Olympiad problems

2020 Azerbaijan Senior NMO, 1

Tags: inequalities , algebra

$x,y,z\in\mathbb{R^+}$. If $xyz=1$, then prove the following: $$\sum\frac{x^6+2}{x^3}\geq3(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})$$

2010 Brazil National Olympiad, 3

Tags: geometry , 3d geometry , trigonometry , vector , linear algebra , matrix , inequalities

What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak? Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

2006 China Team Selection Test, 3

Tags: number theory

For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that: \[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \] then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.

2003 Iran MO (3rd Round), 21

Tags: circumcircle , geometry

Let $ ABC$ be a triangle. $ W_a$ is a circle with center on $ BC$ passing through $ A$ and perpendicular to circumcircle of $ ABC$. $ W_b,W_c$ are defined similarly. Prove that center of $ W_a,W_b,W_c$ are collinear.

2007 Ukraine Team Selection Test, 4

Tags: function , algebra

Find all functions $f: \mathbb Q \to \mathbb Q$ such that $ f(x^{2}\plus{}y\plus{}f(xy)) \equal{} 3\plus{}(x\plus{}f(y)\minus{}2)f(x)$ for all $x,y \in \mathbb Q$.

2012 Iran MO (2nd Round), 2

Tags: induction , combinatorics

Suppose $n$ is a natural number. In how many ways can we place numbers $1,2,....,n$ around a circle such that each number is a divisor of the sum of it's two adjacent numbers?

2023 Caucasus Mathematical Olympiad, 8

Tags: geometry

Let $ABC$ be an acute-angled triangle, and let $AA_1, BB_1, CC_1$ be its altitudes. Points $A', B', C'$ are chosen on the segments $AA_1, BB_1, CC_1$, respectively, so that $\angle BA'C = \angle AC'B = \angle CB'A = 90^{o}$. Let segments $AC'$ and $CA'$ intersect at $B"$; points $A", C"$ are defined similarly. Prove that hexagon $A'B"C'A"B'C"$ is circumscribed.

2021 USMCA, 29

Tags:

Three circles $\Gamma_A, \Gamma_B, \Gamma_C$ are externally tangent. The circles are centered at $A, B, C$ and have radii $4, 5, 6$ respectively. Circles $\Gamma_B$ and $\Gamma_C$ meet at $D$, circles $\Gamma_C$ and $\Gamma_A$ meet at $E$, and circles $\Gamma_A$ and $\Gamma_B$ meet at $F$. Let $GH$ be a common external tangent of $\Gamma_B$ and $\Gamma_C$ on the opposite side of $BC$ as $EF$, with $G$ on $\Gamma_B$ and $H$ on $\Gamma_C$. Lines $FG$ and $EH$ meet at $K$. Point $L$ is on $\Gamma_A$ such that $\angle DLK = 90^\circ$. Compute $\frac{LG}{LH}$.

Gheorghe Țițeica 2025, P4

Tags: number theory

[list=a] [*] Prove that for any positive integers $a,b,c$, there exists a positive integer $N$ such that $$(N+a^2)(N+b^2)(N+c^2)$$ is a perfect square. [*] Prove that there exist five distinct positive integers $a,b,c,d,e$ for which there exists a positive integer $N$ such that $$(N+a^2)(N+b^2)(N+c^2)(N+d^2)(N+e^2)$$ is a perfect square. [/list] [i]Luminița Popescu[/i]

2006 USAMO, 1

Tags: inequalities , floor function , modular arithmetic , pigeonhole principle , number theory

Let $p$ be a prime number and let $s$ be an integer with $0 < s < p.$ Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and \[ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} \] if and only if $s$ is not a divisor of $p-1$. Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.

Kvant 2019, M2587

Tags: algorithm , combinatorics

In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. For what is the smallest amount of rubles you can rearrange chips in reverse order?

PEN O Problems, 28

Tags:

Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x$, $y$ taken from two different subsets, the number $x^{2}-xy+y^{2}$ belongs to the third subset.

2013 Korea National Olympiad, 1

Tags: circumcircle , geometry

Let $P$ be a point on segment $BC$. $Q, R$ are points on $AC, AB$ such that $PQ \parallel AB $ and $ PR \parallel AC$. $O, O_{1}, O_{2} $ are the circumcenters of triangle $ABC, BPR, PCQ$. The circumcircles of $BPR, PCQ $ meet at point $K (\ne P)$. Prove that $OO_{1} = KO_{2} $.

