Olympiad problems

2007 Moldova National Olympiad, 12.7

Find the limit \[\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}\]

VI Soros Olympiad 1999 - 2000 (Russia), 11.1

Tags: algebra , polynomial

The game involves two players $A$ and $B$. Player $A$ sets the value of one of the coefficients $a, b$ or $c$ of the polynomial $$x^3 + ax^2 + bx + c.$$ Player $B$ indicates the value of any of the two remaining coefficients . Player $A$ then sets the value of the last coefficients. Is there a strategy for player A such that no matter how player $B$ plays, the equation $$x^3 + ax^2 + bx + c = 0$$ to have three different (real) solutions?

2011 JHMT, 10

Tags: geometry

Given a triangle $ABC$ with $BC = 5$, $AC = 7$, and $AB = 8$, find the side length of the largest equilateral triangle $P QR$ such that $A, B, C$ lie on $QR$, $RP$, $P Q$ respectively.

2006 Romania National Olympiad, 3

Tags: geometry , 3d geometry

Let $ABCDA_1B_1C_1D_1$ be a cube and $P$ a variable point on the side $[AB]$. The perpendicular plane on $AB$ which passes through $P$ intersects the line $AC'$ in $Q$. Let $M$ and $N$ be the midpoints of the segments $A'P$ and $BQ$ respectively. a) Prove that the lines $MN$ and $BC'$ are perpendicular if and only if $P$ is the midpoint of $AB$. b) Find the minimal value of the angle between the lines $MN$ and $BC'$.

1978 IMO Longlists, 20

Tags: geometric transformation , reflection , geometry

Let $O$ be the center of a circle. Let $OU,OV$ be perpendicular radii of the circle. The chord $PQ$ passes through the midpoint $M$ of $UV$. Let $W$ be a point such that $PM = PW$, where $U, V,M,W$ are collinear. Let $R$ be a point such that $PR = MQ$, where $R$ lies on the line $PW$. Prove that $MR = UV$. [u]Alternative version:[/u] A circle $S$ is given with center $O$ and radius $r$. Let $M$ be a point whose distance from $O$ is $\frac{r}{\sqrt{2}}$. Let $PMQ$ be a chord of $S$. The point $N$ is defined by $\overrightarrow{PN} =\overrightarrow{MQ}$. Let $R$ be the reflection of $N$ by the line through $P$ that is parallel to $OM$. Prove that $MR =\sqrt{2}r$.

2022 Greece Team Selection Test, 2

Tags: geometry , circumcircle

Consider triangle $ABC$ with $AB<AC<BC$, inscribed in triangle $\Gamma_1$ and the circles $\Gamma_2 (B,AC)$ and $\Gamma_2 (C,AB)$. A common point of circle $\Gamma_2$ and $\Gamma_3$ is point $E$, a common point of circle $\Gamma_1$ and $\Gamma_3$ is point $F$ and a common point of circle $\Gamma_1$ and $\Gamma_2$ is point $G$, where the points $E,F,G$ lie on the same semiplane defined by line $BC$, that point $A$ doesn't lie in. Prove that circumcenter of triangle $EFG$ lies on circle $\Gamma_1$. Note: By notation $\Gamma (K,R)$, we mean random circle $\Gamma$ has center $K$ and radius $R$.

2012 Today's Calculation Of Integral, 775

Tags: calculus , integration , calculus computations

Let $a$ be negative constant. Find the value of $a$ and $f(x)$ such that $\int_{\frac{a}{2}}^{\frac{t}{2}} f(x)dx=t^2+3t-4$ holds for any real numbers $t$.

1962 AMC 12/AHSME, 8

Tags:

Given the set of $ n$ numbers; $ n > 1$, of which one is $ 1 \minus{} \frac {1}{n}$ and all the others are $ 1.$ The arithmetic mean of the $ n$ numbers is: $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ n \minus{} \frac {1}{n} \qquad \textbf{(C)}\ n \minus{} \frac {1}{n^2} \qquad \textbf{(D)}\ 1 \minus{} \frac {1}{n^2} \qquad \textbf{(E)}\ 1 \minus{} \frac {1}{n} \minus{} \frac {1}{n^2}$

2020 Latvia Baltic Way TST, 7

Tags: combinatorics

Natural numbers from $1$ to $400$ are divided in $100$ disjoint sets. Prove that one of the sets contains three numbers which are lengths of a non-degenerate triangle's sides.

2024 Ukraine National Mathematical Olympiad, Problem 4

Tags: function , algebra , functional equation

Find all functions $f:\mathbb{R} \to \mathbb{R}$, such that for any $x, y \in \mathbb{R}$ holds the following: $$f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x)$$ [i]Proposed by Mykhailo Shtandenko[/i]

2024 Sharygin Geometry Olympiad, 9.8

Tags: geo , geometry

Let points $P$ and $Q$ be isogonally conjugated with respect to a triangle $ABC$. The line $PQ$ meets the circumcircle of $ABC$ at point $X$. The reflection of $BC$ about $PQ$ meets $AX$ at point $E$. Prove that $A, P, Q, E$ are concyclic.

