Olympiad problems

2016 Tournament Of Towns, 1

Tags: logarithm , algebra

On a blackboard the product $log_{( )}[ ]\times\dots\times log_{( )}[ ]$ is written (there are 50 logarithms in the product). Donald has $100$ cards: $[2], [3],\dots, [51]$ and $(52),\dots,(101)$. He is replacing each $()$ with some card of form $(x)$ and each $[]$ with some card of form $[y]$. Find the difference between largest and smallest values Donald can achieve.

2018 Greece Team Selection Test, 1

Tags: inequalities

If $x, y, z$ are positive real numbers such that $x + y + z = 9xyz.$ Prove that: $$\frac {x} {\sqrt {x^2+2yz+2}}+\frac {y} {\sqrt {y^2+2zx+2}}+\frac {z} {\sqrt {z^2+2xy+2}}\ge 1.$$ Consider when equality applies.

1969 IMO Longlists, 69

Tags: inequality , three variable inequality , algebra

$(YUG 1)$ Suppose that positive real numbers $x_1, x_2, x_3$ satisfy $x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}$ Prove that: $(a)$ None of $x_1, x_2, x_3$ equals $1$. $(b)$ Exactly one of these numbers is less than $1.$

2010 Today's Calculation Of Integral, 589

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 \frac{x}{\{(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\plus{}(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx$.

Kvant 2022, M2727

Tags: geometry , area

A convex quadrilateral $ABCD$ is given. Let $O_a$ be the circumcenter of the triangle $DBC$, and define $O_b,O_c$ and $O_d$ similarly. The points $O_a, O_b, O_c, O_d$ are the vertices of a convex quadrilateral. Prove that its area is equal to half of the absolute value of the difference between the areas of $AO_bCO_d$ and $BO_cDO_a$. [i]Proposed by V. Dubrovsky[/i]

2020 Regional Olympiad of Mexico Center Zone, 4

Tags: square , geometry , circles

Let $\Gamma_1$ be a circle with center $O$ and $A$ a point on it. Consider the circle $\Gamma_2$ with center at $A$ and radius $AO$. Let $P$ and $Q$ be the intersection points of $\Gamma_1$and $\Gamma_2$. Consider the circle $\Gamma_3$ with center at $P$ and radius $PQ$. Let $C$ be the second intersection point of $\Gamma_3$ and $\Gamma_1$. The line $OP$ cuts $\Gamma_3$ at $R$ and $S$, with $R$ outside $\Gamma_1$. $RC$ cuts $\Gamma_1$ into $B$. $CS$ cuts $\Gamma_1$ into $D$. Show that $ABCD$ is a square.

2023 239 Open Mathematical Olympiad, 1

Tags: combinatorics , winning strategy

There are $n{}$ wise men in a hall and everyone sees each other. Each man will wear a black or white hat. The wise men should simultaneously write down on their piece of paper a guess about the color of their hat. If at least one does not guess, they will all be executed. The wise men can discuss a strategy before the test and they know that the layout of the hats will be chosen randomly from the set of all $2^n$ layouts. They want to choose their strategy so that the number of layouts for which everyone guesses correctly is as high as possible. What is this number equal to?

Cono Sur Shortlist - geometry, 2020.G4

Tags: bisects segment , circumcircle , geometry

Let $ABC$ be a triangle with circumcircle $\omega$. The bisector of $\angle BAC$ intersects $\omega$ at point $A_1$. Let $A_2$ be a point on the segment $AA_1$, $CA_2$ cuts $AB$ and $\omega$ at points $C_1$ and $C_2$, respectively. Similarly, $BA_2$ cuts $AC$ and $\omega$ at points $B_1$ and $B_2$, respectively. Let $M$ be the intersection point of $B_1C_2$ and $B_2C_1$. Prove that $MA_2$ passes the midpoint of $BC$. [i]proposed by Jhefferson Lopez, Perú[/i]

1963 AMC 12/AHSME, 40

Tags: geometry , 3d geometry

If $x$ is a number satisfying the equation $\sqrt[3]{x+9}-\sqrt[3]{x-9}=3$, then $x^2$ is between: $\textbf{(A)}\ 55\text{ and }65 \qquad \textbf{(B)}\ 65\text{ and }75\qquad \textbf{(C)}\ 75\text{ and }85 \qquad \textbf{(D)}\ 85\text{ and }95 \qquad \textbf{(E)}\ 95\text{ and }105$

2003 Moldova National Olympiad, 12.2

Tags: calculus , derivative , polynomial , algebra

For every natural number $n\geq{2}$ consider the following affirmation $P_n$: "Consider a polynomial $P(X)$ (of degree $n$) with real coefficients. If its derivative $P'(X)$ has $n-1$ distinct real roots, then there is a real number $C$ such that the equation $P(x)=C$ has $n$ real,distinct roots." Are $P_4$ and $P_5$ both true? Justify your answer.

