Olympiad problems

2018 Stanford Mathematics Tournament, 3

Tags: geometry

A triangle has side lengths of $7$, $8$, and $9$. Find the radius of the largest possible semicircle inscribed in the triangle.

2011 Federal Competition For Advanced Students, Part 1, 2

Tags: inequalities

For a positive integer $k$ and real numbers $x$ and $y$, let \[f_k(x,y)=(x+y)-\left(x^{2k+1}+y^{2k+1}\right)\mbox{.}\] If $x^2+y^2=1$, then determine the maximal possible value $c_k$ of $f_k(x,y)$.

2002 Tournament Of Towns, 5

Tags: floor function , number theory

An infinite sequence of natural number $\{x_n\}_{n\ge 1}$ is such that $x_{n+1}$ is obtained by adding one of the non-zero digits of $x_n$ to itself. Show this sequence contains an even number.

1989 IMO Longlists, 39

Tags: algebra , function , polynomial , calculus , integration , combinatorics

Alice has two urns. Each urn contains four balls and on each ball a natural number is written. She draws one ball from each urn at random, notes the sum of the numbers written on them, and replaces the balls in the urns from which she took them. This she repeats a large number of times. Bill, on examining the numbers recorded, notices that the frequency with which each sum occurs is the same as if it were the sum of two natural numbers drawn at random from the range 1 to 4. What can he deduce about the numbers on the balls?

2022 Saudi Arabia BMO + EGMO TST, 2.1

Tags: geometry , circumcenter

Let $ABC$ be an acute-angled triangle. Point $P$ is such that $AP = AB$ and $PB \parallel AC$. Point $Q$ is such that $AQ = AC$ and $CQ \parallel AB$. Segments $CP$ and $BQ$ meet at point $X$. Prove that the circumcenter of triangle $ABC$ lies on the circumcircle of triangle $PXQ$.

2019 Thailand TST, 3

Tags: tournament graphs , combinatorics

Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.

1954 AMC 12/AHSME, 12

Tags: analytic geometry , graphing lines , slope

The solution of the equations \begin{align*} 2x-3y&=7 \\ 4x-6y &=20 \\ \end{align*} is: $ \textbf{(A)}\ x=18, y=12 \qquad \textbf{(B)}\ x=0, y=0 \qquad \textbf{(C)}\ \text{There is no solution} \\ \textbf{(D)}\ \text{There are an unlimited number of solutions} \qquad \textbf{(E)}\ x=8, y=5$

2003 China Team Selection Test, 3

Tags: inequalities , geometry , trigonometry , ratio

(1) $D$ is an arbitary point in $\triangle{ABC}$. Prove that: \[ \frac{BC}{\min{AD,BD,CD}} \geq \{ \begin{array}{c} \displaystyle 2\sin{A}, \ \angle{A}< 90^o \\ \\ 2, \ \angle{A} \geq 90^o \end{array} \] (2)$E$ is an arbitary point in convex quadrilateral $ABCD$. Denote $k$ the ratio of the largest and least distances of any two points among $A$, $B$, $C$, $D$, $E$. Prove that $k \geq 2\sin{70^o}$. Can equality be achieved?

2014 IFYM, Sozopol, 4

Tags: algebra , polynomial

Find all polynomials $P,Q\in \mathbb{R}[x]$, such that $P(2)=2$ , $Q(x)$ has no negative roots, and $(x-2)P(x^2-1)Q(x+1)=P(x)Q(x^2 )+Q(x+1)$.

1978 AMC 12/AHSME, 27

Tags: number theory , least common multiple

There is more than one integer greater than $1$ which, when divided by any integer $k$ such that $2 \le k \le 11$, has a remainder of $1$. What is the difference between the two smallest such integers? $\textbf{(A) }2310\qquad\textbf{(B) }2311\qquad\textbf{(C) }27,720\qquad\textbf{(D) }27,721\qquad \textbf{(E) }\text{none of these}$

2024 Princeton University Math Competition, A6 / B8

Tags: number theory

Let Pascal’s triangle be constructed where each $\tbinom{n}{i}$ is written inside its own cell in row $n.$ Colby colors the cells red for $1 \le n \le 63$ when $\tbinom{n}{i}$ is divisible by $4.$ How many cells does he color red?

