brain

HSK 5 | adjective/verb | severely insufficient in supply; a state of dire shortage

HSK 5 | verb/noun | to report (news); a news report or coverage

HSK 4 | noun | a widespread outbreak of infectious disease

HSK 3 | conjunction | introduces the single condition that is the only exception to a rule or the only way an outcome can occur

HSK 3 | noun/adjective | a deep sense of contentment and well-being

HSK 5 | adverb | used to single out something as standing out above others

左 (sa/hidari): left. JLPT N5 kanji.

HSK 3 | verb | to be acquainted with someone; to recognize or identify

HSK 1 | adverb/noun | the day after today

HSK 4 | verb | to disappear or vanish from sight or existence

HSK 8 | noun | thermodynamics; the branch of physics dealing with heat and energy

Master formal Japanese time and occasion markers used in official speeches, ceremonies, and business announcements — 〜にあたって, 〜に際して, 〜に先立って, and more.

HSK 4 | noun | doctor, physician

HSK 3 | adjective / verb | convenient, to be convenient, to make things easy

HSK 4 | noun | the rate at which something moves or happens

HSK 3 | noun | the post office; a government postal service location

HSK 5 | noun/verb | marketing; to market (products or services)

HSK 1 | verb | to have a fondness or preference for something

The previous solution failed at the structural point where it tried to characterize tetrahedra admitting a sphere tangent to all six edge-lines.

The problem is a construction problem on a fixed circumcircle.

Consider the equation

We seek all real numbers $x$ satisfying

Let $n$ be a natural number ending in 6 such that moving the 6 to the front produces a number four times as large.

We are asked to construct a triangle $ABC$ when the side lengths

The problem is a pure proof problem.

The statement asks for a lower bound on $a^2+b^2+c^2$ in terms of the area $S$ of a triangle.

The problem asks for conditions on the parameters $a$ and $b$ under which the system admits three distinct positive numbers $x,y,z$.

The problem concerns a right circular cone inscribed in a sphere and a cylinder circumscribed about the same sphere.

The previous solution failed at two genuinely important points.

The reviewers identified two genuine problems in the previous proof.

4b.(IRE 4) A set of 1985 points is distributed around the circumference

A, B, C, D are four points in the plane, with C, D on the

Points A, B, C divide the circumcircle Ωof the triangle ABC

Let P be a set of 7 different prime numbers and C a set of

Let lpha be the positive root of the equation x2 = 1991x + 1. For

An n \times n chessboard (n \geq2) is numbered by the numbers

In town A, there are n girls and n boys, and each girl knows each

Given a convex polygon A1A2 . . . An with area S, and a point

In a certain city, age is reckoned in terms of real numbers

Show that the set of positive integers that cannot be repre-

A brick staircase with three steps of width 2 is made of twelve

Prove that the product of the radii of three circles exscribed to a given triangle does not exceed … times the product…

Let A be a set of N residues (mod N 2). Prove that there

Let Ox, Oy, Oz be three rays, and G a point inside the trihe-

Find the number of partitions of the set {1, 2, . . ., n} into three

On a circle, 2n −1 (n \geq3) different points are given. Find

Two congruent equilateral triangles ABC and A′B′C′ in the

Let a and b be two positive integers such that ab+1 divides

Let a, b be integers that are not perfect squares. Prove that if

In a convex quadrangle with area 32 cm2, the sum of the

Prove that a convex pentagon (a five-sided polygon) ABCDE

(FIN 2‘) Let E be a finite set of points such that E is not contained in

The function f(x, y) is a homogeneous polynomial of the nth

Let the sides of two rectangles be … and … with

I 5 (GBR 3) Let Ar, Br, Cr be points on the circumference of a given

Let a, b, c be positive real numbers with product 1. Prove

Let D be the interior of the circle C and let A \inC. Show

For a polynomial P of degree 2000 with distinct real co-

Determine the least possible value of f(1998), where f is a

Given an oriented line … and a fixed point … on it, consider all trapezoids … one of whose bases … lies on …, in the…

Let a1, a2, . . . , an be positive numbers and q a given real

Find all pairs of integers x, y \geq1 satisfying the equation

Let p be a prime and A = {a1, . . . , ap−1} an arbitrary subset

An infinite sequence a0, a1, a2, . . . of real numbers satisfies

Find the minimum value of

A circle is called a separator for a set of five points in a plane

Consider two segments of length … (…) and a segment of length ….

Let a1, a2, . . . be an infinite sequence of real numbers for

Let n \geq2 be a natural number and let the real numbers

An acute triangle ABC is given. Points A1 and A2 are taken

Let n \geq2 be an integer. Find the maximal cardinality of a set

Given an integer n > 1, denote by Pn the product of all

Find all polynomials P(x) with real coefficients that

Given a point … and lengths …, prove that there exists an equilateral triangle … for which …, …, …, if and only if …,…

Let a and b be natural numbers and let q and r be the

Prove that the set {1, 2, . . ., 1989} can be expressed as the

In a plane two different points O and A are given. For

A rectangular box can be filled completely with unit cubes.

Let … be a convex hexagon such that …, …, and …. Let …, …, … be the circumradii of triangles …, …, … respectively, and…

Let ABCD be a regular tetrahedron and M, N distinct points

Consider pairs of sequences of positive real numbers a1 \geq

Let {f(n)} be a strictly increasing sequence of positive

Let … have orthocenter …, and let … be a point on its circumcircle, distinct from …, …, …. Let … be the foot of the…

Let ABC be a triangle with ∡BAC = 60◦. Let AP bisect

In the plane we have n rectangles with parallel sides. The

Let … be the apothem (distance from the center to one of the sides) of a regular …-gon (…) inscribed in a circle of…

1b.(TUR 5) Find the smallest positive integer n such that

Every point with integer coordinates in the plane is the

A finite set of unit circles is given in a plane such that the area

Two ships sail on the sea with constant speeds and fixed directions. It is known that at … the distance between them…

Given k parallel lines and a few points on each of them, find

Find all positive integers n such that

For any integer r \geq1, determine the smallest integer h(r) \geq1

Let n and k be positive integers. There are given n circles

Let the sequence …, …, be generated as follows:

Find all positive integer solutions x, y, z of the equation 3x +

Let O be the circumcenter of an acute-angled triangle ABC

Find all functions f from the reals to the reals such that

Prove that every positive rational number can be repre-

A5 (NET 2)IMO5 Let A1A2A3A4A5A6 be a regular hexagon. Each of its