figure do all such points $M$ form?

2004 Romania National Olympiad, 3

Tags: function , inequalities , calculus , algebra , logarithm

Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$. Prove that: \[ 2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} . \] [i]Radu Gologan[/i]

2020 MBMT, 27

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The perfect square game is played as follows: player 1 says a positive integer, then player 2 says a strictly smaller positive integer, and so on. The game ends when someone says 1; that player wins if and only if the sum of all numbers said is a perfect square. What is the sum of all $n$ such that, if player 1 starts by saying $n$, player 1 has a winning strategy? A winning strategy for player 1 is a rule player 1 can follow to win, regardless of what player 2 does. If player 1 wins, player 2 must lose, and vice versa. Both players play optimally. [i]Proposed by Jacob Stavrianos[/i]

2022 Caucasus Mathematical Olympiad, 5

Tags: combinatorics

Let $S$ be the set of all $5^6$ positive integers, whose decimal representation consists of exactly 6 odd digits. Find the number of solutions $(x,y,z)$ of the equation $x+y=10z$, where $x\in S$, $y\in S$, $z\in S$.

2008 Paraguay Mathematical Olympiad, 2

Tags: parabola , absolute value , conic

Find for which values of $n$, an integer larger than $1$ but smaller than $100$, the following expression has its minimum value: $S = |n-1| + |n-2| + \ldots + |n-100|$

1997 Polish MO Finals, 3

Tags: combinatorics

Given any $n$ points on a unit circle show that at most $\frac{n^2}{3}$ of the segments joining two points have length $> \sqrt{2}$.

2024 Harvard-MIT Mathematics Tournament, 20

Tags: guts

Compute $\sqrt[4]{5508^3+5625^3+5742^3},$ given that it is an integer.

PEN L Problems, 13

Tags: induction , algebra , polynomial , quadratic , difference of squares , special factorizations , linear recurrence

The sequence $\{x_{n}\}_{n \ge 1}$ is defined by \[x_{1}=x_{2}=1, \; x_{n+2}= 14x_{n+1}-x_{n}-4.\] Prove that $x_{n}$ is always a perfect square.

2014 JBMO Shortlist, 2

Tags: diophantine equation , modular arithmetic

Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

2014 Kosovo National Mathematical Olympiad, 1

Tags: algebra

Let $a$ and $b$ be the solutions to $x^2-x+q=0$, find $a^3+b^3+3(a^3b+ab^3)+6(a^3b^2+a^2b^3)$.

2015 Chile TST Ibero, 3

Tags: geometry

Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.

2007 Bulgarian Autumn Math Competition, Problem 8.2

Tags: geometry , locus of points , equal area

Let $ABCD$ be a convex quadrilateral. Determine all points $M$, which lie inside $ABCD$, such that the areas of $ABCM$ and $AMCD$ are equal.

II Soros Olympiad 1995 - 96 (Russia), 11.2

Tags: geometry , 3d geometry , tetrahedron

Is it possible that the heights of a tetrahedron (that is, a triangular pyramid) would be equal to the numbers $1$, $2$, $3$ and $6$?

2016 IMO Shortlist, A4

Tags: algebra , functional equation

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2023 Iran Team Selection Test, 5

Tags: algebra , polynomial , function

Find all injective $f:\mathbb{Z}\ge0 \to \mathbb{Z}\ge0 $ that for every natural number $n$ and real numbers $a_0,a_1,...,a_n$ (not everyone equal to $0$), polynomial $\sum_{i=0}^{n}{a_i x^i}$ have real root if and only if $\sum_{i=0}^{n}{a_i x^{f(i)}}$ have real root. [i]Proposed by Hesam Rajabzadeh [/i]

2012 Today's Calculation Of Integral, 840

Tags: calculus , integration , function , trigonometry , algebra , domain , inequalities

Let $x,\ y$ be real numbers. For a function $f(t)=x\sin t+y\cos t$, draw the domain of the points $(x,\ y)$ for which the following inequality holds. \[\left|\int_{-\pi}^{\pi} f(t)\cos t\ dt\right|\leq \int_{-\pi}^{\pi} \{f(t)\}^2dt.\]

2009 USAMTS Problems, 2

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The ordered pair of four-digit numbers $(2025, 3136)$ has the property that each number in the pair is a perfect square and each digit of the second number is $1$ more than the corresponding digit of the first number. Find, with proof, all ordered pairs of five-digit numbers and ordered pairs of six-digit numbers with the same property: each number in the pair is a perfect square and each digit of the second number is $1$ more than the corresponding digit of the first number.

2014 Contests, 1

Tags: algebra

A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$ 1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ . 2)for each $n$, prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$ 3)Describe the possible values of $a_{1435}$ 4)Prove that the values that you got in (3) are correct