2023 Turkey EGMO TST, 2

Tags: number theory , prime number

Find all pairs of $p,q$ prime numbers that satisfy the equation $$p(p^4+p^2+10q)=q(q^2+3)$$

1957 AMC 12/AHSME, 28

Tags: logarithm

If $ a$ and $ b$ are positive and $ a\not\equal{} 1,\,b\not\equal{} 1$, then the value of $ b^{\log_b{a}}$ is: $ \textbf{(A)}\ \text{dependent upon }{b} \qquad \textbf{(B)}\ \text{dependent upon }{a}\qquad \textbf{(C)}\ \text{dependent upon }{a}\text{ and }{b}\qquad \textbf{(D)}\ \text{zero}\qquad \textbf{(E)}\ \text{one}$

2018 USAMTS Problems, 5:

Tags:

Acute scalene triangle $\triangle{}ABC$ has circumcenter $O$ and orthocenter $H$. Points $X$ and $Y$, distinct from $B$ and $C$, lie on the circumcircle of $\triangle{}ABC$ such that $\angle{}BXH=\angle{}CYH=90^{\circ{}}$. Show that if lines $XY$, $AH$, and $BC$ are concurrent, then $OH$ is parallel to $BC$.

2021 STEMS CS Cat A, Q2

Tags: combinatorics

Given is an array $A$ of $2n$ numbers, where $n$ is a positive integer. Give an algorithm to create an array $prod$ of length $2n$ where $$prod[i] \, = \, A[i] \times A[i+1] \times \cdots \times A[i+n-1],$$ ($A[x]$ means $A[x \ \text{mod}\ 2n]$) in $O(n)$ time [b]withou[/b]t using division. Assume that all binary arithmetic operations are $O(1)$

II Soros Olympiad 1995 - 96 (Russia), 11.9

Tags: number theory

Let us denote by $b(n)$ the number of ways to represent $n$ in the form $$n = a_0+a_1 \cdot 2 +a_2 \cdot 2^2+...+ a_k \cdot 2^k,$$ where the coefficients at, $r = 1$,$2$,$...$, $k$ can be equal to $0$, $1$ or $2$. Find $b(1996)$.

2012 NZMOC Camp Selection Problems, 4

Tags: number theory , multiple , divide , divisible , prime

A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

2006 All-Russian Olympiad Regional Round, 10.8

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

A convex polyhedron has $2n$ faces ($n\ge 3$), and all faces are triangles. What is the largest number of vertices at which converges exactly $3$ edges at a such a polyhedron ?

2023 HMNT, 6

Tags: number theory

The pairwise greatest common divisors of five positive integers are $$2, 3, 4, 5, 6, 7, 8, p, q, r$$ in some order, for some positive integers $p, q, r$. Compute the minimum possible value of $p + q + r$.

2019 ELMO Shortlist, G2

Tags: geometry

Carl is given three distinct non-parallel lines $\ell_1, \ell_2, \ell_3$ and a circle $\omega$ in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line $\ell$ and a point $P$, constructs a new line passing through $P$ parallel to $\ell$. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle $\omega$ whose sides are parallel to $\ell_1,\ell_2,\ell_3$ in some order. [i]Proposed by Vincent Huang[/i]

2018 MIG, 15

Tags:

Gordon has the least number of coins (half-dollars, quarters, dimes, nickels, pennies) needed to make $99\cent$. He randomly chooses one. What is the probability that it is a penny? $\textbf{(A) } \dfrac15\qquad\textbf{(B) } \dfrac13\qquad\textbf{(C) } \dfrac12\qquad\textbf{(D) } \dfrac23\qquad\textbf{(E) } \dfrac34$

1904 Eotvos Mathematical Competition, 1

Tags: geometry , regular , pentagon

Prove that, if a pentagon (five-sided polygon) inscribed in a circle has equal angles, then its sides are equal.

2011 Bogdan Stan, 4

Tags: vector , algebra , geometry

Show that among any seven coplanar unit vectors there are at least two of them such that the magnitude of their sum is greater than $ \sqrt 3. $ [i]Ion Tecu[/i] and [i]Teodor Radu[/i]

1969 IMO Longlists, 67

Tags: inequality , algebra

Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.

2002 Manhattan Mathematical Olympiad, 4

Tags: geometry , parallelogram

A triangle has sides with lengths $a,b,c$ such that \[ a^2 + b^2 = 5c^2 \] Prove that medians to the sides of lengths $a$ and $b$ are perpendicular.