2022 Princeton University Math Competition, 7

Tags: algebra

Pick $x, y, z$ to be real numbers satisfying $(-x+y+z)^2-\frac13 = 4(y-z)^2$, $(x-y+z)^2-\frac14 = 4(z-x)2$, and $(x+y-z)^2 -\frac15 = 4(x-y)^2$. If the value of $xy+yz +zx$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.

1989 USAMO, 3

Tags: algebra , inequalities , polynomial

Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.

1976 Poland - Second Round, 3

Tags: geometry , combinatorial geometry , 3d geometry , sphere

We consider a spherical bowl without any great circle. The distance between points $A$ and $B$ on such a bowl is defined as the length of the arc of the great circle of the sphere with ends at points $A$ and $B$, which is contained in the bowl. Prove that there is no isometry mapping this bowl to a subset of the plane. Attention. A spherical bowl is each of the two parts into which the surface of the sphere is divided by a plane intersecting the sphere.

2019 IFYM, Sozopol, 3

Tags: geometry

We are given a non-obtuse $\Delta ABC$ $(BC>AC)$ with an altitude $CD$ $(D\in AB)$, center $O$ of its circumscribed circle, and a middle point $M$ of its side $AB$. Point $E$ lies on the ray $\overrightarrow{BA}$ in such way that $AE.BE=DE.ME$. If the line $OE$ bisects the area of $\Delta ABC$ and $CO=CD.cos\angle ACB$, determine the angles of $\Delta ABC$.

2024 ISI Entrance UGB, P1

Tags: integration , calculus , limit , limit as integration of sum

Find, with proof, all possible values of $t$ such that \[\lim_{n \to \infty} \left( \frac{1 + 2^{1/3} + 3^{1/3} + \dots + n^{1/3}}{n^t} \right ) = c\] for some real $c>0$. Also find the corresponding values of $c$.

2009 Kosovo National Mathematical Olympiad, 1

Tags: algebra

Find the graph of the function $y=x+|1-x^3|$.

1995 Austrian-Polish Competition, 7

Tags: number theory , squarefree , perfect square , diophantine , diophantine equation

Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal: (i) $|x|$ is a square of an integer; (ii) $y$ is a squarefree number.

2002 India IMO Training Camp, 1

Tags: geometric transformation , trapezoid , geometry , homothety , power of a point , radical axis

Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$. Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$. Let $\Gamma$ be the circle which is tangent to the line $l$, tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$. The diameter of $\Gamma$ through $D$ meets $l$ in $E$. Show that $AB=DE$.

Kyiv City MO 1984-93 - geometry, 1987.7.1

Tags: geometry , geometric inequality , inradius

The circle inscribed in the triangle $ABC$ touches the side BC at point $K$. Prove that the segment $AK$ is longer than the diameter of the circle.

2022 AMC 8 -, 13

Tags:

How many positive integers can fill the blank in the sentence below? "One positive integer is $\underline{~~~~~}$ more than twice another, and the sum of the two numbers is 28." $\textbf{(A)} ~6\qquad\textbf{(B)} ~7\qquad\textbf{(C)} ~8\qquad\textbf{(D)} ~9\qquad\textbf{(E)} ~10\qquad$

2010 Contests, 1

Tags: algebra , pigeonhole principle , system of equations

Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations \[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \]

1998 Vietnam Team Selection Test, 1

Tags: function , algebra , limit , polynomial

Let $f(x)$ be a real function such that for each positive real $c$ there exist a polynomial $P(x)$ (maybe dependent on $c$) such that $| f(x) - P(x)| \leq c \cdot x^{1998}$ for all real $x$. Prove that $f$ is a real polynomial.

2020 Adygea Teachers' Geometry Olympiad, 1

Tags: congruent triangles , congruent , geometry

In planimetry, criterions of congruence of triangles with two sides and a larger angle, with two sides and the median drawn to the third side are known. Is it true that two triangles are congruent if they have two sides equal and the height drawn to the third side?

1969 IMO Shortlist, 55

Tags: 3d geometry , tetrahedron , triangle inequality , geometry

For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.