2002 Tournament Of Towns, 7

Tags: ceiling function , combinatorics

[list] [*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this? [*] Previous problem for the grid of $5\times 5$ lattice.[/list]

2009 Croatia Team Selection Test, 4

Tags: quadratic , diophantine equation , number theory

Prove that there are infinite many positive integers $ n$ such that $ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.

MathLinks Contest 5th, 6.3

Tags: algebra , inequalities

Let $x, y, z$ be three positive numbers such that $(x + y-z) \left( \frac{1}{x}+ \frac{1}{y}- \frac{1}{z} \right)=4$. Find the minimal value of the expression $$E(x, y, z) = (x^4 + y^4 + z^4) \left( \frac{1}{x^4}+ \frac{1}{y^4}+ \frac{1}{z^4} \right) .$$

2021 Bundeswettbewerb Mathematik, 3

Tags: geometry

Consider a triangle $ABC$ with $\angle ACB=120^\circ$. Let $A’, B’, C’$ be the points of intersection of the angular bisector through $A$, $B$ and $C$ with the opposite side, respectively. Determine $\angle A’C’B’$.

2001 Greece National Olympiad, 1

Tags: ratio , geometry

A triangle $ABC$ is inscribed in a circle of radius $R.$ Let $BD$ and $CE$ be the bisectors of the angles $B$ and $C$ respectively and let the line $DE$ meet the arc $AB$ not containing $C$ at point $K.$ Let $A_1, B_1, C_1$ be the feet of perpendiculars from $K$ to $BC, AC, AB,$ and $x, y$ be the distances from $D$ and $E$ to $BC,$ respectively. (a) Express the lengths of $KA_1, KB_1, KC_1$ in terms of $x, y$ and the ratio $l = KD/ED.$ (b) Prove that $\frac{1}{KB}=\frac{1}{KA}+\frac{1}{KC}.$

2021 AIME Problems, 8

Tags:

Find the number of integers $c$ such that the equation $$\left||20|x|-x^2|-c\right|=21$$ has $12$ distinct real solutions.

2005 iTest, 5

Tags: combinatorics

Find the sum of the answers to all even numbered Short Answer problems, with the exception of #26, rounded to the nearest tenth. [i](.7 points)[/i]

2025 NEPALTST, 2

Tags: number theory

Find all integers $n$ such that if \[ 1 = d_1 < d_2 < \cdots < d_{k-1} < d_k = n \] are the divisors of $n$, then the sequence \[ d_2 - d_1,\, d_3 - d_2,\, \ldots,\, d_k - d_{k-1} \] forms a permutation of an arithmetic progression. [i](Kritesh Dhakal, Nepal)[/i]

2022 Harvard-MIT Mathematics Tournament, 4

Tags: inequalities , algebra

Suppose $n \ge 3$ is a positive integer. Let $a_1 < a_2 < ... < a_n$ be an increasing sequence of positive real numbers, and let $a_{n+1} = a_1$. Prove that $$\sum_{k=1}^{n}\frac{a_k}{a_{k+1}}>\sum_{k=1}^{n}\frac{a_{k+1}}{a_k}$$

2017 Princeton University Math Competition, 15

Tags: number theory

How many ordered pairs of positive integers $(x, y)$ satisfy $yx^y = y^{2017}$?

2018 Swedish Mathematical Competition, 6

Tags: polynomial , algebra

For which positive integers $n$ can the polynomial $p(x) = 1 + x^n + x^{2n}$ is written as a product of two polynomials with integer coefficients (of degree $\ge 1$)?

2007 Balkan MO Shortlist, A6

Tags: algebra , functional equation

Find all real functions $f$ defined on $ \mathbb R$, such that \[f(f(x)+y) = f(f(x)-y)+4f(x)y ,\] for all real numbers $x,y$.

1968 Putnam, A6

Tags: polynomial

Find all polynomials whose coefficients are all $\pm1$ and whose roots are all real.

1975 IMO, 3

Tags: geometry , trigonometry , triangle , angle

In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$. Prove that [b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and [b]b.)[/b] $QR = RP.$

2022 Cyprus TST, 4

Tags: combinatorics , extremal combinatorics

Let \[M=\{1, 2, 3, \ldots, 2022\}\] Determine the least positive integer $k$, such that for every $k$ subsets of $M$ with the cardinality of each subset equal to $3$, there are two of these subsets with exactly one common element.