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HSK 1 | greeting phrase | asks about someone's wellbeing

Learn how to use 〜といっても (to ittemo) to qualify or soften a previous statement — 'even though I say ~, it's not as extreme as you think.' Includes structure, nuance, examples, and comparisons.

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.

Let … be nonnegative real numbers, not all zero.

Prove that there is no positive integer n such that for k =

Let n be an integer, n \geq3. Let a1, a2, . . . , an be real numbers

I 1 (USA 4)IMO1 Alice, Betty, and Carol took the same series of exam-

A cyclic quadrilateral ABCD is given. The lines AD and

Let f(n) be the least number of distinct points in the plane

B4 (BRA 1) A box contains p white balls and q black balls. Beside the

Find all solutions … of the equation

Knowing that the system

Three distinct points A, B, C are fixed on a line in this order.

Let n \geq5 be a given integer. Determine the largest integer

Let u1, u2, . . . , um be m vectors in the plane, each of length

A sequence of integers a1, a2, a3, . . . is defined as follows: a1 = 1,

(ROM 1′) Let n be a composite natural number and p a proper divisor

Prove that there exist two strictly increasing sequences (an)

Let S be the set of all the odd positive integers that are not

Let n be a positive integer. A sequence of n positive integers

Find all nondecreasing functions f : R oR such that

Twenty-one girls and twenty-one boys took part in a

A magician has one hundred cards numbered 1 to 100.

From a collection of n persons q distinct two-member teams

Find the real values of p for which the equation

On an infinite chessboard, a solitaire game is played as

Let a, b, c, d be four nonnegative numbers satisfying a+b+c+d =

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

(GBR 1a)IMO6 For all positive integral n, un+1 = un(u2

Show that for any vectors a, b in Euclidean space,

The point M inside the convex quadrilateral ABCD is such

(a) For which values of n > 2 is there a set of n consecutive

Angles of a given triangle ABC are all smaller than 120◦.

A sequence of real numbers u1, u2, u3, . . . is determined by u1

Let Sn be the number of sequences (a1, a2, . . . , an), where ai \in

The set {a0, a1, . . . , an} of real numbers satisfies the following

Let a0, a1, . . . , ak (k \geq1) be positive integers. Find all positive

A rectangular box can be filled completely with unit cubes.

Given a real number a > 1, construct an infinite and

Let N be the number of integral solutions of the equation

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

Let a, b, and c be positive real numbers such that abc = 1.

Let ABC be a triangle such that ngleA = 90◦and ngleB < ngleC. The

An eccentric mathematician has a ladder with n rungs that he

Let 1 = a0 \leqa1 \leqa2 \leq\cdot \cdot \cdot \leqan \leq\cdot \cdot \cdot be a sequence of

B1 (CAN 2)

Let a1 \geq\cdot \cdot \cdot \geqan \geqan+1 = 0 be a sequence of real numbers.

Find all finite sequences (x0, x1, . . . , xn) such that for every

For any positive integer x define

Ten gangsters are standing on a flat surface, and the distances

Let [x] denote the greatest integer less than or equal to x.

Let P be a cubic polynomial given by P(x) = ax3+bx2+cx+

Find all solutions of the following system of n equations in n

Let p be a prime number. Prove that there exists a prime

Let a, b, c be positive numbers with \sqrta+

A flock of 155 birds sit down on a circle C. Two birds Pi, Pj are

Let {ak}\infty

B3 (GBR 1) Let ABC be a triangle, and let P be a point inside it such

Find, with proof, the point P in the interior of an acute-angled

Given a tetrahedron ABCD whose all faces are acute-

Let ABCD be a tetrahedron having each sum of opposite sides

Let P1(x) = x2 −2, Pj(x) = P1(Pj−1(x)), j = 2, 3, . . . .

Let S be a convex quadrilateral ABCD and O a point inside

Let a, b, and c be given positive real numbers. Determine all

Let n1, n2 be positive integers. Consider in a plane E two dis-

Given that 1 −1

Let a, b, c be positive integers satisfying (a, b) = (b, c) =

Let … be a finite set and let …, … be bijective functions from … onto itself. Let

Let A0, A1, . . . , An be points in a plane such that

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

Let n and k be positive integers such that n/2 < k \leq2n/3.

Let … denote the set of nonnegative integers. Find a bijective function … from … into … such that for all …,

Let Q+ be the set of positive rational numbers. Construct

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

5a.(FRA 3) Let K and K′ be two squares in the same plane, their sides

Find all bases of logarithms in which a real positive number

Let f(n) be a function defined on the set of all positive integers

B6 (FIN 3) Four distinct circles C, C1, C2, C3 and a line L are given in

Let … denote the integer part of …, i.e., the greatest integer not exceeding …. If … is a positive integer, express as…

A set S of points in space will be called completely sym-

C4 (GBR 2)IMO4 Prove that if n is a positive integer such that the

Show that there exists a finite set A \subsetR2 such that for

Let a0, a1, a2, . . . be an arbitrary infinite sequence of positive

Let ABC be a triangle such that ngleACB = 2ngleABC. Let D be

Let ABC be a triangle and L the line through C parallel to

ABCD is a quadrilateral with BC parallel to AD. M is the

In a contest, there are m candidates and n judges, where

Let ABC be a triangle for which there exists an interior

The length of a finite sequence is defined as the number of