1998 National Olympiad First Round, 5

Tags: trigonometry

$ ABCD$ is a cyclic quadrilateral. If $ \angle B \equal{} \angle D$, $ AC\bigcap BD \equal{} \left\{E\right\}$, $ \angle BCD \equal{} 150{}^\circ$, $ \left|BE\right| \equal{} x$, $ \left|AC\right| \equal{} z$, then find $ \left|ED\right|$ in terms of $ x$ and $ z$. $\textbf{(A)}\ \frac {z \minus{} x}{\sqrt {3} } \qquad\textbf{(B)}\ \frac {z \minus{} 2x}{3} \qquad\textbf{(C)}\ \frac {z \plus{} x}{\sqrt {3} } \qquad\textbf{(D)}\ \frac {z \minus{} 2x}{2} \qquad\textbf{(E)}\ \frac {2z \minus{} 3x}{2}$

2007 Argentina National Olympiad, 4

Tags: combinatorics , number theory

$10$ real numbers are given $a_1,a_2,\ldots ,a_{10} $, and the $45$ sums of two of these numbers are formed $a_i+a_j $, $1\leq i<j\leq 10$ . It is known that not all these sums are integers. Determine the minimum value of $k$ such that it is possible that among the $45$ sums there are $k$ that are not integers and $45-k$ that are integers.

2007 ITest, 35

Tags: inequalities

Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits.

2006 All-Russian Olympiad Regional Round, 8.3

Tags: combinatorics

Four drivers took part in the round-robin racing. Their cars started simultaneously from one point and moved at constant speeds. It is known that after the start of the race, for any three cars there was a moment when they met. Prove that after the start of the race there will be a moment when all 4 cars meet. (We consider races to be infinitely long in time.)

1998 Belarus Team Selection Test, 1

Tags: midpoint , geometry , perpendicular bisector , angle bisector , angle

Let $O$ be a point inside an acute angle with the vertex $A$ and $H, N$ be the feet of the perpendiculars drawn from $O$ onto the sides of the angle. Let point $B$ belong to the bisector of the angle, $K$ be the foot of the perpendicular from $B$ onto either side of the angle. Denote by $P,F$ the midpoints of the segments $AK,HN$ respectively. Known that $ON + OH = BK$, prove that $PF$ is perpendicular to $AB$. Ya. Konstantinovski

2011 Princeton University Math Competition, A4

Tags: algebra

Suppose the polynomial $x^3 - x^2 + bx + c$ has real roots $a, b, c$. What is the square of the minimum value of $abc$?

1979 Chisinau City MO, 180

Tags: trinomial , algebra , derivative

It is known that for $0\le x \le 1$ the square trinomial $f (x)$ satisfies the condition $|f(x) | \le 1$. Show that $| f '(0) | \le 8.$

2022 AMC 12/AHSME, 4

Tags: quadratic

For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots? $\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$

2004 AMC 12/AHSME, 15

Tags: ratio , quadratic , algebra

Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run $ 100$ meters. They next meet after Sally has run $ 150$ meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? $ \textbf{(A)}\ 250 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 350 \qquad \textbf{(D)}\ 400\qquad \textbf{(E)}\ 500$

2019 LIMIT Category C, Problem 1

Tags: linear algebra , matrix

Which of the following are true? $\textbf{(A)}~\forall A\in M_n(\mathbb R),A^t=X^{-1}AX\text{ for some }X\in M_n(\mathbb R)$ $\textbf{(B)}~\forall A\in M_n(\mathbb R),I+AA^t\text{ is invertible}$ $\textbf{(C)}~\operatorname{tr}(AB)=\operatorname{tr}(BA),\forall A,B\in M_n(\mathbb R)\text{ but }\exists A,B,C\text{ such that }\operatorname{tr}(ABC)\ne\operatorname{tr}(BAC)$ $\textbf{(D)}~\text{None of the above}$

2022 MOAA, 5

Tags: number theory

Find the smallest positive integer that is equal to the sum of the product of its digits and the sum of its digits.

2013 International Zhautykov Olympiad, 2

Tags: modular arithmetic , number theory

Find all odd positive integers $n>1$ such that there is a permutation $a_1, a_2, a_3, \ldots, a_n$ of the numbers $1, 2,3, \ldots, n$ where $n$ divides one of the numbers $a_k^2 - a_{k+1} - 1$ and $a_k^2 - a_{k+1} + 1$ for each $k$, $1 \leq k \leq n$ (we assume $a_{n+1}=a_1$).